# Circulation Distribution, Entropy Production and by Jiang D.-Q., Qian M.

By Jiang D.-Q., Qian M.

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Extra info for Circulation Distribution, Entropy Production and Irreversibility of Denumerable Markov Chains

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We deﬁne inductively a sequence of random variables {fn (ω) : n ≥ 0} as below: def 1) f0 (ω) = 1; 2) For each n ≥ 0,  pξ (ω)ξn+1 (ω)  fn (ω) pξn (ω)ξ , def n (ω) n+1 fn+1 (ω) =  fn (ω) pi1 i2 ···pis−1 is pis i1 pi i ···pi i pi i s s−1 2 1 1 s if ln+1 (ω) ≥ ln (ω), −1 , if ηn (ω) = [ηn+1 (ω), [i1 , · · · , is ]]. 5 Large Deviations and Fluctuation Theorem fn (ω) = 41 pi1 i2 · · · pil−1 il . 3, we have πξ (ω) pξ0 (ω)ξ1 (ω) · · · pξn−1 (ω)ξn (ω) eWn (ω) = 0 πξn (ω) pξn (ω)ξn−1 (ω) · · · pξ1 (ω)ξ0 (ω) = πξ0 (ω) πξn (ω) c∈C∞ wc wc− wc,n (ω) · fn (ω), and Wn (ω) 1 πξ (ω) = log 0 + n n πξn (ω) c∈C∞ wc wc,n (ω) 1 log + log fn (ω).

Provided the free energy function def c(λ, β, γ) = lim n→+∞ 1 log EeλWn + n β,Φn + γ,Ψn exists and is diﬀerentiable, it holds that {µn : n > 0}, the family of the distributions of { n1 (Wn , Φn , Ψn ) : n > 0}, has a large deviation property with rate function 42 1 Denumerable Markov Chains λz + β, u + γ, v − c(λ, β, γ) . I(z, u, v) = sup λ,β,γ And we have the following generalized ﬂuctuation theorem. 10. If for each n > 0, Φn (rω) = Φn (θ−n ω) and Ψn (rω) = −Ψn (θ−n ω), ∀ω ∈ Ω, then it holds that c(λ, β, γ) = c(−(1 + λ), β, −γ), I(z, u, v) = I(−z, u, −v) − z.

However, in case one can divide over, the above equality can be written as P P Wn n =z Wn n = −z = enz . e. in detailed balance), then I(0) = 0 and I(z) = +∞, ∀z = 0, so in this case the ﬂuctuation theorem gives a trivial result. However, if the Markov chain ξ is not reversible, then for z > 0 in a certain range, the sample entropy production rate Wnn has a positive probability to take the value z as well as the value −z, but the ﬂuctuation theorem tells that the former probability is greater, which accords with the second law of thermodynamics.