Universality in the diffusion of knots

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社会矛盾频发英语作文

In recent years,the frequency of social conflicts has been on the rise,posing a significant challenge to the stability and harmony of society.There are several factors contributing to this phenomenon,and addressing these issues requires a multifaceted approach.Firstly,economic disparities are one of the primary causes of social conflicts.The widening gap between the rich and the poor has led to feelings of injustice and resentment among the less fortunate.This economic inequality can manifest in various forms,such as unemployment,low wages,and lack of access to basic services.Secondly,the rapid pace of urbanization has also contributed to social tensions.As more people migrate to cities in search of better opportunities,the strain on urban resources and infrastructure becomes evident.This can lead to issues such as overcrowding, housing shortages,and environmental degradation,which in turn can exacerbate social conflicts.Moreover,cultural and ethnic differences can be another source of social friction.In a diverse society,misunderstandings and prejudices can arise,leading to tensions between different groups.It is essential to promote tolerance and understanding to bridge these cultural divides and foster a sense of unity.Additionally,the role of social media in amplifying social conflicts cannot be overlooked. The internet has become a platform for people to express their grievances and opinions, which can sometimes escalate into heated debates and even offline confrontations.It is crucial to regulate the spread of misinformation and hate speech online to prevent further escalation of conflicts.To address these social conflicts,a comprehensive strategy is ernments should work towards reducing economic disparities by implementing policies that promote fair distribution of wealth and opportunities.This includes investing in education, healthcare,and social welfare programs to ensure that everyone has access to basic services.Furthermore,urban planning should be carried out with a focus on sustainability and inclusivity.This involves creating affordable housing,improving public transportation, and preserving green spaces to accommodate the growing urban population while maintaining a high quality of life.Education and awareness campaigns are also vital in promoting cultural understanding and respect for diversity.Schools,community centers,and media outlets should play arole in fostering dialogue and appreciation for different cultures and traditions.Lastly,the regulation of social media platforms is necessary to prevent the spread of harmful content that can incite violence and hatred.This includes monitoring and removing posts that promote discrimination,as well as encouraging positive and constructive discussions online.In conclusion,social conflicts are a complex issue that requires a concerted effort from all sectors of society.By addressing the root causes and implementing effective solutions, we can work towards building a more harmonious and inclusive society for everyone.。

Large quantum gravity effects Unexpected limitations of the classical theory

a rXiv:g r-qc/9618v18Oct1996Large quantum gravity eﬀects:Unexpected limitations of the classical theory Abhay Ashtekar ∗Center for Gravitational Physics and Geometry Physics Department,Penn State,University Park,PA 16802,USA Abstract 3-dimensional gravity coupled to Maxwell (or Klein-Gordon)ﬁelds is ex-actly soluble under the assumption of axi-symmetry.The solution is used to probe several quantum gravity issues.In particular,it is shown that the quantum ﬂuctuations in the geometry are large unless the number and fre-quency of photons satisfy the inequality N (¯h Gω)2<<1.Thus,even when there is a single photon of Planckian frequency,the quantum uncertainties in the metric are signiﬁcant.Results hold also for a sector of the 4-dimensional theory (consisting of Einstein Rosen gravitational waves).I.This work has several motivations.The ﬁrst stems from the fact that recently there has been considerable mathematical progress in various approaches to quantum gravity.Hence,it is now important to devise criteria for the physical viability of potential solutions.One obvious demand is that the theory should admit “a suﬃcient number of”semi-classical states.However,the precise meaning of this requirement is not obvious.What exactly is the set of classical solutions that should admit quantum analogs?Indeed,even for theHydrogen atom problem the answer is subtle from the perspective of the classical theory.In the present case,we do not have direct experimental data to guide us.Therefore,a natural strategy is to try to develop intuition by analyzing exactly soluble models.Such models can also provide insights in to a number of other issues.For example,since the modes that are signiﬁcant at inﬁnity in the Hawking process have trans-Planckian frequencies near the horizon,there has been considerable interest in the role of high frequency ﬁelds in quantum gravity.There have been suggestions that an adequate description of such modes may violate local Lorentz invariance [1].It has also been suggested that the usual counting of states at high frequencies is inadequate [2]and that the correct counting may lead to a “holographic picture”in which physics within a region can be captured by states residing at its boundary[3].Finally,there exists a mathematically well-deﬁned theory called semi-classical gravity inwhich the gravitationalﬁeld is treated classically but the matterﬁelds are quantized[4].One hopes that this theory would be phenomenologically satisfactory in the low energy-regime. Is this indeed the case?Can one derive this theory from full quantum gravity?What does it“miss out”?The purpose of this letter is to analyze such issues in the context of an exactly soluble model:3-dimensional(Lorentzian)gravity coupled to Maxwellﬁelds,restricted by the condi-tion that there be a rotational symmetry.As is well-known,in3dimensions,a Maxwellﬁeld F ab is dual to a scalarﬁeldΦvia F ab=ǫab c∇cΦand Maxwell equations reduce to the wave equation,g ab∇a∇bΦ=0.Furthermore,the duality mapping sends the stress-energy tensor T ab(F)to T ab(Φ).Therefore,mathematically,the Einstein-Maxwell system(without further ﬁelds and interactions)is equivalent to the Einstein-Klein-Gordon system.For simplicity, we will use the scalarﬁeld formulation.Since this system is exactly soluble both classically (see e.g.[5])and quantum mechanically[6,7]under the assumption of axi-symmetry,we can explicitly analyze the status of various quantum issues in the solution.We willﬁnd that quantum eﬀects can be unexpectedly large so that a very large number of solutions of the classical theory are spurious:they do not appear in the classical limit of the quantum theory. The model is completely equivalent[5]to a“midi-superspace”consisting of Einstein-Rosen waves in4-dimensional source-free general relativity.Hence,our results apply also to this sector of4-dimensional quantum gravity[8].Finally,it should not be diﬃcult to extend the discussion to the treatment of CGHS dilatonic black holes given in[9].In order to compare and contrast various eﬀects and better understand their origin,let us proceed in steps,“switching¯h and G on and oﬀ”as needed.(However,throughout we set c=1.)Let us begin with quantumﬁeld theory without gravity(G=0).Then we just have an axi-symmetric quantum Maxwellﬁeld in3-dimensional Minkowski space-time(M,ηab), represented by the operator-valued distributionˆΦ(R,T)= ∞0dω[f+ω(R,T)A(ω)+f−ω(R,T)A†(ω)].(1.1) Here f+ω(R,T)=J o(ω,R)exp−iωT(with J o,the Bessel function)are the positive frequency solutions,A†and A,the creation and annihilation operators,satisfying the commutation relations[A(ω),A†(ω′)]=¯hδ(ω,ω′).In the rest frame used in the expansion(1.1),the Hamiltonian is given by:ˆHo= ∞0dωωA†(ω)A(ω).(1.2)The system is Poincar´e invariant and other Poncar´e generators can be expressed similarly. The Hilbert space is the Fock space.Given any classical solution,˜C(R,T)= ∞0dω[f+ω(R,T)C(ω)+f−ω(R,T)¯C(ω)],(1.3) there is a(normalized)coherent state,|Ψc>which remains peaked around˜C for all times:|Ψc>=(exp−1¯h∞dωC(ω)A†(ω)|0>(1.4)However,quantum ﬂuctuations in physical quantities are often negligible compared to the expectation value of that quantity only when the the expected number of photons,N =<Ψc |ˆN|Ψc >≡12 R 0dRR [(∂T ˜C (R,T ))2+(∂R ˜C (R,T ))2],(1.8)which is precisely the (Minkowskian)energy of the scalar ﬁeld ˜C(R,T )in a box of radius R at time T .Thus,the problem of solving the coupled Einstein-Maxwell system reduces to that of solving the wave equation on the ﬂat space (M,ηab ).In the Hamiltonian language,one can fully gauge ﬁx the system,show that the true degree of freedom can be made to reside in the scalar ﬁeld,focus on its dynamics and treat the metric as a secondary quantity which can be expressed,at the end,as a functional of the scalar ﬁeld[7].As in the previous case (with ¯h =0,G =0),there is no natural frequency scale and the general description in insensitive to the detailed proﬁle of C (ω).The asymptotic metric,for example,depends only on the total (Minkowskian)energy H 0(˜C )and not to where C (ω)is peaked.Suppose the ﬁeld ˜C (R,T )has initial data of compact support.Then,a neighborhood of inﬁnity is source-free and in that region the metric is locally ﬂat,with Γ(R,T )=H o (˜C ),a constant (given by the total energy of ˜C).Nonetheless,when GH o (˜C )≥1,the asymptoticgeometry is quite diﬀerent from that of the globallyﬂat metricηab(whereΓ=0).In particular,there is a deﬁcit angle2π(1−exp−GH o/2)at inﬁnity.[11].As a consequence, we no longer have Poincar´e invariance even at inﬁnity[11–13].There is a preferred rest-frame given by∂/∂T selected by the total energy-momentum of the system.Perhaps the most dramatic diﬀerence from the Minkowskian situation is that the total Hamiltonian of the system(including gravity)is now bounded above[12,13]:it is given by1H=(1−e−4GˆH o),(1.10)4Gwhich are both positive deﬁnite and self-adjoint.(The other possibility is to further nor-mal order these operators.However,then the spectrum of the resulting Hamiltonian is unbounded below as well as above and resemblance with(1.9)is lost.Furthermore,our main results on quantumﬂuctuations would not be aﬀected by this change.) The question now is:Are there semi-classical states which approximate solutions (˜C(R,T),g ab)of Einstein-Maxwell equations throughout the space-time?Since we wish to approximate,in particular,the solution˜C to the wave equation,the obvious candidates are,again,the coherent states|Ψc>.Let us compute the expectation values of operators of interest and see if they give us back the classicalﬁelds.Fortunately,these expectation values as well asﬂuctuations around them can be expressed in closed forms.We have: <Ψc|ˆΦ(R,T)|Ψc>=˜C(R,T);<Ψc|ˆg RR|Ψc>=e1[1−e14G(1−e−4G dωω|C(ω)|2).(1.12)4G(In particular,the expectation values have an explicit¯h dependence.)However,since we now have a natural length scale,we can analyze limits.In the low frequency limit,G¯hω<<1,the¯h dependence disappears in the leading order and we recover the classical values provided the frequency is so small that N(G¯hω)2<<1.In this case,we can also evaluate the quantum corrections systematically to any desired order:<Ψc|ˆg RR|Ψc>=g RR[1+1GN(G¯hωo)2e−4GH o(˜C)+...,(1.13) where in the last terms,we have evaluated the integrals approximately for the case when C(ω)is sharply peaked at a frequencyωo.In the high frequency regime,G¯hω>>1,there is always a signiﬁcant disagreement with the classical expressions:<Ψc|ˆg RR|Ψc>=e14G [1−e−14G[1−e−N],(1.14)Note that now¯h does not disappear even in the leading order terms and the departures from the classical values occur even in the asymptotic metric.Let us analyze the quantum uncertainties.For brevity,we will now focus just on the metric operator.The exact expression turns out to be:(△ˆg RR¯h dω|C(ω)|2(1−e G¯hω)2−1].(1.15)In the low frequency regime,G¯hω<<1,we therefore have:(△ˆg RR<ˆg RR>)2≈e N(e2G¯hωo).(1.17)Thus,in the high frequency regime,the quantumﬂuctuations in the metric are huge even when the state is sharply peaked at the given classical scalarﬁeld˜C and they become worse as the number N of expected photons increases.(Recall that the opposite holds for theˆΦﬁeld.)Furthermore,even if we have a single photon of Planck frequency–i.e.,a little“blip”in C(ω)peaked at the Planck frequency with N=1–Eq(1.15)implies that (△ˆg/<ˆg>)2>103!Next,consider the low frequency regime.Now,for the quantum uncertainties in the Maxwellﬁeld to be negligible,we need N>>1and for the uncertainties in the metric to be negligible,we need the frequency to be so low that N(G¯hω)2<<1.Only in this regime does the geometry given by classical equations approximate the predictions of the quantum theory.To summarize,the quantum theory does have inﬁnitely many states with semi-classical behavior.This by itself is a non-trivial property,given the non-linearities of the Einsteinequations.However,all these states are peaked at a very restricted class of classical solutions: the corresponding Maxwellﬁeld proﬁles C(ω)have to satisfy the two inequalities:N> >1and N(G¯hω)2<<1.Thus,most classical solutions–including the one representing the gravitationalﬁeld caused by a“blip”of(trans-)Planckian frequency in the Maxwell ﬁeld–do not arise as classical limits of the quantum theory;they fail to serve as leading order approximations even in regions where the curvature is small.This is a subtle failure of the classical theory;non-linearities of Einstein’s equations magnify the small quantum uncertainties in the Maxwellﬁeld to hugeﬂuctuations in the metric.It would have been diﬃcult to guess this process before hand and arrive at the two speciﬁc restrictions from general considerations.We will conclude by summarizing some ramiﬁcations of these results.1.Semi-classical gravity:Recall[4]that a solution to semi-classical gravity consists of a quadruplet(M,g ab,ˆΦ,|Ψ>)consisting of a metric g ab(of hyperbolic signature)on a manifold M,a quantum matterﬁeldˆΦsatisfying aﬁeld equation on(M,g ab)and a state |Ψ>of theﬁeld such that the semi-classical Einstein equation G ab=8πG<Ψ|ˆT ab|Ψ>is satisﬁed.In4-dimensions,it is diﬃcult toﬁnd exact solutions to this theory and several important issues about the nature of the approximation involved remain open.What is the situation in the present model?Now,it is straightforward to show that the theory admits an inﬁnite number of solutions.Indeed,given any(axi-symmetric)classical solution(˜C,g ab) to Einstein-Maxwell equations on M=R3,the quadruplet(M,g ab,ˆΦ,|Ψc>)is an exact solution to semi-classical gravity.While this abundance serves to show that the theory is mathematically rich,it also brings out its physical limitations.For,unless the proﬁles C(ω) of the Maxwellﬁeld satisfy N(G¯hω)2<<1,the solution to semi-classical gravity is spurious; it does not approximate the situation in the full theory.Very roughly,it knows about bothG and¯h separately,but does not exploit the Planck length which requires both at once.2.High frequency behavior:We saw that something peculiar does happen at Planck-ian and trans-Planckian frequencies.However,local Lorentz invariance is not broken;the ﬁeld equation governing˜C does not,for example,involve an additional vectorﬁeld.The frequency refers to the asymptotic rest-frame(∂/∂T).The relativistic dispersion relation is not modiﬁed,nor is there a high frequency cut-oﬀ.Similarly,within this model,there is no hint of the behavior suggested by the holographic hypothesis.It is sometimes conjectured that in a full quantum theory the trans-Planckian strangeness will trickle down and even aﬀect the vacuum since the vacuumﬂuctuations involve all modes.There is no evidence of such a behavior either.The vacuum|0>is stable although the vacuumﬂuctuations involve modes of arbitrarily high frequencies.Furthermore,the vacuum is an eigenstate both ofˆH (with eigenvalue zero)and ofˆg RR=−ˆg T T(with eigenvalue one).The peculiarity that does arise is of a rather diﬀerent nature:classical geometry simply fails to be a good approximation when C(ω)has a signiﬁcant high frequency component and one has to take the quantum nature of geometry–with all itsﬂuctuations–seriously.3.3versus4dimensions:All our discussion is based on a speciﬁc model.What are its main limitations?The model refers to3-dimensional gravity which has several peculiarities. First,since Newton’s constant in three dimensions has physical dimensions of inverse mass (rather than length times inverse mass)we can not associate a Schwarzschild radius to a given mass(without introducing a cosmological constant).This explains intuitively why we did not encounter gravitational collapse.If we consider a spherical scalarﬁeld in4dimensions,on the other hand,the theory has two sectors;one leading to black holes(strong data)and the other leading to simply scattering(weak data).The present model can only give us insights in to the second sector.Hence,it would be imprudent to draw from it deﬁnitive conclusions about the Bekenstein bound[2]and related conjectures[3].A second diﬀerence is that(in the asymptoticallyﬂat context)while in3dimensions the metric outside sources isﬂat,in4-dimensional general relativity it falls oﬀas1/r.Therefore,in4-dimensional asymptoticallyﬂat situations,the eﬀect of trans-Planckian frequencies will decay and the ﬂuctuations will be signiﬁcant only near(but possibly in a macroscopic region around) the sources.In spite of these limitations,the model does provide concrete evidence for large quantum gravity eﬀects in domains that have been generally ignored.Furthermore, they are not artifacts of an unusual quantization procedure;we used only traditional Fock-space methods.Therefore,with appropriate modiﬁcations,features encountered here should persist in more sophisticated models.Indeed,as already remarked,the model arises directly from symmetry reduction of4-dimensional general relativity[5]and similar reductions exist also in string theory[14].Details as well as several other issues will be discussed in[15].Acknowledgements:I would like to thank Chris Beetle,Eanna Flanagan,Donald Marolf,Monica Pierri and Thomas Thiemann for discussions.This work was supported in part by the NSF grant95-14240,by the Eberly research funds of Penn State University and by the Erwin Schr¨o dinger International Institute for Mathematical Sciences,Vienna.REFERENCES[1]S.Corley,T.Jacobson,Phys.Rev.D541568(1996).[2]J.Bekenstein,Phys.Rev.D49,1912(1994).[3]L.Susskind,J.Math.Phys.36,6397(1995);G.’t Hooft,Class.Quantum Grav.11,621(1994);C.Thorne,in Sakharov Conference in Physics,Moscow,Vol.91,pp447-54.[4]R.Wald,Quantumﬁeld theory in curved space-times and black hole thermodynamics,(Chicago U.P.,Chicago,(1994))and references therein.[5]A.Ashtekar,J.Bicak and B.S.Schmidt,gr-qc/960842.[6]M.Allen,Class.Quantum Grav.4,149(1987).[7]A.Ashtekar and M.Pierri,J.Math.Phys.(in press);gr-qc/9606085.[8]K.Kuchaˇr,Phys.Rev.D4,955(1971).[9]K.Kuchaˇr,J.Romano and M.Varadarajan,gr-qc/9608011.[10]B.K.Berger,P.T.Chru´s ciel,V.Moncrief,Ann.Phys.237,322(1995).[11]S.Deser,R.Jackiw and G.’t Hooft,Ann.Phys.(New York),152,220(1984).[12]M.Henneaux,Phys.Rev.D29,2766(1984).[13]A.Ashtekar and M.Varadarajan,Phys.Rev.D40,4944(1994);M.Varadarajan,Phys.Rev.D52,2020(1995).[14]A.Sen,Nucl.Phys.B434,179(1995).[15]A.Ashtekar and C.Beetle,(in preparation)。

因果论的英语作文

因果论的英语作文Title: Causality: Understanding the Foundation of Existence。

Causality, the fundamental principle that every event has a cause, is a concept deeply embedded in various disciplines ranging from philosophy to science. Its essence permeates through our understanding of the universe, shaping our perceptions and guiding our actions. In this essay, we delve into the significance of causality, its implications across different domains, and its role in shaping our worldview.At its core, causality asserts that every effect is preceded by a cause, creating a chain of interconnected events that govern the unfolding of existence. This principle underpins the scientific method, serving as the basis for experimentation, observation, and inference. Through causal relationships, scientists decipher the mechanisms governing natural phenomena, unraveling themysteries of the cosmos.In the realm of philosophy, causality provokes contemplation on the nature of reality and human agency. Philosophers grapple with questions of determinism versus free will, pondering whether our actions are predetermined by preceding causes or if we possess the autonomy to shape our destinies. While some advocate for a deterministic worldview, arguing that every action is the inevitable consequence of prior events, others espouse the belief in free will, asserting our capacity to make choices independent of external influences.Furthermore, causality extends its influence to the domains of ethics and morality. The concept of moral responsibility hinges on the idea that our actions have consequences, and we are accountable for the effects we generate. Whether in interpersonal relationships or societal dynamics, understanding the causal link between actions and outcomes fosters a sense of accountability and empowers individuals to act with integrity and foresight.In the realm of psychology, causality informs our understanding of human behavior and cognition.Psychologists explore the causal factors that shape personality development, cognitive processes, and emotional responses. By identifying the underlying causes of behavior, therapists and counselors can intervene effectively to promote mental well-being and facilitate personal growth.Moreover, causality plays a pivotal role in the domainof economics and social sciences. Economists analyze causal relationships within complex systems of production, consumption, and exchange, seeking to understand thefactors driving economic growth, inequality, and market fluctuations. Similarly, sociologists investigate thecausal mechanisms behind social phenomena such as stratification, migration, and cultural change, sheddinglight on the intricate interplay of individual actions and societal structures.In everyday life, we constantly navigate a web ofcausal relationships, making decisions based on our understanding of cause and effect. From mundane tasks likeplanning our daily routines to momentous choices that shape our future trajectories, we rely on our intuition of causality to navigate the complexities of existence.However, despite its pervasive influence, causality is not without its complexities and limitations. In the realmof quantum mechanics, for instance, the principle of causality is challenged by phenomena such as quantum entanglement and indeterminacy, where events occurseemingly without a discernible cause or in apparent violation of classical causal laws. Similarly, in chaotic systems, small perturbations can lead to disproportionately large effects, rendering deterministic predictions untenable.Nevertheless, even in the face of these challenges, causality remains a cornerstone of our understanding of the universe. It provides a framework through which weinterpret our experiences, formulate hypotheses, and make predictions about the future. Whether unraveling the mysteries of the cosmos, navigating the complexities of human behavior, or pondering the nature of existence itself,causality serves as a guiding principle that illuminates the pathways of knowledge and understanding.In conclusion, causality stands as a timeless principle that underlies our perception of reality and informs our endeavors across diverse domains. From science to philosophy, from ethics to economics, its influence reverberates through every facet of human inquiry and experience. As we continue to probe the depths of existence and unravel the mysteries of the universe, the principle of causality remains steadfast, guiding our exploration of the intricate tapestry of cause and effect that defines the fabric of reality.。

Universality in the Recombination of Cesium-133 Atoms

a r X i v :0711.1908v 1 [c o n d -m a t .o t h e r ] 13 N o v 2007Universality in the Recombination of Cesium-133AtomsL.Platter ∗Department of Physics,The Ohio State University,Columbus,OH 43210andDepartment of Physics and Astronomy,Ohio University,Athens,OH 45701,USAJ.R.Shepard †Department of Physics,University of Colorado,Boulder,CO 80309,USA(Dated:February 2,2008)We demonstrate the implications of Eﬁmov physics in the recently measured recombination rate of 133Cs atoms.By employing previously calculated results for the energy dependence of the re-combination rate of 4He atoms,we obtain three independent scaling functions that are capable of describing the recombination rates over a large energy range for identical bosons with large scat-tering length.We benchmark these universal functions by successfully comparing their predictions with full atom-dimer phase shift calculations with artiﬁcial 4He potentials yielding large scattering lengths.Exploiting universality,we ﬁnally use these functions to determine the 3-body recombina-tion rate of 133Cs atoms with large positive scattering length,compare our results to experimental data obtained by the Innsbruck group and ﬁnd excellent agreement.PACS numbers:21.45.+v,34.50.-s,03.75.NtIntroduction -In atomic physics the term universality refers to phenomena which are a result of a two-body scattering length a much larger than the range R of the underlying potential and do not depend on any further parameters describing the two-body interaction.The non-relativistic three-body system also exhibits univer-sal properties if a ≫R ,but an additional three-body parameter is needed for the theoretical description of ob-servables.Therefore,one three-body observable can be used (e.g.the atom-dimer scattering length a 3)to pre-dict all other low-energy observables of such systems.A particularly interesting signature of universality in the three-body system is a tower of inﬁnitely many bound states (Eﬁmov states )in the limit a =±∞with an ac-cumulation point at the scattering threshold and a geo-metric spectrum :E (n )T=(e −2π/s 0)n −n ∗¯h 2κ2∗/m,(1)where κ∗is the binding wavenumber of the branch ofEﬁmov states labeled by n ∗.The three-body system dis-plays therefore discrete scaling symmetry in the univer-sal limit with a scaling factor factor e π/s 0.In the case of identical bosons,s 0≈1.00624and the discrete scaling factor is e π/s 0≈22.7.These results were ﬁrst derived in the 1970’s by Vitaly Eﬁmov [1,2]and were rederived in the last decade in the framework of eﬀective ﬁeld theories (EFT)[3,4].Recently,experimental evidence for Eﬁmov physics was found by the Innsbruck group [5].Using a mag-netic ﬁeld to control the scattering length via a Feshbachresonance,they measured the recombination rate of cold 133Cs atoms and observed a resonant enhancement in the three-body recombination rate at a ≈−850a 0which occurs because an Eﬁmov state is close to the 3-atom threshold for that value of a .The three-body recombi-nation rate for atoms with large scattering length at non-zero temperature has been calculated with a number of diﬀerent models or based on the universality of atoms with large scattering lengths [6,7,8,9,10].However,a striking way to demonstrate universality is to describe observables of one system with information which has been extracted from a completely diﬀerent system.In [11],the authors considered Eﬁmov’s radial laws which parameterize the three-atom S-matrix in terms of six uni-versal functions which depend only on a dimensionless scaling variable,x =(ma 2E/¯h 2)1/2,and phase factors which only contain the three-body parameter.In this work,simplifying assumptions justiﬁed over a restricted range of x were made to reduce the six universal functions required to parameterize the atom-dimer recombination rate to just a single function.This function was then extracted from microscopic calculations of the recombi-nation rates for 4He atoms by Suno et al.[12].In a recent paper,Shepard [13]calculated the recombination rates from atom-dimer elastic scattering phase shifts for four diﬀerent 4He potentials and was able to obtain two uni-versal functions.Here,we relax some of the simplifying assumptions made in [11]and extract a set of three independent universal functions capable of parameterizing the three-body re-2 combination rate.We test the performance of these uni-versal functions using“data”generated from phase shiftcalculations[13]employing artiﬁcial short-range4He po-tentials.Finally,we use the new universal functions tocalculate the scattering length and temperature depen-dent recombination rate for133Cs atoms as measured bythe Innsbruck group[5]and comment on our results.Recombination-Three-body recombination is a pro-cess in which three atoms collide to form a diatomicmolecule(dimer).If the scattering length is positiveand large compared to the range of the interaction,wehave to diﬀerentiate between deep and shallow dimers.Shallow dimers have an approximate binding energy ofE shallow≃¯h2/(ma2)≪¯h2/(mR2).The binding energyof deep dimers cannot be expressed in terms of the ef-fective range parameters and E deep>¯h2/(mR2).If theunderlying interaction supports deep bound states,re-combination processes can occur for either sign of a.In acold thermal gas of atoms,recombination processes leadto a change in the number density of atoms n Adx4|S AD,AAA|2,(7)where k denotes a numerical constant.The matrix el-ement S AD,AAA is related to the S-matrix element forelastic scattering by the optical theorem|S AD,AAA|2=1−|S AD,AD|2.(8)Eﬁmov’s radial law then gives the dependence on univer-sal functions and the three-body parameter a∗0asS AD,AD=s22(x)+s212(x)e2is0ln(a/a∗0)1−e−4πs0e2iδ2.(10)It follows that|s11(0)|≃0.002.Theﬁrst simplifyingassumptions being made in[11]was that this functionremains small(i.e.;≪1)for all x and can be ignored.Then the energy dependent recombination rate can bewritten asK(0)(E)=C max sin[s0ln(aa∗0)](h2(x)+ih4(x))2¯h a412H(x)for two representative values of y=−1(dotted line) and y=2(dot-dashed line).therefore be possible to extract all four h i’s from these results.In the present work,we have attempted to do so,however,it turns out that this is numerically not possible. The most probable reason for this is is the eﬀect of rangecorrections which is R/a∼0.1for typical helium poten-tials.We therefore write h3and h4as h3(x)=y(x)H(x) and h4(x)=(1−y(x))H(x).Assuming that y(x)is aslowly varying function,we let y(x)→y,a constant,andobtainh3(x)=yH(x),h4(x)=(1−y)H(x).(13) Byﬁtting the resulting expression for diﬀerent choices toy to the recombination rates of all four4He potentials. It turns out that our results are mostly insensitive to thechoice of y apart from an overall scaling factor for thefunction H(x)which means,in eﬀect,that we are unable discriminate between h3and h4in ourﬁtting procedure.Our results for the functions h1,h2and H are displayed in Fig.1.As required by the threshold condition,all threefunctions vanish at threshold.It should be noted that forx<1the function H(x)∼0and is small compared to h1 and h2for x>1which conﬁrms the assumptions madeabout the imaginary part of the amplitude in Eq.(11)made in[11].To test our new parameterizations we have generated three artiﬁcial potentials(which we call I,IIand III)characterized by diﬀerent three-body parameters a∗0(with a/a∗0=1.32,1.16and0.99,respectively)but having approximately the same ratio of R/a as the real 4He potentials used in this work.We have calculated the recombination rates for these potentials and use these re-sults to benchmark our universal functions.Our resultsare displayed in Fig.2.Weﬁnd that the new set of func-tions is capable of describing the recombination rates of these potentials over a relatively large range of x.For comparison we also display the corresponding results for the case that H(x)is set to0which demonstrates that FIG.2:The exact and predicted results for the potentials I(circles and thick solid line),II(squares and thick dashed line)and III(triangles and thick dot-dashed line).The thin solid,dashed and dot-dashed line denote the corresponding results for potentials I,II and III obtained with the functions h1and h2only and the function H set to zero.adding H(x)amounts to a signiﬁcant improvement of the description of observables at large energies.Results for Cesium-We now use the three universal functions h1(x),h2(x)and H(x)discussed in the previ-ous sections to calculate three-body recombination rates for ultracold133Cs atoms.133Cs atoms can recombine into deep and shallow dimers.As mentioned above,a deep dimer is so strongly bound that it cannot be de-scribed within the EFT for short-range interactions as the binding energy is larger than¯h2/(mR2).We account for such processes by letting ln a∗0→ln a∗0−iη∗/s0as also discussed above.We then calculate the temperature dependent recombination rate by calculatingα(T)=∞0dE E2e−E/(k B T)K3(E)√4FIG.3:The3-body recombination lengthρ3for133Cs for a∗0=210a0and three diﬀerent values of the parameterη∗:0 (solid line),0.01(dashed line),and0.06(dotted lines)plot-ted together with the experimental results of the Innsbruck experiment(triangles)[5].universal functions s ij vary only weakly with energy pos-sibly explains the good qualitative results of the descrip-tion of the recombination in terms of a small number of real functions.We have tested the quality of our parame-terizations with artiﬁcialﬁnite range potentials which are appreciably diﬀerent from the original Helium potentials but which display universal eﬀects in three-body sector.We have found that our three real universal functions can describe the recombination of these artiﬁcial poten-tials reasonably well which gives further evidence that the assumptions made in[11]were well justiﬁed.Finally, we have used these functions to compute the recombi-nation length for133Cs atoms for diﬀerent values of the parameterη∗which approximately accounts for the eﬀect of deep dimer states and have compared our results withexperimental data obtained by the Innsbruck group[5]. Although our results show very good agreement withthe data,sensitivity toη∗is insuﬃcient to permit a pre-cise determination of this parameter.Overall,we con-sider our results to be an excellent example of how few-body systems with large scattering length exhibit uni-versal features.The low-energy properties of4He atoms allow us to compute accurately the low-energy properties of a gas of a completely diﬀerent element,133Cs,which atﬁrst glance has little in common with4He. Nevertheless,we point out that the results cannot be thought of as complete treatment of the problem at hand. For example,not only did we make the assumption that some of the universal functions(speciﬁcally,some of the s ij’s)do not contribute signiﬁcantly to the recombination coeﬃcients,we also extracted the functions from data sets obtained withﬁnite range potentials.Although the impact of range corrections is known to be small for real-istic Helium atom-atom potentials as R/a∼0.1,it needs to be pointed out that range corrections are expected to be sizable for large enough energies.To obtain all uni-versal functions s ij relevant to the recombination rate, a calculation in the limit R→0seems therefore to be necessary.This will also allow a precise determination of the energy-dependence of K deep(E)[18].Furthermore,it is already understood how to include range corrections systematically in the framework of eﬀectiveﬁeld theory [14,15,16].Indeed,this approach has already been used to calculate range corrections to the recombination rate into a shallow dimer[17].Thus,further eﬀort should be devoted to include these eﬀects in the calculation of the energy-dependent recombination rate.We are thankful to Eric Braaten for useful discussions and comments on the manuscript.This work was sup-ported in part by the Department of Energy under grant DE-FG02-93ER40756,by the National Science Founda-tion under Grant No.PHY–0354916.∗Electronic address:lplatter@†Electronic address:James.Shepard@[1]V.Eﬁmov,Phys.Lett.33B,563(1970).[2]V.N.Eﬁmov,Sov.J.Nucl.Phys.12,589(1971).[3]P.F.Bedaque,H.W.Hammer and U.van Kolck,Phys.Rev.Lett.82,463(1999).[4]E.Braaten and H.-W.Hammer,Phys.Rept.428,259(2006).[5]T.Kraemer,M.Mark,P.Waldburger,J.G.Danzl,C.Chin,B.Engeser,A.nge,K.Pilch,A.Jaakkola,H.-C.N¨a gerl,and R.Grimm,Nature440,315(2006).[6]J.P.D’Incao,H.Suno,and B.D.Esry,Phys.Rev.Lett.93,123201(2004).[7]M.D.Lee,T.Koehler and P.S.Julienne,Phys.Rev.A76,012720(2007).[8]S.Jonsell,Europhys.Lett.76,8(2006).[9]M.T.Yamashita,T.Frederico,and L.Tomio,Phys.Lett.A363,468(2007).[10]P.Massignan and H.T.C.Stoof,“Eﬁmov states near aFeshbach resonance,”arXiv:cond-mat/0702462.[11]E.Braaten,D.Kang and L.Platter,Phys.Rev.A75,052714(2007).[12]H.Suno,B.D.Esry,C.H.Greene,and J.P.Burke,Phys.Rev.A65,042725(2002).[13]J.R.Shepard,Phys.Rev.A75,062713(2007).[14]P. F.Bedaque,G.Rupak,H.W.Griesshammer andH.W.Hammer,Nucl.Phys.A714,589(2003).[15]H.W.Hammer and T.Mehen,Phys.Lett.B516,353(2001).[16]L.Platter and D.R.Phillips,Few Body Syst.40,35(2006).[17]H.W.Hammer,hde and L.Platter,Phys.Rev.A75,032715(2007).[18]E.Braaten,H.-W.Hammer,D.Kang and L.Platter,inpreparation.。

Universality of the Plastic Instability in Strained Amorphous Solids翻译

摘要：通过对比金属玻璃和Lennard-Jones 玻璃体的外部张力响应，我们发现了一个在无热准静态限定情况下，定量的具有普适性的塑性不稳定基本原理。

显微镜下这两种类型的玻璃体不同于人们的想象，后者被确定是由于二元交互作用产生的，前者是由于不能被忽视的电子气体导致的多个相互作用产生的。

尽管两者有如此巨大的差异，但是却有着相同的saddle-node bifurcation （分叉）。

因此统计弹性塑性稳态中的应力跟能量降落就具有普适性，具有相同的系统规模指数。

在二元交互作用的非晶固体简单模型中塑性不稳定性的基本原理是近几年才研究清楚的。

一个非晶固体的基本塑性不稳定性是研究无热状态，准静态形变的最好例子，例如消除热波动的影响和应力的比例。

相反，一些图片表明塑性不稳定性出现在‘‘weak’’之前被称为‘‘shear’’（剪切）转换区，塑性不稳定性是随着Hessian 矩阵H 的一个特征值消失而出现的。

212(r ,r ,r )N ij i jU H r r ∂=∂∂…… （1） U 代表系统的总势能，它是一个关于N 个粒子组成的系统中的颗粒位置的函数，除了三个明显的0特征值与Goldstone 模型相关外，其他所有的Hessian 矩阵特征值都是正数，并且是一个对称矩阵。

塑性不稳定性表现出二元玻璃体的普遍性质：在没有外部应力作用下（γ=0）除了Goldstone 模型以外所有的H 特征值都是确定的，lowlying 特征值与扩展特征函数相关联。

随着外部应力的增加，某一个特征值也会逐渐降为零，同时相关的特征函数也变成局部的，此时的特征值用λp 表示，在γp 处到达零时通过 saddle-node bifurcation ，这种方式具有普遍性。

P λ∝上面这个约束有很多有趣的结论。

一个最直接的结论就是系统最小值与凹点之间的势垒E 最终倾向于零32()P E γγ∝-这些通过在弹性塑性稳态下统计的应力跟能量降落的约束得到的结论并不明显（可能说是更有趣）。

言行皆是经典的英语作文

In the realm of language and communication,the essence of classic speech and writing lies in their ability to convey profound thoughts,emotions,and ideas with elegance and precision.The art of crafting such expressions is not only a reflection of ones linguistic prowess but also a testament to the depth of understanding and the ability to connect with the audience on a deeper level.The Power of WordsClassic speech and writing are characterized by their power to inspire,motivate,and even transform.The words chosen are not merely a means of communication but are carefully selected to evoke a specific response or emotion.For instance,the famous I have a dream speech by Martin Luther King Jr.is a timeless example of how powerful language can unite and inspire people towards a common goal.Elegance and PrecisionThe elegance of classic speech and writing is often found in the simplicity and precision of the language used.It avoids unnecessary jargon and complex sentence structures, making the message clear and accessible to a wide audience.For example,William Shakespeares plays are celebrated for their poetic language and profound insights into the human condition,delivered through concise and powerful dialogue.Cultural and Historical SignificanceClassic expressions often hold cultural and historical significance,reflecting the values, beliefs,and experiences of a particular time and place.They serve as a window into the past,allowing us to understand and appreciate the context in which they were created. For example,the works of Charles Dickens provide a vivid portrayal of19thcentury England,offering insights into the social issues of the time.Universality and TimelessnessOne of the hallmarks of classic speech and writing is their universality and timelessness. They transcend cultural and temporal boundaries,resonating with people across different ages and societies.The words of Confucius,for instance,continue to influence moral and ethical thinking in various cultures,demonstrating the enduring appeal of wellcrafted expressions.The Art of StorytellingClassic speech and writing often employ the art of storytelling to engage the audience and convey complex ideas in a relatable manner.Storytelling allows for the exploration of themes and emotions in a way that is both entertaining and enlightening.The narratives of Homers epic poems,The Iliad and The Odyssey,have captivated readers for centuries with their rich storytelling and profound moral lessons.Influence and InspirationThe influence of classic speech and writing is farreaching,inspiring generations of writers,speakers,and thinkers.They serve as a benchmark for excellence in communication,setting a high standard for clarity,creativity,and impact.The speeches of Winston Churchill during World War II,for example,are remembered for their ability to rally a nation and inspire courage in the face of adversity.ConclusionIn conclusion,classic speech and writing are distinguished by their power to move, inspire,and endure.They are crafted with care and consideration,reflecting a deep understanding of language and its potential to shape our world.By studying and appreciating these classics,we can learn valuable lessons about effective communication and the enduring power of wellchosen words.。

衍射的应用英语作文

衍射的应用英语作文Diffraction: A Fascinating Phenomenon in Our Lives.You know, diffraction is this cool thing in physicsthat happens when light or sound waves bend around obstacles. It's like magic, but science!Have you ever noticed those rainbow-colored spots on a CD when you shine a light on it? That's diffraction in action! The tiny grooves on the CD surface scatter thelight in all directions, creating those pretty patterns.In music, diffraction plays a role too. Have you ever noticed that sound seems to travel differently in different environments? That's because walls, floors, and ceilings can diffract sound waves, changing how they travel through a space.Astronomers use diffraction to their advantage too. Telescopes rely on diffraction to gather light from distantstars and galaxies, allowing us to peer into the depths of space. Without diffraction, we'd miss out on so much of the universe's wonders.But diffraction isn't just for scientists and astronomers. It's a part of our daily lives too. From the way light dances through a window's blinds to the way our voices echo in a large room, diffraction is constantly shaping our perception of the world.So, next time you see those rainbow spots on a CD or hear a distant voice echoing in a hallway, remember: it's all thanks to diffraction, that fascinating phenomenon that makes our world just a little bit more magical.。

用初中英语简要介绍双缝实验

用初中英语简要介绍双缝实验The Double-Slit ExperimentThe double-slit experiment is a fundamental experiment in quantum mechanics that demonstrates the wave-particle duality of light and other quantum particles. It was first performed by the English physicist Thomas Young in 1801, and it has since become one of the most famous experiments in the history of science.The basic setup of the double-slit experiment is as follows. A source of light, such as a laser or a monochromatic light source, is directed towards a barrier that has two narrow slits cut in it. The light passing through the slits is then projected onto a screen or a detector. When the light passes through the two slits, it creates an interference pattern on the screen, with alternating bright and dark regions.This interference pattern is a clear demonstration of the wave-like nature of light. If light were simply a stream of particles, one would expect to see two separate bright spots on the screen, corresponding to the two slits. However, the interference pattern shows that the light is behaving like a wave, with the waves from the two slits interfering with each other.The double-slit experiment can also be performed with other quantum particles, such as electrons or atoms. When these particles are directed towards the double slit, they also exhibit an interference pattern, indicating that they too have a wave-like nature.The wave-particle duality of quantum particles is a fundamental concept in quantum mechanics. It means that particles can exhibit both wave-like and particle-like properties, depending on the experiment being performed. This is a departure from the classical view of the world, where objects were either waves or particles, but not both.The double-slit experiment has been used to demonstrate the wave-particle duality of various quantum particles, including electrons, neutrons, atoms, and even large molecules. In each case, the interference pattern observed on the screen is a clear indication of the wave-like nature of the particles.One of the most interesting aspects of the double-slit experiment is the role of the observer. When the experiment is set up to detect which slit the particle goes through, the interference pattern disappears, and the particles behave like classical particles. This suggests that the act of measurement or observation can affect the behavior of quantum particles.This is a concept known as the "observer effect" in quantum mechanics, and it has profound implications for our understanding of the nature of reality. It suggests that the very act of observing or measuring a quantum system can alter its behavior, and that the observer is not a passive participant in the experiment.The double-slit experiment has also been used to explore the concept of quantum entanglement, which is another fundamental concept in quantum mechanics. Quantum entanglement occurs when two or more quantum particles become "entangled" with each other, such that the state of one particle is dependent on the state of the other.In the double-slit experiment, the interference pattern can be used to demonstrate the phenomenon of quantum entanglement. For example, if two particles are entangled and then directed towards the double slit, the interference pattern observed on the screen will depend on the state of the entangled particles.Overall, the double-slit experiment is a powerful and versatile tool for exploring the fundamental nature of reality at the quantum level. It has been used to demonstrate the wave-particle duality of light and other quantum particles, the observer effect, and the phenomenon of quantum entanglement. As such, it remains one ofthe most important and influential experiments in the history of science.。

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a r X i v :0807.0304v 1 [c o n d -m a t .s o f t ] 2 J u l 2008Universality in the diﬀusion of knotsNaoko Kanaeda ∗and Tetsuo DeguchiDepartment of Physics,Graduate School of Humanities and Sciences,Ochanomizu University,2-1-1Ohtsuka,Bunkyo-ku,Tokyo 112-8610,Japan †(Dated:July 2,2008)We have evaluated a universal ratio between diﬀusion constants of the ring polymer with a given knot K and a linear polymer with the same molecular weight in solution through the Brownian dynamics under hydrodynamic interaction.The ratio is found to be constant with respect to the number of monomers,N ,and hence the estimate at some N should be valid practically over a wide range of N for various polymer models.Interestingly,the ratio is determined by the average crossing number (ACN)of an ideal conformation of knotted curve K ,i.e.that of the ideal knot.The ACN of ideal knots should therefore be fundamental in the dynamics of knots.PACS numbers:82.35.Lr,05.40.Fb,05.20.-yNovel knotted structures of polymers have recently been found in various research ﬁelds such as DNA,proteins and synthetic polymers.1,2,3,4The topology of a ring polymer is conserved under thermal ﬂuctu-ations in solution and represented by a knot.5,6,7,8,9Topological constraints may lead to nontrivial sta-tistical mechanical and dynamical properties of ring polymers 6,10,11,12,13,14,15,16,17,18,19.Recent progress in experiments of ring polymers should be quite remarkable.Diﬀusion constants of linear,re-laxed circular and supercoiled DNAs have been measured quite accurately.23Furthermore,hydrodynamic radius of circular DNA has also been measured.24Ring polymers of large molecular weights are synthesized not only quite eﬀectively 20but also with small dispersions and of high purity.21,22Circular DNAs with various knot types are derived,and they are separated into knotted species by gel electrophoresis.We should remark that synthetic ring polymers with nontrivial knots have not been syn-thesized and separated experimentally,yet.However,it is highly expected that ring polymers of nontrivial knot types should be synthesized and their topological eﬀects will be conﬁrmed experimentally in near future.In the paper we discuss diﬀusion constant D K of a ring polymer with ﬁxed topology K in good solution for vari-ous knot types.We evaluate it numerically via the Brow-nian dynamics with hydrodynamic interaction in which bond crossing is eﬀectively prohibited through the ﬁnite extensible non-linear elongational (FENE)potential.25We evaluate diﬀusion constant D L of a linear poly-mer with the same molecular weight,and derive ratio D K /D L .The ratio should corresponds to a universal amplitude ratio of critical phenomena and play a sig-niﬁcant role in the dynamics of knotted ring polymers.According to the renormalization group arguments,ratio D K /D L should be universal if the number of monomers,N ,is large enough.26,27,28Moreover,we expect that ring polymers of diﬀerent knot types can be separated exper-imentally with respect to their topologies by making use of the diﬀerence among the sedimentation coeﬃcients,which can be calculated from the diﬀusion constants.29Through simulation we ﬁnd that ratio D K /D L is ap-proximately constant with respect to N for various knots.Thus,if we evaluate ratio D K /D L at some value of N ,it is practically valid for other values of N unless N is inﬁnitely large.We can therefore predict the diﬀusion constant D K of a polymer model at some value of N ,multiplying the ratio D K /D L by the estimate of D L .Here we remark that the value of D L may depend on the number N and on some details of the model.Furthermore,we show numerically that ratio D K /D L is a linear function of the average crossing number (ACN)of the ideal knot of K ,an ideal conﬁguration of knotted curve K ,which will be deﬁned shortly.Since the ratio D K /D L is almost independent of N ,it follows that the linear ﬁtting formula should be valid practically in a wide range of ﬁnite values of N .Thus,the ideal knot of a knotted curve K should play a fundamental role in the dynamics of ﬁnite-size knotted ring polymers in solution.In the inﬁnite N limit,however,we expect that ratio D K /D L should approach the same universal value for diﬀerent knots.Let us deﬁne the ideal knot,brieﬂy.For a given knot K it is given by the trajectory that allows maximal radial expansion of a virtual tube of uniform diameter centered around the axial trajectory of the knot K .30,31We de-ﬁne the ACN of a knotted curve as follows:We take its projection onto a plane,and enumerate the number of crossings in the knot diagram on the plane.Then we take the average of the crossing number over all normal directions of the planes,and we call it ACN of the given knotted curve.Throughout the paper,we employ the symbols of knots following Rolfsen’s textbook.32Let us formulate the simulation method.The ring polymer is modeled as a cyclic bead-and-spring chain with N beads connected by N FENE springs with forcegiven by F (r )=−H r /(1−r 2/r 2max ),where r =|r |.We denote by b the unit of distance,which gives the average distance between neighboring monomers approximately.We set constants H and r max by H =30k B T/b 2and r max =1.3b .We employ the predictor-corrector version 35of the Ermak-McCammon algorithm for generating time evolu-2 knot type D R2GTABLE I:Estimates of diﬀusion constants D L and D K,andthe mean square radius of gyration radius R2G and for alinear polymer of N=45and ring polymers of N=45withvarious knot types.tion of a ring polymer in solution.The hydrodynamicinteraction is taken into account through the Ronte-Prager-Yamakawa tensor33where the bead friction isgiven byζ=6πηs a with the bead radius a=0.257band a dimensionless hydrodynamic interaction parame-ter h∗=(ζ/6πηs)6t( r G(t)− r G(0))2 (1)Here r G(t)denotes the position vector of the center ofmass of the polymer at time t.Making use of(1)wehave evaluated diﬀusion constants D L and D K.The estimates of diﬀusion constants D L and D K atN=45are listed in Table1together with those of themean square radius of gyration R2G .The data of D Land D K are plotted against N in Figure1.Theﬁttingcurves to them are given by D=bN−ν.The estimatesof the exponent are close to that of the Self-AvoidingWalks.Furthermore,theχ2values are good.Thus,theformula gives goodﬁtting curves,although the numberof segments,N,is not very large.Numerically weﬁnd that ratio D K1/D K2of two diﬀer-ent knots K1and K2is almost constant with respect toN.For instance,the graph of ratio D31/D0versus N forthe data is almostﬂat,as shown in Figure2.Here0and 31denote the trivial and the trefoil knot,respectively. Thus,the estimate of D K/D0evaluated at a value of N, say N=45,for some knot K should also be valid at otherﬁnite values of N unless N is inﬁnitely large,since it is almost independent of N.For the diﬀusion constant of a ring polymer,D R,the ratio D R/D L should correspond to a universal amplitude ratio and should be universal if N is large enough.26,27,28 Here there is no topological constraint in the ring poly-mer model and D R does not mean D K of a knot K.In the previous simulation25it has been found that ratio D0/D L is given by about1.1and almost independent of N within the range investigated.From the numerical observations and the RG argu-ments,we now introduce two conjectures:(A)D0/D L should be given by1.1in the large N limit;(B)ratio D K/D0for a nontrivial knot K should remain almost the same value in a wide range ofﬁnite values of N,i.e. the N-dependence should be very small,and it should ap-proach1in the inﬁnite N limit.The latter part of conjec-ture B should be favorable to the idea of knot localization discussed through computer simulations of self-avoiding lattice polygons.17Quite interestingly weﬁnd that ratio D K/D L can be approximated by a linear function of the average crossing number(ACN)of ideal knots,i.e.the ideal representa-tions of the corresponding knots.In Figure3simulation data of D K/D L are plotted against ACN of ideal knots. Weﬁnd that the data points areﬁtted well by the fol-lowing formula:D K/D L=a+b ACN.(2) Here,the estimates of a and b are given in Figure3.Thus, the diﬀusion constant D K of a knot K can be estimated in terms of the ACN of the ideal knot of K.We thus summarize the simulation results as follows: ratio D K/D0for a knot K should be almost constant with respect to N in a wide range of N and can be ex-pressed by the linear function of ACN of ideal knots, while it approaches1when N becomes inﬁnitely large. In other words,it is aﬁnite-size eﬀect that D K/D0is dif-ferent from1,and theﬁnite-size value of D K/D0should be observed in experiments.Eq.(2)should be useful in separating synthetic ring polymers into various knotted species by making use of the diﬀerence among sedimen-tation coeﬃcients.Ideal knots should play a fundamental role in the dy-namics of knotted ring polymers in solution.In fact, we have shown it for the diﬀusion constants.In exper-iments of gel electrophoresis drift velocities of diﬀerent knots formed on the same DNA molecules was shown to be simply related to the ACN of ideal knots.38The two independent results suggest the importance of the ACN of ideal knots in the dynamics of knotted ring polymers. In fact,the physical situations are rather diﬀerent.For a lattice model,electrophoretic drift velocities of knotted ring polymers in gel are approximately expressed as a function of the ACN of ideal knots.39,40Drift veloc-ities in a dilute solvent have also been evaluated for the same model.However,the ratios of the drift velocity of the trefoil knot31to that of knots41,51and71,respec-tively,are about3percentage larger than the ratios of the diﬀusion constant of knot31to those of the same speciesof knots,respectively.The inconsistency should be re-lated to the renormalization of the Monte-carlo steps for each knot.Here we recall that there is no parameter to be adjusted in ourﬁrst-principle simulation.In conclusion,we have evaluated universal ratios among the diﬀusion constants of knotted ring polymers in good solution for several knots,where bond crossing is eﬀectively prohibited in the Brownian dynamics under hydrodynamic interaction.The universal ratio of diﬀu-sion constants D K/D L is almost constant with respect to the number of polymer segments,N.Moreover,it is found that the ratio D K/D L is determined by the ACN of the ideal knot of K.Through the linear relation,we can estimate the diﬀusion constant of a given knot. 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