Derivation of the Boltzmann Principle ͒ Michele Campisia Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany ͒ Donald H

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Derivation of the Boltzmann principle ͒ Michele Campisia Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany ͒ Donald H. Kobeb Department of Physics, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427 ͑Received 22 September 2009; accepted 4 January 2010͒ W We derive the Boltzmann principle SB =kB ln based on classical mechanical models of thermodynamics. The argument is based on the heat theorem and can be traced back to the second half of the 19th century in the works of Helmholtz and Boltzmann. Despite its simplicity, this argument has remained almost unknown. We present it in a contemporary, self-contained, and accessible form. The approach constitutes an important link between classical mechanics and statistical mechanics. © 2010 American Association of Physics Teachers. ͓DOI: 10.1119/1.3298372͔

I. INTRODUCTION struct a one-dimensional mechanical model of thermodynamics in Sec. III according to the work of Helmholtz.3 The One of the most intriguing equations of contemporary concepts of ergodicity and microcanonical probability distri- physics is Boltzmann’s celebrated principle bution emerge naturally in this model and are introduced in S = k ln W, ͑1͒ Sec. IV. In Sec. V we generalize the one-dimensional model B B to more realistic Hamiltonian systems of N-particles in three where kB is Boltzmann’s constant. Despite its unquestionable dimensions. At this stage the ergodic hypothesis is assumed success in providing a means to compute the thermodynamic and Eq. ͑2͒ is derived. In Sec. VI we point out that the entropy of isolated systems based on counting the number W mechanical entropy of Eq. ͑2͒ does not change during quasi- of available microscopic states, its theoretical justification static processes in isolated systems, in agreement with the remains obscure and vague in most statistical mechanics second law of thermodynamics. Non-quasi-static processes textbooks. In this respect Khinchin has commented:1 “All that can lead to an increase in entropy have been treated existing attempts to give a general proof of this postulate elsewhere.5,6 In Sec. VII the Boltzmann principle is derived. must be considered as an aggregate of logical and math- A summary and some remarks concerning the validity of the ematical errors superimposed on a general confusion in the ergodic hypothesis are given in Sec. VIII. definition of the basic quantities.” The lack of a crystal-clear We present the line of reasoning and main results in the proof of Boltzmann’s principle puts physics students, teach- text, while proofs and problems are provided in Appendices ers, and all physicists in the uncomfortable position of being B–E. forced to accept the relation as a postulate which is necessary to link thermodynamic entropy to microscopic dynamics. II. CLAUSIUS ENTROPY Recent studies in the history and foundations of statistical mechanics2 have drawn attention to the fact that a similar We first review the first and second laws of thermodynam- relation, ics in the formulation given by Clausius,7 which gives the definition of thermodynamic entropy. S = k ln ⌽, ͑2͒ B The differential form of the first law of thermodynamics emerges naturally from classical mechanics if the ergodic is8 hypothesis is made, the properties that entropy should satisfy dE = ␦Q + ␦W, ͑3͒ are appropriately set, and the basic quantities are consistently defined. The quantity ⌽ is the volume in phase space en- where dE is the change in internal energy, ␦Q is the energy closed by a hypersurface of constant energy E. added to the system due to heating, and ␦W is the work done Equation ͑2͒ is valid for both large and small systems and on the system during an infinitesimal transformation. The coincides with the Boltzmann formula for large systems. first law is the energy conservation law applied to a system Hence, the derivation of Eq. ͑2͒ provides the missing link for in which there is an exchange of energy by both work and Eq. ͑1͒. The basic argument underlying the derivation of Eq. heating. It is essential to realize that ␦Q and ␦W are inexact ͑2͒ can be traced to as early as the second half of the 19th differentials, whereas dE is an exact differential. The internal century in the work of Helmholtz and Boltzmann.3,4 energy E is a state variable, a quantity that characterizes the The purpose of this article is to provide an accessible and thermodynamic equilibrium state of the system. In contrast, contemporary presentation of the original argument of Helm- W and Q characterize thermodynamic energy transfers only holtz and Boltzmann3,4 and of its recent developments.2 We and are not properties of the state of the system.9 derive Boltzmann’s principle from classical mechanics with A differential is exact if and only if the integral along a one simple guiding principle—the heat theorem ͑see state- path in the system’s state space depends only on the end ment 1͒ and one central assumption, namely, the ergodic hy- points. In contrast, the corresponding integral of an inexact pothesis. differential with the same end points depends on the path In Sec. II we briefly review the basics of thermodynamics. taken. A summary of the formal definition of differential We give concise formulations of the first and second laws of forms ͑more precisely, 1-form͒ and their major properties is thermodynamics and introduce the heat theorem. We con- given in Appendix A.

608 Am. J. Phys. 78 ͑6͒, June 2010 http://aapt.org/ajp © 2010 American Association of Physics Teachers 608 The second law of thermodynamics can be conveniently ϕ(x; V1) (a) ϕ(x; V2) (b) summarized in three statements. Statement 1. The differential ␦Q/T is exact. This statement E2 ϕ is one of the most important statements of thermodynamics. E1 Although ␦Q is not an exact differential, an exact differential is obtained when it is divided by the absolute temperature T. An equivalent statement is that there exists a state function S (c) (d) such that E1,V1 E2,V2

␦Q p = dS. ͑4͒ T The function S is called the thermodynamic entropy of the x−(E1,V1) 0 x+(E1,V1) x−(E2,V2) 0 x+(E2,V2) system. x x Statement 1 can also be stated in an equivalent integral form. The integral of ␦Q/T along a path connecting a state A Fig. 1. Point particle in the U-shaped potential ␸͑x;V͒=mV2x2 /2. ͑a͒ Shape ͑ ͒ ͑ ͒ to a state B in the system’s state space does not depend on of the potential for a V=V1. b Shape of the potential for a V=V2. c Phase ␸͑ ͒ ͑ ͒ 11 space orbit corresponding to the potential x;V1 at energy E1. d Phase the path, but only on its end points A and B. This statement ␸͑ ͒ ͑ space orbit corresponding to the potential x;V2 at energy E2. The two means that there exists a state function S the thermodynamic quantities E,V, uniquely determine one “state,” that is, one closed orbit in entropy͒ such that phase space. B ␦Q ͵ = S͑B͒ − S͑A͒. ͑5͒ T A A crucial point in statements 2 and 3 is that they are for From Eq. ͑3͒, the heat transfer is ␦Q=dE−␦W. In general, thermally isolated systems. Thermally isolated systems are ␭ those that are not in contact with a thermal bath, by means of work can be performed by changing external parameters i, such as the volume, magnetic field, and the electric field. The which either their temperature or energy can be controlled. ␦ ͚ ␭ Thus, these processes are those in which only the external work W done is given by − iFid i, where Fi denotes the corresponding conjugate forces, pressure, magnetization, and parameter V is varied in a controlled way and the variable E electric polarization, respectively. Therefore, the first law can is uncontrolled. In statement 2 a quasi-static process is one in be written as which the change in the parameter V is so slow that at any instant of time the system is almost in equilibrium. In state- ␦ ␭ ͑ ͒ Q = dE + ͚ Fid i. 6 ment 3, this requirement is relaxed. i

Without loss of generality, we will restrict ourselves in the III. ONE-DIMENSIONAL MECHANICAL MODELS following to only one external parameter V with conjugate OF THERMODYNAMICS force P:12 In this section we construct a one-dimensional classical ␦Q = dE + PdV. ͑7͒ mechanical analog of Clausius thermodynamic entropy. This In this case, statement 1 can be expressed as construction dates back to Helmholtz2,3,10 and is based on the ͑ ͒ ͑ ͒/ ͑ ͒ heat theorem 8 . dE + PdV T = exact differential = dS. 8 Consider a particle of mass m and coordinate x moving in Equation ͑8͒ is known in literature as the heat theorem.10 a U-shaped potential ␸͑x͒, as illustrated in Fig. 1. To allow V͒dV, the heat for the possibility of doing work on the particle by means ofץ/Sץ͑+E͒dEץ/Sץ͑=Because by definition dS theorem can be expressed in equivalent terms as there exists an external intervention, we assume that the potential ␸ de- a function S͑E,V͒ such that13 pends on some externally controllable parameter V so that ␸=␸͑x;V͒. The Hamiltonian of this system is S Pץ S 1ץ = , = . ͑9͒ H͑x,p;V͒ = K͑p͒ + ␸͑x;V͒, ͑12͒ V Tץ E Tץ where K͑p͒=p2 /2m is the kinetic energy and p is the mo- Any inexact differential, such as ␦Q=dE+ PdV, does not mentum. Given this mechanical system, we need to specify enjoy the same property: It is impossible to find a function of its internal energy E, “temperature” T, and the “force” P .V= Pץ/Qץ E=1 andץ/Qץ state Q͑E,V͒ such that conjugate to the external parameter V. The following two statements, regarding the function S, For the internal energy, we take the energy E given by the complete Clausius’s form of the second law. Hamiltonian. For a fixed V, the particle’s energy E is a con- Statement 2. For a quasi-static process occurring in a ther- stant of motion. For simplicity, we define the zero of energy mally isolated system, the entropy change between two equi- in such a way that the minimum of the potential is 0, regard- librium states is zero, less of the value of V. ⌬S =0. ͑10͒ Once V and E are specified, the orbit of the particle in phase space is fully determined. We call E and V the sys- Statement 3. For a non-quasi-static process occurring in a tem’s “thermodynamic” state variables.14 The particle moves thermally isolated system, the entropy change between two between the two turning points xϮ͑E,V͒ and forms closed equilibrium states is non-negative, orbits in phase space with a period ␶͑E,V͒͑see Fig. 1͒. ⌬S Ն 0. ͑11͒ Now that the state variables are fixed, we have to define

609 Am. J. Phys., Vol. 78, No. 6, June 2010 M. Campisi and D. H. Kobe 609 the corresponding temperature and conjugate force. In agree- simple to model a real thermodynamic system composed of ment with the common understanding that temperature for as many as 1023 particles. Thus, it is necessary to generalize classical systems is a measure of the kinetic energy of the the Helmholtz theorem to multidimensional systems. particles, we take the temperature to be proportional to the kinetic energy averaged over a period ␶͑ ͒ 2 E,V IV. ERGODICITY AND THE MICROCANONICAL T͑E,V͒ϵ ͵ K͑p͑t;E,V͒͒dt. ͑13͒ ␶͑ ͒ ENSEMBLE kB E,V 0

The Boltzmann constant kB ensures the correct dimensions To generalize our model to more degrees of freedom, we for the temperature T. For the conjugate force, we take the need ergodicity and the microcanonical ensemble. The one- V,15 dimensional example of Sec. III can be used to introduceץ/␸ץ− time average of these important concepts. ␶͑E,V͒ ␸͑x͑t;E,V͒;V͒ The time average of a phase function f͑x,p͒ over the orbitץ 1 P͑E,V͒ϵ− ͵ dt. ͑14͒ ץ ͒ ␶͑ E,V 0 V specified by E and V is ␶ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 1 In Eqs. 13 and 14 x t;E,V and p t;E,V are the solution ͗f͘ ϵ ͵ f͑x͑t͒,p͑t͒͒dt. ͑20͒ t ␶ of Hamilton’s equations of motion with a fixed V and arbi- 0 ͑ ͒ trary initial condition x0 ,p0 such that H x0 ,p0 ;V =E. Having identified the mechanical analogs of internal en- For simplicity, we have dropped the explicit dependence on ␶ ͑ ͒ ͑ ͒ ͗ ͘ / ergy, external parameter, temperature, and conjugate force E, V of , x t , p t , and f t. Because p=mv=mdx dt, the with the quantities E, V, T, and P, respectively, we can now differential dt is ask whether a mechanical analog of the entropy S exists. To dx answer this question, we must find, according to statement 1, dt = m . ͑21͒ as expressed in Eq. ͑9͒, a function S͑E,V͒ such that p͑x͒ S͑E,V͒ P͑E,V͒ Using this differential, we obtainץ S͑E,V͒ 1ץ = , = . ͑15͒ V T͑E,V͒ 2m x+ dxץ E T͑E,V͒ץ ͗f͘ = ͵ f͑x,p͑x͒͒. ͑22͒ t ␶ p͑x͒ That such a function exists is given by the following theo- x− rem. Helmholtz’s theorem. A function S͑E,V͒ satisfying Eq. The factor 2 is due to the particle going from x− to x+ in a half period ␶/2. Now consider the integral ͑15͒ exists and is given by ͑ ͒ x+ E,V ␦͑ 2/ ␸͑ ͒ ͒ ͑ ͒ ͑ ͒ ͵ ͱ ͑ ␸͑ ͒͒ ͑ ͒ ͵ dp p 2m + x;V − E , 23 S E,V = kB log 2 2m E − x;V dx. 16 ͑ ͒ x− E,V where ␦ denotes the Dirac delta function. If we use the ex- The proof of the theorem is given in Appendix B and Ref. pression ␦͑f͑p͒͒=͚ ␦͑p−p ͒/͉fЈ͑p ͉͒, where the p ’s are the ͑ ͒ i i i i 10, pp. 45–46. The entropy S E,V can be rewritten com- zeroes of f͑p͒ and ͐␦͑p−p ͒dp=1, we obtain for the integral, pactly as i ͵ ␦͑ 2/ ␸͑ ͒ ͒ / ͑ ͒ ͑ ͒ ͑ ͒ Ͷ ͑ ͒ p 2m + x;V − E dp =2m p x . 24 S E,V = kB log pdx, 17 Then, Eq. ͑22͒ becomes where p=ͱ2m͑E−␸͑x;V͒͒ is the momentum of the particle when it is located at x: The integral ͛pdx is called the re- 1 ͗f͘ = ͵ dx͵ dp␦͑p2/2m + ␸͑x;V͒ − E͒f͑p,x͒, ͑25͒ duced action.15 It is the area ⌽ in phase space enclosed by t ␶ the orbit of energy E and parameter V, where it is unnecessary to specify the integration limits xϮ ͑ ͒ ⌽͑ ͒ ͑ ͒ S E,V = kB log E,V , 18 because they are implied by the Dirac delta function. The period ␶ is given by where ␶ x+ dx ␶ = ͵ dt =2͵ ⌽͑E V͒ ͵ dpdx ͑ ͒ ͑ ͒ , = . 19 0 x p x H͑x,p;V͒ՅE − Helmholtz’s theorem shows that there exists a mechanical = ͵ dx͵ dp␦͑p2/2m + ␸͑x;V͒ − E͒. ͑26͒ counterpart of the entropy given by the logarithm of the phase space volume enclosed by the curve of constant energy Hence, we arrive at H͑x,p;V͒=E. S ͑ ͒ That there exists a function satisfying Eq. 15 is a re- ͗ ͘ ␳͑ ͒ ͑ ͒ ͑ ͒ markable result that shows there is a consistent one- f t = ͵ dx͵ dp x,p;E,V f p,x , 27 dimensional mechanical model of thermodynamics. Al- though this model suggests the deep connection between where we have introduced the phase space probability den- classical mechanics and thermodynamic entropy, it is too sity function

610 Am. J. Phys., Vol. 78, No. 6, June 2010 M. Campisi and D. H. Kobe 610 S͑E,V͒ P ͑E,V͒ץ S͑E,V͒ 1ץ 1 ␳͑x,p;E,V͒ = ␦͑p2/2m + ␸͑x;V͒ − E͒. ͑28͒ N = , N = N ? ͑36͒ ͒ ͑ ץ ͒ ͑ ץ ͒ ␶͑ E,V E TN E,V V TN E,V From Eq. ͑26͒, it is clear that ␳ is properly normalized. The The following theorem gives the answer. ͑ ͒ function ␳͑x,p;E,V͒ is the microcanonical distribution. Generalized Helmholtz theorem. A function SN E,V sat- Equation ͑27͒ shows that the time average of a phase space isfying Eq. ͑36͒ exists and is given by function f͑x,p͒ over a period is equal to its microcanonical ͑ ͒ ⌽ ͑ ͒ ͑ ͒ SN E,V = kB log N E,V , 37 average. This property is called ergodicity. All one- dimensional systems with a U-shaped potential are ergodic. where

⌽ ͑ ͒ϵ ͑ ͒ N E,V ͵ ¯ ͵ dqdp. 38 ͑ ͒Յ HN q,p E V. MULTIPARTICLE MECHANICAL MODELS OF THERMODYNAMICS The proof, which is based on the equipartition theorem, is given in Appendix C and Ref. 2. Unlike the temperature and We now extend the treatment for one degree of freedom to conjugate force, SN is not a time average of some phase systems of N-particles in three dimensions with 3N degrees function f͑q,p͒. of freedom. The Hamiltonian for an interacting system of This theorem shows that ergodic systems constitute ideal N-particles of mass m is mechanical models of thermodynamics. The state variables E and V define their thermodynamic state. In addition, their ͑ ͒ ͑ ͒ ␸ ͑ ͒ ͑ ͒ HN q,p;V = KN p + N q;V , 29 temperature and conjugate force can be defined as functions ͑ ͑ ͒ ͚3N 2 / ␸ of the state variables. Surprisingly, the heat differential dE where KN p = i=1pi 2m is the kinetic energy and N is the ͒/ ͕ ͖3N + PNdV TN is exact, allowing for a consistent and logical potential energy. The coordinates q= qi i=1 and their conju- ͕ ͖3N definition of entropy SN. gate canonical momenta p= pi i=1 are 3N-dimensional vec- tors. In analogy with one-dimensional systems with a U-shaped VI. ADIABATIC INVARIANCE potential, we define the microcanonical probability distribu- According to the generalized Helmholtz theorem, SN is tion as consistent with the first law of thermodynamics and state- 1 ment 1 of the second law of thermodynamics. Is this con- ␳ ͑q,p;E,V͒ = ␦͑E − H ͑q,p;V͒͒, ͑30͒ N ⍀ ͑ ͒ N struction also consistent with statements 2 and 3 of the sec- N E,V ond law of thermodynamics? ⍀ ͑ ͒ For S to be consistent with statement 2, it is necessary for where N E,V is the normalization N it to satisfy the following condition. If the parameter V is varied very slowly in time from a V =V͑t ͒ to V =V͑t ͒, the ⍀ ͑ ͒ ͵ ͵ ␦͑ ͑ ͒͒ ͑ ͒ 0 0 f f N E,V = E − HN q,p;V dqdp. 31 ¯ corresponding change in the entropy SN is zero. The time of variation must be much slower than any time scale of the The microcanonical average ͗f͘␮ of a phase function f͑q,p͒ system’s dynamics. By allowing for a time dependence of V, is defined as the Hamiltonian function of the system is also time dependent, and hence energy is not conserved. Consider the initial ͗ ͘ ϵ ͵ ͵ ␳ ͑ ͒ ͑ ͒ ͑ ͒ system at t=t0 with phase space point q0 ,p0. The system f ␮ q,p;E,V f q,p dqdp. 32 ¯ N then evolves under the time-dependent Hamiltonian ͑ ͑ ͒͒ ͑ ͒ ␸ ͑ ͑ ͒͒ ͑ ͒ Continuing the analogy with one-dimensional systems, we HN q,p;V t = KN p + N q;V t 39 make the following crucial assumption. to a new phase space point q ͑q ,p ͒,p ͑q ,p ͒, where we Ergodic hypothesis. For a given E and V, the time average f 0 0 f 0 0 ͗ ͘ ͑ ͒ have made explicit the dependence on the initial condition of f t of any function f q,p is uniquely determined and is the evolved phase space point. The energy at time tf is Ef equal to its microcanonical average ͗f͘␮, ͑ ͑ ͒ ͑ ͒ ͑ ͒͒ 16 =HN qf q0 ,p0 ,pf q0 ,p0 ;V tf . It is known that for er- ͗ ͘ ͗ ͘ ͑ ͒ godic systems the energy Ef reached at the end of a very f t = f ␮. 33 slow change depends only on the initial energy E0 ͑ ͑ ͒͒ In analogy with Eqs. ͑13͒ and ͑14͒, we define the tempera- =HN q0 ,p0 ;V t0 . The final energy Ef is determined by ture as solving the equation ⌽ ͑ ͒ ⌽ ͑ ͒ ͑ ͒ 2 N E0,V0 = N Ef,Vf . 40 T ͑E,V͒ϵ ͗K ͘ , ͑34͒ N N t ⌽ ͑ ͒ 3NkB Thus, the quantity N E,V does not change when V is var- which is twice the average kinetic energy per degree of free- ied infinitely slowly in time. In classical mechanics this property is called adiabatic invariance. Because the entropy dom. The conjugate force is defined as ͑ ͒ ⌽ ͑ ͒ ⌽ ͑ ͒ is SN E,V =kB log N E,V , and N E,V is an adiabatic .␸ invariant, it is evident that S is also an adiabatic invariantץ P ͑E,V͒ϵ− ͳ N ʹ . ͑35͒ N Therefore, the entropy does not change if V is varied very ץ N V t slowly in time and thus SN complies with statement 2. A ⌽ We ask, in agreement with statement 1 as expressed in Eq. proof of adiabatic invariance of N is given in Appendix D ͑ ͒ ͑ ͒ 9 , does there exist a function SN E,V such that and Ref. 16, pp. 27–30.

611 Am. J. Phys., Vol. 78, No. 6, June 2010 M. Campisi and D. H. Kobe 611 It is also possible to prove that, in an averaged sense, SN compliance with the equipartition theorem and the fact that it complies with statement 3 as well.5,6 In this case the average is a positive and increasing function of the energy.27,28 The change in entropy must be considered because the final en- entropy in Eq. ͑37͒ also appears in Gibbs seminal book.29 ergy is not uniquely determined by the initial energy for fast However, its connection with the heat theorem has not been transformations. Depending on the initial conditions, differ- recognized previously. ent final energies can be obtained and hence different final The most crucial point in the derivation of the Boltzmann entropies. principle is the introduction of the ergodic hypothesis. Al- though this hypothesis is generally believed to hold for real VII. THE BOLTZMANN PRINCIPLE macroscopic systems, its proof is a formidable challenge which has been achieved only in few special cases.30 A proof For a system composed of a very large number N of par- that a gas of elastically colliding hard spheres is ergodic was ticles that interact via short-range forces, the phase space announced in 1963 by Sinai.31 However, the full proof was ⌽ ͑ ͒ volume N E approaches an exponential dependence on E, not published and the problem is still open ͑see Ref. 32 for a ͔͒ ͑ ͓ ץ/ ⌽ץ ⌽ ͑ ͒ϰ E ⍀ N E e . Because N = N E see Eq. 31 , we also more detailed discussion͒. Nevertheless, the ergodicity of ⌽ ϰ⍀ ͑ ͒ have N N see Ref. 17,p.148. hard spheres systems is plausible as indicated by recent nu- ⍀ ͑ ͒ The quantity N in Eq. 31 is the measure of the shell of merical simulations ͑see Sec. IV of Ref. 33͒. ͑ ͒ constant energy HN q,p;V =E. As such, it is proportional to In regard to these difficulties, we note that the present the number W of microstates compatible with a given energy derivation of the Boltzmann principle does not use the fact E. According to semiclassical theory each microstate occu- that the average of any arbitrary phase function is equal to its pies a volume h3N of phase space, where h is Planck’s microcanonical average, as required by the ergodic hypoth- constant.18 By introducing an arbitrary energy scale ⌬E, the esis. It only uses the fact that the time average of K and ͑ ץ/␸ץ W W ⍀ ⌬ / 3N͒ number of microstates is given by = N E h . Thus, − V is equal to their microcanonical averages see the for very large N we obtain the proportionalities, proof of the generalized Helmholtz theorem in Appendix C͒. Thus, the ergodic hypothesis can be replaced by the follow- ⌽ ϰ⍀ ϰ W ͑ ӷ ͒ ͑ ͒ N N N 1 . 41 ing hypothesis. We take the logarithm and obtain Weak ergodic hypothesis. For given E and V, the time V͘ are uniquely determined and areץ/␸ץ͗ averages ͗K͘ and Ӎ W ͑ ӷ ͒ ͑ ͒ t t SN kB ln = SB N 1 , 42 equal to their microcanonical averages, that is, ͑ ͒ ͗ ͘ ͗ ͘ ͑ ͒ except for an irrelevant constant. Equation 42 shows that K t = K ␮, 44 for large ergodic systems composed of many particles inter- ͒ ͑ ͘ ץ/␸ץ͗ ͘ ץ/␸ץ͗ -acting via short range forces, the differential of the Boltz V t = V ␮. 45 mann entropy is equal to the differential ␦Q/T. Hence, it can be identified with the Clausius entropy, which gives a proof In summary, ergodicity is too stringent a condition and the of the Boltzmann principle. weaker ergodic condition for only the temperature and pressure is sufficient to obtain the Boltzmann principle.34 VIII. CONCLUSIONS Given an ergodic system, it is possible to specify its thermodynamic state by means of the total energy E and the ACKNOWLEDGMENTS external parameter V. Given the state E,V, we can unam- biguously define the quantities T ͑E,V͒ and P ͑E,V͒. Once The authors wish to thank the Texas Section of the Ameri- N N can Physical Society for the Robert S. Hyer Recognition for these are identified with the system temperature and conju- Exceptional Research award presented at its Fall 2008 meet- gate force, we can ask whether, as prescribed by the heat ing. They also thank Cosimo Gorini for reading the manu- theorem, the combination script. They are grateful to Professor Randall B. Shirts for dE + P dV providing the translation of Ref. 25 and for his comments. N ͑43͒ T Valuable remarks from anonymous referees are also grate- N fully acknowledged. is an exact differential. Surprisingly, the answer is positive, ͑ ͒ meaning that there exists a function SN E,V that can be identified with the thermodynamic entropy of the system. The generalized Helmholtz theorem says that this function is APPENDIX A: DIFFERENTIAL FORMS: BRIEF ⌽ ͑ ͒ REVIEW OF DEFINITIONS AND MAIN RESULTS given by the logarithm of the volume N E,V of phase space enclosed by the hypersurface of energy H͑q,p;V͒=E. A differential form ␻ ͑more precisely, a 1-form͒ in a con- For macroscopic systems, this entropy coincides with the nected subset A of R2 is formally written as Boltzmann entropy, thus revealing the rationale of the Bolt- ␻ M͑ ͒ N͑ ͒ ͑ ͒ zmann principle. = x1,x2 dx1 + x1,x2 dx2, A1 ͑ ͒ The entropy in Eq. 37 is sometimes referred to as the M͑ ͒ N͑ ͒ A Hertz entropy.19,20 Hertz21,22 derived it from the requirement where x1 ,x2 , x1 ,x2 are two functions on and ͑ ͒ 2 35 of adiabatic invariance ͑see also Refs. 16, 23, and 24͒, x1 ,x2 are the coordinates in R . ␺ ͓ ͔→A whereas we have derived it from the heat theorem. The fun- Given a curve : s0 ,s1 , ͑ ͒ damental character of the entropy in Eq. 37 is also recog- ␺͑s͒ = ͑␺ ͑s͒,␺ ͑s͒͒, ͑A2͒ nized in Ref. 25, where its property of being a canonical 1 2 invariant is emphasized, and in Ref. 26, which highlights its the integral of ␻ along the curve ␺ is defined as

612 Am. J. Phys., Vol. 78, No. 6, June 2010 M. Campisi and D. H. Kobe 612 s1 ⌽ ͵ ␻ ͵ ͓M͑␺͑ ͒͒␺Ј͑ ͒ N͑␺͑ ͒͒␺Ј͑ ͔͒ ͑ ͒ ͗ ͘ ͑ ͒ = t 1 t + t 2 t dt, A3 2 K t = , B3 ␺ ␶ s0 ␺Ј ␺ from which we obtain where 1,2 are the derivatives of 1,2. S kץ A differential form ␻ is said to be exact if there exist a = B ͑B4͒ ͘ ͗ ץ function G:A→R such that E 2 K t ␻ = dG, ͑A4͒ using Eq. ͑B2͒. Similarly, we also obtain ␸ץ S kץ ,that is =− B ͳ ʹ . ͑B5͒ ץ ͘ ͗ ץ G V 2 K t V tץ Gץ ͑x ,x ͒ = M͑x ,x ͒, ͑x ,x ͒ = N͑x ,x ͒, ͑A5͒ ͒ ͑ ͒ ͑ 2 1 2 1 ץ 2 1 2 1 ץ x1 x2 From Eqs. 13 and 14 , we obtain S 1ץ where G is called a primitive for the differential form. = , ͑B6͒ E Tץ Let ⌺͑A,B͒ be the set of all curves connecting the point ϵ͑ ͒ ϵ͑ ͒ A A a1 ,a2 to the point B b1 ,b2 in . A differential form S Pץ is exact if and only if for any two points A and B in A and = . ͑B7͒ V Tץ curves ␺ and ␾ in ⌺͑A,B͒ it satisﬁes

͵ ␻ = ͵ ␻. ͑A6͒ ␺ ␾ APPENDIX C: PROOF OF THE GENERALIZED HELMHOLTZ THEOREM The integral of an exact differential form along any curve ␺ ␺ The proof of generalized Helmholtz theorem makes use of connecting A to B does not depend on the curve , but only ͑ ͒ on the ending points, and is given by the multidimensional version of Eq. B1 , ⌽ץ ⍀ = N , ͑C1͒ Eץ ␻ = ͵ dG = G͑B͒ − G͑A͒. ͑A7͒ N ͵ ␺ ␺ which can be proved, in a way similar to Eq. ͑B1͒,byex- ⌽ ͐␪͑ ͑ ͒͒ The following statement also holds: A differential form is pressing N as E−H q,p;V dqdp and using the relation exact if and only if its integral along any closed curve is ␦͑x͒=d␪͑x͒/dx. The equipartition theorem,17 zero. If the functions M and N are of class C1 ͑that is, they ͗ ͘ ⌽ are differentiable͒, then a necessary condition for the form ␻ 2 K ␮ N = ͑C2͒ ⍀ to be exact is that 3N N N is the generalization of Eq. ͑B3͒ to many dimensions. We useץ Mץ = . ͑A8͒ Eqs. ͑C2͒ and ͑C1͒ with Eq. ͑37͒ and obtain ץ ץ x2 x1 S 3Nkץ In this case the differential form ␻ is said to be closed. N = B . ͑C3͒ ͘ ͗ ץ E 2 KN ␮ In a similar way, we obtain ␸ץ S 3Nkץ APPENDIX B: PROOF OF HELMHOLTZ’S N =− B ͳ N ʹ . ͑C4͒ ץ ͘ ͗ ץ THEOREM V 2 KN ␮ V ␮ For a one-dimensional system conﬁned in a U-shaped po- By using Eqs. ͑34͒ and ͑35͒ with the ergodic hypothesis, we tential the period ␶ of the orbit is equal to the derivative with ﬁnally arrive at respect to energy of the phase space area ⌽ enclosed by the S 1ץ 15 orbit N = , ͑C5͒ E Tץ ⌽ Nץ ␶ = . ͑B1͒ S Pץ Eץ N = N . ͑C6͒ V Tץ A simple way to prove this relation is by expressing the area N as ⌽͑E,V͒=͐␪͑E−H͑x,p;V͒͒dpdx, where ␪͑x͒ is the Heavi- side step function ͓␪͑x͒=1 if xՆ0, ␪͑x͒=0 if xϽ0͔.Ifwe APPENDIX D: PROOF OF ADIABATIC ⌽ take the derivative with respect to E and use the relation INVARIANCE OF N ␦͑x͒=d␪͑x͒/dx, we obtain ␶ ͓see Eq. ͑26͔͒. By using Eqs. ⌽ ͑ ͒ ͑B1͒ and ͑18͒, we have To prove that N in Eq. 38 is an adiabatic invariant, we use the time-dependent Hamiltonian ␶ ץ S ͑ ͑ ͒͒ ͑ ͒ ␸͑ ͑ ͒͒ ͑ ͒ = k . ͑B2͒ HN q,p;V t = K p + q;V t . D1 E B ⌽ץ We take the total time derivative of the Hamiltonian ͑ ͒ ͑ ͑ ͒͒ ͑ ͒ From Eq. 22 , we obtain the relation HN q,p;V t in Eq. D1 ,

613 Am. J. Phys., Vol. 78, No. 6, June 2010 M. Campisi and D. H. Kobe 613 ,A. Khinchin, Mathematical Foundations of Statistical Mechanics ͑Dover 1 ͒ ͑ ץ ͒ ͑ dHN q,p;V HN q,p;V dV = , ͑D2͒ New York, 1949͒, p. 142. V dt 2 M. Campisi, “On the mechanical foundations of thermodynamics: Theץ dt generalized Helmholtz theorem,” Stud. Hist. Philos. Mod. Phys. 36, 275– where the terms involving q˙ and p˙ cancel by Hamilton’s ͑ ͒ 36 290 2005 . equations. The derivative dV/dt changes slowly in time, 3 H. Helmholtz, “Principien der statik monocyklischer Systeme,” ץ/ ץ / but dHN dt and HN V can change rapidly because of their Borchardt-Crelles Journal für die reine und angewandte Mathematik 97, dependence on q͑t͒ and p͑t͒. To eliminate the fast variables 111–140 ͑1984͒; also in Wissenshafltliche Abhandlungen, edited by G. ͑ ͒ q,p, we take the average of Eq. ͑D2͒ with respect to the Wiedemann Johann Ambrosious Barth, Leipzig, 1985 , Vol. 3, pp. 142– 162, 179–202. microcanonical ensemble, which gives 4 L. Boltzmann, “Über die Eigenschaften monocyklischer und anderer .damit verwandter Systeme,” Crelles Journal 98, 68–94 ͑1884͒; also L ץ dHN HN dV ͳ ʹ ͳ ʹ ͑ ͒ Wissenschaftliche Abhandlungen = . D3 Boltzmann, in , edited by F. Hasenohrl V ␮ dt ͑J. A. Barth, Leipzig, 1909͒, Vol. 3, pp. 122–152, reissued Chelsea, Newץ dt ␮ 36 York, 1969. By Liouville’s theorem the average on the left-hand side of 5 ͑ ͒ M. Campisi, “Statistical mechanical proof of the second law of thermo- Eq. D3 is dynamics based on volume entropy,” Stud. Hist. Philos. Mod. Phys. 39, ͑ ͒ dH dE 181–194 2008 . ͳ N ʹ = . ͑D4͒ 6 M. Campisi, “Increase of Boltzmann entropy in a quantum forced har- dt ␮ dt monic oscillator,” Phys. Rev. E 78, 051123-1–10 ͑2008͒. 7 ͑ ͒ J. Uffink, “Bluff your way in the second law of thermodynamics,” Stud. The microcanonical average in Eq. 33 on the right-hand Hist. Philos. Mod. Phys. 32 ͑3͒, 305–394 ͑2001͒. side of Eq. ͑D3͒ is 8 H. B. Callen, Thermodynamics ͑Wiley, New York, 1960͒. 9 D. Chandler, Introduction to Modern Statistical Mechanics ͑Oxford U. P., H ͑q,p,V͒ץ dH ͳ N ʹ ͵ ͵ ␳ ͑ ͒ N Oxford, 1987͒. = ¯ N q,p,E,V dqdp 10 ͑ ,V G. Gallavotti, Statistical Mechanics: A Short Treatise Springer-Verlagץ dV ␮ Berlin, 1999͒. ⌽ 11 A path in the space of state variables corresponds to a transformation thatץ 1 =− N , ͑D5͒ V leads the system from A to B through a sequence of almost equilibriumץ ⍀ N states. Consequently, the change ␦Q is understood to be a quasi-static ⌽ ⍀ ͑ ͒ ͑ ͒ change. where N and N are given in Eqs. 31 and 38 , respec- 12 tively. If we substitute Eqs. ͑D4͒ and ͑D5͒ into Eq. ͑D3͒ and For the common case of work due to expansion or compression the ex- .ternal parameter V is the volume and its conjugate force P is the pressure ץ/ ⌽ץ ⍀ use N = N E, we obtain In general, V and P stand for any conjugate pair of “displacement” and -⌽ force depending on the nature of the work done on the system ͑for exץ ⌽ץ ⌽ d N N dE N dV ϵ + =0, ͑D6͒ ample, magnetic field and magnetization͒. 13 ץ ץ dt E dt V dt In other words, the field Fជ ϵ͑1/T, P/T͒ is a conservative vector field. ⌽ That is, there exists a potential function S͑E,V͒ such that Fជ =ٌជ S, where which shows that N is constant and therefore an adiabatic ជ invariant. ٌ is the gradient operator in the space ͑E,V͒. For this reason, the entropy can be understood as a thermodynamic potential. 14 The thermodynamic state variables ͑E,V͒ should not be confused with APPENDIX E: SUGGESTED PROBLEM the mechanical state variables ͑p,x͒. 15 L. Landau and E. Lifshitz, Mechanics ͑Pergamon, Oxford, 1960͒. Consider the following Hamiltonian of a one-dimensional 16 V. L. Berdichevsky, Thermodynamics of Chaos and Order ͑Addison- harmonic oscillator with angular frequency V ͑see Fig. 1͒: Wesley Longman, Essex, 1997͒. 17 ͑ ͒ 2 2 2 K. Huang, Statistical Mechanics, 2nd ed. Wiley, New York, 1987 . p mV x 18 H͑x,p;V͒ = + . ͑E1͒ L. Landau and E. Lifshitz, Statistical Physics, 2nd ed. ͑Pergamon, Ox- 2m 2 ford, 1969͒. 19 S. Hilbert and J. Dunkel, “Nonanalytic microscopic phase transitions and ͑ ͒ ⌽͑ ͒ temperature oscillations in the microcanonical ensemble: An exactly solv- a Calculate the area E,V enclosed by the trajectory of able 1d-model for evaporation,” Phys. Rev. E 74, 011120-1–7 ͑2006͒. energy E and angular frequency V. Use Eq. ͑B1͒ and 20 A. Adib, “Does the Boltzmann principle need a dynamical correction?,” check that the period of the orbit is, as expected, given J. Stat. Phys. 117, 581–597 ͑2004͒. by ␶͑E,V͒=2␲/V. 21 P. Hertz, “Über die mechanischen Grundlagen der Thermodynamik,” ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ Ann. Phys. 33, 225–274 ͑1910͒. b Use Eqs. 13 and 14 to show that kBT E,V =E and 22 ͑ ͒ / P. Hertz, “Über die mechanischen grundlagen der thermodynamik,” Ann. P E,V =−E V. Phys. 33, 537–552 ͑1910͒. ͑c͒ Show that the differential form dE+ PdV, with P͑E,V͒ 23 A. Münster, Statistical Thermodynamics ͑Springer-Verlag, Berlin, 1969͒, as in part ͑b͒ is not exact. ͓Hint: Show that Eq. ͑A8͒ is Vol. 1. 24 not satisfied.͔ Show that the integral of dE+ PdV over H. H. Rugh, “Microthermodynamic formalism,” Phys. Rev. E 64 ͑5͒, ͑ ͒ ͑ ͒ 055101-1–4 ͑2001͒. the rectangular path with corners E0 ,V0 , E0 ,V1 , 25 ͑ ͒ ͑ ͒ A. Schlüter, “Zur Statistik klassischer Gesamtheiten,” Z. Naturforsch. A E1 ,V1 , E1 ,V0 , and E0 E1, V0 V1, is not zero. 3A, 350–360 ͑1948͒. An English translation is available at ͑d͒ Consider the differential form ␻=͑1/T͒dE+͑P/T͒dV, ͗people.chem.byu.edu/rbshirts/research/schluter1948translation.doc͘. 26 with P͑E,V͒ and T͑E,V͒ as in part ͑b͒. Find a primi- E. M. Pearson, T. Halicioglu, and W. A. Tiller, “Laplace-transform tech- ͑ ͒ ␻ nique for deriving thermodynamic equations from the classical microca- tive function S E,V for . Show that, apart from an nonical ensemble,” Phys. Rev. A 32 ͑5͒, 3030–3039 ͑1985͒. additive constant, it is S͑E,V͒=log ⌽͑E,V͒, as dictated 27 P. Talkner, P. Hänggi, and M. Morillo “Microcanonical quantum fluctua- by Theorem 1. Check that Eq. ͑A8͒ is satisfied. tion theorems,” Phys. Rev. E 77, 051131-1–6 ͑2008͒. ⌽ ⍀Ն ץ ⌽Ն 28 The conditions 0, E = 0 ensure that the temperature derived ⌽͒−1 ⌽/⍀ ץ͑ from the Hertz entropy, that is, T= E log = is positive definite. ͒ a Electronic mail: [email protected] 29 J. Gibbs, Elementary Principles in Statistical Mechanics ͑Yale U. P., New ͒ b Electronic mail: [email protected] Haven, 1902͒, reprinted by Dover, New York, 1960.

614 Am. J. Phys., Vol. 78, No. 6, June 2010 M. Campisi and D. H. Kobe 614 30 J. L. Lebowitz and O. Penrose, “Modern ergodic theory,” Phys. Today 26 condition is a one-dimensional particle in a symmetric double well po- ͑ ͒ ͑ ͒ 2 , 23–29 1973 . tential. For energies below a certain critical value Ec, the curve of con- 31 ␥ ␥ Ya. G. Sinai, “On the foundation of the ergodic hypothesis for a dynami- stant energy splits into two disjoint curves l and r, with the phase space ͑ cal system of statistical mechanics,” Sov. Math. Dokl. 4, 1818–1822 trajectory covering only one of them. For energy above Ec there is only ͑ ͒ ͒ ␥ ␥ 1963 . one trajectory and ergodicity holds. Because l and r are mirror images 32 J. Uffink, “Compendium of the foundations of classical statistical phys- of each other, the time averages of even functions of x,suchasK and .V, are not affected by which of the two curves the motion followsץ/␸ץ− ics,” in Philosophy of Physics, edited by J. Butterfield and J. Earman ͑Elsevier, Amsterdam, 2007͒͗philsci-archive.pitt.edu/archive/ Thus, such averages can be calculated as microcanonical averages over ͘ ␥ ഫ␥ 00002691/ . the curve l r. 33 35 ͑ ͒ ͑ ͒ M. Campisi, P. Talkner, and P. Hänggi, “Finite bath fluctuation theorem,” In the main text the role of x1 ,x2 is played by the state variables E,V . Phys. Rev. E 80, 031145-1–9 ͑2009͒. 36 H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. 34 The simplest example of a nonergodic system that satisfies this weaker ͑Addison-Wesley, San Francisco, 2002͒, pp. 337, 419–421.

Dribble Glasses. More properly, these two glasses in the collection of Union College in Schenectady, New York, are examples of Tantalus cup or the automatic siphon. When the water level reaches the bend in the internal glass tube, the siphon starts and the glass empties itself of water. ͑Photograph and Notes by Thomas B. Greenslade, Jr., Kenyon College͒

615 Am. J. Phys., Vol. 78, No. 6, June 2010 M. Campisi and D. H. Kobe 615