The derivation of Markov processes that violate detailed balance

Abstract

Time-reversal symmetry of microscopic laws dictates that the equilibrium distribution of a stochastic process must obey the detailed balance. On the other hand, cyclic Markov processes that do not admit equilibrium distributions with detailed balance, are often used to model open systems driven out of equilibrium by external agents. I show that for a Markov model without detailed balance, an extended Markov model that explicitly includes the degrees of freedom for the driving agent can be constructed, such that the original cyclic Markov model for the driven system can be recovered as an approximation at early times, by summing over the degrees of freedom for the driving agent. In the process, the widely accepted formula for the entropy production in a cyclic Markov model is explicitly expressed as a time derivative of an entropy component in the extended model. I also find an analytic formula for the entropy component that is hidden in the cyclic Markov model.

16 pages, 4 figures. Revised version submitted to a journal. Numerical computation for a model with smooth transition to equilibrium added. Requirement for the derivation of Schnakenberg entropy production formula is weakened. The relations with the quantities in previous works explained in the appendices. A figure showing the full equilibrium distribution of the extended three-state model added

References in corpus (11)

• Thermodynamic uncertainty relation for biomolecular processes
• Probability currents as principal characteristics in the statistical mechanics of non-equilibrium steady states
• Integral fluctuation theorem for the housekeeping heat
• Stochastic approach to equilibrium and nonequilibrium thermodynamics
• Physical insight into the thermodynamic uncertainty relation using Brownian motion in tilted periodic potentials
• A possible classification of nonequilibrium steady states
• Fluctuation Relation for Heat
• Hidden entropy production by fast variables
• A derivation of the master equation from path entropy maximization
• Non-equilibrium Dynamics, Thermalization and Entropy Production
• Strong fluctuation theorem for nonstationary nonequilibrium systems

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