Abstract
We study driven systems with possible population inversion and we give optimal bounds on the relative occupations in terms of released heat. A precise meaning to Landauer’s blowtorch theorem (Phys Rev A 12:636–638, 1975) is obtained stating that nonequilibrium occupations are essentially modified by kinetic effects. Towards very low temperatures we apply a Freidlin-Wentzel type analysis for continuous time Markov jump processes. It leads to a definition of dominant states in terms of both heat and escape rates.
Article PDF
Similar content being viewed by others
References
Schnakenberg J.: Network theory of microscopic and macroscopic behavior of master equation systems. Rev. Mod. Phys. 48, 571–585 (1976)
Jiang, D.Q., Qian, M., Qian, M.P.: Mathematical Theory of Nonequilibrium Steady States—On The Frontier of Probability and Dynamical Systems. Lecturer Notes in Mathematics, vol. 1833. Springer, Berlin
Altaner B., Grosskinsky S., Herminghaus S., Katthän L., Timme M., Vollmer J.: Network representations of non-equilibrium steady states: cycle decompositions, symmetries and dominant paths. Phys. Rev. E. 85, 041133 (2012)
Landauer R.: Inadequacy of entropy and entropy derivatives in characterizing the steady state. Phys. Rev. A. 12, 636–638 (1975)
Shubert B.: A flow-graph formula for the stationary distribution of a Markov chain. IEEE Trans. Syst. Man. Cybernet. 5, 565 (1975)
Miekisz J.: Stochastic stability in spatial games. J. Stat. Phys. 117, 99–110 (2004)
Miekisz, J.: Evolutionary Game Theory and Population Dynamics. In: Capasso, V. Lachowicz, M. (eds.) Multiscale Problems in the Life Sciences, from Microscopic to Macroscopic. Lecture Notes in Mathematics, vol. 1940, pp. 269–316 (2008)
Bollobas B.: Modern Graph Theory. Springer, Berlin (1998)
Katz S., Lebowitz J.L., Spohn H.: Stationary nonequilibrium states for stochastic lattice gas models of ionic superconductors. J. Stat. Phys. 34, 497–537 (1984)
Tasaki, H.: Two theorems that relate discrete stochastic processes to microscopic mechanics. arXiv:0706.1032v1 [cond-mat.stat-mech]
Maes C., Netočný K., Wynants B.: On and beyond entropy production: the case of Markov jump processes. Markov Process. Relat. Fields 14, 445–464 (2008)
Freidlin M.I., Wentzell A.D.: Random Perturbations of Dynamical Systems. Grundlehren der Mathematischen Wissenschaften, vol. 260. Springer, New York (1998)
Maes C., Wynants B.: On a response formula and its interpretation. Markov Process. Relat. Fields 16, 45–58 (2010)
Maes C., Netočný K., Wynants B.: Monotone return to steady nonequilibrium. Phys. Rev. Lett. 107, 010601 (2011)
Donsker M.D., Varadhan S.R.: Asymptotic evaluation of certain Markov process expectations for large time I. Comm. Pure Appl. Math. 28, 1–47 (1975)
Bricmont J., Slawny J.: Phase transitions in systems with a finite number of dominant ground states. J. Stat. Phys. 54, 89–161 (1989)
Ellison G.: Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution. Rev. Econ. Stud. 67, 17–45 (2000)
Escher, M.C.: Waterfall (1961)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Denis Bernard.
Rights and permissions
About this article
Cite this article
Maes, C., Netočný, K. Heat Bounds and the Blowtorch Theorem. Ann. Henri Poincaré 14, 1193–1202 (2013). https://doi.org/10.1007/s00023-012-0214-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-012-0214-8