  We define the {\it violation fraction} $\nu$ 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 $\tau$. In the
large-$\tau$ and large-$N$ limit we show that, for ergodic and microreversible
systems, the mean value of $\nu$ scales as
$\langle\nu(N,\tau)\rangle\sim\big(\tau N^{\frac{1}{1+\alpha}}\big)^{-1}$. The
exponent $\alpha$ is positive and generally depends on the protocol for the
external driving forces, being $\alpha=1$ 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 $\langle \nu \rangle$ with $N$
and $\tau$ agrees with our result. Finally, we discuss how this scaling law may
break down in the vicinity of a continuous phase transition.

Statistical Mechanics