Universal trade-off between irreversibility and intrinsic timescale in thermal relaxation with applications to thermodynamic inference

TL;DR

The paper derives a general lower bound on the instantaneous entropy production rate (EPR) using the KL divergence to the instantaneous stationary state and the Log-Sobolev constant, strengthening the second law. This bound leads to a universal trade-off between the intrinsic relaxation timescale and dissipation, producing an inverse speed limit for state transformations and a quantum version tighter than the classical one, valid even for non-Markovian coarse-grained dynamics. The results provide a practical framework for thermodynamic inference, enabling estimation of EPR from coarse-grained observations in molecular dynamics and Brownian-particle systems. Collectively, the work offers new tools for predicting relaxation behavior, optimizing rapid equilibration, and bounding dissipation in both classical and quantum open systems.

Abstract

We establish a general lower bound for the entropy production rate (EPR) based on the Kullback-Leibler divergence and the Logarithmic-Sobolev constant that characterizes the time-scale of relaxation. This bound can be considered as an enhanced second law of thermodynamics. When applied to thermal relaxation, it reveals a universal trade-off relation between the dissipation rate and the intrinsic relaxation timescale. From this relation, a thermodynamic upper bound on the relaxation time between two given states emerges, acting as an inverse speed limit over the entire time region. We also obtain a quantum version of this upper bound, which is always tighter than its classical counterpart, incorporating an additional term due to decoherence. Remarkably, we further demonstrate that the trade-off relation remains valid for any generally non-Markovian coarse-grained relaxation dynamics, highlighting its significant applications in thermodynamic inference. This trade-off relation is a new tool in inferring EPRs in molecular dynamics simulations and practical experiments.

Figures (4)

• Figure 1: Illustration of the trade-off relation and the lower bound for entropy production in a two-state mode. Here, we set $\beta=1/(k_{B}T)=1$. (a) The two-state model coupled to a heat reservoir with temperature $T$. (b) The trade-off relation between the EPR $\dot{\sigma}(t)$ at $t=0$ and the relaxation time scale $1/(2\lambda_{\text{LS}})$ for relaxation processes with different $\Delta E$, where the distance to equilibrium $\sigma_{\text{tot}}^{t}\equiv D\left[\boldsymbol{p}(t)\vert\vert\boldsymbol{p}^{\text{eq}}\right]$ is fixed to be $0.5$. (c) and (d) We demonstrate Eq. (\ref{['tradeoff1']}) and (\ref{['tradoff2']}) for this model, in which $\Delta E=5.0$ and the initial distribution is chosen to be $(p_{u},p_{d})=(0.99,0.01)$.
• Figure 2: Application of the coarse-grained trade-off relation to molecular dynamics simulation. (a) The two-dimensional interacting Brownian particles model. The total particle number is chosen to be 100. The interacting potential is the spring potential, with a strength of $\kappa=0.01$. The stiffness of the external harmonic potential field is $k=0.1$. The initial distribution is such that every particle is in one of the four spatial regions divided artificially. (b) The comparison between the true EPR (EPR, the dashed black line) obtained from an approximate analytical expression, and our lower bound (the red line).
• Figure 3: The coarse-grained trade-off relations for the interacting Brownian particles system with more coarse-grained states: (a) The lower bound is calculated by dividing the space uniformly into 16 regions (16 states). (b) The lower bound is calculated by dividing the space uniformly into 25 regions (25 states).
• Figure 4: Demonstration of the coarse-grained trade-off relation in an interacting active particles system. There are 4096 active particles. (a) A snapshot of the simulation, showing the system self-assembling into three coarse-grained states: the stripe state (green part), the trimer state (red part), and the disorder state (blue part). (b) The evolution of the probability distributions of the three coarse-grained states during relaxation. (c) The non-adiabatic EPR calculated from detailed knowledge (orange curve) and the coarse-grained lower bound (blue curve) using only the statistics of the coarse-grained states.