Non-extensive study of Rigid and Non-rigid Rotators

a r X i v :c o n d -m a t /0303523v 3 [c o n d -m a t .s t a t -m e c h ] 1 A p r 2004Non-extensive Study of Rigid and Non-rigid RotatorsG˘o khan B.Ba˘g cıa ,Ramazan Sever a ∗,Cevdet Tezcan baDepartment of Physics,Middle East Technical University,06531,Ankara,TurkeybDepartment of Electrical and Electronics Engineering,BaskentUniversity,06530,Ankara,Turkey(February 2,2008)AbstractThe isotropic rigid and non-rigid rotators in the framework of Tsallis statistics are studied in the high and low temperature limits.The generalized partition functions,internal energies and heat capacities are calculated.Classical re-sults of the Boltzmann-Gibbs statistics have been recovered as non-extensivity parameter approaches to 1.It has also been observed that non-extensivity parameter q behaves like a scale parameter in the low temperature regime of the rigid rotator model.PACS:05.20.-y;05.30.-d;05.70.;03.65.-wTypeset using REVT E XI.INTRODUCTIONThe non-extensive generalization of the standard Boltzmann-Gibbs statistics was pro-posed in1988by C.Tsallis[1-4].This non-extensive generalization begins with the supposi-tion of a new,fractal inspired entropy S q≡k(1− i p q i)/(q−1)where q is any real numberand k is a positive constant which becomes Boltzmann constant in the limit q→1.In this case,the new entropy S q takes the form-k B i p i ln p i,which is simply the standard entropy formula.This new non-extensive statistics has been studied a great deal to list its properties and it is also applied to many well-known examples of the Boltzmann-Gibbs statistics such as Ehrenfest theorem[5],Fokker-Planck equations[6]and quantum statistics[7].Wefirst calculate the partition function of the isotropic rigid rotator for both high energy and low energy limits and then obtain the generalized internal energy and specific heat expressions for both cases.This forms the Section II.The usual Maxwell-Boltzmann results are obtained in the limit q→1.In Section III,we turn our attention to non-rigid rotator and follow the same procedure for it as in the Section II.Results and discussions are given in Section IV.II.ISOTROPIC RIGID ROTATOREnergy levels of a rigid rotator are[8]E j=j(j+1)¯h2/2µa2.(1) where j=0,1,2,...,µ=m1m2/(m1+m2)reduced mass of the nuclei and a being the equilib-rium distance between them.The degeneracy of each level is g j=2j+1.In non-extensive formalism(NEXT),the partition function is given byZ q=∞j=0(2j+1)[1−β(1−q)j(j+1)¯h2Z q=∞j=0(2j+1)[1−(1−q)j(j+1)θT≪1and[1-(1−q)j(j+1)θT]1/(1−q).(4)For the interval1 q 2,solution to the above integral isZ q=T2−q.(5)As q goes to1,we getZ q→1=T∂βZ1−qq−1β2[α(2−q),1 q 2(8)whereα≡1β=k B T.(9) It is also possible to calculate the specific heat in this statistics asC q=∂U q2−q )1−q1W i=1p q i=U q.(12) This choice of constraint remedied all previous difficulties.Since we have studied all ther-modynamical quantities with the previous internal energy constraint,we must reconsider them with this new constraint.There are two ways to do this:Firstly,we can recalculate all thermodynamical quantities with this new internal energy constraint by forming the new partition function.Another method is tofind the relation between the temperatuıre param-eters of the old and new calculations.If one has all thermodynamical functions calculated with the old constraint and the relation between temperature parameters is known,it is possible to modify all previous calculations carried out with the constraint W i=1p q iεi=U q.This is the method we will follow,because we already have the solutions with respect to old constraint.We begin by writing all previous calculations in terms of intermediate variable t′where t′≡1/(β′ε).From now on,The superscript(2)will refer to calculations done by old con-straint.The partition function of the rigid rotator in the high temperature limit becomesZ(2)q(β′)=T′(2−q)=t′Z(2)qq.(14)or it is explicitly written asj [p(2)j(β′)]q=(2−q)q1−(1−q)β′U(2)q(β′)/ j[p(2)j(β′)]q.(18)Substituting Eq.(17)into the Eq.(18),we gett′=(2−q)(q+1)/q t1/q.(19) From Ref.[9],we haveZ′(3)q(β)=Z(2)q(β′).(20) By mere substitution,we getZ′(3)q(β)=(2−q)1/q t−1/q.(21) We also havep(3)j(β)=p(2)j(β′).(22)Thus,we formed p(3)j(β).Next,we use the relationj[p(3)j(β)]q=¯(Z q(3))1−q,(23) to obtain the partition function(¯Z q(3))1−q=(2−q)1/q t(1−q)/q.(24) Therefore,one may obtain internal energy and heat capacities by using the following relationβ∂U(3)q∂β(ln q¯Z q(3)).(25)WefindU(3)q=(2−q)1/q t1/q,(26) where ln q x≡x1−q−1dT =kT]1/(1−q).(28)This is nothing but the well-known partition function of the rigid rotator in low temperature limit in MB statistics if(1-q)is small enough.This is the case if we make q closer to1.For such a choice,the partition function takes the formZ(2)q−→1∼=1+3exp(−2θ/T).(29)The generalized internal energy function for this case becomesU(2)q=6α]1/(1−q)]−q[1−2(1−q)βZ q→1exp(−2θ/T),(31)with1/α=k Bθin the q→1limit.Calculation of the specific heat readsC(2)q=12qα]1/(1−q)]−q[1−2(1−q)β1−q−36qα]1/(1−q)]−q−1[1−2(1−q)β1−q.(32)To modify the equations above in accordance with the third constraint,we substitute the Eq.(19)into the Eq.(3)and take thefirst two terms to get the low temperature limit partition functionZ(3)q∼=1+3[1−2(1−q)(2−q)−(q+1)/qθ1/q T−1/q]1/(1−q).(33)Using this partition function,we immediately get the specific heat value asC(3)q=6kα2k B T2exp(−2θ/T).(35)This limiting value of heat capacity is obtained by observing that1of the rigid rotator in the low temperature regime attains the same shape as the classical one but with a narrower width and a shift in the peak to the left.By increasing q,specific heat function attains the same shape as the classical one but with a narrower width and a shift in peak to the left.III.ISOTROPIC NON-RIGID ROTATORIn this case,energy levels are given by[10]E(nonrigid) j =¯h22I20kR20j2(j+1)2.(36)where I0=µR20and k is spring constant.We have neglected the third order term j3(j+1)3 in the above expression.It is better to rewrite equation above in the simple form¯E j (nonrigid)=E(nonrigid)jor simplyZ q= ∞0dx[ax2+bx+1]1/(1−q).(41) wherea=β(1−q)D and b=−β(1−q)B.(42) The integral above can be rewritten in factorial formZ q=(mn)1m )1/(1−q).(x+12[1±(1−4aβ),(44)whereµ=1nandβ=1Γ(µ)Γ(γ−µ) 1dttµ−1(1−t)γ−µ−1(1−xt)−ρ,(45)where the parameters satisfyν>0andµ>ν−σ.Here we see thatν=1and q>1. Thus,we simply getµ=σ=1(3−q)n2F1(1,1/(q−1);2/(q−1);1−m q−1−1)=q−1The resulting internal energy and heat capacity expressions may be found by using Equations(7)and(10)respectively together with the partition function above.Variation of heat capacity in the high temperature limit is plotted in Fig.3.In all the plots related to non-rigid rotator,we have used10.397and4.1×10−5as B and D parameters respectively for HCl molecule[11].ii)Low temperature limitAs in the case for isotropic rigid rotator in the low temperature limit,we look atfirst few terms of the summation in the Eq.(39)Z q∼=1+3[1−2B(1−q)β+4D(1−q)β]11−q[1+3(1+2(B−2D)(q−1)β)1the high and low temperature limits of the isotropic rigid rotator.We have concluded that study of extensive systems through non-extensive thermostatistics bears importance[13].V.ACKNOWLEDGEMENTSWe thank S.Abe for his helpful comments.One of us(G.B.B.)acknowledges C.Tsal-lis for creative environment in the International School and Workshop on Non Extensive Thermodynamics and Physical Applications held in Villasimius/Italy in May23-302001.REFERENCES[1]C.Tsallis,J.Stat.Phys.52(1988)479.[2]C.Tsallis,in:New Trends in Magnetism,Magnetic Materials and their Applications,eds.J.L.Mor´a n-Lopez and J.M.S´a nchez(Plenum Press,New York,1994),p.451. [3]C.Tsallis,Some comments on Boltzmann-Gibbs statistical mechanics,Chaos,Solitonsand Fractals6(1995)539.[4]E.M.F.Curado and C.Tsallis,J.Phys.A24(1991)L69;Corrigenda:J.Phys.A24(1991)3187;25(1992)1019.[5]A.R.Plastino and A.Plastino,Phys.Lett.A177(1993)177.[6]D.A.Stariolo,Phys.Lett.A185(1993)262.[7]F.B¨u y¨u kkili¸c and D.Demirhan,Phys.Lett.A181(1993)24; F.B¨u y¨u kkili¸c, D.Demirhan and A.Gulec,Phys.Lett.A197(1995)209.[8]R.K.Pathria,Statistical Mechanics,Second Edition,(1996),pp144-150.[9]C.Tsallis,Renio S.Mendes and A.R.Plastino,Physica A261(1998)534.[10]G.Herzberg,Molecular Spectra and Molecular Structure Vol.1(1967),pp103-106.[11]R.J.Silbey,R.Alberty,Physical Chemistry,pp469,Third Edition,John Wiley(2001).[12]S.Curilef and C.Tsallis,Physica A215(1995)542.[13]H.Touchette,Physica A305(2002)84.FIGURESFIG.1.Specific heat of the rigid rotator as a function of reduced temperature t=T/θin the high temperature limit.00.20.40.60.811.21.400.51 1.522.533.54C _q /kt q=1.25q=1.50q=1.75FIG.2.Specific of the rigid rotator heat as a function of reduced temperature t=T/θin the low temperature limit.00.20.40.60.811.21.41.61.800.51 1.522.533.54C _q /k t q=1.00q=1.25q=1.50ature limit.00.20.40.60.811.21.400.51 1.52 2.53 3.54C _q /kt q=1.25q=1.50q=1.75ture limit for q=1.25,1.50and 1.75respectively starting from above.05101520T 00.10.20.30.40.50.6C _q。