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

Abstract

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 $⟨\nu (N,\tau )⟩\sim (\tau N^{\frac{1}{1+\alpha }}{)}^{-1}$. 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 $⟨\nu ⟩$ 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.

8 pages, 2 figures, Final version, as accepted for publication in Phys. Rev. E

References in corpus (9)

• Stochastic thermodynamics, fluctuation theorems, and molecular machines
• Path-integral analysis of fluctuation theorems for general Langevin processes
• Symmetries of generating functionals of Langevin processes with colored multiplicative noise
• Non-equilibrium work fluctuations for oscillators in non-Markovian baths
• Jarzynski Relation, Fluctuation Theorems, and Stochastic Thermodynamics for Non-Markovian Processes
• Unifying approach for fluctuation theorems from joint probability distributions
• Fluctuation relations for anomalous dynamics
• Non-adiabatic entropy production for non-Markov dynamics
• Symmetry for the duration of entropy-consuming intervals

Cited by in corpus (1)

• Symmetry for the duration of entropy-consuming intervals