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Duration of local violations of the second law of thermodynamics along single trajectories in phase space

We define the {\it violation fraction} $ν$ as the cumulative fraction of time that the entropy change is negative during single realizations of processes in phase space. This quantity depends both on the number of degrees of freedom $N$ and the duration of the time interval $τ$. In the large-$τ$ and large-$N$ limit we show that, for ergodic and microreversible systems, the mean value of $ν$ scales as $\langleν(N,τ)\rangle\sim\big(τN^{\frac{1}{1+α}}\big)^{-1}$. The exponent $α$ is positive and generally depends on the protocol for the external driving forces, being $α=1$ for a constant drive. As an example, we study a nontrivial model where the fluctuations of the entropy production are non-Gaussian: an elastic line driven at a constant rate by an anharmonic trap. In this case we show that the scaling of $\langle ν\rangle$ with $N$ and $τ$ agrees with our result. Finally, we discuss how this scaling law may break down in the vicinity of a continuous phase transition.