Entropy, Probability and Dynamics

ENTROPY, PROBABILITY AND DYNAMICS E. G. D. Cohen The Rockefeller University New York, NY 10021 Abstract Boltzmann’s struggle with a derivation of the Second Law of Ther- modynamics is sketched. So is his first derivation of the connection be- tween entropy and probability in 1877. Planck’s derivation and quan- tum mechanical modifications of Boltzmann’s connection between en- tropy probability are given next. Then Einstein’s objections to a purely probabilistic rather than a dynamical interpretation of entropy are discussed. Finally, the dynamical basis of the Sinai-Ruelle-Bowen distribution for very chaotic systems is sketched and appears to be an example of Einstein’s dynamical interpretation of entropy. I. Introduction Boltzmann had a strong disposition for mechanics and his first papers were all devoted to purely mechanical derivations of the Second Law of Ther- modynamics. The most important one was that based on the 1872 Boltzmann equation, i.e. on the dynamics of binary collisions, where he says at the end: “One has therefore rigorously proved that, whatever the distribution of the kinetic energy at the initial time might have been, it will, after a very long time, always necessarily approach that found by Maxwell”[1]. He seems to have overlooked entirely that the Stoszzahl Ansatz, i.e. the assumption of molecular chaos used in his equation, was a statistical assump- tion which had no dynamical basis. It is therefore, in my opinion, ironic that perhaps his most famous achieve- ment may well have been the relation of 1877 between entropy and probabil- ity, which was devoid of any dynamical feature[2]. 1 arXiv:0807.1268v2 [physics.hist-ph] 10 Jul 2008 Whenever later dynamical results were obtained, e.g. by Helmholtz, when he introduced his monocycles in 1884[3], Boltzmann immediately jumped at it and extended it to what we now call the origins of ergodic theory, i.e. a dynamical theory[4]. Also his summarizing lectures in the two volume “Lec- tures on Gas Theory”[5] mostly discuss dynamical approaches and reference to the entropy-probability relation can mainly be found on a few pages of the first book in connection with the statistical interpretation of his H-function (ref.5,I,p.38). However, in this lecture I do want to concentrate on his work on entropy and probability first and end with a revival of the dynamical approach as pro- posed by Einstein and as later used, in my view, in the dynamical approach to phase space probabilities in the Sinai-Ruelle-Bowen (SRB) distribution. The sudden switch which Boltzmann made from a purely dynamical to a purely probabilistic approach, might well have been due to the critical at- tacks of many of his colleagues on the Stoszzahl Ansatz, as exemplified by Loschmidt’s Reversibility Paradox[6a] and Zermelo’s Recurrence Paradox[6b]. II. Boltzmann’s original derivation of S ∼logW This was done in a paper of 1887[2]: “On the relation between the Second Law of the Mechanical Theory of Heat and Probability Theory with respect to the laws of thermal equilibrium.” I will sketch first the simplified procedure Boltzmann follows in Chapter I of this paper. The crucial statement here is: “For an ideal gas in thermal equilibrium the probability of the number of “complexions” of the system is a maximum.” Boltzmann introduced the notion of “complexions” as follows: a) Assume discrete kinetic energy values of each molecule, which are repre- sented in an arithmetic series: ε, 2ε, 3ε, ..., pε, where each molecule can only have a finite number, p, of kinetic energies ε. b) Before each binary collision the total kinetic energy of the two colliding molecules is always contained in the above series and “by whatever cause” the same is true after the collision. He says: “There is no real mechanical system [to which this collision assumption is applicable], but the so-defined problem is mathematically much easier to deal with and [in addition] it goes over in the problem we want to solve, when the kinetic energies of the molecules become continuous and p →∞”. 2 Assume that the possible kinetic energies of the N molecules are dis- tributed in all possible ways at constant total kinetic energy E. Then each such distribution of the total kinetic energy over the molecules is called a complexion. What is the number P of complexions, where wj molecules possess a kinetic energy jε (j = 1, ...., N)? This number P [which Boltz- mann calls “the permutability” or “thermodynamic probability”] indicates how many complexions correspond to a given molecular distribution or state of the system. A distribution can be represented by writing down first as many j’s as there are molecules with a kinetic energy jε (wj) etc. Obviously P = N!/Πp j=1wj!. The most probable distribution is that for which the {wj} are such that P = max or Πp j=1wj! or also log Πp j=1wj! a minimum. With the constraints Pp j=1 wj = N and Pp j=1(jε)wj = E and Stirling’s approximation, Boltzmann finds then for the probability that the kinetic en

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