Non-Equilibrium Thermodynamics and Topology of Currents

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.

30 pages, 4 figures

References in corpus (10)

• Spin Transfer Torques
• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Fluctuation theorems for stochastic dynamics
• An Extension of the Fluctuation Theorem
• Path-integral analysis of fluctuation theorems for general Langevin processes
• Directed flow in non-adiabatic stochastic pumps
• Paths of fluctuation induced switching
• Pumping-Restriction Theorem for Stochastic Networks
• Relevance of initial and final conditions for the Fluctuation Relation in Markov processes
• Statistics of Entropy Production in Linearized Stochastic System

Cited by in corpus (22)

• Stochastic thermodynamics, fluctuation theorems, and molecular machines
• Nonequilibrium Markov processes conditioned on large deviations
• Variational and optimal control representations of conditioned and driven processes
• Effective interactions and large deviations in stochastic processes
• A formal view on 2.5 large deviations and fluctuation relations
• Level 2 large deviation functionals for systems with and without detailed balance
• Large-deviation theory for a Brownian particle on a ring: a WKB approach
• Large Fluctuations of the Macroscopic Current in Diffusive Systems: A Confirmation of the Additivity Principle
• A field-theoretic approach to nonequilibrium work identities
• Transition times in the low-noise limit of stochastic dynamics
• Mathematical, Thermodynamical, and Experimental Necessity for Coarse Graining Empirical Densities and Currents in Continuous Space
• Level 2.5 large deviations for continuous time Markov chains with time periodic rates
• Coarse Graining Empirical Densities and Currents in Continuous-Space Steady States
• Stochastic Optimal Control as Non-equilibrium Statistical Mechanics: Calculus of Variations over Density and Current
• Non-Equilibrium Statistical Physics of Currents in Queuing Networks
• Quantization and Fractional Quantization of Currents in Periodically Driven Stochastic Systems II: Full Counting Statistics
• Haldane model on the Sierpiński gasket
• Geometric Universality of Currents
• Topological field theory of dynamical systems
• Current fluctuations of a self-interacting diffusion on a ring
• Stochastic Dynamics of Extended Objects in Driven Systems: I. Higher-Dimensional Currents in the Continuous Setting
• Stochastic Dynamics of Extended Objects in Driven Systems II: Current Quantization in the Low-Temperature Limit