Thermodynamic Uncertainty Relation in Interacting Many-Body Systems

The thermodynamic uncertainty relation (TUR) has been well studied for systems with few degrees of freedom. While, in principle, the TUR holds for more complex systems with many interacting degrees of freedom as well, little is known so far about its behavior in such systems. We analyze the TUR in the thermodynamic limit for mixtures of driven particles with short-range interactions. Our main result is an explicit expression for the optimal estimate of the total entropy production in terms of single-particle currents and correlations between two-particle currents. Quantitative results for various versions of a driven lattice gas demonstrate the practical implementation of this approach.

Figure 1

Three different models of the driven lattice gas I–III. In model II(a) there is only one particle (the blue one) driven by the electric field, whereas in model II(b) all particles are driven.

Figure 2

(a) Particle currents of models II(a) and II(b), (b) diffusion coefficients and correlations of model III, (c) quality factors of all models against $N$, and (d) different quality factors ${\mathcal{Q}}_{\mathit{J}}$, ${\mathcal{Q}}_{\sigma }$, and ${\mathcal{Q}}_{J_{\mathrm{tot}}}$ for model III. For model I, $q_1=1.0$ and $K_{11}=0.8$. For models II(a) and II(b), $q_1=0$, $q_2=1.0$, $K_{11}=0.8$, and $K_{12}=1.2$ and $q_1=1.0$, $q_2=2.5$, $K_{11}=0.8$, and $K_{12}=1.2$, respectively. For model III in (c), $q_1=1.0$, $q_2=2.5$, $K_{11}=0.8$, $K_{12}=1.2$, and $K_{22}=0.4$, whereas in (d) $q_1=1.5$, $q_2=5.0$, $K_{11}=0.8$, $K_{12}=1.2$, and $K_{22}=0.4$.