By Da-Quan Jiang, Min Qian, Ming-Ping Qian
This quantity offers a scientific mathematical exposition of the conceptual difficulties of nonequilibrium statistical physics, corresponding to entropy construction, irreversibility, and ordered phenomena. Markov chains, diffusion methods, and hyperbolic dynamical platforms are used as mathematical types of actual platforms. A measure-theoretic definition of entropy creation price and its formulae in a number of instances are given. It vanishes if and provided that the desk bound process is reversible and in equilibrium. additionally, within the instances of Markov chains and diffusion methods on manifolds, it may be expressed by way of circulations on directed cycles. concerning entropy creation fluctuations, the Gallavotti-Cohen fluctuation theorem is carefully proved.
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Extra info for Mathematical Theory of Nonequilibrium Steady States: On the Frontier of Probability and Dynamical Systems
Example text
Hill [224] using a diagram method where his concepts of cycle flux and detailed balance correspond respectively to the circulation distribution and reversibility of Markov chains. Our results about the reversibility of Markov chains with continuous time parameter can be summarized in the following theorem. 10. Suppose that ξ is an irreducible, positive recurrent and stationary Markov chain with continuous parameter, whose state space S is finite, transition density matrix Q = (qij )i,j∈S and invariant probability distribution Π = (πi )i∈S .
1) where 1, i = j, 0, i = j. δij = Denote D = (dij )i,j∈S , where dij = 1, i = j, q − qiji , i = j. 2) Let D(H) be the determinant of D with rows and columns indexed by the ˜ index set H, and D(H) be the determinant of Q with rows and columns ˜ indexed by H. D(∅) and D(∅) are understood as 1. Set T0 (ω) = 0, T1 (ω) = inf{t > 0 : ξt (ω) = ξ0 (ω)}, Tk+1 (ω) = inf{t > Tk (ω) : ξt (ω) = ξTk (ω) (ω)}, ∀k ∈ N, and ξ˜n (ω) = ξTn (ω) (ω), ∀n ≥ 0. It is well known that Tn are all stopping times and P( lim Tn = +∞) = 1.
3(1 − pq) The entropy production rate of ξ is given by ep = (p − q) log w(1,2,3) p . = w(1,2,3) − w(3,2,1) log w(3,2,1) q The Markov chain ξ is reversible if and only if p = q = 21 , or equivalently, its entropy production rate ep vanishes. For a system which may be described by the model of a Markov chain, the movement of the system is actually a process in which it continuously completes those possible cycles of its state space including loops which contain only one state, and the so-called “back and forth” cycles which contain only two states.
