Inverse thermodynamic uncertainty relation and entropy production

Abstract

Nonequilibrium current fluctuations represent one of the central topics in nonequilibrium physics. The thermodynamic uncertainty relation (TUR) is widely acclaimed for rigorously establishing a lower bound on current fluctuations, expressed in terms of the entropy production rate and the average current. In this study, we focus on an upper bound for the fluctuations, referred to as the inverse thermodynamic uncertainty relation (iTUR). We derive a universal iTUR expression in terms of the entropy production rate for continuous-variable systems governed by overdamped Langevin equations, as well as for discrete-variable systems described by Markov jump processes. The iTUR establishes a no-go theorem prohibiting perpetual superdiffusion in systems with a finite entropy production rate and a finite spectral gap. The divergence of the variance of any current becomes possible only when the spectral gap of the symmetrized time-evolution operator closes or the entropy production rate diverges. As a relevant experimental scenario, we apply the iTUR to the phenomenon of giant diffusion, emphasizing the pivotal roles of the spectral gap and entropy production.

Figures (2)

• Figure 1: Diffusion coefficient, iTUR, and TUR as functions of the external force $F$ in a tilted periodic potential. Parameters are $V_0 = 1$, $L = 1$, and $D_0 = 0.5$$(T= 0.5)$. The dashed line represents $\tilde{D}_\infty$, showing a significant increase near the critical force $F_{\mathrm{c}} = 2\pi$, indicative of giant diffusion. The solid line denotes the iTUR upper bound, closely aligning with $\tilde{D}_\infty$, while the dotted line represents the TUR lower bound. The inset shows the behavior of the entropy production rate $\sigma_{\text{st}}$ and spectral gap $\lambda$ with $F$. The left y-axis corresponds to the spectral gap, while the right y-axis represents the entropy production rate.
• Figure 2: Dependence of the diffusion coefficient $D_\infty$ and the iTUR upper bound on the entropy production rate $\sigma_{\text{st}}$ for fixed spectral gaps $( \lambda = 5, 10, 100 )$. Dashed lines represent $D_\infty/D_0$, and solid lines denote the iTUR upper bound.