Maximum entropy principle and the form of source in non-equilibrium statistical operator method
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Maximum entropy principle and the form of sourcein non-equilibrium statistical operator method
V Ryazanov
Institute for Nuclear Research, pr.Nauki, 47, Kiev, UkraineE-mail: [email protected]
Abstract.
It is supposed that the exponential multiplier in the method of thenon-equilibrium statistical operator (Zubarev‘s approach) can be considered as adistribution density of the past lifetime of the system, and can be replaced by anarbitrary distribution function. To specify this distribution the method of maximumentropy principle as in [Sch¨onfeldt J-H, Jiminez N, Plastino A R, Plastino A, Casas M2007
Physica A aximum entropy principle in NSO
1. Introduction
Among possible approaches to the description of non-equilibrium systems the
Non-equilibrium Statistical Operator Method (NESOM) especially demonstrated its efficiency[1, 2, 3]. NESOM provides a very promising technique that implies in a far-reachinggeneralization the statistical methods developed by Boltzmann and Gibbs. NESOM wasinitially built on intuitive and heuristic arguments, apparently it can be incorporatedwithin an interesting approach to the rationalization of statistical mechanics, ascontained in the maximization of (informational statistical) entropy (MaxEnt for short)and Bayesian methods. NESOM appears as a very powerful, concise, based on soundprinciples, and elegant formalism of a broad scope to deal with systems arbitrarily farfrom equilibrium. The non-equilibrium statistical operator (NSO) introduced in [1, 2, 3]has a form ln ̺ ( t ) = Z ∞ p qzub ( u ) ln ̺ q ( t − u, − u ) du, ln ̺ q ( t,
0) = − Φ( t ) − X n F n ( t ) P n ; (1)ln ̺ q ( t, t ) = e {− t H/i ~ } ln ̺ q ( t, { t H/i ~ } ; Φ( t ) = ln Sp exp {− X n F n ( t ) P n } , where p q ( u ) = p qzub ( u ) = ε e − εu , u = t − t , (2) H is Hamiltonian, ln ρ ( t ) is the logarithm of the NSO, ln ρ q ( t , t ) is the logarithmof the quasi-equilibrium (or relevant) distribution; the first time argument indicatesthe time dependence of the values of the thermodynamic parameters F n ; the secondtime argument t in ln ρ q ( t , t ) denotes the time dependence through the Heizenbergrepresentation for dynamical variables P n on which ln ρ q ( t,
0) can depend [1, 2, 3].In [1, 2, 3] p q ( u ) = p qzub ( u ) = ε exp {− εu } ; after the thermodynamic limitingtransition N → ∞ , V → ∞ , N/V = const , ε →
0. From the complete group ofsolutions of Liouville equation (symmetric in time) the subset of retarded ”unilateral”in time solutions is selected by means of introducing a source K in the Liouville equationfor ln ρ ( t ) ( L is Liouville operator; iL = −{ H, ̺ } = P k (cid:20) ∂H∂p k ∂̺∂q k − ∂H∂q k ∂̺∂p k (cid:21) ; p k , and q k are pulses and coordinates of particles; { . . . } are Poisson brackets) ∂ ln ρ ( t ) ∂t + iL ln ρ ( t ) = − ε (ln ρ ( t ) − ln ρ q ( t, K zub . (3)In [4, 5] a convenient redefinition of the source term is proposed. Althoughinfinitesimally small, the source term introduced by Zubarev into the Liouville equationis shown to influence the macroscopic behaviour of the system in the sense that thecorresponding evolution equations do not coincide exactly with those obtained froman initial-value problem which corresponds to a definite experimental situation and aphysical set of macroobservables.In [6, 7] it was noted that in place of the function p q ( u ) = p qzub ( u ) = ε e − εu in(1) arbitrary (but having certain properties [8]) weigth functions w ( t, t ) can be used. aximum entropy principle in NSO w ( t, t ) (Abel’s kernel (2) in Zubarev’s approach) in thetime-smoothing or quasi-average procedure that has been introduced selects the subsetof retarded solutions from the total group of solutions of the Liouville equation. Thisconsideration is related to the question: how to obtain an irreversible behavior in theevolution of the macroscopic state of the system? In the MaxEnt-NESOM approach theirreversibility is incorporated from the outset using an ad hoc non-mechanical hypothesis.MaxEnt-NESOM yields information on the macrostate of the system at time t , whena measurement is performed, including the evolutionary history (in the interval fromthe initial time of preparation t up to time t ) by which the system came into thatstate (which introduced a generalization of Kirkwood‘s time-smoothing formalism [9]).Functions w ( t, t ) are typically kernels [1, 2, 3, 8] that appear in the mathematical theoryof convergence of integrals. In [10, 11, 12] other interpretation of the functions w ( t, t ),denoted as p q ( u ), is given. With the change of function w ( t, t ) = p q ( u ) the form ofsource (3) in the Liouville equation also changes. For an arbitrary function p q ( u ) itlooks like (4).In [10, 11, 12] it was noted that the function p q ( u ) = p qzub ( u ) = ε e − εu inNESOM [1, 2, 3] for the non-equilibrium distribution function can be interpreted asthe exponential probability distribution of the lifetime Γ of a system. Γ is a randomvariable of lifetime (time span) from the moment t of its birth till the current moment t ; ε − = h t − t i = h Γ i , where h Γ i = R up q ( u ) du is the average lifetime of the system. Thistime period can be called the time period of getting information about system from itspast. Instead of the exponential distribution p qzub ( u ) (2) in (1) used in [1, 2, 3] any othersample distribution p q ( u ) could be taken; integration by parts in time is performed at R p q ( u ) du | u =0 = − R p q ( u ) du | u →∞ = 0. If p q ( u ) = p qzub ( u ) = εe − εu by (2), ε = 1 / h Γ i the expression for NSO passes in (1) from [1, 2, 3].The same interpretation of the distribution p q ( u ) is given in [2], where this valueis understood as the distribution of the initial moment of time t . Since the random(past) lifetime is equal to Γ = u = t − t , the distribution of the past lifetime u coincideswith the distribution of the initial time values t . The moment t will be the momentof the first passage in the inverse time, if the moment t is taken as initial. In [2]the uniform distribution for an initial moment t is chosen, which after the transitionfrom Abel integration to Ces`aro integration passes to the exponential distribution p q ( u ) = p qzub ( u ) = ε e − εu . Such distribution serves as the limiting distribution of thelifetime [13], the first-passage time of a certain level. In the general case it is possibleto choose a lot of functions for the obvious type of distribution p q ( u ), which was notedin [10, 11, 12].In [13] the lifetimes of the system are introduced as random moments of the first-passage time till the moment when a random process describing system reaches a certainlimit, for example, a zero value. In [13] approximate exponential expressions for the aximum entropy principle in NSO K = K zub = − ε [ln ρ ( t ) − ln ρ q ( t, ε →
0, which in the spirit of [10] correspondsto the infinitely large lifetime value of an infinitely large system. For a system with finitesize this source is not equal to zero. In [8] this term enters the modified Liouville operatorand coincides with the form of Liouville equation suggested by Prigogine [16] (theBoltzmann-Prigogine symmetry), when the irreversibility is introduced in the theoryat the microscopic level.In [10] a new interpretation of the method of the NSO is given, in which theoperation of taking the invariant part [1, 2] or the use of an auxiliary ”weight function”(in the terminology of [6, 7, 8]) in NSO are treated as averaging the quasi-equilibriumstatistical operator over the distribution of past lifetime of a system. This approachagrees with the approach of the general theory of random processes, the renewal theory,and also with the conception of Zubarev work [2] where the NSO is conceived as someaveraging over the initial moment of time.The statistical operator depends on the information-gathering interval ( t , t ), butit must be borne in mind that this is the formal point consisting in that (as Kirkwoodpointed out) that the description to be built must contain all the previous historyin the development of the macrostate of the system. In [6, 7, 8] several basic stepsfor the construction of the NESOM formalism are indicated: a third basic step hasjust been introduced, namely, the inclusion of the past history (other terms usedare retro-effects or historicity) of the macrostate of the dissipative system. A fourthbasic step needs now to be considered, which is a generalization of Kirkwood’s time-smoothing procedure: the one that accounts for the past history and future dissipativeevolution. The time-smoothing procedure introduces a kind of Prigogine’s dynamicalcondition for dissipativity. The procedure introduces a kind of evanescent history asthe system macrostate evolves toward future from the initial condition at time t . Thefunction w ( t ; t ) [6, 7, 8] introduces the time-smoothing procedure. In principle, anykernel provided by the mathematical theory of convergence of trigonometrical seriesand transform integrals provides is acceptable for these purposes. Kirkwood, Green,Mori [9, 17, 18] and others have chosen what in mathematical terminology is Fej`er (orCes`aro-1) kernel. Meanwhile Zubarev introduced the one consisting in Abel’s kernel for w in Eq. (1) - which apparently appears to be the best choice, either mathematically butmostly physically: that is, taking w ( t ; t ) = ε e ε ( t − t ) , where ε is a positive infinitesimalvalue which tends to zero after the calculation of averages has been performed, and with aximum entropy principle in NSO t going to −∞ . Therefore a process with fading memory is introduced. In Zubarev’sapproach this fading process occurs in an adiabatic-like form towards the remote past: astime evolves memory decays exponentially with lifetime ε − [8]. The approach suggestedin [10, 11, 12] and in the present work enables to use a family of functions w ( t, t ) = p q ( u )and makes clear both their physical sense and those physical situations in which one oranother function w ( t, t ) = p q ( u ) can be used.Besides the Zubarev’s form of NSO [1, 2, 3], the NSO formulation in the Green-Mori form [17, 18] is known, where one assumes the auxiliary weight function [6, 7, 8]to be equal to W ( t, t ) = 1 − ( t − t ) /τ ; w ( t, t ) = dW ( t, t ) /dt = 1 /τ ; τ = t − t .After averaging one sets τ → ∞ . This choice at p q ( u = t − t ) = w ( t, t ) coincides withthe uniform lifetime distribution. The source in the Liouville equation takes the form K = ln ρ q /τ . In [1] this form of NSO is compared to the Zubarev’s form. One couldname many examples of explicitely setting the function p q ( u ). Each and every definitionimplies some specific form of the source term K in the Liouville equation, some specificform of the modified Liouville operator and NSO [10, 11, 12]. Thus the whole family ofNSO is defined.It is possible to make different assumptions about the form of the function of p q ( u ), getting different expressions for the source in the Liouville equation and for non-equilibrium characteristics of the system. It is possible to show [12] that certain choicesof the function of p q ( u ) result in the changes in non-equilibrium characteristics in thelimit of infinitely large average lifetimes as well. In [10, 11, 12] an analogy is tracedto the passage to the thermodynamic limit of systems of infinite size. So explicit formof the function p q ( u ) is important for describing non-equilibrium systems by the NSOmethod.Setting the form of the function p q ( u ) reflects not only the internal properties ofa system, but also the influence of the environment on an open system, the particularcharacter of its interaction with the environment [8]. In [2] a physical interpretation ofthe exponential distribution for the function p q ( u ) is given: a system evolves freely like anisolated system governed by the Liouville operator. Besides that the system undergoesrandom transitions, and the phase point representing the system switches from onetrajectory to another one with an exponential probability under the influence of the”thermostat”; the average intervals between successive push events increase infinitely.This takes place if the parameter of the exponential distribution tends to 0 after thetransition to the thermodynamic limit. Real physical systems have finite sizes. Theexponential distribution describes completely random systems. The influence of theenvironment on a system can have organized character as well, for example, this is thecase of systems in a stationary non-equilibrium state with input and output fluxes. Thecharacter of the interaction with the environment can also vary; therefore different formsof the function p q ( u ) can be used.The adequate choice of the function p q ( u ) is important for correct description ofthe non-equilibrium properties of statistical systems. To find the type of function p q ( u ),it is necessary to resort to some general principles, such as MaxEnt principle. In this aximum entropy principle in NSO
2. Maximum entropy principle for Liouville equations with source
In this paper we apply the maximum entropy principle for the determination of thefunction p q ( u ). The same approach was applied in [19] for the evolution equations withsource terms. In [7, 10, 11, 12] a general form for the source in the Liouville equationfor ln ρ ( t ) (3) is obtained. For our case the source term has the following form K = p q (0) ln ρ q ( t,
0) + Z ∞ ∂p q ( u ) ∂u ln ρ q ( t − u, − u ) du. (4)In [19] in the Liouville equation the distribution function ρ (z,t) is written in a form ρ ( −→ z , t ) = N f ME ( −→ z , t ) = NZ exp {− M X i =1 λ i A i } , (5)where A i ( ~z ) are M appropriate quantities that are functions of the phase space point ~z ; the quantities A i ( ~z ) correspond to the values P i from (1). The partition function Z is given by Z = Z exp {− M X i =1 λ i A i } d N z. (6)The function f ME ( ~z, t ) is normalized to unity: Z f ME ( −→ z , t ) d N z = 1; (7) Z ρ ( −→ z , t ) d N z = N ( t ); dNdt = Z Kd N z ; (8) ∂ρ∂t + −→ w ∇ ρ = dρdt = K ; ∇−→ w = 0 . The probability distribution f ME ( ~z, t ) is the one that maximizes the entropy S [ f ]under the constraints imposed by normalization and relevant mean values h A i i = R A i ρd N z (or a i = h A i i /N ). The re-scaled mean values a i and the associated Lagrangemultipliers λ i are related by the Jayne’s relations [20, 21] λ i = ∂S∂a i , a i = h A i i N = − ∂∂λ i (ln Z ) , (9) S = − Z f ln f d N z = ln Z + X i λ i a i . If we choose for ln ρ ( z, t ) the functionln ρ = ln ̺ ( t ) = Z ∞ p q ( u ) ln ̺ q ( t − u, − u ) du = Z t −∞ p q ( t − t ) ln ρ q ( t , t − t ) dt (10) aximum entropy principle in NSO p qzub → p q , K zub → K , and, following [1], choose λ i = p q ( t − t ) F i ( t ) , (11)then ln ρ ( ~z, t ) = Z t −∞ p q ( t − t ) ln ρ q ( t , t − t ) dt = (12) − Z t −∞ ( X i λ i A i + p q ( t − t ) ln Z ) dt = ln f ′ ME + ln N , where ln f ′ ME = − Z t −∞ X i λ i A i dt − ln Z λ ;ln N = ∆ Z = ln Z λ − Z t −∞ p q ( t − t ) ln Z dt ; (13) Z λ = Z exp[ − Z t −∞ X i λ i A i dt ] d N z, Z = Z exp[ − X i F i A i ] d N z ( F i are taken from (1)). The values Z λ and Z in the terminology of [1] are related to thepartition functions for a non-equilibrium and relevant statistical operator accordingly.In [19] for the Liouville equation of the kind (3) with constant sources equation onegets for dλ i /dt dλ i dt = ( M X i =1 C ji λ j ) − N ∂∂a i Z K ln f ME d N z, where the Zubarev-Peletminskiy selection rule [22, 23, 1, 7, 8] ~w ~ ∇ A i = M X j =1 C ij A j , ( i = 1 , . . . , M ) , d~zdt = ~w ( ~z ); ∂ρ∂t + ~ ∇ ( ρ ~w ) = K (14)is used; i, j = 1 , ... ; the C ij are c-numbers. In other representations the quantities A i can depend on the space variable, that is, when considering local densities of dynamicalvariables, and then the C ij can depend on the space variable as well or be differentialoperators.If more complex shape of the source (4) is considered, the equation for dλ i /dt takeson the form dλ i dt = M X j =1 C ji λ j ! − ∂∂a i (cid:18) N Z K ln f ME d N z (cid:19) − (15) X j λ j ∂∂a i (cid:18) N Z A j Kd N z (cid:19) − ln Z ∂∂a i ˙ NN ! . aximum entropy principle in NSO ∂∂a i and ∂∂λ i taking into account (9)-(13) by the functionaldifferentiation of the kind ∂∂a i → δδa i = N δδ h A i i , ∂∂λ i → δδλ i , which takes off the integration over time. For example δδ h A i i ln Z λ = − p q ( t − t ) X k ∂F k ∂ h A i i h A k i ; δ ln Z λ δλ i = −h A i i . The relations (5-8) and (9) are thus hold. If to take into account that R . . . ρ q d N z = R . . . ρd N z = h . . . i in the NSO method, then ∂∂a i N = ∂∂a i ˙ NN ! = 0 . (16)Let us consider the integrals in the rhs of (15) of the form R KB ( z ) dz , B being anarbitrary function of the dynamic variables z , and the source term K taken from (4);for Eq. (8), (14) K = [ p q (0) ln ρ q ( t,
0) + Z t −∞ ∂p q ( t − t ) ∂t ln ρ q ( t , t − t ) dt ] ρ . Assumethat p q ( u ) does not depend on z . Integrating by parts and assuming p q ( u ) u →∞ →
0, weget: Z KBd N z = − Z ∞ p q ( u ) ddu h B ln ρ q ( t − u, − u ) i du ; u = t − t . (17)Taking into account (16) and the fact that the operation ∂∂a i → δδ a i eliminates theintegration by time, the equation (15) takes on the form F i ( t ) dp q ( t − t ) dt = − p q ( t − t ) C i − p q ( t − t ) r − p q ( t − t ) r , (18)where C i = P j C ji F j ( t ), r = − ∂∂ h A i i ddt h (ln ρ ( t ) − ln N ) ln ρ q ( t , t − t ) i ; (19) r = − X j F j ( t ) r j ; r j = ∂∂ h A i i ddt h A j ( t ) ln ρ q ( t , t − t ) i . (20)An unknown function p q ( u ) enters the expression (19) through the terms ln ρ ( t ) andln N . To get rid of this dependence, we use the averaging theorem. For the expressionsfor ln ρ ( t ) and ln N in (19) we take all terms besides p q ( u ) out of the time integration.For each of these function however a different effective average time value should beused. The remaining integrals over p q ( u ) are equal to unity. We get:ln N ≃ ln Z ( c ) − ln Z ( c ) , (21) aximum entropy principle in NSO ρ ( t ) ≃ ln ρ q ( c , c − t ) = − ( X m F m ( c ) A m ( c − t ) + ln Z ( c )) . (22)Let us make another approximation and change the order of the operations ∂/∂ h A i i and d/dt in the expressions (19)-(20). The value t R t r dt = D ( t ) − D ( t ) enters theexpression (18), where D ( t ) = − ∂∂ h A i i h (ln ρ ( t ) − ln N ) ln ρ q ( t , t − t ) i = F i ( t )[ln Z ( c ) − ln Z ( c ) − ln Z ( c )] − (23) F i ( c ) ln Z ( t ) + X m,n ( h A m A n i − h A m ih A n i ) (cid:20) F n ( t ) h A m A i i − h A m ih A i i −− F m ( c ) h A i A n i − h A i ih A n i (cid:21) − X m,n F m ( c ) F n ( t ) X k h A k A m A n i − h A m A n ih A k ih A i A k i − h A i ih A k i ; r j = ∂∂t "X m F m ( t ) X k h A k A j A m i − h A j A m ih A k ih A i A k i − h A i ih A k i + δ ij ln Z ( t ) − X m h A j A m i − h A j ih A m ih A i A m i − h A i A m i . (24)The values of the correlators h A i i , h A i A k i , h A j A k A m i are averaged with ρ ( t ) andare t -dependent. In deriving (23), (24) we used the relations like ∂ ln Z ( c ) ∂ h A i = X n ∂ ln Z ( c ) ∂F n ( c ) ∂F n ( c ) ∂ h A i = X n h A n ih AA n i − h A ih A n i ; A n ( − c ) = e − icLA n ; ∂F m ∂ h A i i = 1 h A i A m i − h A i ih A m i . One can proceed with the expression (24) using the relations ∂F i ∂t = X k ∂F i ∂ h A k i ∂ h A k i ∂t = X k ∂F k ∂ h A i i ∂ h A k i ∂t ; ∂F i ∂ h A k i = ∂F k ∂ h A i i = ∂ S∂ h A k i ∂ h A i i ; ∂ ln Z∂t = − X m ∂F m ∂t h A m i . The solution to (18) has the form p q ( t − t ) = p q (0) M p q (0) t Z t F i r M dt . (25) aximum entropy principle in NSO C i and F i ( t ) does not depend on t , M ( t ) = exp (cid:26) − C i F i ( t − t ) − F i Z tt r dt (cid:27) =exp (cid:26) − C i F i ( t − t ) − F i ( D ( t ) − D ( t )) (cid:27) , (26) M ( t ) = 1, where R tt r dt is written in (23).Integrating by parts the second term in the denominator of (25) write it in thefollowing form: p q ( t − t ) = p q (0) M ( t )1 − L ; (27) L = p q (0) 1 F i "(cid:18)Z r dt (cid:19) | t − (cid:18)Z r dt (cid:19) | t M ( t ) − Z tt (cid:18)Z r dt (cid:19) (cid:18) C i F i + 1 F i dD ( t ) dt (cid:19) M ( t ) dt , where Z r dt = X j X m F m ( t ) F j ( t ) X k h A k A j A m i − h A j A m ih A k ih A i A k i − h A i ih A k i + (28) F i ln Z ( t ) − X m X j F j ( t ) h A j A m i − h A j ih A m ih A i A m i − h A i ih A m i ; dD ( t ) dt = X m,n (cid:26) F n ( t ) h A m A n i − h A m ih A n ih A m A i i − h A m ih A i i (cid:20) ddt ln( h A m A n i − h A m ih A n i ) − (29) − ddt ln( h A m A i i − h A m ih A i i ) (cid:21) + F m ( c ) h A m A n i − h A m ih A n ih A i A n i − h A i ih A n i × (cid:20) ddt ln( h A m A n i − h A m ih A n i ) − ddt ln( h A i A n i − h A i ih A n i ) (cid:21) −− F m ( c ) F n ( t ) X k h A k A m A n i − h A m A n ih A k ih A i A k i − h A i ih A k i (cid:20) ddt ln( h A k A m A n i − h A m A n ih A k i ) −− ddt ln( h A i A k i − h A i ih A k i ) (cid:21)(cid:27) . The expression for M ( t ) is given in (26), and p q (0) is determined from the conditionsfor the norm R ∞ p q ( u ) du = 1 and for the average lifetime h Γ i = R ∞ up q ( u ) du .If one either considers the stationary case or assumes a weak time dependence inthe correlators in (23)-(24), D ( t ) ≃ D ( t ) , r ≃
0, and (25)-(27) take on the form p q ( t − t ) = p q (0) exp {− C i F i ( t − t ) } (30)with C i /F i = p q (0) = 1 / h Γ i . In (30) an exponential distribution for p q ( u ), used in[1, 2, 3] is obtained. However generally the correlators in (23)-(24) are time dependent.Applying the full form of (27) for concrete systems it is possible to state the conditions aximum entropy principle in NSO L in the denominator of (27) is small, since p q (0) ≈ / h Γ i ≪
1. In thedenominator of (27) it stands in the combination with , 1 − L ≈ p q = p q (0) M [1 + L + L + . . . ] , and the distribution (27) is close to theexponential distribution (30) used in [1, 2, 3, 7, 8]. This results agrees with the resultsof [13], where the exponential distribution for the lifetime is shown to be a limitingdistribution. But the expression for h Γ i is explicitely given in (30), and in (23)-(29). Itwas already pointed above that the situations where L is comparable to 1 can arise aswell.For the maximum entropy principle with the Shannon measure of the informationentropy the exponential distribution used in [1, 2, 3], is basic. Choosing anothermeasures for the information entropy (e.g. Tsallis and Renyi measures [24, 25]) changesthe function f ME (5), which yields another forms of the lifetime distributions. For theNSO method the functions f ME are in the form (5), and the information entropy isgiven by the Shannon measure, hence basic distribution is the exponential one.The expressions (25), (27) for p q ( u ) depend on F i , the functions C i = P j C ji F j , r k , k = 1 ,
2, in (18)-(19) depend on the index i , the function p q ( u, i ) depends on i aswell. One can get an i -independent function p q ( u ) by symmetrizing the distribution, forexample, using the operation p q ( u ) = " M Y i =1 p q ( u, i ) M . Such formulation of maxent principle, as in [19], gives the distribution for thelifetime, related to the exponential distribution which serves in this case as the baseone. Distributions of other type can be obtained using some other form of maxentprinciple.
3. Another approach to the method of maximum entropy
Yet another approach to the determination of the type of function of distribution oflifetime is related to the method of maximal entropy inference (”maxent”), developedin [26] for the determination of the distribution of superstatistics. We note a formalsimilarity between the superstatistics method where the averaging is performed overthe parameter β , (for example, the inverse temperature) p ( E ) ∝ Z ∞ f ( β ) e − βE dβ , and the NSO method where the averaging is performed over the past life spans u = t − t ,ln ρ ( t ) = Z ∞ p q ( u ) ln ρ q ( t − u, − u ) du ; u = t − t , aximum entropy principle in NSO p q ( u ). The analogy here is not merely formal. The principalassumption of [26], that is the separation of the time scales is essential for the NSOmethod as well [1, 2, 3]. In the approach of superstatistics [28, 29, 30] the system issplit into cells and local fluctuations of the value β are considered; the fluctuations ofthe value u = t − t affect the complete system.Closely related is also the research of Crooks [31]. He studies general non-equilibrium systems, without assuming that the system can be divided into differentcells that are at local equilibrium. Crooks claims that instead of trying to obtain theprobability distribution of the entire non-equilibrium system, one has to try to estimatethe “metaprobability,” the probability of the microstate distribution. Crooks also usesthe maximum entropy principle but sets in (31) λ = 0. The main difference is thatCrooks does not assume a local equilibrium in the cells, hence his approach, thoughbeing an interesting theoretical construction, does not give a straightforward physicalinterpretation to the fluctuating parameter β . However such an approach can be appliedto the fluctuations of the value u = t − t . In the approach of [26] one obtains a localfluctuating temperature that coincides with the thermodynamic temperature and whichcan in principle be measured. The work of Crooks is used by Naudts [32] to describeequilibrium systems. The author shows that some well-known results of the equilibriumstatistical mechanics can be reformulated in a very general context with the use of theconcepts introduced in [26, 29, 31].In [26] following expression is obtained for the distribution function of thesuperstatistics f ( β ; λ i ) f ( β ; λ i ) = Z ( β ) − λ /V Z ( λ i ) exp( − βλ E ( β ) V − λ g ( β )) , where λ i are Lagrange multipliers, V being an arbitrary constant (taking out a commonfactor out of the definition of λ and λ will turn out to be useful in the following).Using the well-known formula S β = ln Z ( β ) + βE ( β ) with Z ( λ i ) being a normalizationconstant that is fixed by the condition h i β = 1.The same approach with said limitations used for the function p q ( u ), yields p q ( u ; λ i ) = Z ( t − u ) − λ /V Z ( λ i ) exp( − λ P m F m ( t − u ) h A m i V − λ g ( u )) , (31)where g ( u ) is an arbitrary function of u , which is determined by the physical peculiaritiesof the behaviour of the system in one or another period of its history. Expression (31)is obtained by the optimization of the entropy S ( λ i ) = R p q ( u ) ln p q ( u ) du with theconstraints for entropy Z p q ( u ) S ( u ) du = Z p q ( u ) Z ρ q ( t − u, − u ) ln ρ q ( t − u, − u ) dzdu = Z p q ( u )[ − X m F m ( t − u ) h A m i − ln Z ( t − u )] du aximum entropy principle in NSO R p q ( u ) P m F m ( t − u ) h A m i du . Similarly to [26], one can set the functions g ( u ) in a different fashion. For example, for g ( u ) = u, λ = λ = 0 , λ = 1 / h Γ i ,where h Γ i is the average span of past life of a system (till the present time moment),we get the exponential distribution used in [1, 2, 3]. Setting g ( u ) = ln u withappropriate corresponding values of λ one gets the power-like distribution for p q ( u );setting g ( u ) = (ln u ) with corresponding λ results in the log-normal distribution. Thussetting the function g ( u ) and λ accordingly it is possible to obtain various distributionsfor the lifetime considered, for example, in [33].It is possible to examine more difficult cases when the behaviour of the systemchanges at different stages of its evolution, when, for example the function g ( u ) = ln u yields the power-like function p q ( u ) at u < c , and g ( u ) gives an exponential shape of p q ( u ) at u > c .
4. Conclusion
For the determination of the lifetime distribution in the NSO method the method ofmaximum entropy principle as in [19] is used. The obtained distribution is close toexponential p qzub ( u ) (2), but does not coincide with it. It is possible to find conditionsat which this difference is essential. Using other variants of the maximum entropyprinciple, as in [26], it is possible to obtain other distributions except exponential one,in particular, power-like and log-normal distributions, transitory ones between them, aswell as distributions of other classes.In the interpretation of [2] it is the random value t in u = t − t that fluctuates.In [2] the limiting transition is performed for the parameter ε, ε → p q ( u ) = ε exp {− εu } after the thermodynamic limiting transition. In theinterpretation of [10] it corresponds to the average lifetime of a system tending to infinity: h Γ i = h t − t i = 1 /ε → ∞ . But the average intervals between successive random jumpsgrow infinitely, getting larger than the lifetime of a system. Therefore the source termin the Liouville equation turns to . If however the distribution p q ( u ) changes overthe interval of the lifetime, the influence of the environment which caused this change,remains within the life span even if the lifetime tends to infinity [12].There are numerous experimental confirmations for such change of the lifetimedistribution p q ( u ) over the interval of the system lifetime. The examples thereto arethe transition to chaos and the transition from laminar to turbulent flow which areaccompanied by the change of the distribution of p q ( u ). In [34, 35] the passage fromGaussian to non-Gaussian behaviour of the distribution of the first-passage time forsome time moment is demonstrated. Besides the real systems possess finite sizes andfinite lifetime. Therefore influence of surroundings on them is always present.Slow change of the function g ( u ) = u = t − t corresponds in the interpretation of[2] to the slow change of the random value t on a temporal scale. Accordingly slowchange of the function g ( u ) = ln u corresponds to the slow change of ln( t − t ). Settingother functions g ( u ), for example, g ( u ) = (ln u ) and so on is explained on the same aximum entropy principle in NSO References [1] Zubarev D N 1974
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