Local Equivalence of Reversible and General Markov Kinetics

Abstract

We consider continuous--time Markov kinetics with a finite number of states and a given positive equilibrium distribution P*. For an arbitrary probability distribution $P$ we study the possible right hand sides, dP/dt, of the Kolmogorov (master) equations. We describe the cone of possible values of the velocity, dP/dt, as a function of P and P*. We prove that, surprisingly, these cones coincide for the class of all Markov processes with equilibrium P* and for the reversible Markov processes with detailed balance at this equilibrium. Therefore, for an arbitrary probability distribution $P$ and a general system there exists a system with detailed balance and the same equilibrium that has the same velocity dP/dt at point P. The set of Lyapunov functions for the reversible Markov processes coincides with the set of Lyapunov functions for general Markov kinetics. The results are extended to nonlinear systems with the generalized mass action law.

arXiv commentSignificantly extended version, 21 pages

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Cited by in corpus (5)

• General H-theorem and entropies that violate the second law
• Maxallent: Maximizers of all Entropies and Uncertainty of Uncertainty
• New universal Lyapunov functions for non-linear reaction networks
• Semistability of complex balanced kinetic systems with arbitrary time delays
• Generalized Mass Action Law and Thermodynamics of Nonlinear Markov Processes