Abstract
The maximum entropy formalism developed by Jaynes determines the relevant ensemble in nonequilibrium statistical mechanics by maximising the entropy functional subject to the constraints imposed by the available information. We present an alternative derivation of the relevant ensemble based on the Kullback–Leibler divergence from equilibrium. If the equilibrium ensemble is already known, then calculation of the relevant ensemble is considerably simplified. The constraints must be chosen with care in order to avoid contradictions between the two alternative derivations. The relative entropy functional measures how much a distribution departs from equilibrium. Therefore, it provides a distinct approach to the calculation of statistical ensembles that might be applicable to situations in which the formalism presented by Jaynes performs poorly (such as non-ergodic dynamical systems).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957)
Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948). 623–656
Jaynes, E.T.: Information theory and statistical mechanics. In: Statistical Physics. W. A. Benjamin Inc, New York (1963)
Kawasaki, K., Gunton, J.D.: Theory of nonlinear transport processes: nonlinear shear viscosity and normal stress effects. Phys. Rev. A 8, 20482064 (1973)
Grabert, H.: Projection Operator Techniques in Nonequilibrium Statistical Mechanics, pp. 29–32. Springer, Berlin (1982)
Zubarev, D.: Statistical Mechanics of Nonequilibrium Processes, pp. 89–98. Wiley, Berlin (1996)
Gaveau, B., Schulman, L.S.: A general framework for non-equilibrium phenomena: the master equation and its formal consequences. Phys. Lett. A 229, 347–353 (1997)
Qian, H.: Relative entropy: free energy associated with equilibrium fluctuations and nonequilibrium deviations. Phys. Rev. E 63, 042103 (2001)
Kawai, R., Parrondo, J.M.R., Van der Broeck, C.: Dissipation: the phase-space perspective. Phys. Rev. Lett. 98, 080602 (2007)
Shell, M.S.: The relative entropy is fundamental to multiscale and inverse thermodynamic problems. J. Chem. Phys. 129, 144108 (2008)
Vaikuntanathan, S., Jarzynski, C.: Dissipation and lag in irreversible processes. Europhys. Lett. 87, 60005 (2009)
Horowitz, J., Jarzynski, C.: Illustrative example of the relationship between dissipation and relative entropy. Phys. Rev. E 79, 021106 (2009)
Roldán, E., Parrondo, J.M.R.: Entropy production and Kullback–Leibler divergence between stationary trajectories of discrete systems. Phys. Rev. E 85, 031129 (2012)
Crooks, G. E., and Sivak, D. A.: Measures of trajectory ensemble disparity in nonequilibrium statistical dynamics. J. Stat. Mech.: Theory Exp. 2011, P06003 (2011)
Crooks, G. E.: On thermodynamic and microscopic reversibility. J. Stat. Mech.: Theory Exp. 2011, P07008 (2011)
Sivak, D.A., Crooks, G.E.: Near equilibrium measurements of nonequilibrium free energy. Phys. Rev. Lett. 108, 150601 (2012)
Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)
Acknowledgments
We would like to express our gratitude to the anonymous reviewers of this article for their insightful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Meléndez, M., Español, P. Gibbs–Jaynes Entropy Versus Relative Entropy. J Stat Phys 155, 93–105 (2014). https://doi.org/10.1007/s10955-014-0954-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-014-0954-6