We show here how to use pieces of thermodynamics’ first law to generate probability distributions for generalized ensembles when only level-population changes are involved. Such microstate occupation modifications, if properly constrained via first law ingredients, can be associated not exclusively to heat and acquire a more general meaning.Instituto de Física La Plat

Curado, E.M.F.

Plastino, Ángel Luis

Physica A Statistical Mechanics and its Applications

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ARTICLE IN PRESS0378-4371/$ - sedoi:10.1016/j.phCorrespondE-mail addrPhysica A 365 (2006) 24–27www.elsevier.com/locate/physaGenerating statistical distributions withoutmaximizing the entropyA. Plastinoa,, E.M.F. CuradobaFacultad de Ciencias Exactas, Universidad Nacional de La Plata, IFLP-CONICET, Argentine National Research Council,C.C. 727, 1900 La Plata, ArgentinabCentro Brasileiro de Pesquisas Fisicas (CBPF), Rua Xavier Sigaud 150, Urca, Rio de Janeiro, BrazilAvailable online 31 January 2006AbstractWe show here how to use pieces of thermodynamics’ first law to generate probability distributions for generalizedensembles when only level-population changes are involved. Such microstate occupation modifications, if properlyconstrained via first law ingredients, can be associated not exclusively to heat and acquire a more general meaning.r 2006 Elsevier B.V. All rights reserved.Keywords: Thermodynamics; Microscopic probability distribution; First law; Second law; MaxEnt1. IntroductionThe first law of thermodynamics is one of physics’ most important statements. Together with thesecond law, the two constitute strong pillars of our understanding of Nature. In statistical mechanicsan underlying microscopic substratum is added that is able to explain not only these laws but thewhole of thermodynamics itself [1–4], one of whose basic ingredients is a microscopic probabilitydistribution (PD) that controls the population of microstates of the system under consideration [1].We will be concerned here only with changes that affect exclusively microstate-population. The way thesechanges are related to changes in a system’s extensive quantities provides the essential content of the first law[2]. In this effort we show that the above mentioned PD establishes a link between this aspect of the firstlaw, on the one hand, and the Maximum Entropy principle (MaxEnt), on the other, according to the schemegiven below. Hypothesis: for a given a concave entropic form (or information measure) S together with (1) a mean internal energy U, (2) mean values An  hAni; ðn ¼ 1; . . . ;MÞ of M additional extensivequantities An, and (3) a temperature T,e front matter r 2006 Elsevier B.V. All rights reserved.ysa.2006.01.022ing author.esses: plastino@fisica.unlp.edu.ar (A. Plastino), evaldo@cbpf.br (E.M.F. Curado).ARTICLE IN PRESSA. Plastino, E.M.F. Curado / Physica A 365 (2006) 24–27 25 Thesis: then, for any system described by(1) a microscopic PD fpig, and(2) assuming a reversible process via pi ! pi þ dpi,(3) one can verify that:(1) if the PD fpig maximizes S this entails dU ¼ T dS PMn¼1 gn dAn, or, alternatively,(2) if dU ¼ T dS PMn¼1 gn dAn, this predetermines a unique PD that maximizes S.It should be remarked that, curiously enough, this uniqueness of the PD does not demand (at this stage)concavity (or convexity) of the entropy with regard to the distribution of probabilities, a requirement thatarises a posteriori, in further developing the theory of statistical mechanics [1]. The transit from (1) to (2) hasbeen studied, for instance, in Refs. [5,6] (by no means an exhaustive list!). Succinctly, given a specific S-form,dU ¼ T dS XMn¼1gn dAn 3 MaxEnt prob. distr. fpig.2. The proofConsider a rather general information measure of the formS ¼ kXipi f ðpiÞ, (1)where for simplicity’s sake, Boltzmann’s constant is denoted just by k. The sum runs over a set of quantumnumbers, collectively denoted by i (characterizing levels of energy i), which specify an appropriate basis inHilbert’s space and P ¼ fpig is an (as yet unknown) un-normalized PD such thatPi pi ¼ constant, the‘‘constant’’ being set eventually equal to unity (often it is preferably, for practical purposes, to postponenormalization until the pertinent computation is finished).Let f be an arbitrary smooth function of the pi, in such a way pi f ðpiÞ is a concave function. Further, considerM quantities An that represent mean values of extensive physical quantities An. These take, for the state i, thevalue ani with probability pi. Also, we suppose that g is another arbitrary smooth, monotonic function of the pisuch that gð0Þ ¼ 0 and gð1Þ ¼ 1. We do not need to require the conditionPi gðpiÞ ¼ 1. The mean energy U andthe An are given byU ¼XiigðpiÞ; An ¼Xiani gðpiÞ. (2)Assume now that the probability-set P changes in the fashionpi ! pi þ dpi withXidpi ¼ 0 ðnormalization!Þ, (3)which in turn generates corresponding changes dS; dAn, and dU in, respectively, S, An, and U.The essential point that we are introducing in this effort is that we want to make sure that, in the abovedescribed circumstances, the following condition, related to the first law, is obeyed:dU  T dS þXMn¼1dAnln ¼ 0, (4)with T the temperature. As a consequence of (4), a little algebra yields, up to first order in the dpi, theconditionXi½Cð1Þi þ Cð2Þi dpi XKi dpi ¼ 0,ARTICLE IN PRESSA. Plastino, E.M.F. Curado / Physica A 365 (2006) 24–2726Cð1Þi ¼XMn¼1ln ani þ i" #g0ðpiÞ; Cð2Þi ¼ kT ½ f ðpiÞ þ pi f0ðpiÞ, (5)where the primes indicate derivative with respect to pi. Eq. (5) should hopefully yield one and just oneexpression for the pi.2.1. Equality of the Ki in Eq. (5)We proceed to show now that all the Ki are equal. As the dpi are linked by the relationXiKi dpi ¼ 0, (6)we can write, if we are dealing with N micro-states,dpN ¼ XN1i¼1dpi. (7)Substituting (7) in (6) we obtainXN1i¼1ðKi  KN Þdpi ¼ 0. (8)Now, since the N  1 ‘‘population-variations’’ dpi are independent, this entails that each term in (8) shouldvanish by itself, which implies thatKi ¼ KN for all i ¼ 1; . . . ;N  1. (9)Thus, Ki ¼ constant (in the sense of being independent of i) ¼ K for all i.2.2. The role of KInterestingly enough, we do not need to give a specific value to K for our present purposes, although it willbecome clear below that it is related to the probabilities-normalization constant. We only need to ascertain theK-role, in the following sense. We have, on account of (9),K ¼ Cð1Þi þ Cð2Þi ðfor any iÞ,Cð1Þi ¼XMn¼1ln ani þ i" #g0ðpiÞ,Cð2Þi ¼ kT ½ f ðpiÞ þ pi f0ðpiÞ, (10)so that, if we redefine things in the fashionTð1Þi ¼ f ðpiÞ þ pi f0ðpiÞ,Tð2Þi ¼ bXMn¼1ln ani þ i !g0ðpiÞ  K" #ðb  1=kTÞ, (11)we can recast (10) asTð1Þi þ Tð2Þi ¼ 0 ðfor any iÞ, (12)a relation whose importance will be presently become manifest.ARTICLE IN PRESSA. Plastino, E.M.F. Curado / Physica A 365 (2006) 24–27 272.3. The MaxEnt route revisitedAssume now that you wish to extremize S subject to the constraints of fixed valued for (i) U and (ii) the M valuesAn. This is achieved via Lagrange multipliers (1) b and (2) Mgn. We also need a normalization Lagrange multiplier x.dfpig S  bU XMn¼1gn An  xXipi" #¼ 0, (13)leading to, with gn ¼ bln, to0 ¼ dpmXipi f ðpiÞ XibgðpiÞXMn¼1ln ani þ i !þ xpi" # !, (14)so that0 ¼ f ðpiÞ þ pi f0ðpiÞ  bg0ðpiÞXMn¼1ln ani þ i !þ x" #) if x  bK ,0 ¼ f ðpiÞ þ pi f0ðpiÞ  b g0ðpiÞXMn¼1ln ani þ i !þ K" #) 0 ¼ Tð1Þi þ Tð2Þi . (15)Clearly, (12) and the last equality of (15) are one and the same equation! The equivalence stated in the Abstract is thusproven.3. DiscussionWe have here endeavored to show that appropriate manipulation of some ingredients of the first law ofthermodynamics can be used to generate the equilibrium microscopic probability distribution (PD) thatdescribes a system within the framework of a generalized ensemble [1], and that such an approach is analternative to the MaxEnt-one. We were exclusively concerned with changes that affect exclusively microstate-population and, more specifically, with the way these modifications are related to internal variation of thesystem’s extensive quantities [2]. We started with (1) a given concave entropic form (or an informationmeasure (IM)) S, (2) a mean internal energy U, and M mean values An  hAni ðn ¼ 1; . . . ;MÞ of M extensivequantities An, (3) a temperature T, and demonstrated that, for any system described by a microscopicprobability distribution (PD) fpig, assuming a reversible process via pi ! pi þ dpi that is forced to verifythe relation dU ¼ T dS PMn¼1 gn dAn, we got an equation that yields a unique PD that maximizes S. By wayof contrast, MaxEnt starts from S and, extremizing it with appropriate constraints, allow one to find thesystem’s PD. In other words, dU ¼ T dS PMn¼1 gn dAn3MaxEnt prob. distr. fpig. An alternative route tomicroscopic PDs, with some first law flavor, has thus been found in the present communication.References[1] R.K. Pathria, Statistical Mechanics, Pergamon Press, Exeter, 1993;H.B. Callen, Thermodynamics, Wiley, New York, 1960;E.A. Desloge, Thermal Physics, Holt, Rhinehart and Winston, New York, 1968.[2] F. Reif, Statistical and Thermal Physics, McGraw-Hill, New York, 1965.[3] J.J. Sakurai, Modern Quantum Mechanics, Benjamin, Menlo Park, CA, 1985.[4] E.T. Jaynes, Phys. Rev. 106 (1957) 620;E.T. Jaynes, Phys. Rev. 108 (1957) 171;E.T. Jaynes, in: R.D. Rosenkrantz (Ed.), Papers on Probability, Statistics and Statistical Physics, Reidel, Dordrecht, Boston, 1987;A. Katz, Principles of Statistical Mechanics, The Information Theory Approach, Freeman and Co., San Francisco, 1967;A. Plastino, A.R. Plastino, Braz. J. Phys. 29 (1999) 50.[5] R.S. Mendes, Physica A 242 (1997) 299.[6] E.M.F. Curado, Braz. J. Phys. 29 (1999) 36.

Generating statistical distributions without maximizing the entropy

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Generating statistical distributions without maximizing the entropy

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