THERMODYNAMICSEdited by Tadashi MizutaniThermodynamicsEdited by Tadashi MizutaniPublished by InTechJaneza Trdine 9, 51000 Rijeka, CroatiaCopyright 2011 InTechAll chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source.Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Ana NikolicTechnical Editor Teodora SmiljanicCover Designer Martina SiroticImage Copyright Khotenko Volodymyr, 2010. Used under license from Shutterstock.comFirst published January, 2011Printed in IndiaA free online edition of this book is available at www.intechopen.comAdditional hard copies can be obtained from orders@intechweb.org Thermodynamics, Edited by Tadashi Mizutanip.cm. ISBN 978-953-307-544-0free online editions of InTech Books and Journals can be found atwww.intechopen.comPart 1Chapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9PrefaceIXFundamentals of Thermodynamics1New Microscopic Connections of Thermodynamics3A. Plastino and M. CasasRigorous and General Definition of Thermodynamic Entropy23Gian Paolo Beretta and Enzo ZanchiniHeat Flow, Work Energy, Chemical Reactions, and Thermodynamics: A Dynamical Systems Perspective51Wassim M. Haddad, Sergey G. Nersesov and VijaySekhar ChellaboinaModern Stochastic Thermodynamics73A. D. Sukhanov and O. N. GolubjevaOn the Two Main Laws of Thermodynamics99Martina Costa Reis and Adalberto Bono Maurizio Sacchi BassiNon-extensive Thermodynamics of Algorithmic Processing the Case of Insertion Sort Algorithm121Dominik Strzaka and Franciszek GrabowskiLorentzian Wormholes Thermodynamics133Prado Martn-Moruno and Pedro F. Gonzlez-DazFour Exactly Solvable Examples in Non-Equilibrium Thermodynamics of Small Systems153Viktor Holubec, Artem Ryabov, Petr ChvostaNonequilibrium Thermodynamics for Living Systems: Brownian Particle Description177Ulrich ZrcherContentsContents VIApplication of Thermodynamics to Science and Engineering193Mesoscopic Non-Equilibrium Thermodynamics: Application to Radiative Heat Exchange in Nanostructures195Agustn Prez-Madrid, J. Miguel Rubi, andLuciano C. LapasExtension of Classical Thermodynamics to Nonequilibrium Polarization205Li Xiang-Yuan, Zhu Quan, He Fu-Cheng and Fu Ke-XiangHydrodynamical Models of Superfluid Turbulence233D. Jou, M.S. Mongiov, M. Sciacca, L. Ardizzone and G. GaetaThermodynamics of Thermoelectricity275Christophe GoupilApplication of the Continuum-Lattice Thermodynamics293Eun-Suok OhPhonon Participation in Thermodynamics andSuperconductive Properties of Thin Ceramic Films317Jovan P. etraji, Vojkan M. Zori, Nenad V. Deli,Dragoljub Lj. Mirjani and Stevo K. JaimovskiInsight Into Adsorption Thermodynamics349Papita Saha and Shamik ChowdhuryIon Exchanger as Gibbs Canonical Assembly365Heinrich Altshuler and Olga AltshulerMicroemulsions: Thermodynamic and Dynamic Properties381S.K. Mehta and Gurpreet KaurThe Atmosphere and Internal Structure of Saturns Moon Titan, a Thermodynamic Study407Andreas Heintz and Eckard BichInteroperability between Modelling Tools (MoT) with Thermodynamic Property Prediction Packages (Simulis Thermodynamics) and Process Simulators (ProSimPlus) Via CAPE-OPEN Standards425Ricardo Morales-Rodriguez, Rafiqul Gani, Stphane Dchelotte, Alain Vacher and Olivier BaudouinPart 2Chapter 10Chapter 11Chapter 12Chapter 13Chapter 14Chapter 15Chapter 16Chapter 17Chapter 18Chapter 19Chapter 20PrefaceProgress of thermodynamics has been stimulated by the ndings of a variety of elds of science and technology. In the nineteenth century, studies on engineering problems, e ciencyofthermalmachines,leadtothediscoveryofthesecondlawofthermo-dynamics.Followingdevelopmentofstatisticalmechanicsandquantummechanics allowed us to understand thermodynamics on the basis of the properties of constitu-entmolecules.Thermodynamicsandstatisticalmechanicsprovideabridgebetween microscopic systems composed of molecules and quantum particles and their macro-scopic properties. Therefore, in the era of the mesoscopic science, it is time that various aspects of state-of-the-art thermodynamics are reviewed in this book. In modern science a number of researchers are interested in nanotechnology, surface science, molecular biology, and environmental science.In order to gain insight into the principles of various phenomena studied in such elds, thermodynamics should oer solid theoretical frameworks and valuable tools to analyse new experimental observa-tions.Classicalthermodynamicscanonlytreatequilibriumsystems.However,ther-modynamics should be extended to non-equilibrium systems, because understanding of transport phenomena and the behaviour of non-equilibrium systems is essential in biological and materials research. Extension of thermodynamics to a system at the me-soscopic scale is also important due to recent progress in nanotechnology. The princi-ples of thermodynamics are so general that the application is widespread to such elds as solid state physics, chemistry, biology, astronomical science, materials science, infor-mation science, and chemical engineering. These are also major topics in the book. Therstsectionofthebookcoversthefundamentalsofthermodynamics,thatis, theoreticalframeworkofthermodynamics,foundationsofstatisticalmechanicsand quantum statistical mechanics, limits of standard thermodynamics, macroscopic uc-tuations, extension of equilibrium thermodynamics to non-equilibrium systems, astro-nomical problems, quantum uids, and information theory. The second section covers applicationofthermodynamicstosolidstatephysics,materialsscience/engineering, surfacescience,environmentalscience,andinformationscience.Readerscanexpect coverages from theoretical aspects of thermodynamics to applications to science and engineering.Thecontentshouldbeofhelptomanyscientistsandengineersofsuch eld as physics, chemistry, biology, nanoscience, materials science, computer science, and chemical engineering.Tadashi MizutaniDoshisha University, Kyoto JapanPart 1 Fundamentals of Thermodynamics 0New Microscopic Connections of ThermodynamicsA. Plastino1and M. Casas21Facultad de C. Exactas, Universidad Nacional de La PlataIFLP-CONICET, C.C. 727, 1900 La Plata2Physics Departament and IFISC-CSIC, University of Balearic Islands07122 Palma de Mallorca1Argentina2Spain1. IntroductionThis is a work that discusses the foundations of statistical mechanics (SM) by revisiting itspostulates in the case of the two main extant versions of the theory. A third one will herewe added, motivated by the desire for an axiomatics that possesses some thermodynamicavor, whichdoesnothappenwithneitherofthetwomainSMcurrentformulations,namely, thoseofGibbs(1; 2), basedontheensemblenotion, andofJaynes, centeredonMaxEnt (3; 4; 5).Onehastomentionattheoutsetthatwerationallyunderstandsomephysicalproblemwhen we are able to place it within the scope and context of a specic Theory. In turn, wehaveatheorywhenwecanbothderivealltheknowninterestingresults andsuccessfullypredict newones startingfromasmall set of axioms. Paradigmaticexamples arevonNeumanns axioms for QuantumMechanics, Maxwells equations for electromagnetism,Euclids axioms for classical geometry, etc. (1; 3).Boltzmanns main goal in inventing statistical mechanics during the second half of the XIXcenturywastoexplainthermodynamics. However, hedidnotreachtheaxiomaticstagedescribed above. The rst successful SM theory was that of Gibbs (1902) (2), formulated on thebasis of four ensemble-related postulates (1). The other great SM theory is that of Jaynes (4),based upon the MaxEnt axiom (derived from Information Theory): ignorance is to be extremized(with suitable constraints).Thermodynamics (TMD) itself has also been axiomatized, of course, using four macroscopicpostulates (6). Now, the axioms of SM and of thermodynamics belong to different worldsaltogether. The former speak of either ensembles (Gibbs), which are mental constructs,or of observers ignorance (Jaynes), concepts germane to thermodynamics language, thatrefers to laboratory-parlance. In point of fact, TMD enjoys a very particular status in the wholeof science, as the one and only theory whose axioms are empirical statements (1).Of course, there is nothing to object to the two standard SM-axiomatics referred toabove. However, anatural questionemerges: wouldit bepossibletohaveastatisticalmechanics derived from axioms that speak, as far as possible, the same language as that ofthermodynamics? To what an extent is this feasible? It is our intention here that of attemptinga serious discussion of such an issue and try to provide answers to the query, following ideasdeveloped in (7; 8; 9; 10; 11; 12; 13).12 Thermodynamics2. Thermodynamics axiomsThermodynamics can be thought of as a formal logical structure whose axioms are empiricalfacts, which gives it a unique status among the scientic disciplines (1). The four postulateswe state below are entirely equivalent to the celebrated three laws of thermodynamics (6):1. For every system there exists a quantity E, called the internal energy, such that a uniqueEvalue is associated to each of its states. The difference between such values for twodifferent states in a closed system is equal to the work required to bring the system, whileadiabatically enclosed, from one state to the other.2. Thereexistparticularstatesofasystem, calledtheequilibriumones, thatareuniquelydetermined by E and a set of extensive (macroscopic) parametersA, = 1, . . . , M. Thenumber and characteristics of the A depends on the nature of the system (14).3. For every system there exists a state function S(E, A) that (i) always grows if internalconstraints are removed and (ii) is a monotonously (growing) function ofE. S remainsconstant in quasi-static adiabatic changes.4. S and the temperature T = [ES ]A1,...,AM vanish for the state of minimum energy and are 0for all other states.Fromthesecondand3rd. Postulates wewill extract andhighlight thefollowingtwoassertions, that are essential for our purposes Statement 3a) for every system there exists a state function S, a function of E and the AS = S(E, A1, . . . , AM). (1) Statement 3b) S is a monotonous (growing) function of E, so that one can interchange theroles of E and S in (1) and writeE = E(S, A1, . . . , AM), (2)Eq. (2) clearly indicates thatdE =ES dS +EAdA dE = TdS +PdA, (3)with P generalized pressures and the temperature T dened as (6)T =

ES

[ A]. (4)Eq. (3) will play a central role in our considerations, as discussed below.If we know S(E, A1, . . . , An) (or, equivalently because of monotonicity,E(S, A1, . . . , An)) wehaveacompletethermodynamicdescriptionof asystem. It isoftenexperimentally more convenient to work with intensive variables.Let dene S A0. The intensive variable associated to the extensive Ai, to be called Pi is:P0 T = [ES]A1,...,An, 1/T = Pj j/T = [EAj]S,A1,...,Aj1,Aj+1,...,An4 ThermodynamicsNew Microscopic Connections of Thermodynamics 3Any one of the Legendre transforms that replaces any s extensive variables by their associatedintensive ones (,s will be Lagrange multipliers in SM)Lr1,...,rs= E jPj Aj, (j = r1, . . . , rs)containsthesameinformationaseither Sor E. ThetransformLr1,...,rsisafunctionofn sextensiveandsintensivevariables. ThisiscalledtheLegendreinvariantstructureofthermodynamics.3. Gibbs approach to statistical mechanicsIn 1903 Gibbs formulated the rst axiomatic theory for statistical mechanics (1), that revolvesaround the basic physical concept of phase space. Gibbs calls the phase of the system toits phase space (PS) precise location, given by generalized coordinates and momenta. Hispostulates refer to the notion of ensemble (a mental picture), an extremely great collectionof N independent systems, all identical in nature with the one under scrutiny, but differingin phase. One imagines the original system to be repeated many times, each of them withadifferent arrangement of generalizedcoordinatesandmomenta. Liouvillescelebratedtheoremof volume conservationinphase space for Hamiltonianmotionapplies. Theensemble amounts to a distribution of N PS-points, representative of the true system. Nis so large that one can speak of a density D at the PS-point = q1, . . . , qN; p1, . . . , pN, withD = D(q1, . . . , qN; p1, . . . , pN, t) D(), with t the time, and, if we agree to call d the pertinentvolume element,N =

dD; t. (5)If a system were to be extracted randomly from the ensemble, the probability of selecting onewhose phase lies in a neighborhood of would be simplyP() = D()/N. (6)Consequently,

Pd = 1. (7)Liouvilles theoremfollows fromthe fact that, since phase-space points can not bedestroyed, ifN12 =

21Dd, (8)thendN12dt= 0. (9)An appropriate analytical manipulation involving Hamiltons canonical equations of motionthen yields the theorem in the form (1)D +NiDpi pi +NiDqi qi = 0, (10)entailing what Gibbs calls the conservation of density-in-phase.5 New Microscopic Connections of Thermodynamics4 ThermodynamicsEquilibrium is simply the statementD = 0, i. e.,NiDpi pi +NiDqi qi = 0. (11)3.1Gibbs postulates for statistical mechanicsThe following statements wholly and thoroughly explain in microscopic fashion the corpus ofequilibrium thermodynamics (1). The probability that at time t the system will be found in the dynamical state characterizedby equals the probability P() that a system randomly selected from the ensemble shallpossess the phase will be given by (6). All phase-space neighborhoods (cells) have the same a priori probability. D depends only upon the systems Hamiltonian. The time-average of a dynamical quantity F equals its average over the ensemble, evaluatedusing D.4. Information theory (IT)The IT-father, Claude Shannon, in his celebrated foundational paper (15), associates a degreeof knowledge (or ignorance) to any normalized probability distributionp(i), (i = 1, . . . , N),determined by a functional of the pi called the information measureI[pi], giving thusbirth to a new branch of mathematics, that was later axiomatized by Kinchin (16), on the basisof four axioms, namely, I is a function ONLY of the p(i), I is an absolute maximum for the uniform probability distribution, I is not modied if an N + 1 event of probability zero is added, Composition law.4.1CompositionConsider two sub-systems [1, p1(i)] and [2, p2(j)] of a composite system [, p(i, j)]with p(i, j) = p1(i) p2(j). Assume further that the conditional probability distribution (PD)Q(j[i) of realizing the eventj in system 2 for a xed ievent in system 1. To this PD oneassociates the information measure I[Q]. Clearly,p(i, j) = p1(i) Q(j[i). (12)Then Kinchins fourth axiom states thatI(p) = I(p1) +ip1(i) I

Q(j[i)

. (13)An important consequence is that, out of the four Kinchin axioms one nds that ShannonssmeasureS = Ni=1p(i) ln[p(i)], (14)is the one and only measure complying with them.6 ThermodynamicsNew Microscopic Connections of Thermodynamics 55. Information theory and statistical mechanicsInformation theory (IT) entered physics via Jaynes Maximum Entropy Principle (MaxEnt) in1957 with two papers in which statistical mechanics was re-derived ` a la IT (5; 17; 18), withoutappeal to Gibbs ensemble ideas. Since ITs central concept is that of information measure(IM) (5; 15; 17; 19), a proper understanding of its role must at the outset be put into its properperspective.InthestudyofNature, scientictruthisestablishedthroughtheagreementbetweentwoindependent instances that can neither bribe nor suborn each other: analysis (pure thought) andexperiment (20). The analytic part employs mathematical tools and concepts. The followingscheme thus ensues:WORLD OF MATHEMATICAL ENTITIES LABORATORYThemathematicalrealmwascalledbyPlato ToposUranus(TP). Scienceingeneral, andphysics inparticular, is thus primarily(although notexclusively, of course)to be regardedas a TPExperiment two-way bridge, in which TP concepts are related to each other in theform of laws that are able to adequately describe the relationships obtaining among suitablechosenvariablesthat describethephenomenononeisinterestedin. Inmanyinstances,althoughnotinall ofthem, theselawsareintegratedintoacomprehensivetheory(e.g.,classical electromagnetism, based upon Maxwells equations) (1; 21; 22; 23; 24).If recourse is made to MaxEnt ideas in order to describe thermodynamics, the above schemebecomes now:IT as a part of TP Thermal Experiment,or in a more general scenario:IT Phenomenon to be described.It should then be clear that the relation between an information measure and entropy is:IM Entropy S.One can then state that an IM is not necessarily an entropy! How could it be? The rst belongstotheToposUranus, becauseitisamathematicalconcept. Thesecondtothelaboratory,because it is a measurable physical quantity. All one can say is that, at most, in some specialcases, an association I M entropy S can be made. As shown by Jaynes (5), this association isboth useful and proper in very many situations.6. MaxEnt rationaleThe central IM idea is that of giving quantitative form to the everyday concept of ignorance (17).If, in a given scenario, N distinct outcomes (i = 1, . . . , N) are possible, then three situations mayensue (17):1. Zero ignorance: predict with certainty the actual outcome.2. Maximum ignorance: Nothing can be said in advance. The N outcomes are equally likely.3. Partial ignorance: we are given the probability distribution Pi; i = 1, . . . , N.Theunderlyingphilosophyof theapplicationof ITideas tophysics viathecelebratedMaximumEntropyPrinciple(MaxEnt) of Jaynes (4) isthat originatedbyBernoulli and7 New Microscopic Connections of Thermodynamics6 ThermodynamicsLaplace (the fathers of Probability Theory) (5), namely: the concept of probability refers to anstate of knowledge. An information measure quanties the information (or ignorance) contentof a probability distribution (5). If our state of knowledge is appropriately represented by aset of, say, M expectation values, then the best, least unbiased probability distribution is theone that reects just what we know, without inventing unavailable pieces of knowledge (5; 17)and, additionally, maximizes ignorance: the truth, all the truth, nothing but the truth.Such is the MaxEnt rationale (17). It should be then patently clear that, in using MaxEnt, oneis NOT maximizing a physical entropy. One is maximizing ignorance in order to obtain theleast biased distribution compatible with the a priori knowledge.6.1Jaynes mathematical formulationAs stated above, Statistical Mechanics and thereby Thermodynamics can be formulated onthebasisofInformationTheoryifthestatisticaloperator isobtainedbyrecoursetotheMAXIMUM ENTROPY PRINCIPLE (MaxEnt). Consequently, we have the MaxEnt principle:MaxEnt: Assume your prior knowledge about the system is given by the values of M expectation values< A1>, . . . , < AM >. Then is uniquely xed by extremizing I( ) subject to the constraints givenby the M conditions< Aj>= Tr[ Aj](entailing the introduction of M associated Lagrange multipliers i) plus normalization of (entailinga normalization Lagrange multiplier .)In the process one discovers that I S, the equilibriumBoltzmanns entropy, if our prior knowledge , . . . , refers to extensivequantities. SuchIvalue, once determined, yields complete thermodynamical information withrespect to the system of interest.7. Possible new axioms for SMBoth Gibbs andMaxEnt are beautiful, elegant theories that satisfactorily account forequilibriumthermodynamics. Whysshouldwebelookingfor still another axiomatics?Precisely because, following Jaynes IT-spirit, one should be endeavoring to use all informationactuallyavailabletousinbuildingupourtheoreticfoundations, andthisisnotdoneinMaxEnt, as we are about to explicitate.Our main argument revolves around the possibility of giving Eq. (3), an empirical statement,the status of an axiom, actually employing thus a piece of information available to us withoutany doubt. This constitutes the rst step in our present discourse. More explicitly, in order toconcoct a new SM-axiomatics, we start by establishing as a theoretic postulate the followingmacroscopic assertion:Axiom (1)dE = TdS +PdA. (15)Since this is a macroscopic postulate in a microscopic axiomatics corpus, it is pertinent nowto ask ourselves which is the minimum amount of microscopic information that we wouldhave to add to such an axiomatics in order to get all the microscopic results of equilibriumstatistical mechanics. Since we know about Kinchins postulates, we borrow from him his8 ThermodynamicsNew Microscopic Connections of Thermodynamics 7rst one. Consequently, we conjecture at this point, and will prove below, that the followingstatements meets the bill:Axiom(2) If there are Amicroscopic accessible states labelledbyi, of microscopicprobability pi, thenS = S(p1, p2, . . . , pA ). (16)In what follows, the number of microstates will also be denoted by W.Now, we will take as a postulate something that we actually know form both quantum andclassical mechanics.Axiom(3) Theinternal energyEandtheexternal parameters Aaretoberegardedas expectation values of suitable operators, respectively the hamiltonianHand {(i.e.,A < {>).ThustheA(andalsoE) will dependontheeigenvaluesof theseoperatorsandontheprobability set. (The energy eigenvalues depend of course upon the {.) The reader willimmediately realize that Axiom (2) is just a form of Boltzmanns atomic conjecture, pureand simple. In other words, macroscopic quantities are statistical averages evaluated using amicroscopic probability distribution (25). It is important to realize that our three new axiomsare statements of fact in the sense that they are borrowed either from experiment or frompre-existent theories. In fact, the 3 axioms do not incorporate any knew knowledge at all!Inordertoprovethat ourabovethreepostulatesdoallowonetobuildupthemightySM-edice we will show below that they are equivalent to Jaynes SM-axiomatics (4).Of course, the main SM-goal is that of ascertaining which is the PD (or the density operator)that best describesthesystemof interest. Jaynesappealsinthisrespect tohisMaxEntpostulate, the only one needed in this SM-formulation. We restate it below for the sake ofxing notation.MaxEnt axiom: assume your prior knowledge about the system is given by the values of Mexpectation valuesA1 < {1>, . . . , AR < {M>. (17)Then, is uniquely xed by extremizing the information measure I() subject tonormalization plus the constraints given by theM conditions constituting our assumedforeknowledgeA =< {>= Tr[{]. (18)This leads, after aLagrange-constrainedextremizingprocess, totheintroductionof MLagrange multipliers , that one assimilates to the generalized pressures P. The truth, thewhole truth, nothing but the truth (17). If the entropic measure that reects our ignorancewere not maximized, we would be inventing information that we do not actually possess.Inperformingthevariational processJaynesdiscoversthat, providedonemultipliestheright-hand-side of the information measure expression by Boltzmanns constant kB, the IMequalstheentropicone. Thus, I S, theequilibriumthermodynamicentropy, withthecaveat that our prior knowledge A1 =< {1>, . . . , AM =< {M> must refer just to extensivequantities. Once is at hand, I() yields complete microscopic information with respect to the systemof interest. Our goal should be clear now. We need to prove that our new axiomatics, encapsulatedby (15) and (16), is equivalent to MaxEnt.9 New Microscopic Connections of Thermodynamics8 Thermodynamics8. Equivalence between MaxEnt and our new axiomaticsWe will here deal with the classical instance only. The quantal extension is of a straightforwardnature. Consider a generic change pi pi + dpi constrained by Eq. ( 15), that is, the changedpi must be of such nature that (15) is veried. Obviously, S, Aj, and E will change with dpiand, let us insist, these changes are constrained by (15). We will not specify the informationmeasure, as several possibilities exist (26). For a detailed discussion of this issue see (27). Inthis endeavor our ingredients are an arbitrary, smooth function f (p) that allows us to express the information measure in thefashionI S(pi) =ipi f (pi), (19)such that S(pi) is a concave function, M quantities A that represent mean values of extensive physical quantities '{`, that take,for the micro-state i, the value aiwith probability pi, another arbitrary smooth, monotonic function g(pi) (g(0) = 0; g(1) = 1). It is in order touse generalized, non-Shannonian entropies that we have slightly generalized mean-valuedenitions using the function g.We deal then with (we take A1 E), using the function g to evaluate (generalized) expectationvalues,A '{` =Wiai g(pi); = 2, . . . , M, (20)E =Wi

i g(pi), (21)where i is the energy associated to the microstate i. The probability variations dpi will nowgenerate corresponding changes dS, dA, and dE in, respectively, S, the A, and E.8.1Proof, part IThe essential point of our present methodology is to enforce obedience todE TdS +W=1dA = 0, (22)withTthetemperatureandgeneralizedpressures. Weusenowtheexpressions(19),(20), and(21)soastocast(22)intermsoftheprobabilities, accordingtoaninnitesimalprobabilities changepi pi + dpi. (23)If we expand the resulting equation up to rst order in the dpi, it is immediately found, after alittle algebra, that the following set of equations ensues (7; 8; 9; 10; 11; 12; 13) (remember thatthe Lagrange multipliers are identical to the generalized pressures P of Eq. (3))C(1)i= [M=1 ai+ i]C(2)i= TSpi10 ThermodynamicsNew Microscopic Connections of Thermodynamics 9i[C(1)i+ C(2)i]dpi iKidpi = 0. (24)We can rearrange matters in the fashionT(1)i= f (pi)+ pi f/(pi)T(2)i= [(M=1 ai+ i) g/(pi) K],( 1/kT), (25)so that we can recast (24) asT(1)i+ T(2)i= 0; ( f or any i), (26)a relation whose importance will become manifest in Appendix I.We wish that Eqs. (24) or (26) should yield one and just one piexpression, which it indeeddoes (7; 8; 9; 10; 11; 12; 13). We do not need here, however, for our demonstration, an explicitexpression for this probability distribution, as will be immediately realized below.8.2Proof, part II: follow Jaynes procedureAlternatively, proceed` alaMaxEnt. ThisrequiresextremizingtheentropySsubject totheusualconstraintsinE, A, andnormalization. Theensuing, easytocarryout Jaynesvariational treatment, can be consulted in (7; 8; 9; 10; 11; 12; 13), that is (we set 1 = 1/T)pi[S 'H` M=2'{` ipi] = 0, (27)(we need also a normalization Lagrange multiplier ) is easily seen to yield as a solution thevery set of Eqs. (24) as well! (see Appendix I for the proof). These equations arise then outof two clearly separate treatments: (I) our methodology, based on Eqs. (15) and (16), and (II),following the MaxEnt prescriptions. This entails that MaxEnt and our axiomatics co-implyeach other, becoming thus equivalent ways of building up statistical mechanics. An importantpoint is to be here emphasized with respect to the functional Sform.The specic form of S[pi] is not needed neither in Eqs. (24) nor in (27)!9. What does all of this mean?We have already formally proved that our axiomatics is equivalent to MaxEnt, and servesthus as a foundation for equilibrium statistical mechanics. We wish now to dwell in deeperfashion into the meaning of our new SM-formulation. First of al it is to be emphasized that, incontrast to both Gibbs and Jaynes postlates, ours have zero new informational content, sincethey are borrowed either from experiment or from pre-existing theories, namely, informationtheory and quantum mechanics. In particular, we wish to dwell to a larger extent on both theinformational and physical contents of our all-important Eqs. (24) or (26).The rst and second laws of thermodynamics are two of physics most important statements.They constitute strong pillars of our present understanding of Nature. Of course, statisticalmechanics (SM) adds an underlying microscopic substratum that is able to explain not onlythese two laws but the whole of thermodynamics itself (6; 17; 28; 29; 30; 31). One of SMsbasic ingredients is a microscopic probability distribution (PD) that controls the population11 New Microscopic Connections of Thermodynamics10 Thermodynamicsofmicrostatesofthesystemunderconsideration(28). Sincewewerehererestrictingourconsiderations to equilibrium situations, what we have been really doing here was to mainlyconcern ourselves with obtaining a detailed picture, from a new perspective (7; 8; 9; 10; 11; 12;13), of how changes in the independent external parameters - thermodynamic parameters -affect this micro-state population and, consequently, the entropy and the internal energy, i.e.,reversible changes in external parameters param changes in the microscopic probabilitydistribution entropic (dS) and internal energy (dU) changes.Weregardedasindependent external parametersbothextensiveandintensivequantitiesdeningthemacroscopicthermodynamicstateof thesystem. It iswell-knownthat theextensive parameters, always known with some (experimental) uncertainty, help to denethe Hilbert space (HS) in which the system can be represented. The intensive parameters areassociated with some physical quantities of which only the average value is known. They arerelated to the mean values of operators acting on the HS previously dened. The eigenvaluesof these operators are, therefore, functions of the extensive parameters dening the HS. Themicroscopic equilibrium probability distribution (PD) is an explicit functionof theintensiveparameters and an implicit function - via the eigenvalues of the above referred to operators(known in average) - of the extensive parameters dening the HS.What is the hard core of the new view-point of (7; 8; 9; 10; 11; 12; 13)?It consists, as will bedetailed below, in enforcing the relation dU= TdS + PdAin an innitesimal microscopic changepi pi + dpi of the probability distribution (PD) that describes the equilibrium properties of anarbitrary system and ascertaining that this univocally determines the PD, and furthermore, that the ensuing pi coincides withthat obtainedfollowingthe maximumentropyprinciple (MaxEnt) tenet of extremizing the entropy S subject to an assumedly known meanvalue U of the systems energy.Consider now only innitesimal macroscopic parameter-changes (as opposite to themicroscopic PD-ones dealt with in (7)), according to the scheme below.Reversiblechangesinparametersparam PD-changes entropic(dS) andinternalenergy (dU) changes + some work effected (W).Forcingnowthat parambeof suchnaturethat dU= TdS + Wonegetsanunivocalexpression for the PD.That is, we study variations in both the (i) intensive and (ii) extensive parameters of the systemand wish to ascertain just how these variations materialize themselves into concrete thermalrelations.9.1Homogeneous, isotropic, one-component systemsFor simplicity, consider just simple, one-component systems (6) composed by a single chemicalspecies, macroscopicallyhomogeneous, andisotropic(6). Themacroscopicequilibriumthermal stateof suchasimple, one-component systemis described, inself-explanatorynotation, byT, V, N(6). Focus attentionuponaquitegeneral informationmeasureSthat, accordingtoKinchinsaxiomsforinformationtheorydependsexclusivelyonoftheprobability distribution pi. We use again the specic but rather general form given above12 ThermodynamicsNew Microscopic Connections of Thermodynamics 11for S, viz.,S = kWi=1pi f (pi), (28)with Wthe number of microscopic states,k = Boltzmanns constant, and the sum runningover a set of quantum numbers, collectively denoted by i (characterizing levels of energy i),that specify an appropriate basis in Hilberts space ( fis an arbitrary smooth function of the pisuch that p f (p) is concave).Remember that the quantity U represents the mean value of the Hamiltonian, and, as bets anhomogeneous, isotropic, one-component system in the Helmholtz free energy representation(6) we have1. as external parameter the volume (V) and the number of particles (N) (exactly known andused to dene the Hilbert space),2. as intensive variable the temperature T, associated with the mean value Uof the internalenergy E, i.e., U = 'E`.The energy eigenvalues of the Hamiltonian i are, obviously, functions of the volume and ofthe number of particles, namely, i =i(V, N). From now on, for simplicity, we take N asxed, and drop thereby the dependence of the energy eigenvalues on N, i.e., i = i(V).The probability distribution (PD) depends, then, on the external parameters in the fashionpi = pi(T, i(V)). (29)Remind that the mean energy U = 'E` is given byU = 'E` =Wi=1g(pi)i. (30)The critical difference between what we attempt to do now and what was related above [Cf.Eq. (23)] is to be found in the following assumption, on which we entirely base our considerationsin this Section:the temperature T and the volume V reversibly change in the fashionT T+ dT and V V+ dV. (31)Asaconsequenceof (31), correspondingchangesdpi, dS, di, anddUaregeneratedin,respectively, pi, S, i, and U. Variations in, respectively, pi, S, and U writedpi =piT dT +Wj=1pijjV dV, (32)dS =Wi=1SpipiT dT +Wi,j=1SpipijjV dV, (33)and, last but not least,dU =Wi=1gpipiT

idT +Wi,j=1gpipijjV

idV +Wi=1g(pi)iV dV, (34)13 New Microscopic Connections of Thermodynamics12 Thermodynamicswhere, forsimplicity, wehaveconsiderednon-degeneratelevels. Clearly, onaccount ofnormalization, the changes in pi must satisfy the relationidpi = 0. (35)Note that if we deal with three thermodynamic parameters and one equation of state we cancompletely describe our system with any two of them (32). Here, we are choosing, as the twoindependent thermodynamic parameters, T and V. It is important to remark that independentthermodynamic parameters do not mean natural parameters. For example, if T and V are nowthe independent thermodynamic parameters, the internal energy can be written as function ofthese parameters, i.e., U(T, V). Clearly, T and V are not the natural parameters of the internalenergy. These are S and V. However, our developments require only independent parameters,that are not necessarily the natural ones (32).9.2Macroscopic considerationsThermodynamics states that, in the present scenario, for a reversible process one hasdU = Q + W = TdS + W, (36)where we have used the Clausius relation Q = TdS. Multiplying Eq. (33) by T we can recastEq. (36) in the microscopic fashion (involving the microstates PD)dU = TWi=1SpipiT dT +Wi,j=1SpipijjV dV+ W, (37)which is to be compared with (34).9.3Changes in the temperatureEqs. (34) and (37) must be equal for arbitrary changes in T and V. We take this equality as thebasis of our future considerations. As T and V can be changed in an independent way, let usrst consider just changes in T. Enforcing equality in the coefcients of dT appearing in Eqs.(34) and (37) we obtain (we are assuming, as it is obvious, that the mechanicalW does notdepend on the temperature)Wi=1gpipiT

idT = TWi=1SpipiT dT, (38)that must be satised together with [Cf. (32)]idpi =ipiT dT = 0. (39)We recast now (38) in the fashion14 ThermodynamicsNew Microscopic Connections of Thermodynamics 13Wi=1

gpi

i TSpi

piT dT iKipiT dT = 0. (40)Since the Wpis are not independent (Wi=1 pi= 1), we can separate the sum in (40) into twoparts, i.e.,W1i=1

gpi

i TSpi

piT dT+

gpW

W TSpW

pWTdT = 0. (41)PickingoutlevelWforspecialattentionisarbitrary. Anyotherilevelcouldhavebeenchosen as well, as the example given below will illustrate. Taking into account now that,from Eq. (39),pWT= W1i=1piT , (42)we see that Eq. (41) can be rewritten asW1i=1gpi

i TSpi

gpW

W TSpW

piT dT = 0. (43)As the W 1 pis are now independent, the term into brackets should vanish, which entailsgpi

i TSpi

gpW

W TSpW

= 0, (44)for all i = 1,, W 1. Let us call the term into parentheses asKW=gpW

W TSpW K = constant. (45)Finally, we cast Eqs. (44) and (45) asgpi

i TSpiK = 0; (i = 1,, W), (46)an equation that we have encountered before [Cf. Eq. (24) with g(x) x] and that should yielda denite expression for any of the Wpis. We did not care above about such an expression,but we do now.15 New Microscopic Connections of Thermodynamics14 ThermodynamicsExample 1 Consider the Shannon orthodox instanceS = kipiln pig(pi) = piS/pi= k[ln pi + 1] = k[i + lnZ 1]]. (47)Here equation (46) yields the well known MaxEnt (and also Gibbs?) resultln pi= [i + lnZ]; i.e.,pi= Z1ei/kTlnZ = 1 K/kT, and, nally, (48)S/pi = k(i K),lnZi= pi;pij= pi (ij pj); TSpipii= (i K)pi, (49)showing, asanticipated, that wecouldhaveselectedanyilevel amongtheWoneswithout affecting the nal result.Thus, changes in the inverse temperature completely specify the microscopic probabilitydensity pMaxEntif theyareconstrainedtoobeytherelationdU= TdS + W, foranyreasonablechoiceof theinformationmeasureS. Thisequivalence, however, cannot beestablished in similar fashion if the extensive variable V also changes. This is our next topic.9.4Changes in the extensive parameterLet us now deal with the effect of changes in the extensive parameters that dene the Hilbertspace in which our system lives and notice that Eq. (37) can be written in the fashion dU =Q + W = TdS + W dU = T

dTWi=1SpipiT+ dVWi,j=1SpipijjV

+ W. (50)That is, there are two ingredients entering TdS, namely,TdS = QT dT + QV dV; with QT = TWi=1SpipiT . (51)Our interest now lies in the second term. What is QV? Clearly we have16 ThermodynamicsNew Microscopic Connections of Thermodynamics 15QV= TWi,j=1SpipijjV. (52)Next, substitute the expression for (g/pi)i given by Eqs. (45) and (46),gpi

i = TSpi+ K;(i = 1, . . . , W), (53)into the second term of the R.H.S. of Eq. (34),Wi,j=1gpipijjV

idV =Wi,j=1[TSpi+ K]pijjV dV= TWi,j=1SpipijjV dV + KWi,j=1pijjV dV=TWi,j=1SpipijjVdV = QV dV, (54)on account of the fact thatKWi,j=1pijjV dV = 0; since(/V)ipi = 0. (55)We recognize in the term QV dV of the last line of (54) the microscopic interpretation of arather unfamiliar volume contribution to Clausius relation Q= TdS (dQ-equations (32)).Notice that we are not explicitly speaking here of phase-changes. We deal with reversibleprocesses. If the change in volume were produced by a phase-change one would reasonablybe tempted to call the term QV dV a latent heat.Thus, associated with a change of state in which the volume is modied, we nd in the termQV dVthe microscopic expression of a heat contribution for that transformation, i.e., theheat given up or absorbed during it. It we wish to call it latent, the reason would be thatitisnotassociatedwithachangeintemperature. Thus, wesawjusthowchangesintheequilibrium PD caused by modications in the extensive parameter dening the Hilbert spaceof the system give also a contribution to the heat part of the dU = TdS + W relation.17 New Microscopic Connections of Thermodynamics16 ThermodynamicsExample 2: In the Shannon instance discussed in Example 1 one has [Cf. (48) and (49)]pii= pi (1 Z1), (56)TSpipii= (i K)pi, (57)QV= i(i K)piiV (1 Z1). (58)Since the origin of the energy scale is arbitrary, in summing over i we can omit the Ktermby changing the energy-origin and one may writeQV= iipiiV (1 Z1). (59)Foe a particle of mass m in an ideal gas (N particles) the energy i is given by (29)

i = V2/3 ni2; =( h)22m;ni2 (n2x, n2y, n2z)nx, ny, nza set of three integersiV= (2/3)i/V. (60)Thus,the microscopic expression for QV turns out to beQV= (2/3V)'E2` (1 Z1), (61)which indeed has dimension of (energy/volume).Finally, for Eq. (34) to become equal to Eq. (50) we have to demand, in view of the abovedevelopments,W = dVig(pi)iV, (62)the quantity within the brackets being the mean value,

EV

=ig(pi)iV , (63)usually associated in the textbooks with the work done by the system.Summing up, our analysis of simple systems in the present Section has shown that byconsideringchangesdTanddVandhowtheyinuencethemicroscopicprobabilitydistribution if these variations are forced to comply with the relation (36) dU = TdS + Wwe ascertain that changes in the intensive parametergivecontributionsonlyrelatedtoheat and lead to theattaining the equilibrium PD (an alternative way to the MaxEnt principle) and changes in the extensive-Hilbert-space-determining parameter lead to two contributions1. one related to heat and2. the other related to work.18 ThermodynamicsNew Microscopic Connections of Thermodynamics 1710. Other entropic formsWe illustrate now our procedure with reference to information measures not of the Shannonlogarithmic form. We use mostly the relationship (46), namely,K = i g/(pi) kT[ f (pi)+ pi f/(pi)] [ f (pi)+ pi f/(pi)] [i g/(pi) K] = 0, 1/kT. (64)10.1Tsallis measure with linear constraintsWe have, for any real number q the information measure (28) built up with (26; 33; 34)f (pi) =(1 pq1i)q 1, (65)and, in the energy-constraint of Eq. (30)g(pi) = pi, (66)so thatf/(pi) = pq2iand Eq. (64) becomes, with = (1/kT),q pq1i= 1 + (q 1)K (q 1) i, (67)which after normalization yields a distribution often referred to as the Tsallis one (33)pi= Z1q

1 (q 1) /

i

1/(q1)Zq=i

1 (q 1) /

i

1/(q1), (68)where / /(1 + (q 1)K).10.2Tsallis measure with non-linear constraintsThe information measure is still the one built up with the functionf (pi) of (65), but we usenow the so-called Curado-Tsallis constraints (35) that arise if one usesU = 'E` =Wi=1g(pi)i, (69)withg(pi) = pqig/(pi) = q pq1i. (70)Eq. (64) leads topi = (1q)1/(q1)[1 (1 q) i]1/(1q), (71)and, after normalization, one is led to the Curado-Tsallis distribution (35)pi= (Zq)1[1 (1 q) i]1/(1q)Zq=i[1 (1 q) i]1/(1q). (72)19 New Microscopic Connections of Thermodynamics18 Thermodynamics10.3 Exponential entropic formThis measure is given in (36; 37) and also used in (38). One hasf (pi) = 1 exp(bpi)piS0, (73)where b is a positive constant and S0 = 1 exp(b), together withg(pi) = 1 ebpiS0g/(pi) = bebpiS0, (74)which, inserted into (64), after a little algebra, leads topi = 1blnbS0 K+ ln(1 iS0)

. (75)which, after normalization, gives the correct answer (37).11. ConclusionsWe have seen that the set of equationsi[C(1)i+ C(2)i]dpi = 0,C(1)i= [M=1P ai+ i] g/(pi)C(2)i= TSpiyields a probability distribution that coincides with the PD provided by either the MaxEnts, SM axiomatics of Jaynes our two postulates (15) and (16).We remind the reader that in our instance the postulates start with1. the macroscopic thermodynamic relation dE = TdS + PdA,, adding to it2. Boltzmanns conjecture of an underlying microscopic scenario ruled by microstateprobability distributions.The two postulates combine then (i) a well-tested macroscopic result with (ii) a by now ununcontestable microscopic state of affairs (which was not the case in Boltzmanns times). Thuswe may dare to assert that the two axioms we are here advancing are intuitively intelligiblefrom a physical laboratory standpoint. This cannot be said neither for Gibbs ensemble nor forJaynes extremizing of the Observers ignorance, their extraordinary success notwithstanding,since they introduce concepts like ensemble or ignorance that are not easily assimilated tolaboratoryequipment. Wemustinsist: thereisnothingwrongwithmakinguseoftheseconcepts, of course. We just tried to see whether they could be eliminated from the axioms ofthe theory.Summingup, wehaverevisitedthefoundationsofstatistical mechanicsandshownthatitispossibletoreformulateitonthebasisofjustabasicthermodynamics relationplusBoltzmanns atomic hypothesis. The latter entails (1) the (obvious today, but not in 1866)existence of a microscopic realm ruled by probability distributions.20 ThermodynamicsNew Microscopic Connections of Thermodynamics 1912. Appendix IHere we prove that Eqs. (24) are obtained via the MaxEnt variational problem (27). Assumenow that you wish to extremize S subject to the constraints of xed valued for i) U and ii) theM values A. This is achieved via Lagrange multipliers (1) and (2) M. We need also anormalization Lagrange multiplier . Recall thatA = '{` =ipi ai , (76)with ai= 'i[{[i` the matrix elements in the chosen basis 'i` of {. The MaxEnt variationalproblem becomes now (U = i pi

i)piS U M=1A ipi = 0, (77)leading, with = , to the vanishing ofpm i

pi f (pi) [pi(M=1 ai+ i) + pi]

, (78)so that the 2 quantities below vanishf (pi) + pi f/(pi) [(M=1 ai+ i) + ]if K,f (pi) + pi f/(pi) pi(M=1ai+ i) + K] 0 = T(1)i+ T(2)i. (79)Clearly, (26) and the last equality of (79) are one and the same equation! Our equivalence isthus proven.13. AcknowledgmentsThis work is founded by the Spain Ministry of Science and Innovation (Project FIS2008-00781)and by FEDER funds (EU).14. References[1] R. B. Lindsay and H. Margenau, Foundations of physics, NY, Dover, 1957.[2] J. Willard Gibbs, Elementary Principles in Statistical Mechanics, NewHaven, YaleUniversity Press, 1902.[3] E.T. Jaynes, ProbabilityTheory: TheLogicof Science, CambridgeUniversityPress,Cambridge, 2005.[4] W.T. Grandy Jr. and P. W. Milonni (Editors), Physics and Probability. Essays in Honor ofEdwin T. Jaynes, NY, Cambridge University Press, 1993.[5] E. T. Jaynes Papers on probability, statistics and statistical physics, editedby R. D.Rosenkrantz, Dordrecht, Reidel, 1987.[6] E. A. Desloge, Thermal physics NY, Holt, Rhinehart and Winston, 1968.[7] E. Curado, A. Plastino, Phys. Rev. E 72 (2005) 047103.[8] A. Plastino, E. Curado, Physica A 365 (2006) 2421 New Microscopic Connections of Thermodynamics20 Thermodynamics[9] A. Plastino, E. Curado, International Journal of Modern Physics B 21 (2007) 2557[10] A. Plastino, E. Curado, Physica A 386 (2007) 155[11] A. Plastino, E. Curado, M. Casas, Entropy A 10 (2008) 124[12] International Journal of Modern Physics B 22, (2008) 4589[13] E. Curado, F. Nobre, A. Plastino, Physica A 389 (2010) 970.[14] The MaxEnt treatment assumes that these macrocopic parameters are the expectationvalues of appropiate operators.[15] C. E. Shannon, Bell System Technol. J. 27 (1948) 379-390.[16] A. Plastino and A. R. Plastino in Condensed Matter Theories, Volume 11, E. Lude na (Ed.),Nova Science Publishers, p. 341 (1996).[17] A. Katz, Principles of Statistical Mechanics, The information Theory Approach, San Francisco,Freeman and Co., 1967.[18] D. J. Scalapino in Physics and probability. Essays in honor of Edwin T. Jaynes edited by W.T. Grandy, Jr. and P. W. Milonni (Cambridge University Press, NY, 1993), and referencestherein.[19] T. M. Cover and J. A. Thomas, Elements of information theory, NY, J. Wiley, 1991.[20] B. Russell, A history of western philosophy (Simon & Schuster, NY, 1945).[21] P. W. Bridgman The nature of physical theory (Dover, NY, 1936).[22] P. Duhem The aim and structure of physical theory (Princeton University Press, Princeton,New Jersey, 1954).[23] R. B. Lindsay Concepts and methods of theoretical physics (Van Nostrand, NY, 1951).[24] H. Weyl Philosophy of mathematics and natural science (Princeton University Press,Princeton, New Jersey, 1949).[25] D. Lindley, Boltzmanns atom, NY, The free press, 2001.[26] M. Gell-MannandC. Tsallis, Eds. NonextensiveEntropy: Interdisciplinaryapplications,Oxford, Oxford University Press, 2004.[27] G. L. Ferri, S. Martinez, A. Plastino, Journal of Statistical Mechanics, P04009 (2005).[28] R.K. Pathria, Statistical Mechanics (Pergamon Press, Exeter, 1993).[29] F. Reif, Statistical and thermal physics (McGraw-Hill, NY, 1965).[30] J. J.Sakurai, Modern quantum mechanics (Benjamin, Menlo Park, Ca., 1985).[31] B. H. Lavenda, Statistical Physics (J. Wiley, NewYork, 1991); B. H. Lavenda,Thermodynamics of Extremes (Albion, West Sussex, 1995).[32] K. Huang, Statistical Mechanics, 2nd Edition. (J. Wiley, New York, 1987). Pages 7-8.[33] C. Tsallis, Braz. J. of Phys. 29, 1 (1999); A. Plastino and A. R. Plastino, Braz. J. of Phys. 29,50 (1999).[34] A. R. Plastino and A. Plastino, Phys. Lett. A 177, 177 (1993).[35] E. M. F. Curado and C. Tsallis, J. Phys. A, 24, L69 (1991).[36] E. M. F. Curado, Braz. J. Phys. 29, 36 (1999).[37] E. M. F. Curado and F. D. Nobre, Physica A 335, 94 (2004).[38] N. Canosa and R. Rossignoli, Phys. Rev. Lett. 88, 170401 (2002).22 Thermodynamics0Rigorous and General Denition ofThermodynamic EntropyGian Paolo Beretta1and Enzo Zanchini21Universit` a di Brescia, Via Branze 38, Brescia2Universit` a di Bologna, Viale Risorgimento 2, BolognaItaly1. IntroductionThermodynamics andQuantumTheory are amongthefewsciences involvingfundamentalconcepts and universal content that are controversial and have been so since their birth, andyetcontinuetounveilnewpossibleapplicationsandtoinspirenewtheoreticalunication.The basic issues in Thermodynamics have been, and to a certain extent still are: the range ofvalidity and thevery formulation of the SecondLaw of Thermodynamics, the meaningandthe denition of entropy, the originof irreversibility, and the unication with QuantumTheory(Hatsopoulos & Beretta,2008). ThebasicissueswithQuantumTheoryhavebeen, andtoacertainextentstillare: themeaningofcomplementarityandinparticularthewave-particleduality,understandingthemanyfacesof themanywonderful experimental andtheoreticalresults on entanglement, and the unication with Thermodynamics (Horodecki et al., 2001).Entropyhasacentralroleinthissituation. Itisastonishingthatafterover140yearssincethetermentropyhasbeenrstcoinedbyClausius(Clausius, 1865), thereisstill somuchdiscussionandcontroversyabout it, not tosayconfusion. Tworecent conferences, bothheldinOctober2007, provideastate-of-the-artscenariorevealinganunsettledandhardtosettleeld: one, entitledMeetingtheentropychallenge(Beretta et al., 2008), focusedonthemanyphysical aspects(statistical mechanics, quantumtheory, cosmology, biology, energyengineering), theother, entitledFacetsof entropy(Harrem oes, 2007), onthemanydifferentmathematical conceptsthat indifferent elds(informationtheory, communicationtheory,statistics, economics, social sciences, optimization theory, statistical mechanics) have all beentermed entropy on the basis of some analogy of behavior with the thermodynamic entropy.Followingthewell-knownStatistical MechanicsandInformationTheoryinterpretationsofthermodynamicentropy, thetermentropyisusedinmanydifferentcontextswherevertherelevant statedescriptionisintermsofaprobabilitydistributionoversomeset ofpossibleevents which characterize the system description. Depending on the context, such events maybemicrostates, oreigenstates, orcongurations, ortrajectories, ortransitions, ormutations, andsoon. Givensuchaprobabilisticdescription,thetermentropyisusedforsomefunctionaloftheprobabilitieschosenasaquantieroftheirspreadaccordingtosomereasonablesetofdeningaxioms(Lieb & Yngvason, 1999). Inthissense, theuseofacommonnameforall the possible different state functionalsthatshare such broad dening features, may havesome unifying advantage from a broad conceptual point of view, for example it may suggestanalogies and inter-breeding developments between very different elds of research sharingsimilar probabilistic descriptions.22 ThermodynamicsHowever, fromthe physics point of view, entropythe thermodynamic entropyis asingle denite propertyof everywell-denedmaterial systemthat canbe measuredineverylaboratorybymeansofstandardmeasurementprocedures. Entropyisapropertyofparamount practical importance, becauseit turnsout (Gyftopoulos & Beretta, 2005) tobemonotonically related to the difference E between the energy E of the system, above thelowest-energy state, andtheadiabaticavailability ofthesystem, i.e.,themaximumworkthesystem can do in a process which produces no other external effects. It is therefore veryimportant that wheneverwetalkormakeinferencesabout physical (i.e., thermodynamic)entropy, we rst agree on a precise denition.In our opinion, one of the most rigorous and general axiomatic denitions of thermodynamicentropy available in the literature is that given in (Gyftopoulos & Beretta, 2005), which extendsto the nonequilibrium domain one of the best traditional treatments available in the literature,namely that presented by Fermi (Fermi, 1937).Inthis paper, the treatment presentedin(Gyftopoulos & Beretta, 2005) is assumedas astartingpoint andthefollowingimprovements areintroduced. Thebasic denitions ofsystem, state, isolatedsystem, environment, process, separablesystem, andparametersofasystemaredeepened, bydevelopingalogicalschemeoutlinedin(Zanchini, 1988; 1992).Operativeandgeneral denitionsoftheseconceptsarepresented, whicharevalidalsointhepresenceofinternal semipermeablewallsandreactionmechanisms. Thetreatment of(Gyftopoulos & Beretta, 2005) is simplied, byidentifyingtheminimal set of denitions,assumptions and theorems which yield the denition of entropy and the principle of entropynon-decrease. Inviewoftheimportantroleofentanglementintheongoingandgrowinginterplay between QuantumTheory and Thermodynamics, the effects of correlations on theadditivityofenergyandentropyarediscussedandclaried. Moreover, thedenitionofareversible process is given with reference to a given scenario;the latter is thelargest isolatedsystem whose subsystems are available for interaction, for the class of processes under exam.Without introducing the quantum formalism, the approach is nevertheless compatible with it(and indeed, it was meant to be so, see, e.g., Hatsopoulos & Gyftopoulos (1976); Beretta et al.(1984; 1985); Beretta(1984; 1987; 2006; 2009)); it is thereforesuitable toprovideabasiclogical frameworkfortherecentscienticrevival ofthermodynamicsinQuantumTheory[quantumheatengines(Scully, 2001; 2002), quantumMaxwelldemons(Lloyd, 1989; 1997;Giovannetti et al., 2003), quantum erasers (Scully et al., 1982; Kim et al., 2000), etc.]as well asfortherecentquestforquantummechanicalexplanationsofirreversibility [see, e.g., Lloyd(2008); Bennett (2008); Hatsopoulos & Beretta (2008); Maccone (2009)].Thepaperis organizedasfollows. InSection 2 wediscuss thedrawbacksofthetraditionaldenitions of entropy. In Section 3we introduce and discuss a full set of basic denitions, suchas those of system, state,process, etc. thatform the necessary unambiguous background onwhich to build our treatment. In Section 4we introduce the statement of the First Law and thedenition of energy. In Section 5 we introduce and discuss the statementof the Second Lawand, throughtheproofofthreeimportanttheorems, webuildupthedenitionofentropy.In Section 6 we briey complete the discussion by proving in our context the existence of thefundamental relation for the stable equilibrium states and by dening temperature, pressure,and other generalized forces.In Section 7we extend our denitions of energy and entropy tothe model of an open system.In Section 8 we prove the existence of the fundamental relationfor the stable equilibrium states of an open system. In Section 9we draw our conclusions and,in particular, we note that nowhere in our construction we use or need to dene the conceptof heat.24 ThermodynamicsRigorous and General Denition of Thermodynamic Entropy 32. Drawbacks of the traditional denitions of entropyIntraditionalexpositionsofthermodynamics, entropyisdenedintermsoftheconceptofheat, which in turn is introduced at the outset of the logical development in terms of heuristicillustrations based on mechanics. For example, in his lectures on physics, Feynman (Feynman,1963) describes heat as one of several different forms of energy related to the jiggling motion ofparticles stuck together and tagging along with each other (pp. 1-3 and 4-2), a form of energywhich really is just kinetic energy internal motion (p. 4-6), and is measured by the randommotions of the atoms (p. 10-8). Tisza (Tisza, 1966) argues that such slogans as heat is motion,in spite of their fuzzy meaning, convey intuitive images of pedagogical and heuristic value.There are at least three problems with these illustrations. First, work and heat are not stored ina system. Each is a mode of transfer of energy fromone systemto another. Second, concepts ofmechanics are used to justify and make plausible a notion that of heat which is beyondthe realm of mechanics; although at a rst exposure one might nd the idea of heat as motionharmless, and even natural, the situation changes drastically when the notion of heat is usedtodene entropy, andthelogical loop is completed when entropy is shown to imply ahostofresultsaboutenergyavailabilitythatcontrastwithmechanics. Third,andperhapsmoreimportant, heatisamodeofenergy(andentropy)transferbetweensystemsthat areveryclose to thermodynamic equilibrium and,therefore, any denition of entropy based on heatis bound to be valid only at thermodynamic equilibrium.The rst problem is addressed in some expositions. Landau and Lifshitz (Landau & Lifshitz,1980) dene heatas thepart of an energy changeof a body thatis not due to work done onit. Guggenheim (Guggenheim, 1967) denes heat as an exchange of energy that differs fromwork and is determined by a temperature difference. Keenan (Keenan, 1941) denes heat asthe energy transferred from one system to a second system at lower temperature, by virtue ofthe temperature difference, when the two are brought into communication. Similar denitionsare adopted in most other notable textbooks that are too many to list.None of these denitions, however, addresses the basic problem. The existence of exchangesof energy that differ from work is not granted by mechanics. Rather,it is one of the strikingresults of thermodynamics, namely, of theexistenceof entropyas apropertyof matter.As pointedout byHatsopoulos andKeenan(Hatsopoulos & Keenan, 1965), without theSecondLawheat andworkwouldbeindistinguishable; moreover, themost general kindof interactionbetweentwosystemswhichareveryfarfromequilibriumisneitheraheatnor aworkinteraction. FollowingGuggenheimit wouldbepossibletostatearigorousdenition of heat, with reference toa very special kind of interactionbetween twosystems,and to employ the concept of heat in the denition of entropy (Guggenheim, 1967). However,Gyftopoulos and Beretta (Gyftopoulos & Beretta, 2005) have shown that the concept of heat isunnecessarily restrictive for the denition of entropy, as it would conne it to the equilibriumdomain. Therefore, in agreement with their approach, we will present and discuss a denitionof entropy where the concept of heat is not employed.Otherproblemsarepresentinmosttreatmentsofthedenitionofentropyavailableintheliterature:1. many basic concepts, such as those of system, state, property, isolated system, environmentof a system, adiabatic process are not dened rigorously;2. onaccount of unnecessaryassumptions(suchas, theuseof theconcept of quasistaticprocess), the denitionholds onlyfor stable equilibriumstates (Callen, 1985), or forsystems which are in local thermodynamic equilibrium (Fermi, 1937);25 Rigorous and General Definition of Thermodynamic Entropy4 Thermodynamics3. in the traditional logical scheme (Tisza, 1966; Landau & Lifshitz, 1980; Guggenheim, 1967;Keenan, 1941; Hatsopoulos & Keenan, 1965; Callen, 1985; Fermi, 1937), someproofsareincomplete.To illustrate the third point, which is not well known, let us refer to the denition in (Fermi,1937), which we consider one of the best traditional treatments available in the literature. Inordertodenethethermodynamictemperature, Fermiconsidersareversiblecyclicenginewhich absorbs a quantity of heat Q2 from a source at (empirical) temperature T2 and suppliesaquantityofheatQ1toasource at(empirical) temperatureT1. Hestatesthatiftheengineperforms n cycles, the quantity of heat subtracted fromthe rst source is nQ2 and the quantityof heat supplied to the second source is nQ1. Thus, Fermi assumes implicitly that the quantityof heat exchanged in a cycle between a source and a reversible cyclic engine is independent ofthe initial state of the source. In our treatment, instead, a similar statementis made explicit,and proved.3. Basic denitionsLevel ofdescription, constituents, amountsofconstituents, deeperlevel ofdescription.Wewillcalllevelofdescriptionaclassofphysicalmodels wherebyallthatcanbesaid aboutthematter containedinagivenregionof space R, at atimeinstant t, canbedescribedbyassumingthat thematter consistsof aset of elementarybuildingblocks, that wecallconstituents, immersedintheelectromagneticeld. Examplesof constituentsare: atoms,molecules, ions, protons,neutrons, electrons. Constituentsmaycombineand/ortransforminto other constituents according to a set of model-specic reaction mechanisms.For instance, at thechemical level of descriptiontheconstituents arethedifferent chemicalspecies,i.e.,atoms, molecules,andions; attheatomiclevelofdescriptiontheconstituentsarethe atomic nuclei and the electrons; at thenuclear level of description they are the protons, theneutrons, and the electrons.The particle-like nature of the constituents implies that a counting measurement procedure isalways dened and, when performed in a region of space delimited by impermeable walls, itisquantizedinthesensethat themeasurementoutcomeisalwaysanintegernumber, thatwecall thenumberof particles. If thecountingisselectiveforthei-thtypeof constituentonly, wecall theresultingnumberofparticlestheamountof constituenti anddenoteitbyni. When a number-of-particle counting measurement procedure is performed in a region ofspacedelimited byatleastoneideal-surfacepatch, someparticlesmaybefoundacrossthesurface. Therefore, an outcome of the procedure must also be the sum, for all the particles inthis boundary situation, of a suitably dened fraction of their spatial extension which is withinthe given region of space. As a result, the number of particles and the amount of constituent i willnot be quantized but will have continuous spectra.A level of description L2is called deeper than a level of description L1if the amount of everyconstituentinL2is conserved for all the physical phenomenaconsidered, whereas the sameisnottruefor theconstituentsinL1. Forinstance, theatomiclevelofdescription isdeeperthan the chemical one (because chemical reaction mechanisms do not conserve the number ofmolecules of each type,whereas they conserve thenumber of nuclei of each type as well asthe number of electrons).Levelsofdescriptiontypicallyhaveahierarchical structurewherebytheconstituentsofagiven level are aggregates of the constituents of a deeper level.Region of space which contains particles of the i-th constituent. We will call region of spacewhichcontainsparticlesofthei-thconstituentaconnectedregion Riofphysicalspace(the26 ThermodynamicsRigorous and General Denition of Thermodynamic Entropy 5three-dimensionalEuclideanspace)inwhichparticlesofthei-thconstituentarecontained.The boundary surface of Ri may be a patchwork of walls, i.e., surfaces impermeable to particlesof the i-th constituent, and ideal surfaces (permeable to particles of the i-th constituent). Thegeometryoftheboundarysurfaceof Rianditspermeabilitytopologynature(walls, idealsurfaces) can vary in time, as well as the number of particles contained in Ri.Collectionofmatter, composition. Wewillcallcollectionofmatter, denotedby CA, asetofparticlesofoneormoreconstituentswhichisdescribedbyspecifyingtheallowedreactionmechanismsbetween different constituentsand,atany time instantt, theset of rconnectedregions of space, RRRA=RA1 , . . . , RAi, . . . , RAr, each of which contains nAiparticles of a single kindof constituent. The regions of space RRRAcan vary in time and overlap. Two regions of spacemaycontainthesame kindof constituentprovided thattheydonotoverlap. Thus,thei-thconstituent could be identical with the j-th constituent, provided that RAiand RAjare disjoint.If, due to time changes, two regions of space which contain the same kind of constituent beginto overlap, from that instant a new collection of matter must be considered.Comment. This method of description allows to consider the presence of internal walls and/orinternalsemipermeable membranes, i.e., surfaces which can be crossed only by some kinds ofconstituentsandnotothers. Inthesimplestcaseofacollectionofmatterwithoutinternalpartitions, the regions of space RRRAcoincide at every time instant.The amount ni of the constituent in the i-th region of space can vary in time for two reasons: matter exchange: during a time interval in which the boundary surface of Riis not entirelya wall, particles may be transferred into or out of Ri; we denote by nAthe set of rates atwhich particles are transferredin or out of each region, assumed positive if inward, negativeif outward; reaction mechanisms: in a portionof space where two or more regions overlap, theallowed reaction mechanisms may transform, according to well specied proportions (e.g.,stoichiometry), particles of one or more regions into particles of one or more other regions.Compatible compositions, setof compatible compositions. We say thattwo compositions,n1Aandn2Aof a given collection of matter CAare compatible if the change betweenn1Aandn2Aor viceversa can take place as a consequence of the allowed reaction mechanisms withoutmatterexchange. WewillcallsetofcompatiblecompositionsforasystemAthesetofallthecompositions of A which are compatible with a given one. We will denote a set of compatiblecompositions for A by the symbol (n0A, A). By this we mean that the set of allowed reactionmechanismsisdenedlikeforchemicalreactionsbyamatrixofstoichiometriccoefcientsA= [()k], with ()krepresenting the stoichiometric coefcient of thek-th constituent in the-th reaction.The set of compatible compositions is a -parameter set dened by the reactioncoordinates A= A1 , . . . , A

, . . . , Athrough the proportionality relationsnA=n0A+A A, (1)where n0Adenotes the compositioncorrespondingtothe value zeroof all the reactioncoordinates A. To x ideas and for convenience, we will select A=0 at time t = 0 so thatn0Ais the composition at time t = 0 and we may call it the initial composition.In general, the rate of change of the amounts of constituents is subject to the amounts balanceequations nA= nA+A A. (2)External forceeld. Let us denote byFaforceeldgivenbythe superpositionof agravitationaleldG,anelectriceldE,andamagneticinductioneldB.Letusdenoteby27 Rigorous and General Definition of Thermodynamic Entropy6 ThermodynamicsAtthe union of all the regions of space RRRAtin which the constituents of CAare contained, at atime instantt, which we also call region of space occupied by CAat time t. Let us denote byAthe union of the regions of space At, i.e., the union of all the regions of space occupied byCAduring its time evolution.We call external force eld for CAat time t, denoted by FAe,t, the spatial distribution of F which ismeasured at time t in Atif all the constituents and the walls of CAare removed and placedfarawayfromAt. Wecall external forceeldfor CA, denotedbyFAe, thespatial andtimedistributionof FwhichismeasuredinAif all theconstituentsandthewallsof CAareremoved and placed far away fromA.System, properties of a system. We will call system A a collection of matter CAdened by theinitial composition n0A, the stoichiometric coefcients Aof the allowed reaction mechanisms,and the possibly time-dependent specication, over the entire time interval of interest, of: the geometrical variables and the nature of the boundary surfaces that dene the regions ofspace RRRAt, the rates nAtat which particles are transferred in or out of the regions of space, and the external force eld distribution FAe,t for CA,provided that the following conditions apply:1. an ensemble of identically prepared replicas of CAcan be obtained at any instant of time t,according to a specied set of instructions or preparation scheme;2. asetofmeasurementprocedures, PA1, . . . , PAn, exists, suchthatwheneachPAiisappliedonreplicas of CAatanygiveninstantoftimet: eachreplicarespondswithanumericaloutcome which may vary from replica to replica; but either the time interval t employedtoperformthe measurement canbe made arbitrarilyshort sothat the measurementoutcomesconsideredforPAiarethosewhichcorrespondtothelimitast 0, orthemeasurement outcomes are independent of the time interval t employed to perform themeasurement;3. thearithmeticmeanPAi

tofthenumericaloutcomesofrepeatedapplicationsofanyoftheseprocedures, PAi, ataninstantt, onanensembleofidenticallyprepared replicas, isavaluewhichisthesameforeverysubensembleofreplicasof CA(thelatterconditionguarantees the so-called statistical homogeneity of the ensemble); PAi

t is called the value ofPAifor CAat time t;4. theset of measurement procedures, PA1, . . . , PAn, iscomplete inthesensethat theset ofvalues {PA1

t, . . . , PAn

t} allows to predict the value of any other measurement proceduresatisfying conditions 2 and 3.Then, each measurement procedure satisfying conditions 2 and 3 is called a property of systemA, and the set PA1, . . . , PAna complete set of properties of system A.Comment. Althoughingeneraltheamountsofconstituents, nnnAt, andthereactionrates, t,areproperties accordingtotheabovedenition, wewilllistthemseparatelyandexplicitlywheneveritisconvenientforclarity. Inparticular, intypicalchemicalkineticmodels, tisassumed to be a function of nnnAtand other properties.State of a system. Given a system A as just dened, we call state of systemA at time t, denotedbyAt, the set of the values at time t of28 ThermodynamicsRigorous and General Denition of Thermodynamic Entropy 7 all the properties of the systemor, equivalently, of a complete set of properties,{P1

t, . . . , Pn

t}, the amounts of constituents, nnnAt, thegeometrical variablesandthenatureof theboundary surfaces of theregions of spaceRRRAt, the rates nAtof particle transfer in or out of the regions of space, and the external force eld distribution in the region of space Atoccupied by A at time t, FAe,t.With respect to the chosen complete set of properties, we can writeAt

P1

t, . . . , Pn

t; nnnAt;RRRAt; nAt; FAe,t

. (3)For shorthand, statesAt1, At2,. . . , are denoted byA1, A2,. . . . Also, when the context allows it,the value PA

t1of property PAof systemA at time t1 is denoted depending on convenienceby the symbol PA1, or simply P1.Closed system, open system. A system A is called a closed system if, at every time instant t, theboundary surface of every region of space RAitis a wall. Otherwise, A is called an open system.Comment. For a closed system, in each region of space RAi, the number of particles of the i-thconstituent can change only as a consequence of allowed reaction mechanisms.Composite system, subsystems. Givena systemCinthe external force eldFCe , wewill saythat Cis the composite of systems AandB, denotedAB, if: (a) thereexists apair of systems AandBsuchthat the external forceeldwhichobtains whenboth AandBareremovedandplacedfar awaycoincides withFCe ; (b) noregionof space RAioverlapswithanyregionofspace RBj; and(c)therC= rA+ rBregionsofspaceof CareRRRC= RA1 , . . . , RAi, . . . , RArA, RB1, . . . , RBj, . . . , RBrB. Then we say thatAandBare subsystems of thecomposite system C, and we write C = AB and denote its state at time t by Ct = (AB)t.Isolated system.We say that a closed system I is an isolated system in the stationary externalforce eldFIe,or simply an isolatedsystem,if,during thewholetimeevolution ofI: (a)onlythe particles ofIare present in I; (b) the external force eld forI, FIe, is stationary, i.e., timeindependent, and conservative.Comment. In simpler words, a systemIis isolated if, at every time instant: no other materialparticle is present in the whole region of space Iwhich will be crossed by systemIduringitstimeevolution; ifsystemI isremoved, onlyastationary(vanishingornon-vanishing)conservative force eld is present in I.Separable closed systems. Consider a composite system AB, with Aand B closedsubsystems.We say that systems A and B are separable at time t if: theforce eld external toAcoincides (where dened) with theforce eld external toAB,i.e., FAe,t = FABe,t; theforce eld externaltoBcoincides (where dened) withtheforce eld external toAB,i.e., FBe,t = FABe,t.Comment.In simpler words, systemA is separable from B at time t, if at that instant the forceeld produced by B is vanishing in the region of space occupied byA and viceversa. Duringthe subsequent time evolution of AB, A and B need not remain separable at all times.Subsystems in uncorrelated states. Consider a composite systemAB such that at time t thestatesAtandBtof thetwo subsystems fully determine the state (AB)t, i.e., the values of all29 Rigorous and General Definition of Thermodynamic Entropy8 Thermodynamicsthe properties of AB can be determined by local measurements of properties of systems A andB. Then, at time t, we say that the states of subsystems A and B are uncorrelated from each other,and we write the state of AB as (AB)t = AtBt. We also say, for brevity, that A and B are systemsuncorrelated from each other at time t.Correlated states, correlation. If at time t the states At and Bt do not fully determine the state(AB)tof the composite systemAB, we say thatAtandBtare states correlated with each other.We also say, for brevity, thatA and B are systems correlated with each other at time t.Comment. Two systems A and B which are uncorrelatedfromeach other at time t1 can undergoan interaction such that they are correlated with each other at time t2> t1.Comment. Correlations between isolated systems. Let us consider an isolated system I = AB suchthat, attimet, systemAisseparableanduncorrelated fromB. Thiscircumstancedoesnotexclude that,attimet, A and/orB(or both)may be correlated with a system C, even if thelatterisisolated, e.g. itisfarawayfromtheregionofspaceoccupiedbyAB. Indeedourdenitions of separability and correlation are general enough to be fully compatible with thenotionofquantumcorrelations,i.e.,entanglement,whichplaysanimportantrole inmodernphysics. In other words, assume that an isolated system U is made of three subsystemsA, B,and C, i.e., U = ABC, with C isolated andAB isolated. The fact thatA is uncorrelated from B,so that according to our notation we may write (AB)t = AtBt, does not exclude thatA and Cmay be entangled, in such a way thatthe statesAtand Ctdo not determine the state ofAC,i.e., (AC)t = AtCt, nor we can write Ut = (A)t(BC)t.Environment of a system, scenario.If for the time span of interest a system A is a subsystemof an isolated systemI =AB, we can chooseAB as the isolated system to be studied. Then,we will call B the environment of A, and we callAB the scenario under whichA is studied.Comment. The chosen scenarioABcontainsassubsystems all andonly thesystems thatareallowed to interact withA; thus all the remaining systems in the universe, even if correlatedwithAB, are considered as not available for interaction.Comment. Asystemuncorrelatedfromits environment inonescenario, maybe correlatedwithits environment inabroader scenario. Consider asystemAwhich, inthescenario AB, isuncorrelatedfromitsenvironmentBattimet. IfattimetsystemAisentangledwithanisolated system C, in the scenario ABC, A is correlated with its environment BC.Process, cycle. WecallprocessforasystemAfromstateA1tostateA2inthescenarioAB,denoted by(AB)1 (AB)2, the change of state from (AB)1to (AB)2of the isolated systemAB which denes the scenario. We call cycle for a systemA a process whereby the nal stateA2 coincides with the initial stateA1.Comment.In every process of any system A, the force eld FABeexternal toAB, where B is theenvironment of A, cannot change. In fact, AB is an isolated system and, as a consequence, theforce eld external toAB is stationary.Thus, in particular, for all the states in which a systemA is separable: the force eld FABeexternal toAB, where B is the environment ofA, is the same; the force eld FAeexternal toA coincides, where dened, with the force eld FABeexternaltoAB, i.e., the force eld produced by B (if any) has no effect on A.Process between uncorrelated states, external effects.A process in the scenario AB in whichtheendstatesofsystemAarebothuncorrelatedfromitsenvironment Biscalledprocessbetweenuncorrelatedstatesanddenotedby A,B12 (A1A2)B1B2. Insuchaprocess, thechangeofstateoftheenvironmentBfromB1toB2iscalledeffectexternaltoA. Traditionalexpositions of thermodynamics consider only this kind of process.30 ThermodynamicsRigorous and General Denition of Thermodynamic Entropy 9Compositeprocess. Atime-orderedsequenceofprocessesbetweenuncorrelatedstatesofasystemAwithenvironment B, A,B1k=(A,B12, A,B23,. . . , A,B(i1)i,. . . , A,B(k1)k) iscalledacomposite process if thenal stateof ABfor processA,B(i1)iis theinitial stateof ABforprocess A,Bi(i+1), for i = 1, 2, . . . , k 1. When the context allows the simplied notation ifori = 1, 2, . . . , k 1 for the processes in thesequence, the compositeprocess may also be denotedby (1, 2,. . . , i,. . . , k1).Reversibleprocess, reverseofareversibleprocess. Aprocessfor AinthescenarioAB,(AB)1 (AB)2, iscalledareversibleprocessifthereexistsaprocess(AB)2 (AB)1whichrestores the initial state of the isolated system AB. The process (AB)2(AB)1 is called reverseofprocess (AB)1 (AB)2. With different words, aprocess of anisolated systemI =ABisreversible if it can be reproduced as a part of a cycle of the isolated systemI. For a reversibleprocess between uncorrelated states, A,B12 (A1 A2)B1B2, the reverse will be denoted byA,B12 (A2 A1)B2B1.Comment. Thereverseprocessmaybeachievedinmorethanoneway(inparticular, notnecessarily by retracing the sequence of states (AB)t, with t1t t2, followed by the isolatedsystem AB during the forward process).Comment.The reversibility in one scenario does not grant the reversibility in another.If the smallestisolatedsystemwhichcontainsAisABandanotherisolatedsystemCexistsinadifferentregion of space, one can choose as environment ofA either B or BC.Thus, the time evolutionof Acanbedescribedbytheprocess(AB)1 (AB)2inthescenarioABorbytheprocess(ABC)1(ABC)2inthescenarioABC. Forinstance, theprocess(AB)1(AB)2couldbeirreversible, howeverbybroadening thescenarioso thatinteractionsbetweenABandCbecome available, a reverse process (ABC)2 (ABC)1may be possible. On the other hand,aprocess(ABC)1 (ABC)2couldbeirreversibleonaccount of anirreversibleevolutionC1 C2 of C, even if the process (AB)1 (AB)2 is reversible.Comment. A reversible process need not be slow. In the general framework we are setting up, it isnoteworthythatnowherewestatenorweneedtheconceptthataprocesstobereversibleneedstobeslowinsomesense. Actually, aswell representedin(Gyftopoulos & Beretta,2005) and clearly understood within dynamical systems models based on linear or nonlinearmasterequations, thetimeevolutionofthestateofasystemistheresultofacompetitionbetween (hamiltonian) mechanisms which are reversible and (dissipative) mechanisms whichare not.So, to design a reversible process in the nonequilibrium domain, we most likely needa fast process, whereby the state is changed quickly by a fast hamiltonian dynamics, leavingnegligible time for the dissipative mechanisms to produce irreversible effects.Weight. We call weight a system M always separable and uncorrelated from its environment,such that: Misclosed, ithasasingle constituentcontainedin asingle region of spacewhoseshapeand volume are xed, it has a constant mass m; inanyprocess, thedifferencebetweentheinitial andthenal stateof Misdetermineduniquely by the change in the position z of the center of mass ofM, which can move onlyalong a straight line whose direction is identied by the unit vector k = z; along the straight line there is a uniform stationary external gravitational force eld Ge=gk, where g is a constant gravitational acceleration.31 Rigorous and General Definition of Thermodynamic Entropy10 ThermodynamicsAs a consequence, the difference in potential energy between any initial and nal states of Mis given by mg(z2z1).Weight process, work in a weight process. A process between states of a closed systemA inwhich A is separable and uncorrelated fromits environment is called a weight process, denotedby (A1 A2)W, if the only effect external toA is the displacement of the center of mass of aweightM between two positions z1andz2. We call work performed byA (or, done by A) in theweight process, denoted by the symbol WA12, the quantityWA12= mg(z2z1) . (4)Clearly, the work done by Ais positive if z2>z1 and negative if z2 0, the energy ofA is lowered and the regions ofspace RRRAoccupied by the constituents of A have no net change. On account of Assumption 1,33 Rigorous and General Definition of Thermodynamic Entropy12 Thermodynamicsit would be possible to perform a weight process 2forA in which the regions of space RRRAoccupied by theconstituentsofAhaveno net change,the weightMis restored to its initialstatesothatthepositiveamountofenergyWA= W> 0issuppliedbacktoA, andthenal state ofA is a nonequilibrium state, namely, a state clearly different fromAse. Thus, thezero-work composite process (1, 2) would violate the denition of stable equilibriumstate.Comment. Kelvin-Planck statement of the Second Law. As noted in (Hatsopoulos & Keenan, 1965)and(Gyftopoulos & Beretta,2005,p.64), theimpossibility ofaperpetualmotion machineofthe second kind (PMM2), which is also known as the Kelvin-Planck statement of the Second Law,is a corollary of the denition of stable equilibrium state, provided that we adopt the (usuallyimplicitly) restriction to normal systems (Assumption 1).Second Law. Among all the states in which a closed systemA is separable and uncorrelatedfrom its environment and the constituents of A are contained in a given set of regions of spaceRRRA, there is a stable equilibrium state for every value of the energy EA.Lemma 1. Uniqueness of the stable equilibriumstate. There can be no pair of different stableequilibrium states of a closed system A with identical regions of space RRRAand the same valueof the energy EA.Proof. SinceAis closed and in any stableequilibrium state it is separable and uncorrelatedfrom its environment, if two such states existed, by the rst law and the denition of energythey could be interconnected by means of a zero-work weight process. So, at least one of themcould be changed to a different state with no external effect, and hence would not satisfy thedenition of stable equilibrium state.Comment. Recall that for a closed system, the composition nnnAbelongs to the set of compatiblecompositions (n0A, A) xed once and for all by the denition of the system.Comment. Statementsof theSecondLaw. Thecombinationof ourstatement of theSecondLawandLemma1establishes, foraclosedsystemwhosematterisconstrainedintogivenregionsofspace, theexistenceanduniquenessofastableequilibriumstateforeveryvalueof theenergy; thispropositionisknownastheHatsopoulos-Keenanstatement of theSecondLaw(Hatsopoulos & Keenan, 1965). Well-knownhistorical statementsof theSecondLaw,inadditiontotheKelvin-Planckstatement discussedabove, areduetoClausius andtoCarath eodory. In(Gyftopoulos & Beretta, 2005, p.64, p.121, p.133)itisshownthateachofthesehistorical statementsisalogical consequenceof theHatsopoulos-Keenanstatementcombined with a further assumption, essentially equivalent to our Assumption 2 below.Lemma 2. Any stable equilibriumstate As of a closed system A is accessible via an irreversiblezero-work weight process from any other stateA1in whichA is separable and uncorrelatedwith its environment and has the same regions of space RRRAand the same value of the energyEA.Proof. Bytherst lawandthedenitionofenergy, AsandA1canbeinterconnectedbyazero-workweight processfor A. However, azero-workweight processfromAstoA1would violate the denition of stable equilibrium state. Therefore, the process must be in thedirection from A1 toAs. The absence of a zero-work weight process in the opposite direction,implies that any zero-work weight process from A1 to As is irreversible.Corollary1. AnystateinwhichaclosedsystemAisseparableanduncorrelatedfromitsenvironmentcanbechangedtoauniquestableequilibriumstatebymeansofazero-workweight process for A in which the regions of space RRRAhave no net change.Proof. The thesis follows immediately from the Second Law, Lemma 1 and Lemma 2.Mutual stable equilibrium states. We say that two stable equilibrium statesAseand Bsearemutual stable equilibriumstates if, when Ais in state Ase and B in state Bse, the composite system34 ThermodynamicsRigorous and General Denition of Thermodynamic Entropy 13ABis in astable equilibrium state. The denition holds also for apair of statesof the samesystem: in this case, system AB is composed of A and of a duplicate of A.Identical copyof a system. We saythat a systemAd, always separable fromAanduncorrelatedwithA, isanidentical copyofsystemA(or, aduplicateof A)if, ateverytimeinstant: the difference between the set of regions of space RRRAdoccupied by the matter of Adand thatRRRAoccupied by the matter ofA is only a rigid translation r with respect to the referenceframe considered, and the composition ofAdis compatible with that ofA; the external force eld forAdat any position r +r coincides with the external force eldfor A at the position r.Thermal reservoir. We call thermal reservoir a system R with a single constituent, contained ina xed region of space, with a vanishing external force eld, with energy values restricted to anite range such that in any of its stable equilibrium states, R is in mutual stable equilibriumwith an identical copy of R, Rd, in any of its stable equilibrium states.Comment. Every single-constituent system without internal boundaries and applied externalelds, andwithanumberof particlesof theorderof onemole(sothat thesimplesystemapproximation as dened in (Gyftopoulos & Beretta, 2005, p.263) applies), when restricted toa xed region of space of appropriate volume and to the range of energy values correspondingto the so-called triple-point stable equilibriumstates, is an excellent approximation of a thermalreservoir.Referencethermalreservoir. Athermalreservoirchosenonceandforall, willbecalledareference thermal reservoir. To x ideas, we will choose as our reference thermal reservoir onehaving water as constituent, with a volume, an amount, and a range of energy values whichcorrespond to the so-called solid-liquid-vapor triple-point stable equilibrium states.Standard weight process. Given a pair of states (A1, A2) of a closed systemA, in whichA isseparable and uncorrelated from its environment, and a thermal reservoir R, we call standardweight process forAR fromA1toA2a weight process for the composite systemAR in whichthe end states of R are stable equilibrium states. We denote by (A1R1 A2R2)swa standardweight process forAR from A1toA2and by (ER)swA1A2the corresponding energy change ofthe thermal reservoir R.Assumption2. Everypairofstates(A1, A2)inwhichaclosedsystemAisseparableanduncorrelatedfromitsenvironment canbeinterconnectedbyareversiblestandardweightprocess for AR, where R is an arbitrarily chosen thermal reservoir.Theorem2. ForagivenclosedsystemAandagivenreservoirR, amongallthestandardweight processes for AR between a given pair of states (A1, A2) in which system A is separableand uncorrelated fromits environment, the energy change (ER)swA1A2of the thermal reservoirR has a lower bound which is reached if and only if the process is reversible.Proof. LetARdenoteastandardweight processfor ARfromA1toA2, andARrevareversibleone; theenergychanges of RinprocessesARandARrevare, respectively,(ER)swA1A2and(ER)swrevA1A2. With thehelpofFigure 1, wewill prove that, regardless oftheinitial state of R:a) (ER)swrevA1A2 (ER)swA1A2;b) if also AR is reversible, then (ER)swrevA1A2= (ER)swA1A2;c) if (ER)swrevA1A2= (ER)swA1A2, then also AR is reversible.Proofofa). LetusdenotebyR1andbyR2theinitialandthenal statesofRinprocessARrev. LetusdenotebyRdtheduplicateofRwhichisemployed inprocess AR, byRd335 Rigorous and General Definition of Thermodynamic Entropy14 ThermodynamicsdR3dR4rev AR AR1A1R2A2Rswrev2 1) (A ARE sw2 1) (A ARE dR3dR4rev AR AR1A1R2A2Rswrev2 1) (A ARE sw2 1) (A ARE Fig. 1. Illustration of the proof ofTheorem 2: standard weightprocesses ARrev (reversible) andAR; Rdis a duplicate of R; see text.1" R2" R' AR" AR1A1' R2A2' Rswrev '2 1) (A ARE swrev "2 1) (A ARE 1" R2" R' AR" AR1A1' R2A2' Rswrev '2 1) (A ARE swrev "2 1) (A ARE Fig. 2. Illustration of the proof ofTheorem 3, part a): reversiblestandard weight processes AR andAR , see text.and by Rd4the initial and the nal states of Rdin this process.Let us suppose, ab absurdo, that(ER)swrevA1A2> (ER)swA1A2. Then, thecomposite process (ARrev, AR)wouldbeaweightprocess for RRdin which, starting from the stable equilibrium state R2Rd3, the energy of RRdis lowered and the regions of space occupied by the constituents of RRdhave no net change,in contrast with Theorem 1. Therefore, (ER)swrevA1A2 (ER)swA1A2.Proof of b). If AR is reversible too, then, in addition to (ER)swrevA1A2(ER)swA1A2, the relation(ER)swA1A2 (ER)swrevA1A2must hold too. Otherwise, the composite process (ARrev,AR)would be a weight process for RRdin which, starting from the stable equilibrium state R1Rd4,theenergy ofRRdislowered andtheregions ofspaceoccupiedbytheconstituentsofRRdhave no net change, in contrast with Theorem 1. The