Critical exponents in stochastic sandpile models

合集下载

半线性椭圆方程

µ − a + β . K•§ (0.12) – k˜‡
ë•©zµ [1] L. Caﬀarelli, R. Kohn, L. Nirenberg, First order interpolation inequality with weights, Compos. Math. 53 (1984) 259–275. [2] K. Chou, C. Chu, On the best constant for a weighted Sobolev–Hardy inequality, J. Lond. Math. Soc. (2) 48 (1993) 137–151. [3] F. Catrina, Z. Wang, On the Caﬀarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of external functions, Comm. Pure Appl. Math. 54 (2) (2001) 229–258. [4] X.J. Huang, X.P. Wu, C.L. Tang, Multiple positive solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents, Nonlinear Anal. 74 (2011) 2602–2611 [5] L. Ding, C.L. Tang, Existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving Hardy term and Hardy-Sobolev critical exponents, J. Math. Anal. Appl. 339 (2008) 1073–1083. [6] L. Huang, X.P. Wu, C.L. Tang, Existence and multiplicity of solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents, Nonlinear Anal. 71 (2009) 1916–1924. [7] P. H. Rabinowitz. Minimax methods in critical point theory with applications to diﬀerential equations, Conference Board of the Mathematical Sciences, vol. 65 Providence, RI: American Mathematical Society, 1986. [8] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987. [9] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (4) (1983) 437–477. [10] D. Kang, G. Li, S. Peng, Positive solutions and critical dimensions for the elliptic problems involving the Caﬀarelli-Kohn-Nirenberg inequalities, J. Jilin. Univ. Sci. 46 (2008) 423–427.

Compositeness, Triviality and Bounds on Critical Exponents for Fermions and Magnets

a rXiv:h ep-th/93127v11D ec1993ILL-(TH)-93-21COMPOSITENESS,TRIVIALITY AND BOUNDS ON CRITICAL EXPONENTS FOR FERMIONS AND MAGNETS Aleksandar KOCI ´C and John KOGUT Loomis Laboratory of Physics,University of Illinois,Urbana,Il 61801Abstract We argue that theories with fundamental fermions which undergo chiral symmetry breaking have several universal features which are qualitatively diﬀerent than those of theories with fundamental scalars.Several bounds on the critical indices δand ηfollow.We observe that in four dimensions the logarithmic scaling violations enter into the Equation of State of scalar theories,such as λφ4,and fermionic models,such as Nambu-Jona-Lasinio,in qualitatively diﬀerent ways.These observations lead to useful approaches for analyzing lattice simulations of a wide class of model ﬁeld theories.Our results imply that λφ4cannot be a good guide to understanding the possible triviality of spinor QED .1.IntroductionThere are two classes of theories in the literature that are used to model the Higgs sector of the Standard Model[1,2].One is based on the self-interactingφ4scalar theory in which the Higgs is elementary.The other is based on strongly interacting constituent fermiﬁelds in which the Higg’s particle is a fermion-antifermion bound state.A recent proposal for the second realization uses Nambu-Jona-Lasinio(NJL)models in which composite scalars emerge as a consequence of spontaneous chiral symmetry breaking[2].In four dimensions both types of models have a trivial continuum limit and are meaningful only as eﬀective theories with aﬁnite cutoﬀ.This restriction places constraints on the low-energy parameters e.g.bounds on the masses.We wish to point out in this letter that triviality in the two models is realized in diﬀerent ways.The diﬀerences between theories of composite and elementary mesons can be expressed in terms of the critical indicesδandη,and several inequalities and bounds on these indices will follow.These results should prove useful in theoretical,phenomenological and lattice simulation studies of a wide class of modelﬁeld theories.We begin with a few comments about the physics in each model.In a NJL model[3],as a result of spontaneous chiral symmetry breaking,the pion-fermion coupling is given by the Goldberger-Treiman relation gπN=M N/fπ,where M N is the fermion mass and fπis the pion decay constant.Being the wavefunction of the pion,fπdetermines its radius as well:rπ∼1/fπ.The Goldberger-Treiman can then be written in the suggestive form gπN∼M N rπ.Thus,the coupling between pions and fermions vanishes as the size of the pion shrinks to zero.The origin of triviality of the Nambu-Jona-Lasinio model is precisely the loss of compositeness of the mesons[3].The force between the fermions is so strong that the constituents collapse onto one another producing pointlike mesons and a noninteracting continuum theory.In a self-interacting scalar theory,likeφ4,the mesons are elementary and the reason for triviality is diﬀerent[4,5].At short distances the interaction is repulsive,so there is no collapse.The structure of the scalars,needed for the interaction to survive the continuum limit,should be built by weakly interacting bosons.In four dimensions, the short rangedλφ4interaction fails to provide a physical size for the mesons.It cannot be felt by the particles because of the short-distance repulsion–they cannot meet where collective behavior can set in and produce macroscopicﬂuctuations.In this way the cutoﬀremains the only scale and the continuum limit is trivial.Consider two simple,soluble examples:the large-N limits of the O(N)σ-and the four-fermi model [6,7].They exhibit a phase transition atﬁnite coupling for2<d<4.Their critical exponents are given in Table1.As is apparent from the Table,the two sets of critical indices evolve diﬀerently when d is reduced below4.Atﬁrst glance this might be surprising since both models break the same symmetry spontaneously and one expects that they describe the same low energy physics.The purpose of this paper is to show that this diﬀerence between the critical exponents is generally valid,irrespective of the approximations employed.As a consequence of this it will be possible to establish a bound on the exponentδwhich for scalar theories isδ≥3,and for fermionic theories isδ≤3.Although in four dimensions the two sets of exponents coincide,they are accompanied by logarithmic corrections due to scaling violations.Itconsequently the scaling violations have opposite signs in the two classes of theories.These bounds are a consequence of diﬀerent realizations of symmetry breaking,the essential diﬀerence being the fact that for scalar theories mesons are elementary,while in the case of the chiral transition in fermionic theories,they are composite.The bounds onδare just another way of expressing this diﬀerence in terms of universal quantities. Finally,we will discuss the implications of these results on triviality in both models in four dimensions.Table1Leading order critical exponents for the spherical and four-fermi modelexponentσ-model four-fermiβ1d−2ν1d−2δd+21d−2η04−d2.Mass ratios and bounds onδTo approach the problem,it is convenient to adopt a particular view of the phase transition[8].Instead of the order parameter we will use mass ratios to distinguish the two phases.While the order parameter is a useful quantity to parametrize the phase diagram,the spectrum carries direct information about the response of the system in it’s diﬀerent phases and its form does not change in the presence of an external symmetry-breakingﬁeld.In what follows we will switch from magnetic to chiral notation without notice.The correspondence is:magneticﬁeld(h)⇔bare mass(m);magnetization(M)⇔chiral condensate(<¯ψψ>); longitudinal and transverse modes⇔(σ,π);h→0⇔chiral limit.Theories that treat scalars as elementary will be referred to as’magnets’and those that give rise to composite mesons as a consequence of spontaneous chiral symmetry breaking will be refered to as’fermions’.Consider the eﬀect of spontaneous symmetry breaking on the spectrum from a physical point of view. In the symmetric phase,there is no preferred direction and symmetry requires the degeneracy between longitudinal and transverse modes(chiral partners).Therefore,in the zero-ﬁeld(chiral)limit the ratio R=M2T/M2L=M2π/M2σ→1.As the magneticﬁeld(bare mass)increases,the ratio decreases(because of level ordering,σis always heavier thenπ).In the broken phase,however,the ratio vanishes in the chiral limit because the pion is a Goldstone boson.This time,the ratio clearly increases away from the chiral limit.dynamics.The value of the ratio at large h is less sensitive to variations in the coupling.The qualitative behavior of the mass ratio is sketched in Fig.1.The important property of the mass ratio,in this context, is that its properties follow completely from the properties of the order parameter[8].This,after all,comes as no surprise since both quantities,M and R,contain the same physics and merely reﬂect two aspects of one phenomenon.The essential ingredients are the Equation of State(EOS)and the Ward identity which follows from it.h a=M a Mδ−1f t/M1/β ,χ−1T=h/M,χ−1L=∂hR =χ−1LOne way to determine the sign of the scaling violations in four dimensions is to proceed in the spirit of the ǫ-expansion i.e.to approach four dimensions from below [9].The transcription to the language of scaling violations is established by the replacement ǫ→1/log Λin the limit ǫ→0.Thus,the extension of the arguments made before for d <4can be made by simply taking the limit d →4.In this way we anticipate that the two inequalities prevail and suggest that the scaling violations have diﬀerent signs in the two theories.The diﬀerence in the sign of the scaling violations in fermions and magnets has a simple explanation and lies at the root of the diﬀerence between the patterns of symmetry breaking in the two systems.Imagine that we ﬁx the temperature to its critical value T =T c and approach the critical point (T =T c ,h =0)in the (T,h )plane from the large-h region.The possible similarity of the two models is related to their symmetry.This is apparent in the chiral (zero ﬁeld)limit where this symmetry is manifest.By going away from this limit chiral symmetry is violated and the two models diﬀer.Consider the behavior of the mass ratio for magnets in a strong magnetic ﬁeld,away from the scaling region.In this regime,the temperature factor can be neglected and the hamiltonian describes free spins in an external ﬁeld (H →h i S i ).The energy of longitudinal excitations is proportional to the ﬁeld-squared,χ−1L ∼h 2,while the transverse mass remains ﬁxed by the symmetry,namely χ−1T =h/M for any value of h .The eﬀect of the external ﬁeld is to introduce a preferred direction and its increase results in ampliﬁcation of the diﬀerence between the longitudinal and transverse dirrections.For large h the ratio scales as R ∼1/h .Therefore,an increase in magnetic ﬁeld reduces the ratio towards zero.The critical isotherm in this case bends down (Fig.3).For fermions,the mesons are fermion-antifermion composites.Close to the chiral limit,they are collec-tive.However,as the constituent mass increases,they turn into atomic states and the main contribution to the meson mass comes from the rest energy of its constituents.In the limit of inﬁnite bare mass,interactions are negligible and M →2m regardless of the channel.Thus,outside of the scaling region,an increase in m drives the ratio to 1(Fig.3).Thus,the scaling violations for magnetis and fermionis have opposite signs .They contain knowledge of the physics away from the chiral limit where the two models are quite diﬀerent and these diﬀerences remain as small corrections close to the chiral limit.To establish the connection between scaling violations and triviality,we introduce the renormalized cou-pling.It is a dimensionless low-energy quantity that contains information about the non-gaussian character of the theory.It is conventionally deﬁned as [10]g R =−χ(nl )∂h 3= 123<φ(0)φ(1)φ(2)φ(3)>c (4)The normalization fators,χ= x <φ(0)φ(x )>c and ξd ,in eq.(3)take care of the four ﬁelds and the three integrations.In a gaussian theory all higher-point functions factorize,so g R ing the hyperscalingg R∼ξ(2∆−γ−dν)/ν(5)where∆=β+γis the gap exponent.Being dimensionless,g R should be independent ofξifξis the only scale.Thus,the validity of hyperscaling requires that the exponent must vanish.It implies the relation, 2∆−γ−dν=0,between the critical indices.In general,it is known that the following inequality[11]holds2∆≤γ+dν(6)The exponent in the expression for g R is always non-positive,so that violations of hyperscaling imply that the resulting theory is non-interacting.Above four dimensions,the exponents are gaussian(γ=1,∆=3/2,ν=1/2).In this case,it is easy to verify the above inequality:3≤1+d/2,which amounts to d≥4.In four dimensions mostﬁeld theoretical models have mean-ﬁeld critical exponents,but with logarithmic corrections that drive g R to zero.Scaling violations in any thermodynamic quantity propagate into the renormalized coupling and,according to eq.(6), these violations lead to triviality.Instead of using the ideas of theǫ-expansion where scaling is always respected and where equalities between exponents hold,we willﬁx d=4and compute the logarithmic corrections to the critical exponents. In order to focus on the problem in question,we analyze two simple models:φ4and(¯ψψ)2theories both in the large-N limit.The results that will be discussed are completely general and the two models are chosen just to make the argument simple.The eﬀective actions for the two models are[12,13]V(M)=−14M42t<¯ψψ>2+<¯ψψ>4log(1/<¯ψψ>)(7b) This is the leading log contribution only.In theﬁrst example,it is clear how log-corrections lead to triviality. The logarithm can simply be thought of as coming from the running coupling–quantum corrections lead to the replacementλ→λR.The vanishing of the renormalized coupling is then manifest from eq.(7a).In the case of fermions,eq.(7b),the details are completely diﬀerent–the analogous reasoning would lead to an erroneous conclusion that the renormalized coupling increases in the infrared.In eq.(7b)the explicit coupling is absent from theﬂuctuating term–it is already absorbed in the curvature.Once the curvature isﬁxed,the eﬀective coupling is independent of the bare one.The vanishing of the renormalized coupling here follows from the wave funciton renormalization constant Z∼1/ln(1/<¯ψψ>)[13].In both cases the renormalized coupling is obtained through the nonlinear susceptibility.For simplicity, we work in the symmetric phase where the odd-point functions vanish.The correlation length is related to the susceptibility byξ2=χ/Z.For magnets the folowing relations holdχ(nl)∼χ4λln(1/M)∼1For fermions,on the other hand,we have1χ(nl)∼χ4ln(1/<¯ψψ>),Z∼(9b)ln(1/<¯ψψ>)In this context the following point should be made.The nonlinear susceptibility is a connected four-point function for the composite¯ψψﬁeld.The free fermionic theory is not gaussian in¯ψψ,so even in freeﬁeld theory g R does not vanish.The fact that g R→0near the critical point indicates that the resulting theory is indeed gaussian in the compositeﬁeld which results in a free bosonic theory in the continuum.The Equation of State(EOS)is obtained from the eﬀective potential by simple diﬀerentiation.To make the connection withδ,we take t=0.The critical EOS for the magnets is[12]M3h∼(11)log(1/M)Thus,on the ratio plot,Fig.2a,the critical isotherm is no longerﬂat,but goes down,as the h-ﬁeld(order parameter)increases.The result of eq.(11)is well known in the literature and has been obtained in the past using theǫ-expansion:δ=3+ǫ[12],where the correspondence with eq.(10)is made after the replacement 1/ǫ→log.Such corrections toδ,as eq.(11),can never occur in the case of the chiral transition.For the four-fermi model the critical EOS[14]reads,m∼<¯ψψ>3log(1/<¯ψψ>)(12) Unlike scalar theories,the log’s appear in the numerator–the right hand side in eq.(12)vanishes slower then the pure power and the”eﬀective”δis smaller then the(pure)mean-ﬁeld value.˜δ=3−1We have seen that the two diﬀerent realizations of spontaneous symmetry breaking can be expressed simply in terms of universal quantities.In this way,many apparently complicated dynamical questions become transparent.In addition,some of our observations lead to practical applications–they can be used for extracting the properties of the continuum limit of theories with newﬁxed points,especially when clear theoretical ideas about the low-energy physics of the theory are missing.Of special importance is the knowledge of the position of the logarithms when triviality is studied on the lattice.It is extremely diﬃcult to establish the presence of the logarithms for aﬁnite system and to disentangle them fromﬁnite size eﬀects. The bounds obtained in this paper establish some criteria in this direction as far as chiral transitions are concerned.Recently,they were proven to be decisive in studies of the chiral transition of QED[15]and in establishing triviality of the NJL model in four dimensions by computer simulations[16].The literature on fermionic QED abounds with loose statements such as”QED is ultimately trivial and reduces toλφ4”.If QED suﬀers from complete screening(the Moscow zero),then we expect the NJL model to describe its triviality.One result of this paper is thatλφ4cannot be a good guide to fermionic QED under any circumstances!There are several physical implications that the two bounds onδimply and we discuss some of them brieﬂy below.The wavefunction renormalization constant respects the Lehmann bound:0≤Z≤1.Roughly speaking, Z is the probability that the scalarﬁeld creates a single particle from the vacuum.The limit Z=1 corresponds to a noninteracting theory whereas the compositeness condition,Z=0,sets an upper bound on the eﬀective coupling[17].The anomalous dimensionηdetermines the scaling of Z in the critical region Z∼ξ−η.It describes the scaling of the correlation function in the massless limit:D(x)∼1/|x|d−2+η.Small ηis associated with weak coupling andη=O(1)with the strong coupling limit of the theory.It is related to the exponentδthrough the hyperscaling relationd+2−ηδ=especially visible from the behavior of the correlation functions in the two theories.In the scalar theory, where anomalous dimensions are small,the scaling of the correlation functions is weakly aﬀected at short distances.They behave almost like free particle propagators,D(x)∼1/|x|d−2.For fermions,however, there are nontrivial changes in scaling due to strongly interacting dynamics at short distances.In four-fermi theory,for example,at large-Nη=4−d and D(x)∼1/|x|2irrespective of d.Another consequence of the diﬀerence in the bounds onδconcerns the physics in the broken phase. As an eﬀect of spontaneous symmetry breaking a trilinear coupling is generated and the decayσ→ππis a dominant decay mode in the Goldstone phase.The pole in theσpropagator is buried in the continuum states and the pion state saturates all the correlation functions.This is especially visible in lower dimensions and persists even atﬁnite h.For fermions,however,this is not necessarily the case for the simple reason that theπ−σmass-squared ratio in the broken phase is bounded by1/δ>1/3,away from the chiral limit and for theσ→ππdecay to occur the masses must satisfy M2π/M2σ≤1/4.Thus,for appropriately chosen couplings and bare fermion masses,the decay becomes kinematically forbidden even in the broken phase.Regarding the usage of perturbation theory,the rules are diﬀerent for magnets and fermions.The applicability of perturbation theory to magnets was noted long ago[18].Its success near two and four dimensions is not surprising.Below four dimensions,φ4posesses an infraredﬁxed point at coupling g c∼O(ǫ). This coupling is an upper bound on the renormalized coupling i.e.g R∼ǫ.So,theǫ-expansion is in eﬀect renormalized perturbation theory.The critical exponents receive corrections of the typeδ=3+O(g R),β= 1/2+O(g R),etc..Thus,inφ4the success of perturbation theory is a consequence of the fact that the infraredﬁxed point moves to the origin as d→4.In the non-linearσ-model the critical coupling is an ultravioletﬁxed point that moves to the origin as d→2.The weak coupling phase is at low-temperatures. Due to the presence of Goldstone bosons,all the correlation functions are saturated with massless states and the entire low-temperature phase is massless.Every point is a critical point in the limit of vanishing magneticﬁeld.Thus,the low-temperature expansion is an expansion in powers of T.Terms of the form exp(−M/T)are absent and there is no danger that they will be omitted by using perturbation theory.In this way,in principle,the critical region can be accessed through perturbation theory[18].Clearly,such reasoning can not be applied to fermions simply because the weak coupling phase is symmetric.Thus,no matter how small the coupling is,perturbation theory omits the Goldstone physics as a matter of principle. It can not produce bound states that accompany the chiral transition and its applicability is questionable in general.Especially,it is diﬃcult to imagine how perturbation theory could give a mass ratio that is constant, independent of the bare parameters,once the bare coupling is tuned to the critical value.Even if this were possible,the renormalized coupling would be sensitive to the variation of the bare mass leading to conﬂicting renormalization group trajectories as a consequence.Finally,we comment on one possible use of theδ<3bound for controllingﬁnite size eﬀects in lattice studies of chiral transitions.As was argued in[8],it is convenient to introduce a plotχ−1πversus<¯ψψ>2. The usefulness of this plot becomes clear if we write the critical EOS.From m∼<¯ψψ>δandχ−1π=m/<¯ψψ>,it follows thatχ−1π∼(<¯ψψ>2)(δ−1)/2(15)concave downwards.On a small lattice the order parameter is smaller and pion mass is bigger then in the thermodynamic limit.Thus,small volume distortions are always in the direction of opposite concavity of the plot and the wrong concavity of this plot is a clear sign of the presence ofﬁnite size eﬀects[8].Acknowledgement We wish to acknowledge the discussions with E.Fradkin,A.Patrascioiou and E.Seiler. This work is supported by NSF-PHY92-00148.References[1]See for example The Standard Model Higgs Boson,Edited by M.Einhorn,(North-Holland,Amsterdam, 1991).[2]Y.Nambu,in New Trends in Physics,proceedings of the XI International Symposium on Elementary Particle Physics,Kazimierz,Poland,1988,edited by Z.Ajduk S.Pokorski and A.Trautman(World Scientiﬁc, Singapore,1989);V.Miransky,M.Tanabashi and K.Yamawaki,Mod.Phys.Lett.A4,1043(1989);W. Bardeen,C.Hill and M.Lindner,Phys.Rev.D41,1647(1990).[3]Y.Nambu and G.Jona-Lasinio,Phys.Rev.122,345(1961).[4]M.Aizenman,Comm.Math.Phys.86,1(1982);C.Arag˜a o de Carvalho,S.Caracciolo and J.Fr¨o hlich, Nuc.Phys.B215[FS7],209(1983).[5]M.L¨u scher and P.Weisz,Nuc.Phys.B290[FS20],25(1987)[6]S.K.Ma,in Phase Transitions and Critical Phenomena Vol.6,eds.C.Domb and M.Green(Academic Press,London,1976).[7]S.Hands,A.Koci´c and J.B.Kogut,Phys.Lett.B273(1991)111.[8]A.Koci´c,J.B.Kogut and M.-P.Lombardo,Nuc.Phys.B398,376(1993).[9]K.Wilson and J.Kogut,Phys.Rep.12C,75(1974).[10]See,for example,C.Itzykson and J.-M.Drouﬀe,Statistical Field Theory(Cambridge University Press, 1989);V.Privman,P.C.Hohenberg and A.Aharony,in Phase Transitions and Critical Phenomena Vol.14, eds.C.Domb and J.L.Lebowitz(Academic Press,London,1991).[11]B.Freedman and G.A.Baker Jr,J.Phys.A15(1982)L715;R.Schrader,Phys.Rev.B14(1976)172;B.D.Josephson,Proc.Phys.Soc.92(1967)269,276.[12]E.Brezin,J.-C.Le Guillou and J.Zinn-Justin,in Phase Transitions and Critical Phenomena Vol.6,eds.C.Domb and M.Green(Academic Press,London,1976).[13]T.Eguchi,Phys.Rev.D17,611(1978).[14]S.Hands,A.Koci´c and J.B.Kogut,Ann.Phys.224,29(1993).[15]A.Koci´c,J.B.Kogut and K.C.Wang,Nucl.Phys.B398(1993)405.[16]S.Kim,A.Koci´c and J.Kogut(unpublished)[17]S.Weinberg,Phys.Rev.130,776(1963).[18]E.Brezin and J.Zinn-Justin,Phys.Rev.B14,3110(1976).Figure captions1.Susceptibility ratio as a function of magneticﬁeld(bare mass)forﬁxed values of the temperature (coupling).2.The behavior of the critical mass ratio,R(t=0,h)=1/δ,for diﬀerent values of d,in a)magnets and b) in the case of chiral transition.3.Critical mass ratio,R(t=0,h),for fermions and magnets in four dimensions over extended range of magneticﬁeld(mass).This figure "fig1-1.png" is available in "png" format from: /ps/hep-th/9312007v1This figure "fig1-2.png" is available in "png" format from: /ps/hep-th/9312007v1This figure "fig1-3.png" is available in "png" format from: /ps/hep-th/9312007v1。

Local scale-invariance in disordered systems

1
Recall that physical ageing as it is understood here comes from reversible microscopic processes, whereas chemical or biological ageing may come from the action of essentially irreversible (bio-)chemical processes.
1 Introduction
Understanding cooperative phenomena far from equilibrium poses one of the most challenging research problems of present-day many-body physics. At the same time, the practical handling of many of these materials has been pushed to great sophistication, and a lot of practical knowledge about them exists since prehistoric times. Paradigmatic examples of such system are glasses. In many cases, they are made by rapidly cooling (‘quenching’) a molten liquid to below some characteristic temperature-threshold. If this cooling happens rapidly enough, normal crystallization no longer takes place and the material remains in some non-equilibrium state. These non-equilibrium states may at ﬁrst and even second sight look very stationary – everyone has probably seen in archaeological museums intact specimens of Roman glass or even older tools from the Paleolithic or old-stone-age – after all, obsidian or ﬁre-stone is a quenched volcanic melt. But since the material is not at equilibrium, at least in principle it is possible (and it does happen very often in practice) that over time the properties of the material change - in other words, the material ages.1 The properties of such non-equilibrium systems depend on the time – their age – since they were brought out of equilibrium and this is colloquially referred to as ageing behaviour.

Critical Exponents of the Four-State Potts Model

β
(12)
For d = 2, from (11) and (13), the exact values for the scaling and critical exponents of the four-state Potts model are yt = 3 , yh = 2
15 ,α 8 2 =3 ,β = 1 ,γ 12 2 =7 , δ = 15, ν = 3 , and η = 6 1 4
<i,j>
(1)
2
where δ (σi , σj ) =
1,
if σi = σj ,
0, if σi = σj ,
and σi , σj = 1, 2, ..., q . The latent heat at the critical temperature Tc is given by [6] L = 2J (1 + q − 2 ) tanh where θ is deﬁned by 2 cosh θ = q 2 . The zero-ﬁeld (spontaneous) magnetization at Tc is [12] M0 = 1 − x2n−1 , q > 4, 2n n=1 1 + x
Abstract
The critical exponents of the four-state Potts model are directly derived from the exact expressions for the latent heat, the spontaneous magnetization, and the correlation length at the transition temperature of the model. PACS numbers: 05.50.+q, 05.70.–a, , 75.10.Hk

Monte Carlo simulations of the critical properties of the restricted primitive model

a r X i v :c o n d -m a t /0409356v 2 [c o n d -m a t .s t a t -m e c h ] 24 S e p 2004Monte Carlo simulations of the critical properties of therestricted primitive modelJean-Michel Caillol Laboratoire de Physique Th´e orique UMR 8267,Bˆa t.210Universit´e de Paris-Sud 91405Orsay Cedex,France ∗(Dated:February 2,2008)Abstract Recent Monte Carlo simulations of the critical point of the restricted primitive model for ionic solutions are reported.Only the continuum version of the model is considered.A ﬁnite size scaling analysis based in the Bruce-Wilding procedure gives critical exponents in agreement with those of the three-dimensional Ising universality class.An anomaly in the scaling of the speciﬁc heat with system size is pointed out.PACS numbers:1I.INTRODUCTIONThe primitive model(PM)for electrolytes,molten salts,collo¨ids,etc is a mixture of M species of charged hard spheres living either on a lattice or within a continuous volume of real space.In this paper we shall focus only on the oﬀ-lattice version of the model.The simplest version of the PM consists of a binary mixture(i.e.M=2)of positive and negative charged hard spheres±q all with the same diameterσ.Under this form the model which is thought to be the prototype of many ionicﬂuids has been christened the restricted primitive model(RPM).A thermodynamic state of the RPM is entirely speciﬁed by a reduced density ρ∗=Nσ3/V(N number of ions,V volume)and a reduced temperature T∗=kTσ/q2(k Boltzmann’s constant).The RPM undergoes a liquid-vapor transition which has been studied extensively these last past years by means of Monte Carlo(MC)simulations and various theoretical ap-proaches.In particular the behavior of the system at its critical point(CP)has been the subject of a huge amount of numerical and theoretical studies.The question is obviously of great importance since it is reasonable to assume that real electrolytes-or at last a large class of them-and the RPM belong to the same universality class which dictates a similar critical behavior.It is perhaps the right place to note that the main feature of ionic solutions is that the pair potential between two ions i and j at a distance r ij=|r i−r j|which reads asq i q jv(r ij)=support this claim.Most numerical studies of the CP of the oﬀ-lattice version of the RPM were performed by the Orsay group and I would like to review our contributions towards a better understanding of the critical properties of this model in the lines below.II.A BRIEF HISTORICAL SUR VEYQuite generally,a single componentﬂuid will undergo a liquid vapor transition if the pair potential which is assumed to represent the molecular interactions is(suﬃciently)attractive at large distances.From this point of view the situation is not so clear in the case of the RPM(cf eq.(1.1))and the very existence of the transition is not guaranteed.Several studies were necessary to clarify this point and a brief historical survey is worthwhile.Theﬁrst evidence that the RPM actually undergoes a liquid-vapor transition can be tracked back to two papers of Chasovkikh and Vorontsov-Vel’Yaminov(CVVY)published as soon as in19766,7.These authors performed isobaric MC simulations and found a tran-sition with a CP located at T∗c=0.095,ρ∗c=0.24.Several years after(in1991)Valleau studied three isotherms of the RPM with his method of the density scaling MC and ob-tained a diﬀerent location for the CP,namely T∗c=0.07,ρ∗c=0.078.Subsequently(in1992) Panagiotopoulos9obtained still diﬀerent results,i.e.T∗c=0.056,ρ∗c=0.04,by performing MC simulations in the Gibbs ensemble(GE),at the moment a powerful new method of simulation which he had invented a little bit earlier10.Subsequent GE simulations using an improved biased MC sampling11yielded Panagiotopoulos and Orkoulas to the new estimate T∗c=0.053,ρ∗c=0.025.Finally,making use myself of the Gibbs ensemble combined with the use of hyperspherical geometries I obtained rather T∗c=0.057,ρ∗c=0.0412,13.Commenting on this striking dispersion of the MC data Prof M.Fisher talked once of the ”sad street of numerical simulations”.This was in1999,at the SCCS conference,St Malo, France and,at this point of the story,I must agree with him retrospectively.However many advances have been done since.Before giving an account of these new achievements some comments are in order.•(i)All the MC studies conﬁrm the existence of a liquid vapor transition for the RPM.It seems to take place at unusually low densities and temperatures.Caillol and Weis give further support for such a low critical temperature14.Moreover it turns out that the coexistence curve is very dissymmetric9,11,12.•(ii)The MC simulation of ionic systems is a numerical challenge due to the long range of Coulomb potential.In order to deal with this,some caution is needed.Thus,in the case of MC simulations performed in a cubic box with periodic boundary conditions (PBC),one must use Ewald potentials in order to obtain the correct physics15,16,17.The point is that the Ewald potential is the solution of Poisson equation in a cubico-periodical geometry17and many properties of ionicﬂuids(electro-neutrality,screening, etc)are a consequence of this fact.In their MC simulations CVVY and Valleau considered truncated Coulomb potentials and very small samples of N=32particles which yields quantitatively wrong results.By contrast the data of Panagiotopoulos et al.9,11are more reliable since Ewald sums have been used.The same remark apply to my simulations which were performed on a4D sphere(a hypersphere for short)by considering interactions obtained by solving Poisson equation in this geometry.This alternative method of simulation is therefore also indisputably correct,moreover it is much more eﬃcient.The rough agreement observed between the simulations of refs.9,11and12both involving the same number of ions,i.e.N=512,is therefore not fortuitous.•(iii)None of the above mentioned studies took correctly into accountﬁnite size ef-fects which are of an overwhelming importance near a CP.These eﬀects aﬀect the behavior ofﬁnite systems as soon as the correlation length of the critical densityﬂuc-tuations is of the same order of magnitude than the size of the simulation box.In the simulations9,11,12some”apparent”critical temperature T∗c has been measured which could be very diﬀerent from its inﬁnite volume limit T∗c(∞).In order to extract from MC simulations the critical behavior of the RPM in the thermo-dynamic limit(i.e.the critical exponents)and also the inﬁnite volume limit of T∗c andρ∗c it is necessary to perform an analysis of the MC data in the framework of theﬁnite size scaling (fss)theory which is part of the renormalization group(RG)theory18,19.In this approach one needs to work in the Grand Canonical(GC)ensemble rather than in the Gibbs ensemble which is ill adapted for a fss analysis.Subsequent MC simulations on the RPM were thus all performed in this ensemble.Panagiotopoulos and coworkers turned their attention to the lattice version of the RPM whereas the Orsay group continued to work on its oﬀ-lattice version.III.FINITE SIZE SCALING ANALYSIS OF MC DATAA.Scalingﬁelds and operatorsStarting with the seminal work of Bruce and Wilding(BW)20,21,22simulation results for the critical behavior ofﬂuids have customarily been analyzed along the lines of the so-called revised scaling theory of Rehr and Mermin23.In this approach oneﬁrst deﬁnes scalingﬁelds and operators aimed at restoring the particle-hole symmetry and therefore to map the the ﬂuid onto a magnetic sytem with Ising-like symmetry.The two relevant scalingﬁelds h(the strong orderingﬁeld)andτ(the weak thermal ﬁeld)are assumed to be linear combinations of deviations from their critical values of the chemical potentialµand the inverse temperatureβ=1/T(reduced values are assumed henceforward).One thus hash=µ−µc+r(β−βc)τ=βc−β+s(µ−µc),(3.1)where r and s are theﬁeld mixing parameters which deﬁne the mapping.Of course relations (3.1)are valid only in the vicinity of the CP.The conjugate scaling operators M and E are then deﬁned as<M>=1∂hlnΞ=1V∂1−sr(<u>−r<ρ>),(3.2)whereΞis the GC partition function of the RPM,ρthe total number density,and u the internal energy per unit volume.Brackets<...>denote GC averages.M is the order parameter(magnetization)of the magnetic system associated with theﬂuid and E its mag-netic energy.E should be invariant under the transformations(M,h)→(−M,−h)for appropriate values of s and r.In this framework the coexistence curve is therefore deﬁned by the eq.h=0.The revised scaling of Rehr and Mermin implies the analyticity of the coexistence chemical potentialµ(T)at T∗c.Although this is the case for some peculiar lattice gas models with ”hidden”symmetries there is no reason that in general,forﬂuid systemsµ(T)should lack asingularity as recognized already by Rehr and Mermin23and emphasized more recently by Fisher and co-workers40,41,42.B.The scaling hypothesisA central role in the subsequent fss analysis is played by the GC joint distribution P L(M,E)∝P L(ρ,u)for the scaling operators M and E.Following BW20,21,22we will assume that,in the immediate vicinity of the CP,P L(M,E)obeys to the following scaling law:P L(M,E)=a−1M a−1EL d−yτL d−y h P(a−1ML d−y hδM,...a−1EL d−yτδE,a M L y h h,a E L yττ,a i L y i,...),(3.3) where L are the linear dimensions of the system(taken as V1/3,where V is the volume of the simulation box,either a cube or a hypersphere).I have denoted byδM≡M−<M>c andδE≡E−<E>c the deviations of the scaling operators from their value at criticality. The cornerstone of this scaling hypothesis is that the function P which enters eq.(3.3)is universal in the sense that it depends only upon the universality class of the model and of the type of geometry considered.The constants a M,a E,and a i are system dependent constants which are deﬁned in such a way that P has unit variance.Finally,the renormalization exponents y h,yτ,and y i which enter eq.(3.3)are deﬁned asy h=d−β/νyτ=1/νy i=−θ/ν(3.4) in terms of the usual critical exponents:•βexponent of the orderingﬁeld,i.e.<δM>∼|τ|βfor T∗<T∗c at h=0•νexponent of the correlation length,i.e.ξ∼|τ|−ν•θWegner’s correction-to-scaling exponent(ﬁrst irrelevant exponent).The scaling hypothesis(3.3)was established on a solid RG basis for Ising-like systems24and received substantial supports from MC studies25.We stress once again that the coexistence curve is determined in this approach by the condition h=0and that,at coexistence,the order parameter distribution P L(M)should be an even function of M.In practice this symmetry requirement can be satisﬁed by tuning the two parameters(µ,s)at a givenβ. We now concentrate our attention on the scaling behavior of the histogram P L(M).C.The matching procedureIntegrating both sides of eq.(3.3)over E oneﬁnds that,along the coexistence line h=0 one hasP L(M)=a−1M L d−y h P(a−1ML d−y hδM,a E L yττ,a i L y i),(3.5)where,in the r.h.s.the dependence of the universal function P upon h has been discardedfor clarity.Let us deﬁne now x=a−1ML d−y hδM,then,assumingτ∼0and L∼∞a Taylor expansion of eq.(3.5)yieldsP L(M)=a−1ML d−y h P∗(x)+a E L yττ P∗1(x)+a′2E L2yττ2 P∗2(x)+...+a i L y i P∗3(x)+... ,(3.6) where the various P∗entering the r.h.s.are universal functions.Note that,for L=∞the normalized orderingﬁeld distribution P L(M)collapses onto an universal function P∗(x)at τ=0.For Lﬁnite but large P L(M)collapses approximately onto P∗(x)at some apparent τL∝L−yτ+y i.Since for h=0one hasτ∝β−βc then the matching of the histogram P L(M) onto the universal function P∗(x)should occur at some apparent temperature T∗c(L)scaling with system size asT∗c(∞)−T∗c(L)∝L−(θ+1)/ν+....,(3.7) where T∗c(∞)denotes the inﬁnite volume limit of the critical temperature.D.Technical detailsTo assess the critical behavior and the critical parameters of the system,we need,in a ﬁrst step,to locate the coexistence curve h=0.At a given temperatureβclose toβc theordering distribution function P L(M)depends solely on the chemical potentialµand the mixing parameter s.At coexistence,the value of(µ,s)can be obtained unambiguously by requiring that P L(M)is symmetric in M−<M>22.Tuning at willµand s at givenβrequires to know the joint histogram P L(M,E)∝P L(ρ,u)for a continuous set of values of µat a givenβ.Moreover,since this analysis must be performed at diﬀerentβone needs in fact to know P L(M,E)for a continuous set of values of(β,µ)in the critical region.This technical diﬃculty is circumvented by using the multiple histogram reweighting proposed by(ρ,u)for a continuous Ferrenberg and Swensen26,27,28.With this method one can obtains Pβ,µL(ρ,u),i=1,...,R obtained set of values of(β,µ)from the knowledge of R histograms Pβi,µiLby performing R distinct MC simulations in the R(neighbor)thermodynamic states(βi,µi).Since the precision of the simulations ofﬂuid systems has still not reached that obtained in the MC simulations of Ising like systems it is impossible to construct ex nihilo theﬁxed point universal distribution P∗(x).In refs.29,30our attempts to match P L(M)on P∗(x) were realized by using the estimate of P∗is(x)made by Hilfer and Wilding32for the3D Ising model.Two new-and better-estimates of P∗is(x)obtained by Tsypin and Bl¨o te33for the 3D Ising model and the spin-1Blume-Capel model were considered in ref.31.The discussion is postponed to next section.IV.RESULTSA.General discussionIt turns out that theﬁeld mixing parameter s of the RPM is practically independent of the temperature and of the size L of the system.Its magnitude,s∼−1.4629,30,31,is much higher than for neutralﬂuids(typically s∼0.02for square well or Lennard-Jonesﬂuids34) which explains the large dissymmetry of the liquid-vapor coexistence curve of the RPM.The collapse of the ordering operator distribution P L(M)onto the universal ordering distribution P∗is(x)given by the Blume-Capel model33is depicted in Fig.1for four diﬀerent values of the volume ranging from V/σ3=5000to V/σ3=40000,i.e.up to a linear size L/σ=34.At volume V/σ3=5000a mismatch is observed at the lowest values of M due to an inadequate sampling of of the low density conﬁgurations at small volume.The overall good agreement leads us to conclude that the universality class of the RPM is that of the3D Ising model.The reduced apparent critical temperature T∗c(L)versus the size L of the system(in reduced units)has been plotted in Fig.2.Depending on the choice made for the universal ordering distribution P∗is(x)one obtains two sets of values of T∗c(L)from which T∗c(∞)can be obtained by making use of eq.(3.7).One obtains T∗c(∞)=0.04917±0.00002using P∗is(x)derived from the Blume-Capel model and T∗c(∞)=0.04916±0.00002using P∗is(x) obtained for the3D-Ising model.The approximate P∗is(x)of Hilfer and Wilding yields slightly diﬀerent results.Note that in all cases we have used the Ising valuesν=0.63035 andθ=0.5336of the critical exponents.The previous analysis merely establishes the compatibility of the MC data with an Ising-like criticality.One can try to go beyond by considering the scaling behavior of the Binder cumulant<δM2>2LQ B(L)=•Conversely,ﬁxing Q c=0.623andθ=0.53one obtainsβc=1/0.04918andν∼0.63±0.03.The variations of Q B(L)as a function ofβfor the diﬀerent volumes is shown in Fig.3. Although there is considerable spread in the intersection points due to correction-to-scaling contributions,the corresponding values of Q c are close to the Ising value Q c=0.623and permit to rule out meanﬁeld behavior(i.e.Q c=0.45738).Further support for Ising criticality is provided by the behavior of<δM2>at T∗c(L). According to the scaling hypothesis(3.6)it should scales as L2β/νwith system size.From the slope of the curve displayed in Fig.4one obtainsβ/ν=0.52in accord with the3D Ising value(0.517)and in clear contrast with the classical value1.In summary,our fss analysis leads to an estimate of the critical exponentsνandβ/νand the Binder cumulant Q c based on the sole knowledge of the critical temperature and the renormalization exponentθ.Within the numerical uncertainties these values are compatible with Ising-like criticality.Our conclusion is that the RPM,as ordinary neutralﬂuids,belongs to the universality class of the Ising model.A complete discussion of our MC data is out of the scope of the present paper and can be found in ref.31.For completeness I give below the values obtained for the critical temperature,chemical potentials and densities(the critical pressure is largely unknown):•T⋆c=0.04917±0.00002•ρ⋆c=0.080±0.005•µ⋆c=−13.600±0.005B.The speciﬁc heatThe revised scaling theory of Rehr and Mermin which is the framework of our fss analysis is however not the most general scaling theory which can be proposed for aﬂuid system lacking the”particle-hole”symmetry.Its main weakness,as recognized already by Rehr and Mermin23,Yang and Yang39,and more recently by Fisher and co-workers40,41,42,is that it assumes the analyticity of the chemical potential at coexistenceµ(T)at the critical point. The more general scaling assumption should yield singularities for bothµ(T)and p(T)asT∗→T∗c.Let us examine the consequences of these singularities on the behavior of the speciﬁc heat capacity at constant volume C V.In the two phase region it can be rewritten as39C V=V T ∂2p∂T2V=C p+Cµ,(4.3) where C p(not to be confused with the speciﬁc heat capacity at constant pressure)and Cµ(not to be confused with the speciﬁc heat capacity at constant chemical potential)denote the two contributions to C V.I stress that,in eq.(4.3)p(T)andµ(T)denote the pressure and the chemical potential at coexistence.The formula can be used for any densityρg(T)<ρ<ρl(T)within the two phase region(ρg(T)andρl(T)being the densities of the gas and the liquid at coexistence respectively).In the revised scaling theory only C p diverges as |T∗−T∗c|−αwhereas one expects a divergence of both C p and Cµ(both as|T∗−T∗c|−α).In Fig.5I display the curves Cµ(T)and C V(T)along the locusχNNN∝<(N−<N>)3>=0,(4.4) for the four volumes considered in our last MC simulations31.Although the peak posi-tions shift correctly as∝L−1/νwith system size,in accord with fss theory18,19,there is no detectable scaling of the heights of the peaks which should scale as Lα/νwith L.These ob-servations corroborate similar results obtained by Valleau and Torrie43,44.In particular Cµdoes not show any anomaly which should challenge the use of eqs.(3.1)for the scalingﬁelds.A possible explanation for the non singular behavior of C V(T)is that the amplitude of the singular term in C V(T)is small in the RPM and that the speciﬁc heat is dominated by its regular part.Note however that,the peak heights in C V(T)/V would scale,assuming Ising value forαonly by a factor2α/ν∼1.12when doubling the linear dimensions of the system. It is possible that such a small eﬀect is not detectable within the statistical uncertainties of our calculations.V.CONCLUSIONIn this paper which resumes my talk at the Lviv NATO workshop I have described recent attempts to elucidate the nature of the critical behavior of the RPM model for ionicﬂuids,prototype of a system governed by long range Coulomb interactions by means of MC simulations.After endeavor over more than a decade we have now reached a point where we can claim conﬁdently that the RPM belongs to the same universality class than the3D Ising model.The critical values of non-universal quantities such as the temperature and the chemical potential were established with a high accuracy whereas the uncertainties on the critical density are more signiﬁcant,and the critical pressure is unknown.The behavior of the constant volume speciﬁc heat gives no indication of the expected Lα/νscaling of the peak height within the range of system sizes considered in the most recent simulations.Recent investigations of Camp and co-workers45where diﬀerences in the behavior of C V in the canonical and the GC ensemble are reported have emphasized this problem.At the moment it is diﬃcult to explain this unexpected behavior of the speciﬁc heat.I have only discussed the properties of the continuous version of the RPM.The phase diagram of the various lattice versions of the model is in fact more complex46,47and was not described here due to a lack of place.I have also excluded from my presentation assymetric versions,either in charge or/and in size,of the continuum or lattice versions of the primitive model.The interested reader should consult recent works of Panagiotopoulos et al.48,49,50 and de Pablo et al.51,52,53.AcknowledgmentsI thank the organizers of the NATO workshop”Ionic Soft Matter”held in Lviv,Ukraine in April2004,Pr D.Henderson and Pr M.Holovko for having invited me to give a talk.It is a pleasure to acknowledge many scientiﬁc discussions with Pr.I.M.Mryglod,O.Patsahan, J.-P.Badiali,P.J.Camp,W.Schr¨o er,and J.Staﬁej.Figure Captions•ﬁgure1:Collapse of the ordering distribution P L(M)onto the universal Ising order-ing distribution P∗is(x)for V/σ3=5000,T∗c(L)=0.004934,s=−1.45;V/σ3=10000, T∗c(L)=0.004926,s=−1.465;V/σ3=20000,T∗c(L)=0.004921,s=−1.47;and V/σ3=40000,T∗c(L)=0.004922,s=−1.43. P∗is(x)(solid circles)if the MC result of Tsypin and Bl¨o te(Ref.33)obtained for the Blume-Capel model.The scaling variable is x=a−1Lβ/νδM where a M is chosen in such a way that P L(x)has unit variance.M•ﬁgure2:The apparent critical temperature T∗c(L)as a function of L−(θ+1)/νwithθ=0.53,ν=0.630obtained by matching the universal ordering distribution calculatedfor the Blume-Capel model(top)and the Ising model(bottom).Extrapolating by linear least squareﬁt to the inﬁnite volume limit yields T∗c(∞)=0.04917±0.00002 (top)and T∗c(∞)=0.04916±0.00002(bottom).•ﬁgure3:Variation of Q B(L)as a function of the inverse temperatureβfor the diﬀerent volumes considered in ref.31.From top to bottom V/σ3=40000,20000,10000, and5000respectively.The symbols are the MC data and the lines theﬁts obtained by means of eq.(4.2).•ﬁgure4:Variations of ln<δM2>at T∗c(L)as a function of ln L.The slope of the linear least squareﬁt is2β/ν=1.04.•ﬁgure5:Variations of the total speciﬁc heat at constant volume C V/V and the contribution Cµ/V with temperature along the locusχNNN=0at volumes V/σ3= 5000,10000,20000,and40000(from left to right).∗Electronic address:Jean-Michel.Caillol@th.u-psud.fr1Weing¨a rtner H.,Schr¨o er W.//Adv.Chem.Phys.,2001,vol.116,p.1.2Fisher M.E.//J.Stat.Phys.,1994,vol.75,p.1.3Stell G.//J.Stat.Phys.,1995,vol.78,p.197.4Fisher M.E.//J.Phys.:Condens.Matter,1996,vol.8,p.9103.5Stell G.//J.Phys.:Condens.Matter,1996,vol.8,p.9329.6Vorontsov-Vel’Yaminov P.N.,Chasovkikh B.P.//High.Temp.,1975,vol.13,p.1071.7Chasovkikh B.P.,Vorontsov-Vel’Yaminov P.N.//High.Temp.,1976,vol.14,p.1174.8Valleau J.//J.Chem.Phys.,1991,vol.95,p.584.9Panagiotopoulos A.Z.//Fluid Phase Equilibria,1992,vol.76,p.97.10Panagiotopoulos A.Z.//Mol.Phys.,1987,vol.61,p.813.11Orkoulas G.,Panagiotopoulos A.Z.//J.Chem.Phys.,1994,vol.101,p.1452.12Caillol J.-M.//J.Chem.Phys.,1994,vol.100,p.2169.13Caillol J.-M.//J.Phys.:Condens.Matter,1994,vol.6,p.A171.14Caillol J.-M.,Weis J.-J.//J.Chem.Phys.,1995,vol.102,p.7610.15Brush S.G.,Sahlin H.L.,Teller E.//J.Chem.Phys.,1966,vol.45,p.2102.16de Leeuw S.W.,Perram S.W.,Smith E.R.//Proc.R.Soc.London A,1980,vol.373,p.27. 17Caillol J.-M.//J.Chem.Phys.,1999,vol.111,p.6528.18Privman V.,ed.,Finite Size Scaling and Numerical Simulation in Statistical Systems,Singapore, World Scientiﬁc,1990.19Cardy J.L.,ed.,Finite Size Scaling,Amsterdam,North Holland,1988.20Bruce A.D.,Wilding N.D.//Phys.Rev.Lett.,1992,vol.68,p.193.21Wilding N.D.,Bruce A.D.//J.Phys.:Condens.Matter,1992,vol.4,p.3087.22Wilding N.D.//Phys.Rev.E,1995,vol.52,p.602.23Rehr J.J.,Mermin N.D.//Phys.Rev.A,1973,vol.8,p.472.24Bruce A.D.//J.Phys.C:Solid State Phys.1981,vol.14,p.3667.25Nicolaides D.,Bruce A.D.//J.Phys.A:Math.Gen.1988,vol.21,p.233.26Ferrenberg A.M.,Swendsen R.R.//Phys.Rev.Lett.,1988,vol.61,p.2635.27Ferrenberg A.M.,Swendsen R.R.//Phys.Rev.Lett.,1989,vol.63,p.1195.28Deutsch H.-P.//J.Stat.Phys.,1992,vol.67,p.1039.29Caillol J.-M.,Levesque D.,Weis J.-J.//Phys.Rev.Lett.,1996,vol.77,p.4039.30Caillol J.-M.,Levesque D.,Weis J.-J.//J.Chem.Phys.,1997,vol.107,p.1565.31Caillol J.-M.,Levesque D.,Weis J.-J.//J.Chem.Phys.,2002,vol.116,p.10794.32Hilfer R.,Wilding N.D.//J.Phys.A,1995,vol.28,p.L281.33Tsypin M.M.,Bl¨o te H.W.J.//Phys.Rev.E,2000,vol.62,p.73.34Caillol J.-M.//J.Chem.Phys.,1998,vol.109,p.4885.35Ferrenberg A.M.,Landau.D.P.//Phys.Rev.B,1991,vol.44,p.5081.36Chen J.H.,Fisher M.E.,Nickel B.G.//Phys.Rev.Lett.,1982,vol.48,p.630.37Bl¨o te H.W.J.,Luijten E.,Heringa J.R.//J.Phys.A,1995,vol.28,p.6289.38Luijten E.,Bl¨o te H.W.J.//Phys.Rev.Lett.,1996,vol.76,p.1557.39Yang C.N.,Yang C.P.//Phys.Rev.Lett.,1964,vol.13,p.303.40Fisher M.E.,Orkoulas G.//Phys.Rev.Lett.,2000,vol.85,p.696.41Orkoulas G.,Fisher M.E.,Ust¨u n C.//J.Chem.Phys.,2000,vol.113,p.7530.42Orkoulas G.,Fisher M.E.,Panagiotopoulos A.Z.//Phys.Rev.E,2001,vol.63,p.051507.43Valleau J.,Torrie G.//J.Chem.Phys.,1998,vol.108,p.5169.44Valleau J.,Torrie G.//J.Chem.Phys.,2002,vol.117,p.3305.45Daub C.D.,Camp P.J.,Patey G.N.//J.Chem.Phys.,2003,vol.118,p.4164.46Luijten E.,Fisher M.E.,Panagiotopoulos A.Z.//J.Chem.Phys.,2001,vol.114,p.5468.47Panagiotopoulos A.Z.//J.Chem.Phys.,2002,vol.116,p.3007.48Panagiotopoulos A.Z.,Fisher M.E.//Phys.Rev.Lett.,2002,vol.88,p.045701-1.49Cheong D.W.,Panagiotopoulos A.Z.//J.Chem.Phys.,2003,vol.119,p.8526.50Romero-Enrique J.M.,Rull L.F.,Panagiotopoulos A.Z.//Phys.Rev.E,2003,vol.66,p.041204. 51Yan Q.,de Pablo J.J.//J.Chem.Phys.,2001,vol.114,p.177.52Yan Q.,de Pablo J.J.//Phys.Rev.Lett.,2002,vol.88,p.095504-1.53Yan Q.,de Pablo J.J.//J.Chem.Phys.,2002,vol.116,p.2697.00.00020.00040.00060.00080.049100.049200.049300.04940T*00.00020.00040.00060.0008L −(θ+1)/ν0.049100.049200.049300.04940T*。

A scale-invariant cohesive crack model

Department of Structural and Geotechnical Engineering, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129 Torino, Italy
Abstract The ®ctitious crack model by Hillerborg is the most widely used model to simulate damage and fracture in concrete structures. Its peculiar capability to capture the evolution of the cracking process is accompanied by its simplicity. However, some aspects of the phenomenon are not considered in the model, for instance the size-dependence of the nominal quantities involved in the cohesive law. This aects the predictive capabilities of the model, when it is used to extrapolate results from small laboratory specimens to full-scale structures. In this paper, a scale-independent cohesive law is put forward, which overcomes these drawbacks and permits to obtain a unique constitutive relationship for softening in concrete. By assuming damage occurring in a fractal band inside the specimen, nominal stress, crack opening displacement and nominal fracture energy become scale dependent. Hence they should be substituted by fractal quantities, which are the true material constants. A mutual relation among their fractal physical dimensions puts a strong restriction to disorder. By varying the scaling exponents of the kinematical quantities, a clear transition from discrete to smeared cracking can be obtained. The fractal cohesive law is eventually applied to some tensile test data, showing perfect agreement between theory and experiments. Ó 2001 Elsevier Science Ltd. All rights reserved.

BEC-BCS crossover, phase transitions and phase separation in polarized resonantly-paired su

arXiv:cond-mat/0607803v2 [cond-mat.supr-con] 23 Jan 2007
Background and Motivation
One of the most in the study of degenerate atomic gases has been the observation1–9 of singlet paired superﬂuidity of fermionic atoms interacting via an s-wave Feshbach resonance10–18 . A crucial and novel feature of such experiments is the tunability of the position (detuning, δ ) of the Feshbach resonance, set by the energy of the diatomic molecular (“closed-channel”) bound state relative to the open-channel atomic continuum, which allows a degree of control over the fermion interactions that is unprecedented in other (e.g., solid-state) contexts. As a function of the magnetic-ﬁeld controlled detuning, δ , fermionic pairing is observed to undergo the Bose-Einstein condensate to Bardeen-Cooper-Schrieﬀer (BEC-BCS) crossover19–29 between the Fermi-surface momentum-pairing BCS regime of strongly overlapping Cooper pairs (for large positive detuning) to the coordinate-space pairing BEC regime of dilute Bosecondensed diatomic molecules (for negative detuning). Except for recent experiments30–35 that followed our original work36 , and a wave of recent theoretical37–83 activity, most work has focused on the case of an equal mixture of two hyperﬁne states (forming a pseudo-spin 1/2 system), observed to exhibit pseudo-spin singlet superﬂuidity near an s-wave Feshbach resonance. Here we present an extensive study of such systems for an unequal number of atoms in the two pairing hyperﬁne states, considerably extending results and calculational details beyond those reported in our recent Letter36 . Associating the two pairing hyperﬁne states with up (↑) and down (↓) projections of the pseudo-spin 1/2, the density diﬀerence δn = n↑ − n↓ between the two states is isomorphic to an imposed “magnetization” m ≡ δn (an easily controllable experimental “knob”), with the chemical potential dif-

Novel type of phase transition in a system of self-driven particles

Abstract
A simple model with a novel type of dynamics is introduced in order to investigate the emergence of self-ordered motion in systems of particles with biologically motivated interaction. In our model particles are driven with a constant absolute velocity and at each time step assume the average direction of motion of the particles in their neighborhood with some random perturbation (η) added. We present numerical evidence that this model results in a kinetic phase transition from no tran |va| = 0) to ﬁnite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous since |va| is found to scale as (ηc − η)β with β ≃ 0.45.
——————–
One of the most interesting aspects of many particle systems is that they exhibit a complex cooperative behavior during phase transition [1]. This remarkable feature of equilibrium systems has been studied in great detail for the last couple of decades leading to a deeper understanding of processes which may take place in an assembly

Continuous Quantum Phase Transitions

a r X i v :c o n d -m a t /9609279v 1 29 S e p 1996Continuous Quantum Phase TransitionsS.L.SondhiDepartment of Physics,Princeton University,Princeton NJ 08544S.M.Girvin Department of Physics,Indiana University,Bloomington IN 47405J.P.Carini Department of Physics,Indiana University,Bloomington IN 47405D.Shahar Department of Electrical Engineering,Princeton University,Princeton NJ 08544February 1,2008AbstractA quantum system can undergo a continuous phase transition at the absolute zero of temperature as some parameter entering its Hamiltonian is varied.These transi-tions are particularly interesting for,in contrast to their classicalﬁnite temperature counterparts,their dynamic and static critical behaviors are intimately intertwined. We show that considerable insight is gained by considering the path integral descrip-tion of the quantum statistical mechanics of such systems,which takes the form of the classical statistical mechanics of a system in which time appears as an extra dimension.In particular,this allows the deduction of scaling forms for theﬁnite temperature behavior,which turns out to be described by the theory ofﬁnite size scaling.It also leads naturally to the notion of a temperature-dependent dephas-ing length that governs the crossover between quantum and classicalﬂuctuations. We illustrate these ideas using Josephson junction arrays and with a set of recent experiments on phase transitions in systems exhibiting the quantum Hall eﬀect.CONTENTSI.Quantum Statistical Mechanics:Generalities6A.Partition Functions and Path Integrals7B.Example:1D Josephson Junction Arrays9C.Quantum-Classical Analogies12D.Dynamics and Thermodynamics15 II.Quantum Phase Transitions16A.T=0:Dynamic Scaling16B.T=0:Finite Size Scaling20C.The Quantum-Classical Crossover and the Dephasing Length23 III.Experiments:QPTs in Quantum Hall Systems25A.Temperature and Frequency Scaling30B.Current Scaling32C.Universal Resistivities35D.Unresolved Issues36 IV.Concluding Remarks,Other Systems37 ACKNOWLEDGMENTS38 Appendix A40 References41A century subsequent to Andrews’s discovery of critical opalescence1in carbon dioxide, continuous phase transitions continue to be a subject of great interest to physicists.The appeal of the subject is twofold.First,the list of systems that exhibit interesting phase transitions continues to expand;it now includes the Universe itself!Second,the formal theory of equilibrium phase transitions has found applications in problems as diverse as constructingﬁeld and string theories of elementary particles,the transition to chaos in dynamical systems,and the long time behavior of systems out of equilibrium.Our purpose in this Colloquium is to give a brief and qualitative account of some basic features of a species of phase transitions,2termed‘Quantum Phase Transitions’(QPTs), that have attracted much interest in recent years.These transitions take place at the absolute zero of temperature,where crossing the phase boundary means that the quantum ground state of the system changes in some fundamental way.This is accomplished by changing not the temperature,but some parameter in the Hamiltonian of the system.This parameter might be the charging energy in Josephson junction arrays(which controls their superconductor-insulator transition),the magneticﬁeld in a quantum Hall sample(which controls the transition between quantized Hall plateaus),doping in the parent compound of a high T c superconductor(which destroys the antiferromagnetic spin order),or disorder in a conductor near its metal-insulator transition(which determines the conductivity at zero temperature).These and other QPTs raise new and fascinating issues for theory and experiment,most notably the inescapable necessity of taking quantum eﬀects into account.Exactly what quantum eﬀects are at issue is a bit subtle.As a corollary of our deﬁnition, allﬁnite temperature3transitions are to be considered“classical”,even in highly quantummechanical systems like superﬂuid helium or superconductors.It is not that quantum me-chanics is unimportant in these cases,for in its absence there would not be an ordered state, i.e.the superﬂuid or superconductor.Nevertheless,suﬃciently close to the critical point quantumﬂuctuations are important at the microscopic scale,but not at the longer length scales that control the critical behavior;in the jargon of statistical mechanics,quantum mechanics is needed for the existence of an order parameter4but it is classical thermal ﬂuctuations that govern it at long wavelengths.For instance,near the superﬂuid‘lambda’transition in4He,the order parameter is a complex-valuedﬁeld which is related to the underlying condensate wave function.However,its criticalﬂuctuations can be captured exactly by doing classical statistical mechanics with an eﬀective Hamiltonian for the order parameterﬁeld(for instance the phenomenological Landau-Ginsburg free energy functional (Goldenfeld,1992)).The physics behind the classical nature ofﬁnite temperature transitions is the following: Phase transitions are quite generally accompanied by a divergent correlation length and cor-relation time,i.e.the order parameter(e.g.the magnetization in a ferromagnet)ﬂuctuates coherently over increasing distances and ever more slowly.The latter implies that there is a frequencyω∗associated with the criticalﬂuctuations that vanishes at the transition.A quantum system behaves classically if the temperature exceeds all frequencies of interest, and since¯hω∗≪k B T c close to the transition,the criticalﬂuctuations will behave classically.This argument also shows that the case of QPTs where T c=0,is qualitatively diﬀerent and that there the criticalﬂuctuations must be treated quantum mechanically.In the following we will describe the language and physical pictures that enable such a treatmentand which have come into common usage among practitioners in theﬁeld in the last few years.Although much of this wisdom,which has its roots in work on quantum Ising models (Young,1975;Suzuki,1976),dates back to the work of Hertz(1976)5,it remains unknown or poorly understood in the wider community and extracting it from the literature remains a daunting task.It is our hope here to communicate this set of ideas to a wider audience with the particular desire to be helpful to newcomers to thisﬁeld,experimentalists and theorists alike.Our discussion is organized as follows.In Section I we introduce the statistical me-chanics of quantum systems and the path integral(Feynman,1972)approach to it,which is an extremely useful source of intuition in these problems.A running theme throughout this discussion is the intertwining of dynamics and thermodynamics in quantum statistical mechanics.In Section I.A we describe the general features of a QPT at T=0and how a non-zero temperature alters the physics.This leads naturally to a discussion of what kind of scaling behavior in experiments is evidence of an underlying QPT in Section I.B.We il-lustrate this using particular examples from phase transitions in quantum Hall systems.We end in Section IV with a brief summary and pointers to work on QPTs in other interesting systems.Readers interested in a highly informative discussion at a higher technical level should consult the recent beautiful review article by Sachdev(1996).I.QUANTUM STATISTICAL MECHANICS:GENERALITIESBefore we discuss what happens in the vicinity of a QPT,let us recall some very general features of the statistical mechanics of quantum systems.The quantities of interest are the partition function of the system,Z(β)=Tr e−βH(1)and the expectation values of various operators,O =1ln Z(β),becomes the ground state energy and the various thermal averages become βground state expectation values.From Z we can get all the thermodynamic quantities of interest.Expectation values of operators of the form O≡A(r t)A(r′t′)are related to the results of dynamical scattering and linear response measurements.For example,A might be the local density(X-ray scattering)or current(electrical transport).A.Partition Functions and Path IntegralsLet us focus for now on the expression for Z.Notice that the operator density matrix, e−βH,is the same as the time evolution operator,e−iH T/¯h,provided we assign the imaginary value T=−i¯hβto the time interval over which the system evolves.More precisely,when the trace is written in terms of a complete set of states,Z(β)= n n|e−βH|n ,(3) Z takes the form of a sum of imaginary time transition amplitudes for the system to start in some state|n and return to the same state after an imaginary time interval−i¯hβ.Thus we see that calculating the thermodynamics of a quantum system is the same as calculating transition amplitudes for its evolution in imaginary time,the total time interval beingﬁxed by the temperature of interest.The fact that the time interval happens to be imaginary is not central.The key idea we hope to transmit to the reader is that Eq.(3)should evoke an image of quantum dynamics and temporal propagation.This way of looking at things can be given a particularly beautiful and practical imple-mentation in the language of Feynman’s path integral formulation of quantum mechanics (Feynman,1972).Feynman’s prescription is that the net transition amplitude between twostates of the system can be calculated by summing amplitudes for all possible paths between them.The path taken by the system is deﬁned by specifying the state of the system at a sequence ofﬁnely spaced intermediate time steps.Formally we writee−βH= e−1δτH|m1 m1|e−1¯hδτH|n .(5)¯hThis rather messy expression actually has a rather simple physical interpretation.Formally inclined readers will observe that the expression for the quantum partition function in Eq.(5) has the form of a classical partition function,i.e.a sum over conﬁgurations expressed in terms of a transfer matrix,if we think of imaginary time as an additional spatial dimension. In particular,if our quantum system lives in d dimensions,the expression for its partition function looks like a classical partition function for a system with d+1dimensions,except that the extra dimension isﬁnite in extent—¯hβin units of time.As T→0the system size in this extra“time”direction diverges and we get a truly d+1dimensional,eﬀective, classical system.Since this equivalence between a d dimensional quantum system and a d+1dimensional classical system is crucial to everything else we have to say,and since Eq.(5)is probably not very illuminating for readers not used to a daily regimen of transfer matrices,it will be very useful to consider a speciﬁc example in order to be able to visualize what Eq.(5)means.B.Example:1D Josephson Junction ArraysConsider a one-dimensional array comprising a large number L of identical Josephson junctions as illustrated in Fig.(1).Such arrays have recently been studied by by Haviland and Delsing.(Haviland and Delsing,1996)A Josephson junction is a tunnel junction con-necting two superconducting metallic grains.Cooper pairs of electrons are able to tunnel back and forth between the grains and hence communicate information about the quantum state on each grain.If the Cooper pairs are able to move freely from grain to grain through-out the array,the system is a superconductor.If the grains are very small however,it costs a large charging energy to move an excess Cooper pair onto a grain.If this energy is large enough,the Cooper pairs fail to propagate and become stuck on individual grains,causing the system to be an insulator.The essential degrees of freedom in this system are the phases of the complex super-conducting order parameter on the metallic elements connected by the junctions7and their conjugate variables,the charges(excess Cooper pairs,or equivalently the voltages) on each grain.The intermediate state|m j at timeτj≡jδτ,that enters the quan-tum partition function Eq.(5),can thus be deﬁned by specifying the set of phase angles {θ(τj)}≡[θ1(τj),θ2(τj),...,θL(τj)].Two typical paths or time histories on the interval [0,¯hβ]are illustrated in Fig.(2)and Fig.(3),where the orientation of the arrows(‘spins’) indicates the local phase angle of the order parameter.The statistical weight of a given path,in the sum in Eq.(5),is given by the product of the matrix elementsj {θ(τj+1)}|e−12 j V2j−E J cos ˆθj−ˆθj+1 ,(7)is the quantum Hamiltonian of the Josephson junction array.Hereˆθj is the operator repre-senting the phase of the superconducting order parameter on the j th grain8;V j≡−i2e∂θj is conjugate to the phase9and is the voltage on the j th junction,and E J is the Josephson coupling energy.The two terms in the Hamiltonian then represent the charging energy of each grain and the Josephson coupling of the phase across the junction between grains.As indicated previously,we can map the quantum statistical mechanics of the array onto classical statistical mechanics by identifying the the statistical weight of a space-time path in Eq.(6)with the Boltzmann weight of a two-dimensional spatial conﬁguration of a classical system.In this case the classical system is therefore a two-dimensional X-Y model,i.e.its degrees of freedom are planar spins,speciﬁed by anglesθi,that live on a two-dimensional square lattice.(Recall that at temperatures above zero,the lattice has aﬁnite width¯hβ/δτin the temporal direction.)While the degrees of freedom are easily identiﬁed,ﬁnding the classical hamiltonian for this X-Y model is somewhat more work and requires an explicit evaluation of the matrix elements which interested readers canﬁnd in the Appendix.It is shown in the Appendix that,in an approximation that preserves the universality class of the problem10,the product of matrix elements in Eq.(6)can be rewritten in the.See(Wallin,et al.,1994).The cosine term in∂θjEq.(7)is a‘torque’term which transfers units of conserved angular momentum(Cooper pairs)from site to site.Note that the potential energy of the bosons is represented,somewhat paradoxically,by the kinetic energy of the quantum rotors and vice versa.10That is,the approximation is such that the universal aspects of the critical behavior such asform e−H XY where the Hamiltonian of the equivalent classical X-Y model is1H XY=to the Josephson coupling E J in the array,CK∼ E J.(9) and has nothing to do with the physical temperature.(See Appendix.)The physics here is that a large Josephson coupling produces a small value of K which favors coherent ordering of the phases.That is,small K makes it unlikely thatθi andθj will diﬀer signiﬁcantly,even when sites i and j are far apart(in space and/or time).Conversely,a large charging energy leads to a large value of K which favors zero-pointﬂuctuations of the phases and disorders the system.That is,large K means that theθ’s are nearly independent and all values are nearly equally likely.12Finally,we note that this equivalence generalizes to d-dimensional],a state of indeﬁnite phase on∂θja site has deﬁnite charge on that site,as would be expected for an insulator.arrays and d+1-dimensional classical XY models.C.Quantum-Classical AnalogiesThis speciﬁc example of the equivalence between a quantum system and a classical system with an extra‘temporal’dimension,illustrates several general correspondences between quantum systems and their eﬀective classical analogs.Standard lore tells us that the classical XY model has an order-disorder phase transition as its temperature K is varied.It follows that the quantum array has a phase transition as the ratio of its charging and Josephson energies is varied.One can thus see why it is said that the superconductor-insulator quantum phase transition in a1-dimensional Josephson junction array is in the same universality class as the order-disorder phase transition of the 1+1-dimensional classical XY model.[One crucial caveat is that the XY model universality class has strict particle-hole symmetry for the bosons(Cooper pairs)on each site.In reality, Josephson junction arrays contain random‘oﬀset charges’which destroy this symmetry and change the universality class(Wallin,et al.,1994),a fact which is all too often overlooked.] We emphasize again that K is the temperature only in the eﬀective classical problem.In the quantum case,the physical temperature is presumed to be nearly zero and only enters as theﬁnite size of the system in the imaginary time direction.The coupling constant K,the fake‘temperature,’is a measure not of thermalﬂuctuations,but of the strength of quantum ﬂuctuations,or zero point motion of the phase variables.13This notion is quite confusing,so the reader might be well advised to pause here and contemplate it further.It may be useful to examine Fig.(4),where we show a space time lattice for the XY model corresponding to a Josephson junction array at a certain temperature,and at a temperature half as large.The size of the lattice constant in the time direction[δτin the path integral in Eq.(5)]and K are the same in both cases even though the physical temperature is not the same.The only diﬀerence is that one lattice is larger in the time direction than the other.In developing intuition about this picture,it may be helpful to see how classical physics is recovered at very high temperatures.In that limit,the time interval¯hβis very short compared to the periods associated with the natural frequency scales in the system and typical time histories will consist of a single static conﬁguration which is the same at each time slice.The dynamics therefore drops out of the problem and a Boltzmann weight exp(−βH classical)is recovered from the path integral.The thermodynamic phases of the array can be identiﬁed from those of the XY model.A small value of K corresponds to low temperature in the classical system and so the quantum system will be in the ordered ferromagnetic phase of the XY model,as illustrated in Fig.(2).There will be long-range correlations in both space and time of the phase variables.14This indicates that the Josephson coupling dominates over the charging energy, and the order parameter is notﬂuctuating wildly in space or time so that the system is in the superconducting phase.For large K,the system is disordered and the order parameter ﬂuctuates wildly.The correlations decay exponentially in space and time as illustrated in Fig.(3).This indicates that the system is in the insulating phase,where the charging energy dominates over the Josephson coupling energy.Why can we assert that correlations which decay exponentially in imaginary time indicate an excitation gap characteristic of an insulator?This is readily seen by noting that the Heisenberg representation of an operator in imaginary time isA(τ)=e Hτ/¯h Ae−Hτ/¯h(10)and so the(ground state)correlation function for any operator can be expressed in terms ofa complete set of states asG(τ)≡ 0|A(τ)A(0)|0 = m e−(ǫm−ǫ0)τ/¯h| 0|A|m |2,(11) whereǫm is the energy of the m th excited state.The existence of aﬁnite minimum excitation gap∆01≡ǫ1−ǫ0guarantees that for long(imaginary)times the correlation function will decay exponentially,15i.e.,G(τ)∼e−∆01τ/¯h.(12) To recapitulate,we have managed to map theﬁnite temperature1D quantum problem into a2D classical problem with oneﬁnite dimension that diverges as T→0.The parameter that controls theﬂuctuations in the eﬀective classical problem does not involve T,but instead is a measure of the quantumﬂuctuations.The classical model exhibits two phases, one ordered and one disordered.These correspond to the superconducting and insulating phases in the quantum problem.In the former the zero-point or quantumﬂuctuations of the order parameter are small.In the latter they are large.The set of analogies developed here between quantum and classical critical systems is summarized in Table I.Besides the beautiful formal properties of the analogy between the quantum path integral and d+1dimensional statistical mechanics,there are very practical advantages to this analogy.In many cases,particularly for systems without disorder,the universality class of the quantum transition is one that has already been studied extensively classically and a great deal may already be known about it.For new universality classes,it is possible to do the quantum mechanics by classical Monte Carlo or molecular dynamics simulations of the appropriate d+1-dimensional model.Finally,there is a special feature of our particular example that should be noted.In this case the quantum system,the1D Josephson junction array(which is also the1D quantumX-Y model),has mapped onto a classical model in which space and time enter in the same fashion,i.e.,the isotropic2D classical X-Y model.Consequently,the dynamical exponent z(to be deﬁned below)is unity.This is not true in general—depending upon the quantum kinetics,the coupling in the time direction can have a very diﬀerent form and the eﬀective classical system is then intrinsically anisotropic and not so simply related to the starting quantum system.D.Dynamics and ThermodynamicsWe end this account of quantum statistical mechanics by commenting on the relation-ship between dynamics and thermodynamics.In classical statistical mechanics,dynamics and thermodynamics are separable,i.e.,the momentum and position sums in the partition function are totally independent.For example,we do not need to know the mass of the particles to compute their positional correlations.In writing down simple non-dynamical models,e.g.the Ising model,we typically take advantage of this simplicity.This freedom is lost in the quantum problem because coordinates and momenta do not commute.16It is for this reason that our path integral expression for Z contains information on the imaginary time evolution of the system over the interval[0,¯hβ],and,with a little bit of care,that information can be used to get the dynamics in real time by the analytic continuation,G(τ)−→G(+it)(13)in Eq.(11).Stating it in reverse,one cannot solve for the thermodynamics without also solv-ing for the dynamics—a feature that makes quantum statistical mechanics more interesting but that much harder to do!Heuristically,the existence of¯h implies that energy scales that enter thermodynamics necessarily determine time scales which then enter the dynamics and vice-versa.Consider the eﬀect of a characteristic energy scale,such as a gap∆,in the spectrum.By the uncertainty principle there will be virtual excitations across this gap on a time scale¯h/∆,which will appear as the characteristic time scale for the dynamics.Close to the critical point,where ∆vanishes,and atﬁnite temperature this argument gets modiﬁed—the relevant uncertainty in the energy is now k B T and the characteristic time scale is¯hβ.In either case,the linkage between dynamics and thermodynamics is clear.II.QUANTUM PHASE TRANSITIONSWe now turn our attention to the immediate neighborhood of a quantum critical point. In this region the mapping of the quantum system to a d+1dimensional classical model will allow us to make powerful general statements about the former using the extensive lore on critical behavior in the latter.Hence most of the following will consist of a reinterpretation of standard ideas in classical statistical mechanics in terms appropriate for d+1dimensions, where the extra dimension is imaginary time.A.T=0:Dynamic ScalingIn the vicinity of a continuous quantum phase transition we willﬁnd several features of interest.First,we willﬁnd a correlation length that diverges as the transition is approached. That diverging correlation lengths are a generic feature of classical critical points,immedi-ately tells us that diverging lengths and diverging times are automatically a generic featureof quantum critical points,since one of the directions in the d+1dimensional space is time. This makes sense from the point of view of causality.It should take a longer and longer time to propagate information across the distance of the correlation length.Actually,we have to be careful—as we remarked earlier,the time direction might easily involve a diﬀerent set of interactions than the spatial directions,leading to a distinct cor-relation“length”in the time direction.We will call the latterξτ,reserving the symbolξfor the spatial correlation length.Generically,at T=0bothξ(K)andξτ(K)diverge as δ≡K−K c−→0in the manner,17ξ∼|δ|−νξτ∼ξz.(14) These asymptotic forms serve to deﬁne the correlation length exponentν,and the dynamical scaling exponent,z.The nomenclature is historical,referring to the extension of scaling ideas from the study of static classical critical phenomena to dynamics in the critical region associated with critical slowing down(Hohenberg and Halperin,1977;Ferrell,1968).In the classical problem the extension was a non-trivial step,deserving of a proper label.As remarked before,the quantum problem involves statics and dynamics on the same footing and so nothing less is possible.For the case of the Josephson junction array considered previously,we found the simplest possible result,z=1.As noted however this is a special isotropic case and in general,z=1.As a consequence of the divergingξandξτ,it turns out that various physical quantities in the critical region close to the transition have(dynamic)scaling forms,i.e.their dependence on the independent variables involves homogeneity relations of the form:O(k,ω,K)=ξd O O(kξ,ωξτ)(15)where d O is called the scaling dimension18of the observable O measured at wavevector k and frequencyω.The meaning of(and assumption behind)these scaling forms is simply that,close to the critical point,there is no characteristic length scale other thanξitself19 and no characteristic time scale other thanξτ.Thus the speciﬁc value of the coupling K does not appear explicitly on the RHS of Eq.(15).It is present only implicitly through the K dependence ofξandξτ.If we specialize to the scale invariant critical point,the scaling form in Eq.(15)is no longer applicable since the correlation length and times have diverged to inﬁnity.In this case the only characteristic length left is the wavelength2π/k at which the measurement is being made,whence the only characteristic frequency is¯ω∼k z.As a result weﬁnd the simpler scaling form:O(k,ω,K c)=k−d O˜O(k z/ω),(16) reﬂecting the presence of quantumﬂuctuations on all length and time scales.20 The utility and power of these scaling forms can be illustrated by the following example. In an ordinary classical system at a critical point in d dimensions where the correlation length has diverged,the correlations of many operators typically fall oﬀas a power law˜G(r)≡ O(r)O(0) ∼118The scaling dimension describes how physical quantities change under a renormalization group transformation in which short wavelength degrees of freedom are integrated out.As this is partly a naive change of scale,the scaling dimension is often close to the naive(“engineering”)dimen-sion of the observable but(except at special,non-generic,ﬁxed points)most operators develop “anomalous”dimensions.See Goldenfeld(1992).19For a more precise statement that includes the role of cutoﬀscales,see Goldenfeld(1992).20Equivalently,we could have argued that the scaling function on the RHS of Eq.(15)must for large arguments x,y have the form O(x,y)∼x−d O˜O(x z y−1)in order for the observable to have a sensible limit as the critical point is approached.so that the Fourier transform diverges at small wavevectors likeG(k)∼k−2+ηd.(18) Suppose that we are interested in a QPT for which the d+1-dimensional classical system is eﬀectively isotropic and the dynamical exponent z=1.Then the Fourier transform of the correlation function for the d+1-dimensional problem isG(k,ωn)∼21A Goldstone mode is a gapless excitation that is present as a result of a broken continuous。

Critical to Mean Field Crossover in Polymer Blends

Adv Polym Sci(2005)183:1–61DOI10.1007/b135882©Springer-Verlag Berlin Heidelberg2005Published online:26July2005Critical to Mean Field Crossover in Polymer BlendsDietmar SchwahnInstitut für Festkörperforschung,Forschungszentrum Jülich GmbH,52425Jülich, Germanyd.schwahn@fz-juelich.de1Introduction (2)2Small Angle Neutron Scattering (8)2.1Experimental Design of a SANS Diffractometer (9)2.2Scattering Cross Section of an Ideal Polymer Mixture (11)2.3Neutron Scattering Contrast (13)3Thermodynamic Model for Polymer Blends (15)3.1Flory–Huggins Theory (15)3.2Random Phase Approximation (16)3.3Determination of a Polymer Blend Phase Diagram by SANS (18)4Mean Field to3D-Ising Crossover in Polymer Blends (21)4.1Theoretical Background (21)4.2Binary Homopolymer Blends (24)4.3Binary Blends of Statistical Copolymers (27)4.3.1Theory (28)4.3.2Experimental Results (29)4.4Polymer Blends in an External Pressure Field (32)4.4.1SANS Results (33)4.4.2Clausius–Clapeyron Equation (35)4.5Binary Blends with Small Additions of a Non-Selective Solvent (38)4.5.1Structure Factor (38)4.5.2Susceptibility and Correlation Length (39)4.5.3Ginzburg Number and Critical Amplitudes (39)5Crossover to the Renormalized3D-Ising Critical Behavior (42)5.1Hidden Variables–Fisher Renormalization (43)5.2SANS Results on Blends (44)5.3Renormalized3D-Ising Critical Behaviorin Blends and Blend-Solvent Systems (45)6Crossover to Isotropic Lifshitz Critical Behaviorin(A/B)Polymer Blend/(A-B)Diblock Copolymer Mixtures (46)6.1Phase Diagramof a(A/B)Polymer Blend/(A-B)Diblock Copolymer Mixture (47)6.2Structure Factor within Mean Field Approximation (49)2 D.Schwahn 6.3Effect of Thermal Composition Fluctuations (51)6.4SANS Results from below the Lifshitz Line (52)7Summary and Outlook (55)References (59)Abstract Crossover phenomena due to thermal compositionﬂuctuations play a multifar-ious role in polymer blends.This is demonstrated in this article by describing results from small angle scattering experiments in particular with neutrons.Scattering methods are a direct tool to measure the strength and correlation length of compositionﬂuctua-tions.We will review the effects of thermalﬂuctuations in binary(A/B)polymer blends under various conditions of external temperatures and pressures and additives,such as non-selective solvents and(A-B)diblock copolymers,and will give an interpretation with the corresponding crossover theories.General conclusions are that the effects from ther-mal compositionﬂuctuations have to be more seriously considered in polymer blends and that the more sophisticated crossover theories are needed for a precise determination of the Flory–Huggins interaction parameter and the phase boundaries.In addition,we discuss observations of crossover to other universality classes such as the3D-Ising case, namely the transition to the renormalized Ising case when the composition of a third component starts toﬂuctuate and to the isotropic Lifshitz critical behavior when an(A-B) diblock copolymer is added.Keywords Polymer blends·Crossover by thermal compositionﬂuctuations·3D-Ising critical behavior·Renormalized Ising critical behavior·Lifshitz critical behavior·Ginzburg criterion·Flory–Huggins parameter·Externalﬁelds of temperature and pressure·Additives of non-selective solvent and diblock copolymers1IntroductionPolymer blends represent materials for a large class of industrial products and are the subject of intensive academic research.Blending of chemically different polymers is an important tool in industrial production for tailoring products with optimized material properties[1].On the other hand polymer mixtures are model systems in statistical physics for studying fundamental aspects of equilibrium and non-equilibrium properties such as phase dia-grams,thermal compositionﬂuctuations far and near the critical point of a second order phase transition,conformational properties of the chains,the kinetics of phase transitions as well as the detailed dynamics of diffusion processes.There are strong activities in thisﬁeld from both theory and ex-periment as can be realized from numerous review articles[2–5].In this article we will focus on the behavior of thermal compositionﬂuctu-ations in binary(A/B)homopolymer blends(A,B denote the monomers)in different externalﬁelds such as temperature and pressure,in their pure state,Critical to Mean Field Crossover in Polymer Blends3 as well as with small additions of a non-selective solvent and symmetric(A-B) diblock copolymers.The externalﬁeld conditions as well as additives can strongly inﬂuence the degree of thermalﬂuctuations and lead to crossover phenomena between different universality classes of meanﬁeld,3D-Ising, renormalized3D-Ising,and isotropic Lifshitz critical behavior.Each univer-sality class is characterized by a unique set of critical exponents describing the divergence of several thermodynamic parameters in the near vicinity of a second order phase transition.Two of those prominent parameters,namely, the correlation length of thermal compositionﬂuctuations and the corres-ponding susceptibility are determined in scattering experiments.Generally, scattering methods are a sensitive tool to measure spatial inhomogeneities, such as thermal compositionﬂuctuations in blends,and neutrons have the particularly important advantages of strong scattering contrast and deep pen-etration into materials[6,7].So,it is quite natural that most of the relevant studies in thisﬁeld have been performed with neutron small angle scattering (SANS)techniques.Thermal compositionﬂuctuations are always present in multi-component systems.If the interaction energy between same components is stronger,clus-tering of(A/A)and(B/B)monomers is preferred,and phase separation will occur at low temperatures.A schematic phase diagram of a symmetric bi-nary polymer blend is depicted in Fig.1.It represents an“upper critical solution temperature”(UCST)system;at high temperatures the blend is ho-mogeneously mixed,while at low temperatures the system decomposes into macroscopically large domains being rich in A or B polymers.Such a pro-Fig.1Schematic phase diagram of binary polymer melt of equal molar volume V A=V B. At high temperature both polymers are miscible;at low temperature below the binodal the sample separates into two macroscopic large domains with compositions determined by the binodal.The spinodal separates metastable and unstable regions which determine the process of phase separation.The disordered regime is separated by the Ginzburg number Gi domain into regions of small and large degrees of thermal compositionﬂuc-tuations4 D.Schwahn cess can be easily followed by eye as the sample becomes turbid from the strong scattering of light by theµm large domains.The binodal is the bor-der line between the one and two-phase regimes,while the spinodal separates the meta-stable and unstable parts of the two-phase regime.The binodal andspinodal meet at the critical point which also represents the highest tem-perature of immiscibility in monodisperse polymer systems[8].In symmetricblends with equal molar volumes V A=V B theory predicts a critical compo-sition ofΦC=50%volume fraction.The one-phase regime is homogeneous on a macroscopic length scale and usually appears transparent to the eye.Onsmaller length scales,however,the composition is heterogeneous due to ther-malﬂuctuations.The occurrence of thermalﬂuctuations is a dynamic processas they are created with a given probability and decay afterwards with a re-laxation rate determined by the interdiffusion constant[3,4].In static elasticneutron scattering experiments one measures the equilibrium mean square deviation δΦ2 from the average compositionΦgiven by the volume frac-tion of one of the components.The range of thoseﬂuctuations is determinedby the correlation lengthξwhich sensitively depends on external parameters such as temperature and pressure:ξis of the order of interatomic distances athigh temperatures when entropy is dominating;it is large at low temperatures and becomes inﬁnite at the critical point.The critical point represents a par-ticular position of the phase boundary where a continuous(second order) phase transition is observed.According to theﬂuctuation–dissipation theorem the equilibrium meansquare deviation of thermal compositionﬂuctuations δΦ2 is related to the ﬁrst derivative of the order parameter with respect to the chemical poten-tial[3,4].If the order parameter is deﬁned as the compositionΦof the component“A”then the conjugateﬁeld is represented by the difference of the chemical potentials∆µ(=µA–µB);so the degree of thermalﬂuctuations is related to∂∆µΦ(∂∆µ≡∂/∂∆µ)which is equivalent to1/∂2Φ∆G,where ∆G represents the Gibbs free energy of mixing with its natural parameters temperature T,pressure P,and compositionΦ.This shows how the ther-modynamic parameters can be determined from measurements of thermal ﬂuctuations.The border line between strong and weak thermalﬂuctuations is esti-mated by the Ginzburg number Gi:=(1–T C/T X),which is evaluated from the Ginzburg criterion[3,4]and which represents a reduced temperature T X be-low which deviations from the meanﬁeld approach are observed(see hatched area in Fig.1).Fluctuations are considered to be weak as long as theﬂuctu-ation modes superimpose linearly.In this weak limit theﬂuctuation modes can be described within the Gaussian approximation,and meanﬁeld theory is a good approximation.Near the critical point thermalﬂuctuations become strong and lead to visible non-linear effects.In this range more sophisticated theories as the3D-Ising model and crossover theories are needed[9,10].Critical to Mean Field Crossover in Polymer Blends5 Fluctuation effects are usually neglected in theoretical descriptions ofpolymer blends which are therefore described within the meanﬁeld the-oretical approach of the Flory–Huggins(FH)theory[2–4].In this modelthe Gibbs free energy of mixing∆G is depicted by a combinatorial en-tropy of mixing being inversely proportional to the molar volume V andby the FH parameterΓ=Γh/T–Γσbeing described by enthalpic and non-combinatorial entropic termsΓh andΓσ,respectively.The general neglect ofthermalﬂuctuations in polymer blends might be explained with the originalestimate of the Ginzburg criterion by deGennes on the basis of incompress-ible FH theory.This relationship represents a universal criterion according to Gi∝1/N(N degree of polymerization)and proposes an extremely small Ginzburg number[3].Since in low molecular liquids a Gi∼=10–2is the ex-pected value,a roughly N times smaller Gi is estimated in polymer blends indicating a small region with strong thermalﬂuctuations[10].Meanwhile, it has been shown that the Ginzburg criterion for polymer blends is non-universal with appreciably larger Ginzburg numbers and therefore larger re-gions where thermalﬂuctuations are signiﬁcant.Theoretical considerations, computer simulations,and scattering experiments show that the compress-ibility is dependent on the non-combinatorial entropyΓσwhich has a strong inﬂuence on the Ginzburg criterion and thereby on the degree of thermal compositionﬂuctuations.The effect of thermalﬂuctuations will be the main emphasis of this article.The inﬂuence of several chain parameters onΓσand therefore on Gi has been identiﬁed.These are polymer asymmetry,different monomer struc-tures,chain stiffness,compressibility,and chain end-effects.The effects of compressibility and monomer structure have been discussed by Lifshitz et al.[11]and Dudowicz et al.[12]on basis of the Lattice Cluster The-ory(LCT),respectively.Considering these effects a one to several orders of magnitude increase of Gi is found.The present theoretical ideas behind an ex-tended Ginzburg criterion have in common the consideration of correlations (or non-random mixing)on length scales between the coil and monomer size. These shorter length scaleﬂuctuations lead to an extended meanﬁeld de-scription and are comprised into a non-combinatorial entropic termΓσof the FH parameter.It will be shown later(Eq.18)in the context of the crossover theories for the susceptibility and correlation length thatΓσstrongly inﬂu-ences the Ginzburg number and thereby the strength of thermalﬂuctuations and the domain over which they must be considered in the analysis of experi-mental data.Another theoretical approach including the effects of thermalﬂuctua-tions has been derived fromﬁeld-theoretical methods(see recent review by Fredrickson et al.[13]).Wang[14]quite recently gave a compact overview and a derivation from a systematic renormalization procedure of an effective FH parameter and Ginzburg criterion.Starting from a“bare”FH parameter as originally derived by Flory,which only considers the microscopic enthalpic6 D.Schwahn interaction,Wang formulated an“effective”FH parameter comprising the ef-fects of local correlations of wavelengths between monomer and chain sizes.The last mentioned parameter is determined from SANS experiments within a meanﬁeld approximation,is comparable with the FH parameter derivedfrom the LC-theory,and also includes molecular conformational asymmetry effects as discussed in[15].In addition,an“apparent”FH parameter is deter-mined which also includes contributions from long wavelengthﬂuctuations that become relevant near the critical point.Fluctuations lead to a reduced ef-fective FH parameter,to a decrease of the critical temperature,and to a largerGi.In the limit of large degree of polymerization,a1/N scaling of Gi is pre-dicted.Monte Carlo simulations very early demonstrated the effect of ther-mal compositionﬂuctuations in low molecular blends.Studies by Saribanet al.[16]exclusively found Ising critical behavior in blends of molar volume up to about16000cm3/mol and no indications of a crossover to meanﬁeld behavior.Such a meanﬁeld crossover was later detected by Deutsch et al.[17]in blends with an order of magnitude larger chains.These results and the techniques of Monte Carlo simulations have been extensively reviewed by Binder in[4].These theoretical considerations and Monte Carlo simulations were ac-companied by scattering experiments,which consistently show that the range of relevant thermalﬂuctuations and thereby Gi is at least of the order of magnitude larger than originally estimated from simple incompressible FH theory[18–26].An important development in the description of thermal ﬂuctuations is the derivation of proper analytic crossover functions for the susceptibility and correlation length which also can be relatively easily han-dled by the experimentalist[10,27–30].These equations allow an analysis of the susceptibility and correlation length over the whole experimental range including meanﬁeld and Ising behavior and therefore lead to much more precisely determined thermodynamic parameters.A Ginzburg criterion is quite naturally derived from these crossover functions and is determined by the ratio of the meanﬁeld and Ising critical amplitudes for the suscep-tibility and/or correlation length.The so determined Ginzburg number is a non-universal function,which sensitively depends on the degree of poly-merization and on the FH parameter non-combinatorial entropy of mixing Γσ[31,32].An increase ofΓσleads to a strong increase of Gi,and only in the limit of negligibleΓσis a universal Ginzburg criterion with the proposed1/N scaling behavior obtained consistent with earlier predictions for incompress-ible polymer blends.High external pressureﬁelds can lead to an appreciable change of the FH interaction parameter,the phase boundaries,and the Ginzburg number. Pressure usually induces an increase of the phase transition temperature, a decrease of the Ginzburg parameter and of the enthalpic(Γh)and abso-lute value of the entropic(Γσ)terms of the FH parameter.An increase ofCritical to Mean Field Crossover in Polymer Blends7 the phase boundary with pressure is expected as it is related to a decrease of the free volume and a corresponding decrease of the entropicΓσ[33–41]. In LCST systems(phase transition occurs at high temperatures)such as the PS/PVME(polystyrene/poly(vinyl methyl ether)blendΓh is constant within 1and120MPa whileΓσis negative and increases with pressure[34].In a few blends,however,an“abnormal”pressure induced decrease of the phase boundary and a corresponding increase of theΓh andΓσFH terms are ob-served[42,43].The shift of the phase boundary with pressure is described by the Clausius–Clapeyron equation in terms of the FH parameter and the Ginzburg number[40].In some blends one actuallyﬁnds a shift of the critical temperature dominated by the Ginzburg number according to∂P Gi[40].The exploitation of the Clausius–Clapeyron equation also allows a check of con-sistency with respect to the underlying theory,such as the dependence of the Ginzburg criterion on the entropic FH termΓσ.In this respect,the study of pressure induced changes of thermalﬂuctuations may lead to relevant insight into the mechanisms governing the polymer blend thermodynamic proper-ties and their interrelations.Mixing a binary polymer blend with a small amount of a third com-ponent usually leads to strong changes of the phase behavior and in some cases even to a crossover to a different universality class.The addition of a non-selective solvent generally leads to improved compatibilization and to a larger Ginzburg number.Furthermore,very near the critical temperature, a crossover from Ising to the renormalized Ising behavior is predicted for those blend-solvent systems.Such crossover phenomena have been inten-sively explored by the group of Nose[44,45]with light scattering.They found this type of crossover very pronounced after adding a selective solvent but only rather weak in a few special cases after adding a non-selective solvent. These observations are consistent with the SANS studies discussed in this article.A much more complex situation is achieved if an(A-B)diblock copoly-mer is mixed with a binary(A/B)homopolymer blend.Generally the diblock copolymer leads to a better compatibilization of the homopolymers similar to the action of surfactant molecules in oil-water mixtures.A complex phase di-agram is obtained that depends upon the diblock content:Several disordered and ordered phases appear at,respectively,high and low temperatures.At low and high diblock content a two-phase region of macroscopically large do-mains and an ordered phase known from diblock copolymer melts emerge, respectively,and intermediate diblock content leads to droplet and bicon-tinuous microemulsion phases[46–52].Bicontinuous microemulsion and ordered phases at high diblock copolymer concentrations are characterized by a periodicity lengthΛwhich in scattering experiments becomes visible as a peak at Q∗=2π/Λ.Within the disordered phases several crossover phe-nomena are observed,namely,from meanﬁeld to3D-Ising and to isotropic Lifshitz critical behavior,as well as to the Brasovskii type of pure diblock8 D.Schwahn copolymers [53].One reason for this complex behavior is that homopoly-mer mixtures and diblock copolymers belong to different universality classes.Diblock copolymers shows much stronger thermal ﬂuctuations which even change the character of the disorder-order phase transition from second to ﬁrst order and lead to a broader critical range with a weaker N proportion-ality of Gi according to Gi ∝1/√N [54].Mean ﬁeld theory predicts that the critical lines of “blend like”and “di-block like”behavior meet at the isotropic critical Lifshitz point and the Lifshitz line (LL)which is deﬁned when Q ∗becomes zero.The isotropic crit-ical Lifshitz point represents a new universality class [54–56].Under special conditions even a tricritical Lifshitz point is predicted [55].In this article we will discuss in some detail SANS experiments on a mixture of a critical bi-nary (A /B)polymer blend with different concentrations of a symmetric (A-B)diblock copolymer of roughly ﬁve times larger molar volume.Under such conditions an isotropic critical Lifshitz point is predicted [55].Near the critical Lifshitz point the phase behavior and the corresponding phase diagram are strongly inﬂuenced by thermal composition ﬂuctuations.The strength of these ﬂuctuations can be understood from a decrease of the surface energy acting as a restoring force for thermal ﬂuctuations.A further peculiarity is,that a critical Lifshitz point can only exist within mean ﬁeld ap-proximation.It is “destroyed”by thermal ﬂuctuations because the Ginzburg criterion and therefore the stabilization of the miscible phase is of different strengths on both sides of the Lifshitz line.A further observation is a change of the LL concentration with temperature near the two-phase regime [48,49];recent renormalization group calculations explain these observations as due to thermal ﬂuctuations [57].In the range near the LL one also observes a mi-croemulsion phase not predicted by the corresponding existing mean ﬁeld theories [47].From these introductory remarks one already gets the impression about the prominent and diversiﬁed role of thermal composition ﬂuctuations on the properties of polymer blends,which will be detailed in the following sections mainly from an experimental point of view.2Small Angle Neutron ScatteringA separate section is devoted to describing small angle neutron scattering (SANS)techniques as this experimental method plays a prominent role in the investigation of polymer blends.Neutrons are a particularly appropriate probe for explorations of polymer properties because of the good scattering contrast conditions and their weak interactions with material in general [6,7].The last mentioned condition relies on the fact that neutrons have no chargeCritical to Mean Field Crossover in Polymer Blends9 and thus allow the exploration of several mm thick samples,making SANS a non-destructive tool.The above mentioned scattering contrast is propor-tional to the difference of the coherent scattering length density of the poly-mer components and is mainly determined from the neutron interaction withthe atomic nuclei.This interaction can lead to appreciably different scattering lengths for isotopes as most prominently found for hydrogen and deuterium.Consequently,a mixture of deuterated and protonated polymer chains shows a strong scattering contrast and thereby makes neutrons a sensitive tool.Be-yond that the contrast variation with H/D content of selected polymers offersdetailed insight into single chain properties in complex systems as the above mentioned(A/B/A-B)polymer blend-diblock copolymer mixtures.2.1Experimental Design of a SANS DiffractometerA schematic lay-out and photography of the SANS experiments at the re-search reactor FRJ2in Jülich are given in Fig.2a and b.Neutrons are released in nuclear reactors byﬁssion reactions of the uranium isotope235or areevaporated in spallation sources from a heavy metal target after being bom-barded with high energy protons.For example about20neutrons are releasedfrom a lead nucleus per1Ge V proton[7].After being released neutrons are moderated to lower kinetic energies.For SANS and other neutron scatter-ing methods as spin-echo and back scattering spectrometers a special“coldsource”moderator of liquid hydrogen or deuterium is installed near the reac-tor core which delivers more than one order of magnitude larger intensity oflong wavelength neutrons between7and15˚A.Neutrons of this wavelength approach a smaller momentum transfer Q,as will be deﬁned in Eq.1,andavoid double Bragg scattering in crystalline materials.These“cold”neutrons are guided to the instrument through neutron guides that operate by total reﬂection.Neutron guides consist of evacuated glass channels with good sur-face quality and are coated with an element having a large coherent scattering length such as the Nickel58isotope.The neutrons entering the instrumentﬁrst pass a velocity selector,then a collimator with two apertures,a sample,and areﬁnally detected in a pos-ition sensitive detector if scattered into a given angular interval.The selector and collimator,respectively,determine the wavelength with a relative distri-bution of typically∆λ/λ∼=0.10and the divergence of the neutron beam.In order to keep multiple scattering low,the sample should scatter only part (ca.10%)of the incident neutrons(intensity I0)whereas the non-scatteredneutrons(intensity I∗0)are absorbed in a beam stopper in front of the detec-tor.The distances of sample to detector as well as of the two apertures can be changed typically from1to20m allowing the measurement of scattering angles from roughly0.1◦to20◦and to the adjustment of an optimized re-lationship between maximum intensity and sufﬁciently good resolution.In10 D.SchwahnFig.2Top:Schematic lay-out of a pin hole small angle neutron diffractometer(SANS). Bottom:Photography of one of the two SANS instruments at the research reactor FRJ-2 at Jülich.Vacuum sample chamber and detector tank are visible.Both Jülich SANS in-struments have been in operation since1987and have up until now been seen worldwide to the most powerful instruments after the two SANS instruments at the ILL in Grenoble (France)order to gain optimized intensity conditions,additional neutron guides can be inserted into the neutron beam axis so that neutrons are always trans-ported through neutron guides until theﬁrst aperture.The scattered intensity is usually given as a function of the momentum transfer Q whose absolute value Q is determined according toQ=(4π/λ)sin(Θ/2)(1) Q is inversely proportional to the neutron wave lengthλand is proportional to the scattering angleΘ.The typical wavelengths between5and10˚A lead to a Q range of10–3<Q[˚A–1]<0.3and correspond in real space to a resolution range between10<R[˚A]<103.The Q range can be extended to smaller Q by other SANS techniques using a focusing mirror optic or perfect Silicon single crystals as monochromator and analysor[58].The neutron intensity I D(Q)scattered by the momentum transfer Q is de-scribed in Eq.2;it is proportional to the incident intensity I0,the sample thickness D,the sample transmission T=I∗0/I0,and the space angle∆ΩD ofa detector element,I D (Q )=I 0DTd Σd Ω(Q )∆ΩD(2)The transmission describes the attenuation of the neutrons non-scattered by the sample,and the macroscopic cross section d Σ/d Ωin units of cm –1is the quantity to be determined.The calibration of the macroscopic cross section d Σ/d Ωin absolute units is usually performed by an additional measurement with a sample of known d Σ/d Ωor by measuring the direct beam in both cases with the same collimator setting [59,60].2.2Scattering Cross Section of an Ideal Polymer MixtureBlending two polymers with zero energy of mixing (FH parameter χ=0)leads to an ideal mixture.Such ideal mixtures do not show any phase behavior and can be realized within good approximation by blending protonated and deuterated polymers of the same species of not too large molar volumes V .In order to determine the structure factor S (Q )of such a melt,we consider the chains on a so-called “Flory”lattice.All lattice points are of the same size and are occupied by the monomers of the two A /B polymers.The basic equation for the macroscopic cross section in SANS within ﬁrst Born approximation is given according to [6,7]d Σ/d Ω(Q )=(1/V S ) ¯b i exp(i Qr i ) 2(3)The scattering amplitude is determined from the sum of the coherent scatter-ing length,¯bi of the monomer at position i over all lattice points multiplied with the corresponding phase factor exp(i Qr i ).The square of the scattering amplitude divided by the sample volume V S gives d Σ/d Ω(Q ).From now on the momentum transfer will be regarded as a scalar as we always will discuss isotropically scattering samples.To proceed,we deﬁne a variable σi with the following meaning,σi =1¯bi =¯b A 0¯b i =¯b B (4)which determines the occupation of the Flory lattice by the monomers of the polymers A and B as represented by their coherent scattering length.The scattering length at position i is then given as ¯bi =σi (¯b A –¯b B )+¯b B =σi ·∆¯b+¯b B and the cross section becomes d Σd Ω Q =∆¯b2V SN ijσi σj exp(i Qr ij ) =K ·S (Q )(5)。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

1 Dipartimento di Fisica and Unit´a INFM, Universit´a di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy
2 The Abdus Salam International Centre for Theoretical Physics (ICTP), P.O. Box 586, 34100 Trieste, Italy
We consider a two-dimensional square lattice of linear size L and associate to each site an integer variable zi (energy). At each time step an energy grain is added on a randomly chosen site (zi → zi + 1). When one of the sites reaches or exceeds the local threshold zc = 2 a “toppling” occurs:: zi = zi − 2 and zj = zj + 1, where j represents two randomly chosen nearest neighbor sites of site i. A toppling can induce nearest-neighbor sites to topple on their turn and so on, until all the lattice sites are below the critical threshold. This process is called an avalanche. Grains are added only when all the sites are below the threshold, which corresponds to a ﬁne tuning of the external driving ﬁeld [2]. In addition, the model is conservative and energy is dissipated only at boundary sites [1, 6].
1ቤተ መጻሕፍቲ ባይዱ
are diﬃcult to interpret, since diﬀerent methods of analysis typically yield diﬀerent results [6, 9, 10, 11, 12, 13]. This is probably due to intrinsic scaling anomalies and ﬁnite size eﬀects present in sandpile models.
Despite the large conceptual impact and the huge eﬀort devoted to the study of sandpile automata in the last ten years, many basic issues, such as the precise values of the critical exponents, the identiﬁcation of universality classes and of the upper critical dimension, still lay unresolved. Theoretically, many approaches [3, 4, 5] point out that diﬀerent sandpile models, such as the Bak, Tang and Wiesenfeld (BTW) [1] and the Manna [6] models, all belong to the same universality class. Theoretical estimates for critical exponents have been provided (especially in Euclidean dimension d = 2) by means of diﬀerent methods [3, 4, 7], and some exact results [8] can be derived from the Abelian structure of the BTW model. Numerical results
2
5
10
σs(q) P(s) Lβsτs
4
0 3
L=512 L=1024
2
L=2048
−10
1
0
−20
0
0.5
1
1.5
q
2
−35
−25
s/L−βs15
−5
Figure 1: (a) Plot of the moments spectrum for the distribution of toppling events s.The linear part has slope 2.73.(b) Data collapse analysis for the avalanche size distribution. The values used for the critical exponents are τs = 1.27 and βs = 2.7.
Sandpile automata [1] are prototypical models to describe avalanche transport processes. All these models show a stationary state that after a suitable tuning of the driving ﬁelds[2] displays a singular response function characterized by power law distributed events. These distributions are typically bounded by upper cut-oﬀs related to the system size. In analogy with critical phenomena, is possible to deﬁne a complete set of scaling exponents describing the large scale behavior of these models.
P (x) = x−τxG(x/xc),
(1)
where x = s, a, t. When the system size L goes to inﬁnity the cutoﬀ xc diverges as xc ∼ Lβx. Under the ﬁnite size scaling (FSS) assumption of Eq. (1), the set of exponents {τx, βx} deﬁnes the universality class of the model. In order to test the above FSS picture and to ﬁnd an accurate estimate of the various critical exponents, we perform numerical simulations of two-dimensional Manna model with open boundary conditions and conservative dynamics. The lattice size ranges from L = 128 to L = 2048, and statistical distributions are obtained averaging over 107 nonzero avalanches. The direct numerical determination of the exponents τx from the power law behavior of the probability distributions contains intrinsic bias due to the lower and upper cut-oﬀs. This makes very diﬃcult to get better than a 10% accuracy. Extrapolations methods have been devised [13], but the estimate of their accuracy is rather diﬃcult. For these reasons, we use the
Here, we present very large scale numerical simulations of the Manna model[6], that is the standard example of a sandpile automaton with stochastic toppling rule. We show that Manna model can be coherently described within a ﬁnite size scaling (FSS) framework. Critical exponents are evaluated with great accuracy and the results are conﬁrmed by data collapse analysis.
3 PMMH-ESPCI, 10 Rue Vauquelin, 75234 Paris CEDEX 05, France
Abstract