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Tsallis statistics generalization of non-equilibrium work relations M. Ponmurugan The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India (Dated: January 9, 2012) We use third constraint formulation of Tsallis statistics and derive theq-statistics generalization ofnon-equilibriumworkrelationssuchastheJarzynskiequalityandtheCrooksfluctuationtheorem whichrelatethefreeenergydifferencesbetweentwoequilibriumstatesandtheworkdistributionof thenon-equilibrium processes. 2 1 PACSnumbers: 05.70.Ln,05.20.-y,05.70.-a,05.40.-a 0 Keywords: FluctuationTheorem,Jarzynskiequality,Tsallisstatistics,non-equilibriumprocess 2 n a Introduction– Recent advances in theory of non- by considering Tsallis statistics [8] as a finite heat bath J equilibrium statistical mechanics have established the statistics [7]. However, this theorem generally contains 6 methods to calculate the free energy differences between two temperatures instead of one as observed usually in the two equilibrium states of the drivensystem from the non-equilibriumworkrelations(Eqs.1and 2). There has ] h non-equilibrium work measurements [1–3]. These meth- notbeenanygeneralizedconnectionestablishedbetween c ods are generally called as non-equilibrium work rela- Tsallis statistics and non-equilibrium work relations at e tions and in particular named as the Jarzynski equality a single temperature. Since the generalized connection m [1] and the Crooks work fluctuation theorem [2]. Con- should exhibit interesting applications in complex sys- - siderasysteminitiallyinequilibriumattemperature(in- tems [9, 10], inthis paper, we derivethe q-statistics gen- t a verse) β = 1/kT (k is the Boltzmann constant), which eralization of Jarzynski equality and the Crooks work t s is externally driven from its initial equilibrium state A fluctuation theorem for the classical system driven be- . t to final equilibrium state B by non-equilibrium process. tween two equilibrium states by a non-equilibrium pro- a Let P [γF] be the probability of the phase space trajec- cess using Tsallis statistics. m F tory γF, for the system drivenbetween the two states in The theory of Tsallis statistics based on a generalized - d forward direction. This satisfies the Crooks work fluctu- form of entropy, Sq, characterized by the index q ∈ R, n ation theorem [2, 4], such that q = 1 recovers the standard theory of Boltz- o mann and Gibbs. This generalized (Tsallis) entropy is [c PF[γF] = eβ(W−∆F), (1) given by the expression [8] P [γR] R 2 1−cq S =k (3) v where W is the work performed on the driven system, q (q−1) 3 ∆F is the free energy difference between the two equi- 5 libriumstates andP [γR]is the probabilityofthe phase where k is the positive (Boltzmann) constant and 1 R spacetrajectoryγR,forthesystemdriveninthereversed 5 w . direction. This is the direct relation between the work c = pq. (4) 0 q i dissipation and the ratio of probabilities for the forward 1 i=1 X 1 andthereversedtrajectoriesanditsintegratedversionis 1 Jarzynski equality [1, 5], Here, w is the total number of microstates of the system : and pi is the probability of the system at microstate i. v hexp[−βW]i=exp[−β∆F]. (2) In the limit q →1 one can recover BG entropy i X w r The averageh...i is over a statistical ensemble of realiza- a tions of a given thermodynamic process. SBG =−k pi ℓn pi. (5) i=1 In Crooks work fluctuation theorem, the probabili- X ties of non-equilibrium forward and reversed trajecto- Preserving the standard variational principle, Tsallis es- ries are related by taking the initial conditions from the tablished the canonical generalized distributions and its Boltzmann-Gibb’s (BG) equilibrium distribution. There refinements [9, 11]. Using Tsallis statistics, Chame and are very few studies on work fluctuation theorem relat- Mello have derived the generalization of the fluctuation ingthenon-equilibriumforwardandreversedtrajectories dissipation theorem [12]. Tsallis nonextensive statisti- takenfrom other statistical distribution [6, 7]. In partic- cal mechanics is also considered as an approach to non- ular, finite bath work fluctuation theorem has been de- equilibriumstationarystatesofsmallorcomplexsystems rived earlier [7], which includes the microcanonical work [9]. However, its equilibrium formulation remains to be fluctuationtheoremandtheCrooksworkfluctuationthe- validforobtainingthermodynamicpropertiesoftheequi- orem as the two limiting cases. This has been obtained librium system [9, 13]. There are four versions of Tsallis 2 statistics [14], in particular, we use the most widely ac- (escorted probability) weighted Hamiltonian eigenvalue cepted third constraint (escorted probability) [9] formu- [11]averagedoverallmicrostateinaninitialequilibrium lation of Tsallis statistics. state A (see, Eqs. 6, 7 and 8) and InTsallisthirdconstraintformulation,thegeneralized equilibrium canonical distribution at β is given by [11] β[H(ΓA)−Uq(A)] 1−1q Z (A)= 1−(1−q) (14) q c (A) 1 β[ǫi−Uq] 1−1q XΓA (cid:20) q (cid:21) p = 1−(1−q) (6) i Z c q (cid:20) q (cid:21) with exp [−β(ǫ −U )/c ] q i q q ≡ , q Zq cq(A) = [P(ΓA)] . (15) XΓA where ǫ is the energy of the ith microstate, U is the i q normalized constrained of internal energy which is given SupposethegivensystemevolvesintimeunderHamil- by [11, 15] tonian dynamics and reaches a different macrostate B whichis to be in equilibrium atsame β. The probability w U = 1 pqǫ (7) distribution for the system in a microstate ΓB is given q c i i by, q i X and Zq is the q-generalized partition function which is 1 β[H(ΓB)−Uq(B)] 1−1q P(Γ )= 1−(1−q) (,16) given by [9] B Z (B) c (B) q (cid:20) q (cid:21) w 1 Z = 1−(1−q)β[ǫi−Uq] 1−q (8) where H(ΓB) is the Hamiltonian for the system in a mi- q c crostateΓ ,U (B),Z (B)andc (B)havesamemeaning i (cid:20) q (cid:21) B q q q X asabovebutforthemacrostateB. Inordertoderivethe The normalization condition of pi leads to the relation non-equilibrium work relations for a given β, we formu- [11] late the problem as follows. Setup– Consider a classical Hamiltonian system in a c =Z1−q. (9) q q macrostateAwhichis initiallyinequilibriumwithreser- voir at inverse temperature β. Let λ be an external Thismodifiedformalismalsobecomesordinarycanonical t protocolapplied in the arbitrarytime interval τ to drive ensemble theoryinthe limit q →1[9]with c =1 and q=1 the system from its initial equilibrium state A to an- w other state B at constant bath temperature β. It is as- Z =exp[βU] exp[−βǫ ]≡exp [β(U −F)], (10) q=1 i sumed that the final state B is not necessarily to be in Xi equilibrium. However, the system in the state B at the where the internal energy constantbathtemperaturerelaxtowardstheequilibrium state B for the same β without doing any work [5]. Let w exp[βU] H(Γ ,λ ) is the Hamiltonian with externally controlled t t U = ǫ exp[−βǫ ] (11) Zq=1 i i time-dependent protocol λt and the phase-space coordi- i X nates of the system, Γ at a particular time t. At t = 0, w t ǫ exp[−βǫ ] = i i i the system Hamiltonian which is in any one of the mi- w P i exp[−βǫi] crostate ΓA is H(Γ0,λ0) = H(ΓA); and at time t = τ, the system Hamiltonian is H(Γ ,λ ). Let γ denote the and the free energy P τ τ entire trajectory of the driven system from t = 0 to τ. w One can obtain the statistical ensemble of possible real- F =−kT ℓn exp[−βǫi]. (12) izations by performing the above process repeatedly. In Xi following the refs [4, 5, 7], the work performed on the system for a given trajectory can be defined as ConsiderasysteminaninitialmacrostateA(forexam- ple closed system of volume V ) which is in equilibrium i W = H(Γ ,λ )−H(Γ ,λ ) (17) at β. The probability for the system in a microscopic τ τ 0 0 phase-space (microstate) ΓA is given by [11, 16] ≡ H(Γτ,λτ)−H(ΓA). 1 β[H(ΓA)−Uq(A)] 1−1q It should be noted that the work defined in Eq.(17) is P(Γ )= 1−(1−q) (,13) A different from the q-dependent work as given in ref.[11] Z (A) c (A) q (cid:20) q (cid:21) (seealsorefs.[16,17]). Differentdefinitionofworkandits where H(Γ ) is the Hamiltonian for the system in a mi- physical meaning has been discussed in detail in refs.[4, A crostate Γ , U (A) is the internal energy which is the 18, 19]. A q 3 During the time interval τ in which the system is The aboveequationprovidesthe consistentusageofβ = driven, we assume that the reservoir should always be 1/kT as the (inverse) reservoir temperature in Tsallis in equilibriumat a givenβ [20]. Insuch a case,the total statistics [17]. heat transferred by the system can be written into two The usage of thermodynamic laws are not mandatory part as for the proof of non-equilibrium work relations [4, 5]. Sincebothinternalenergy,appliedworkandheatarefor- Q = Qd+Qr, (18) q mulated clearly for the above driven system [26], we use where Qd is the q-dependent heat transferred between Eq.(20) and Eq.(22) and obtain the q-generalized non- q the system and the reservoir which should preserve the equilibriumworkrelationsinwhichtheinitialprobability (q-dependent)equilibriumnatureofthereservoirandQr distributions(Eqs. 13and16)aretakenfromthe Tsallis is the heat transferred when the system relaxes towards statistics. therequiredequilibriumstatefromthe finalstateB ata q-generalizedJarzynskiequality–Inordertoobtainthe given β [21]. There is no work performed on the system q-generalized version of Jarzynski equality for the above during relaxation [5]. We can define the heat transfer driven process, one can take the following q-exponential due to relaxation as averageoveranensembleofrealizationsinwhichtheini- tial distribution is taken from Tsallis statistics. Qr = H(Γ )−H(Γ ,λ ). (19) B τ τ −β(W+Qr−∆Uq) −β(W+Qr−∆Uq) Since the energy conservation is also valid for non- e cq(B) = e cq(B) P(Γ ) dΓ .(25) q q A A equilibrium process[20, 22, 23], the above drivensystem * + Z should obey the principle of energy conservation for any microscopic trajectory as Using the q-exponential identity exey = e[x+y+(1−q)xy] q q q W +Q=∆U , (20) [9] and Eq.(13), the integral of the above equation can q be rewritten as where ∆U = U (B)−U (A). The principle of energy q q q conservation as given in Eq.(20) for the non-equilibrium −β(W+Qr−∆Uq) 1 process [23] is different from the q-generalization of first eq cq(B) = Z (A) eMq q+NqdΓA, (26) * + q lawasproposedinrefs.[11,16,17]forequilibriumsystem. Z Therefore,theq-dependentheattransferredbythedriven where, system between the two equilibrium state is Qd = ∆U −W −Qr (21) β[W +Qr−∆Uq] q q Mq = − . (27) c (B) = ∆U −[H(Γ )−H(Γ )] q q B A whichdependsonlyontheinitialandfinalsystemstates. and Since Qd is independent of the non-equilibrium trajecto- q β[H(Γ )−U (A)] ries of the driven system, we impose the condition for N = − A q [1+(1−q)M ]. (28) q q c (A) the system relaxing towards the required (q-dependent) q equilibrium state at a given β as Using Eq.(20) and Eq.(22), one can rewrite βQd q cq(B)=cq(A)"1−(1−q)cq(A)#. (22) 1+(1−q)M = 1+(1−q) βQdq (29) q c (B) q Using Eq.(3), the above equation can be written as c (A)−c (B) q q = 1+ −βQd = 1−cq(B) − 1−cq(A) (23) cq(B) q q−1 q−1 c (A) q 1 = . = [Sq(B)−Sq(A)] cq(B) k σ = q Using Eqs.(17 and 19), one can obtained k where σ =S (B)−S (A) is the q-dependent change in −β q q q M +N = [H(Γ )−U (B)]. (30) q q B q entropy of the equilibrium system [9]. From the above c (B) q condition (Eq. 22), the q-dependent change in (equilibrium) reservoir entropy for the driven non-equilibrium Therefore, Eq.(26) becomes, process is obtained as [23–25] −β(W+Qr−∆Uq) 1 −β[H(ΓB)−Uq(B)] ∆Sqr = βQd. (24) eq cq(B) = Z (A) eq cq(B) dΓA. k q * + q Z 4 Since Hamiltonian dynamics preserve the phase-space microstateΓ isH(ΓR,λR)=H(Γ );andattimet=τ, B 0 0 B volume, dΓ = dΓ [4, 5] and using Eq.(16) we can the system Hamiltonian is H(ΓR,λR). The probability A B τ τ rewrite the above equation as of the phase space trajectory γR, for the system driven in reversedirectionobtained from the initial equilibrium e−β(Wc+qQ(Br−)∆Uq) = Zq(B) P(Γ ) dΓ (31) distribution is given as [4] q B B Z (A) * + q Z P [γR] = Peq(ΓR) (34) Z (B) R B 0 = q . 1 Z (A) 1 β[H(ΓB)−Uq(B)] 1−q q = 1−(1−q) . Z (B) c (B) q (cid:20) q (cid:21) Wehaveobtainedq-generalizedversionofoneofthenon- equilibrium work relation. It should be noted that β ap- Using Eq.(17), Eq (19) and Eq.(20), Eq.(33) can be peared in the q-exponential average to ensure that the rewritten as temperature of the reservoirremainssame, however,one 1 dinogesthneotdkrnivoewnapnryotcheisnsg. aFbuorutthesry,stQemr atnemdpceqr(aBtu)riendtuhre- PF[γF]= Zq1(A)"1−(1−q)β[H(ΓB)c−q(UAq)(B)+Qdq]#1−q . above relation also takes care of the heat exchange due to relaxationof the system towardsthe final equilibrium state [21]. This may provide the possible physicalmean- 1 1 βQd 1−q ing of the above average for the driven non-equilibrium P [γF] = 1−(1−q) q (35) F process instead of thinking as an adhoc method [5, 21]. Zq(A)" cq(A)# In the limit q → 1, expq(x) = exp(x) and cq = 1 [9], 1−1q then using Eqs.(10-12), Eq.(31) becomes β[H(Γ )−U (B)] B q 1−(1−q) . hexp[−β(W +Qr−∆U)]i = exp[β(∆U −∆F)(]32)  cq(A)[1−(1−q)cβqQ(Adq)] hexp[−β(W +Qr)]i = exp[−β∆F],   Using Eq.(22) and Eq.(34), Eq.(35) becomes, where ∆U = U(B)−U(A) is the change in internal en- 1 ergy and ∆F = F(B) − F(A) is the change in equi- P [γF]= Zq(B) 1−(1−q) βQdq 1−q P [γR]. (36) librium free energy of the BG canonical system. Thus, F Z (A) c (A) R q " q # we can obtain more general form of Jarzynski equality whichincludes heatdue to systemrelaxation[21]for the We can get from Eq.(20) that BG canonical system in the limit q →1. If Q is within r the measurements/numerical error for work calculation PF[γF] Zq(B) β[W +Qr−∆Uq] 1−1q = 1+(1−q) (37). inexperiments/simulations,theoriginalJarzynskiequal- P [γR] Z (A) c (A) ity (without the heat term due to relaxation[5, 21]) can R q (cid:20) q (cid:21) Z (B) β(W +Qr−∆U ) provide the reliable estimates of the free energy differ- ≡ q exp q . Z (A) q c (A) ences. q (cid:20) q (cid:21) q-generalized Crooks Work fluctuation theorem– In or- We have obtained the q-generalized version of another dertoderivetheq-generalizedversionoftheCrookswork non-equilibrium work relation. In the limit q → 1, fluctuation theorem, we proceed with the problem anal- exp (x) = exp(x) and c = 1 [9], then using Eqs.(10- q q ogoustoref.[4]asfollows. SinceH(Γ ,λ )=H(Γ ),the 0 0 A 12), Eq.(37) becomes probabilityofthe phase spacetrajectoryγF, for the system driven between the two equilibrium states obtained P [γF] F = exp[β(W +Qr−∆F)]. (38) from the initial equilibrium distribution in the forward P [γR] R direction (A to B) is given as [4] Thus, we can obtain more general form of Crooks work PF[γF] = PAeq(ΓF0) (33) fluctuation relation which includes heat due to system 1 β[H(ΓA)−Uq(A)] 1−1q relaxation [21] for the BG canonical system in the limit = 1−(1−q) . q →1. Z (A) c (A) q (cid:20) q (cid:21) In order to obtain the q-generalizedversionof Jarzyn- Suppose the system is driven from equilibrium state B ski equality from the q-generalized work fluctuation re- to state A using the time reversed protocol, λRt = λFτ−t, lation, one can take the following q-exponential average H(ΓR,λR) is the Hamiltonian for the externally con- over ensemble of realization in forward direction as t t trolledtime-dependentprotocolλR and,ΓR isthephase- t t −β(W+Qr−∆Uq) −β(W+Qr−∆Uq) spacecoordinateofthe systemataparticulartime t. At e cq(B) = e cq(B) P [γF] dγ .(39) q q F F t=0,the systemHamiltonianwhichis in anyone ofthe * + Z 5 Using Eq.(37), the above equation can be rewritten as AcknowledgementsIwouldliketothankS.S.Naina Mohammed and R. Chandrashekar for the valuable e−β(Wc+qQ(Br−)∆Uq) = Zq(B) I , (40) discussions during my learning stage of Tsallis statis- q Z (A) q tics. I also thank P. Renugambal for proof-reading the * + q manuscript. IthankProf. M.Campisifordrawingmyat- where tentiontohisearlierwork[7]. MyspecialthankstoProf. C. Tsallis for his personal appreciation of the present −β(W+Qr−∆Uq) β(W+Qr−∆Uq) I = e cq(B) e cq(A) P [γR] dγ . (41) work. q q q R R Z Since Hamiltonian dynamics preserve the phase-space volume, dγ =dγ [4, 5]. Using the q-exponential iden- F R tity exey = e[x+y+(1−q)xy] [9], we rewrite the integral of q q q email correspondence: [email protected] the above equation as [1] C. Jarzynski, Phys. Rev.Lett. 78, 2690 (1997). [2] G. E. Crooks, Phys. Rev. E 60, 2721 (1999); G. E. Iq = eqβ(W+Qr−∆Uq)DqPR[γR] dγR. (42) Crooks, Phys.Rev.E 61, 2361 (2000). Z [3] J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr, C. where Bustamante, Science296, 1832 (2002); D.Collin, F. Ri- tort,C.Jarzynski,S.B.Smith,I.TinocoJrandC.Bus- 1 1 β(W +Qr−∆Uq) tamante, Nature437, 231 (2005). D = − −(1−q) (.43) q c (A) c (B) c (A)c (B) [4] J. Horowitz and C. Jarzynski, J. Stat. Mech.: Theory (cid:20) q q q q (cid:21) Exp. P11002 (2007). Using Eq.(20)and Eq.(22), the aboveequationbecomes, [5] C.Jarzynski,J.Stat.Mech.: TheoryExp.P09005(2004). [6] C.BeckandE.G.D.Cohen,PhysicaA344,393(2004). 1 1 [c (B)−c (A)] [7] M.Campisi, P.TalknerandP.Hanggi,Phys.Rev.E80, q q Dq = − − =0. (44) 031145 (2009). c (A) c (B) c (A)c (B) (cid:20) q q q q (cid:21) [8] C. Tsallis, J. Stat. Phys. 52, 479 (1988); see, http://tsallis.cat.cbpf.br/biblio.htm for more references. Since e0 = 1, I = P [γR] dγ = 1 and Eq.(40) be- q q R R [9] C. 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