[Paper Review] Nonequilibrium fluctuation response relation in a time scale separated system

Shou‐Wen Wang, Kyogo Kawaguchi+2 more|arXiv (Cornell University)|Oct 1, 2016

Advanced Thermodynamics and Statistical Mechanics2 references3 citations

TL;DR

This paper develops a theoretical framework to analyze fluctuation-response relation (FRR) violation in finite Markov systems with two well-separated time scales, showing that FRR violation manifests as a plateau in the intermediate frequency region, which quantifies hidden entropy production due to slow-fast process coupling. The key result is that FRR violation for fast observables is a robust probe of hidden dissipation, while slow observables exhibit effective temperature close to bath temperature with small deviations due to coupling.

ABSTRACT

We present a theoretical framework to analyze the violation of fluctuation-response relation (FRR) for any observable from a finite Markov system with two well-separated time scales. We find that, generally for both slow and fast observables, a broad plateau exists in the intermediate frequency region, which contributes to a finite hidden entropy production. Assuming that non-equilibrium behavior arises only from coupling of slow and fast processes, we find that, at large observation time scae, the effective temperature for a slow observable deviates only slightly from the bath temperature, accompanied by an emerging well-defined effective potential landscape, while the deviation is significant for a fast observable. Our study also identifies a wider range of applicability of the Harada-Sasa equality in Markov jumping systems.

Motivation & Objective

• To understand how fluctuation-response relation (FRR) violation reveals hidden entropy production in non-equilibrium systems with two time scales.
• To clarify the role of slow-fast process coupling in generating non-equilibrium behavior and dissipation not visible at the coarse-grained level.
• To extend the Harada-Sasa equality to discrete Markov jumping systems and identify its conditions of applicability.
• To quantify effective temperature deviations for slow and fast observables and link them to dissipation and potential landscape emergence.
• To provide a practical method for detecting hidden entropy production via low-frequency FRR violation in fast observables.

Proposed method

• Derives analytical expressions for correlation and response spectra in a finite Markov system with two time scales using perturbative analysis.
• Introduces a decomposition of total entropy production into contributions from fast dynamics, slow dynamics, and their coupling, identifying the coupling term as hidden entropy production.
• Applies the generalized Harada-Sasa equality to Markov jumping systems, requiring diffusive transitions and homogeneous transition rate prefactors along the observable direction.
• Uses orthogonal observables to systematically probe independent dissipation channels and quantify FRR violation plateaus.
• Models effective temperature as the ratio of correlation to response, analyzing its frequency dependence for slow and fast observables.
• Validates the framework using a sensory adaptation model in E. coli, demonstrating the emergence of an effective potential landscape for slow variables.

Experimental results

Research questions

• RQ1 How does FRR violation in a time-scale-separated Markov system reflect hidden entropy production from slow-fast coupling?
• RQ2 What determines the frequency dependence of effective temperature for slow and fast observables in such systems?
• RQ3 Under what conditions can the Harada-Sasa equality be generalized to discrete Markov jumping systems?
• RQ4 Why is FRR violation for fast observables more effective in revealing hidden dissipation than for slow observables?
• RQ5 How does the emergence of an effective potential landscape relate to the effective temperature of slow observables?

Key findings

• A broad plateau in the FRR violation spectrum appears in the intermediate frequency region for both slow and fast observables, directly linked to finite hidden entropy production from slow-fast coupling.
• For slow observables, the effective temperature remains close to the bath temperature across all frequencies, with only small deviations of order ε at low and intermediate frequencies due to coupling.
• For fast observables, the effective temperature significantly deviates from the bath temperature at low frequencies but approaches it at high frequencies, indicating a two-temperature, two-time-scale behavior.
• The generalized Harada-Sasa equality holds in Markov jumping systems when transitions are diffusive and the rate prefactor is homogeneous along the observable direction, enabling measurement of total entropy production from projected dynamics.
• The hidden entropy production rate can be quantified by measuring the FRR violation plateau in the low-frequency region of fast observables, which is a more sensitive probe than slow observables.
• In the timescale separation limit, the slow dynamics exhibit a well-defined effective potential landscape, consistent with the small deviation of effective temperature from bath temperature.