Abstract
The H-theorem is an extension of the Second Law to a time-sequence of states that need not be equilibrium ones. In this paper we review and we rigourously establish the connection with macroscopic autonomy.
If for a Hamiltonian dynamics for many particles, the macrostate evolves autonomously, then its entropy is non-decreasing as a consequence of Liouville's theorem. That observation, made since long, is here rigorously analyzed with special care to reconcile the application of Liouville's theorem (for a finite number of particles) with the condition of autonomous macroscopic evolution (sharp only in the limit of infinite scale separation); and to evaluate the presumed necessity of a semigroup property for the macroscopic evolution.
Similar content being viewed by others
References
W. De Roeck, T. Jacobs, C. Maes, and K. Netočný, An Extension of the Kac ring model. J. Phys. A: Math Gen. 36:11547–11559 (2003).
E. T. Jaynes, The second law as physical fact and as human inference (1990) (unpublished), download from http://www.etjaynescenter.org/bibliography.shtml
E. T. Jaynes, The evolution of Carnot's principle. In: G. J. Erickson, and C. R. Smith (eds.), Maximum-Entropy and Bayesian Methods in Science and Engineering, 1, Kluwer, Dordrecht, pp. 267–281 (1988).
E. T. Jaynes, Gibbs vs Boltzmann entropies. In: Papers on Probability, Statistics and Statistical Physics, Reidel, Dordrecht (1983). Originally in Am. J. Phys. 33:391 (1965).
A. J. M. Garrett, Irreversibility, probability and entropy, P. Grassberger and J.-P. Nadal (eds.), From Statistical Physics to Statistical Inference and Back, pp. 47–75, Kluwer (1994).
P. L. Garrido, S. Goldstein, and J. L. Lebowitz, The Boltzmann Entropy for Dense Fluids Not in Local Equilibrium. Physical Review Letters 92:050602 (2003).
S. Goldstein and J. L. Lebowitz, On the (Boltzmann) Entropy of Nonequilibrium Systems. Physica D 193:53–66 (2004). cond-mat/0304251 and MPArchive 03–167.
C. Maes, Fluctuation relations and positivity of the entropy production in irreversible dynamical systems. Nonlinearity 17:1305–1316 (2004).
C. Maes and K. Netočný, Time-reversal and Entropy. J. Stat. Phys. 110:269–310 (2003).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Roeck, W.D., Maes, C. & Netočný, K. H-Theorems from Macroscopic Autonomous Equations. J Stat Phys 123, 571–584 (2006). https://doi.org/10.1007/s10955-006-9079-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-006-9079-x