Infinite family of second-law-like inequalities

The probability distribution function for an out of equilibrium system may sometimes be approximated by a physically motivated “trial” distribution. A particularly interesting case is when a driven system (e.g., active matter) is approximated by a thermodynamic one. We show here that every set of trial distributions yields an inequality playing the role of a generalization of the second law. The better the approximation is, the more constraining the inequality becomes: this suggests a criterion for its accuracy, as well as an optimization procedure that may be implemented numerically and even experimentally. The fluctuation relation behind this inequality, a natural and practical extension of the Hatano-Sasa theorem, does not rely on the a priori knowledge of the stationary probability distribution.

Figure 1

Optimized $J_{ij}$ for the SSEP model with open boundaries for ${\rho }_0={\rho }_1$ (dotted red line) and for ${\rho }_0\ne {\rho }_1$ (solid blue line). The inset shows similar results for the optimized $h_i$.

Figure 2

Analytical and simulation results for the steady-state density profile $\langle n_i\rangle$. Red squares and orange crosses correspond to the Monte Carlo procedure for densities ${\rho }_0=0.8$, ${\rho }_1=0.2$ using the optimized trial ${\rho }_{\mathrm{opt}}\left(\mathbf{n}\right)$ and the local equilibrium measure ${\rho }_{LE}\left(\mathbf{n}\right)$, respectively, whereas blue diamonds are the analytical results; see Eq. (37). Purple triangles correspond to the Monte Carlo procedure for densities ${\rho }_0={\rho }_1=0.3$ using the equilibrium measure ${\rho }_{\mathrm{eq}}\left(\mathbf{n}\right)$, whereas black circles are the analytical results.

Figure 3

Analytical and simulation results for the two-point correlation function ${\langle n_i{n}_j\rangle }_c$ in the steady state. Red squares and orange crosses correspond to the Monte Carlo procedure for densities ${\rho }_0=0.8$, ${\rho }_1=0.2$ using the optimized trial ${\rho }_{\mathrm{opt}}\left(\mathbf{n}\right)$ and the local equilibrium measure ${\rho }_{LE}\left(\mathbf{n}\right)$, respectively, whereas blue diamonds are the analytical results; see Eq. (38). Purple triangles correspond to the Monte Carlo procedure for densities ${\rho }_0={\rho }_1=0.3$ using the equilibrium measure ${\rho }_{\mathrm{eq}}\left(\mathbf{n}\right)$, whereas black circles are the analytical results.