Reinaldo García-García, Daniel Domínguez
We study the fraction of time that a system spends violating the second law of thermodynamics within a single trajectory in the phase space under driving at a constant rate. We denominate this quantity as {\it violation ratio}, and we denote it by $\mR_-(N,t) =\tau_-(N,t)/t$, where $\tau_-(N,t)$ is the amount of time the system has violated the second law up to time $t$ in a system with $N$ particles. We develop general arguments to predict that for generic ergodic systems its mean value scales, in the large-$t$ - large-$N$ limit, as $<\mR_-(N,t)>\sim(t\sqrt{N})^{-1}$. We determine, for a simple system driven at a constant rate, the precise asymptotic behaviour of the mean violation ratio, both, analytically and numerically, showing a good agreement with our prediction. Finally, we briefly discuss how this simple scaling may break down in the vicinity of a phase transition.
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http://arxiv.org/abs/1212.0844
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