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Non-Equilibrium Thermodynamics and Topology of Currents

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Abstract

In many experimental situations, a physical system undergoes stochastic evolution which may be described via random maps between two compact spaces. In the current work, we study the applicability of large deviations theory to time-averaged quantities which describe such stochastic maps, in particular time-averaged currents and density functionals. We derive the large deviations principle for these quantities, as well as for global topological currents, and formulate variational, thermodynamic relations to establish large deviation properties of the topological currents. We illustrate the theory with a nontrivial example of a Heisenberg spin-chain with a topological driving of the Wess-Zumino type. The Cramér functional of the topological current is found explicitly in the instanton gas regime for the spin-chain model in the weak-noise limit. In the context of the Morse theory, we discuss a general reduction of continuous stochastic models with weak noise to effective Markov chains describing transitions between stable fixed points.

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Authors and Affiliations

  1. Department of Chemistry, Wayne State University, 5101 Cass Avenue, Detroit, MI, 48202, USA

    Vladimir Y. Chernyak & Sergey V. Malinin

  2. Theoretical Division and Center for Nonlinear Studies, LANL, Los Alamos, NM, 87545, USA

    Vladimir Y. Chernyak, Michael Chertkov & Razvan Teodorescu

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  1. Vladimir Y. Chernyak
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  2. Michael Chertkov
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  3. Sergey V. Malinin
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  4. Razvan Teodorescu
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Correspondence to Michael Chertkov.

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Chernyak, V.Y., Chertkov, M., Malinin, S.V. et al. Non-Equilibrium Thermodynamics and Topology of Currents. J Stat Phys 137, 109–147 (2009). https://doi.org/10.1007/s10955-009-9832-z

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  • DOI: https://doi.org/10.1007/s10955-009-9832-z

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