Robustness of the Second Law of Thermodynamics under Generalizations of the Maximum Entropy Method

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1 Robustness of the Second Law of Thermodynamcs under Generalzatons of the Maxmum Entropy Method Sumyosh Abe Stefan Thurner SFI WORKING PAPER: SFI Workng Papers contan accounts of scentfc work of the author(s and do not necessarly represent the vews of the Santa Fe Insttute. We accept papers ntended for publcaton n peer-revewed journals or proceedngs volumes, but not papers that have already appeared n prnt. Except for papers by our external faculty, papers must be based on work done at SFI, nspred by an nvted vst to or collaboraton at SFI, or funded by an SFI grant. NOTICE: Ths workng paper s ncluded by permsson of the contrbutng author(s as a means to ensure tmely dstrbuton of the scholarly and techncal work on a non-commercal bass. Copyrght and all rghts theren are mantaned by the author(s. It s understood that all persons copyng ths nformaton wll adhere to the terms and constrants nvoked by each author's copyrght. These works may be reposted only wth the explct permsson of the copyrght holder. SANTA FE INSTITUTE

2 Robustness of the second law of thermodynamcs under generalzatons of the maxmum entropy method S. ABE 1,2 and S. THURNER 3,4 1 Department of Physcal Engneerng, Me Unversty, Me , Japan 2 Insttut Supéreur des Matéraux et Mécanques Avancés, 44 F. A. Barthold, Le Mans, France 3 Complex Systems Research Group, HNO, Medcal Unversty of Venna, Währnger Gürtel 18-20, A-1090, Austra 4 Santa Fe Insttute, 1399 Hyde Park Road, Santa Fe, NM 8750, USA PACS a Thermodynamcs PACS Ln Nonequlbrum and rreversble thermodynamcs PACS y Classcal statstcal mechancs Abstract It s shown that the laws of thermodynamcs are extremely robust under generalzatons of the form of entropy. Usng the Bregman-type relatve entropy, the Clausus nequalty s proved to be always vald. Ths mples that thermodynamcs s hghly unversal and does not rule out consstent generalzaton of the maxmum entropy method. There s a consensus of a certan knd that among the physcal laws those of thermodynamcs are extremely unversal and wll persst. Ths optmsm mght have ts orgn n the fact that the thermodynamc prncples have remaned unchanged when the shft was made from classcal statstcal mechancs to quantum. Ths s ndeed astonshng f one takes nto account that the formulaton of thermodynamcs was completed n the 19th century, before the establshment of atomsm n physcs. Here, by the term statstcal mechancs, we refer to ordnary equlbrum statstcal mechancs that descrbes normal systems, n whch ergodcty s supposed to be well realzed (though t stll seems vague n quantum theory. Today, t s well known that there exst a number of systems n nature, whch are essentally nonequlbrum and nonergodc, but exhbt defnte statstcal propertes. In partcular, nonequlbrum statonary states of complex systems are of prmary nterest n varous branches of physcal scence. These systems are often characterzed by 1

3 nonexponental dstrbutons ncludng the power-law type. An mportant pont here s that n many such systems the statonary states survve for very long perods of tme, much longer than typcal mcroscopc dynamcal tme scales. Ths makes t legtmate to suppose that such a state may maxmze a certan statstcal measure, whch can be dfferent from the Boltzmann-Gbbs entropy, n general. Here s a pont, at whch generalzaton of the ordnary maxmum entropy method s expected to play a role. Now, the above-mentoned reasonng brngs thermodynamcs a new ssue to be examned. That s, how robust the laws of thermodynamcs are under such a generalzaton. The purpose of the present work s to answer ths queston. We execute t by employng the hghly generalzed entropc measure proposed recently and examnng manly the second law for the generalzed maxmum entropy method based on t. We shall see how the Clausus nequalty s robust under such a generalzaton. There are several works devoted to hghly generalzed entropc measures n the lterature [1-6]. Here, let us employ the one recently proposed n Ref. [5], whch may be the most general form that s consstent wth the maxmum entropy condton. In the unt n whch the Boltzmann constant s unty, t reads wth S G [B] =!# B (" $(B(" + %[B] (1 B(#![B] = % $ d x x d " (x + c, (2 d x whch can be rewrtten as 0 S G [B] =!% d x " (x + c. (3 B(# $ 0 In these equatons, B(! s a normalzed and monotoncally decreasng dstrbuton functon of!, whch s the value of energy of the system n ts th state.! (x s a generalzed logarthm satsfyng! (x < 0 and d!(x / d x > 0 for 0 < x <1, and x! (x " 0 ( x! 0 +, makng S G [B] nonnegatve and concave (ts further propertes wll be specfed later. c s a constant, whch ensures that S G [B 0 ] = 0 for a completely ordered state, B 0 [! ] = "!,! 0 wth the ground-state energy! 0. Ths condton s necessary for the thrd law to hold. Note however that the thrd law s fundamentally concerned wth quantum theory n the low temperature regme, whereas we are consderng a 2

4 classcal system here. A generalzed maxmum entropy method, gven the exstence of a statonary dstrbuton functon, B (!, s formulated as follows: wth! G! B(" B= B = 0 (4 & G! S G [B] "# '% B($ "1* ( + ", & $ B($ "U '% * ( +, (5 where! and! are Lagrange multplers, and U denotes the average energy. The statonary soluton to ths problem s gven by B (! = E("# " $ (! " U, (6 where E(x s a generalzed exponental, whch s the nverse functon of! (x : E(! (x =! (E (x = x. U denotes the average energy evaluated n the statonary state n Eq. (6 n a self-referental manner. In other words,! (x n Eq. (1 s chosen so that t s the nverse functon of the statonary dstrbuton functon.! n Eq. (6 can be elmnated as follows [7]. In addton to Eq. (4, we consstently mpose the followng condton: # " B(! " B(! G B= B Combnng ths wth the normalzaton condton, where = 0. (7 3 B " (! = 1, we obtan! = "# F " $[ B ], (8 F! U " # "1 S G [ B ]. (9 On the other hand,! s found to satsfy! S G [ B ]! U = ", (10 whch s actually nothng else than a generc feature n the varatonal problem n Eqs. (4 and (5 [8]. Thus, the thermodynamc Legendre transformaton structure always exsts, and accordngly F n Eq. (9 s regarded as the free energy n the statonary state. It should however be notced that the zeroth law of thermodynamcs does not hold n the present case, snce t s not possble to dvde the total system nto two ndependent subsystems and the entropc measure s not addtve, n general. Ths makes t hard to nterpret! "1 as a temperature (recall that the temperature specfed by the zeroth law s an

5 equlbrum temperature. Takng nto account the physcal propertes of a nonequlbrum statonary state, t s natural to regard! "1 spatally and temporally. as a local temperature that may vary both The frst law of thermodynamcs reflects the conservaton of energy and s therefore vald also n any nonequlbrum stuaton. Let us look at the defnton of the nternal energy n a generc state, B(! U = "! B(!. (11 Change n a thermodynamcs process s! U = # "! B(" + # B ("! ", (12 where! B(" mples the change of the functonal form of B(! whereas! " s the change of the energy due to a certan external nfluence. Identfyng the quantty of heat and the work respectvely as!' Q = # "! B (", (13!' W = " $ B( #! #, (14 we mmedately obtan the frst law of thermodynamcs!' Q =! U +!' W. (15 To ascertan that the above dentfcatons are legtmate, let us consder a quas-reversble process [9] connectng two statonary states that are nfntesmally dstnct each other. Then, the change of S G s evaluated for the statonary state n Eq. (6 as follows:! S G = "% # ( B ($! B($ =! # " $ B (", (16 provded that the normalzaton condton on B(! has been used. Therefore, we fnd! S G = "! ' Q. (17 Ths relaton for a quas-reversble process justfes the dentfcatons mentoned above. Now let us proceed to study the second law of thermodynamcs. For ths purpose, we 4

6 consder comparng a general dstrbuton functon, B(!, wth the statonary state dstrbuton functon, B (!. In nformaton theory, ths can tradtonally be done by makng use of the Kullback-Lebler relatve entropy (or, dvergence [10,11], whch s ntrnscally assocated wth the ordnary Boltzmann-Gbbs-Shannon entropc measure. However, t s a nontrval problem n our case, snce we are concerned wth a hghly generalzed form of entropy n Eq. (1. For a soluton to ths problem, a clue may be found n the work n Ref. [7]. There, t s ndcated that the relatve entropy assocated wth the ordnary defnton for expectaton values s the one of the Bregman type [12] (see also Ref. [3]. In the present context, the relatve entropy of ths type reads [B B ] = % d x [!( x "!( B (# ], (18 B( # $ B (# whch can be recast n the form wth [B B ] = S G [ B ]! S G [B] + "U! " U! " # F (19! F = F " F = (U " S[B] / # "( U " S[ B ] / #. (20 Therefore, one sees that the physcal meanng of [B B ] s essentally the free energy dfference. Let us recall some fundamental propertes of [B 1 B 2 ]. The followng two are of mportance for our dscusson. One s postve sem-defnteness: that s, [B 1 B 2 ]! 0 and [B 1 B 2 ] = 0 ff B 1 (! = B 2 (!. The other s ts convexty wth respect to the frst argument: [! B 1 + (1"! B 2 B 3 ] #! [B 1 B 3 ] + (1"! [B 2 B 3 ], (21 where!" #(0,1. Now, n connecton wth Eq. (18, take some arbtrary thrd state, B *, satsfyng Then, defne the varaton of the state B(! naturally as follows: { } $ B(". (23! B(" = # B * (" + (1 $ #B (" Under ths varaton, the change of [B B ] wth fxed B (! yelds 5

7 ! B [B B ] = % ["(B(# $ "( B (# ]! B(# = %!(B (" # B(" + $ % " # B(". (24 That s,! B [B B' ] = "! S G + #! ' Q. (25 From Eqs. (22 and (23, we see that the left-hand sde of ths equaton s not postve:! B [B B ] " [# B * +(1 $ # B B ]$ [ B B ] { } # 0. (26 =! [ B* B ] " [ B B ] Consequently, we obtan the Clausus nequalty! "' Q # " S G. (27 In concluson, we have shown that the laws of thermodynamcs reman fundamentally unchanged under generalzng the form of entropy n the maxmum entropy method. In partcular, the second law, expressed n the form of the Clausus nequalty does not change for a very wde class of generalzed entropes. Thus, thermodynamcs s hghly unversal and does not rule out any effort for generalzng the form of entropy. * * * The work of S. A. was supported n part by Grant-n-Ad for Scentfc Research (B of the Mnstry of Educaton. S. T. s grateful to Austran Scence Foundaton Projects P17621 and P REFERENCES [1] ALMEIDA M. P., Physca A 325 ( [2] ABE S., J. Phys. A 36 ( [3] NAUDTS J., Rev. Math. Phys. 16 ( [4] ABE S., KANIADAKIS G. and SCARFONE A. M., J. Phys. A 37 ( [5] HANEL R. and THURNER S., Physca A 380 ( [6] SCHWÄMMLE V., CURADO E. M. F. and NOBRE F. D., Eur. Phys. J. B 58 ( [7] ABE S. and BAGCI G. B., Phys. Rev. E 71 ( [8] YAMANO T., Eur. Phys. J. B 18 (

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