[Paper Review] Inferring dissipation from the violation of Fluctuation-Dissipation Theorem

Shou‐Wen Wang|arXiv (Cornell University)|Oct 28, 2017

Protein Structure and Dynamics1 references4 citations

TL;DR

This paper demonstrates that the Fluctuation-Dissipation Theorem (FDT) is violated at high frequencies in discrete Markov systems with asymmetric perturbation responses, causing a divergent FDT violation integral that invalidates the Harada-Sasa equality. It proposes renormalized FDT violation integrals that accurately predict energy dissipation rates when entropy production per jump is small, enabling experimental inference of dissipation and perturbation asymmetry from high-frequency FDT violation.

ABSTRACT

The Harada-Sasa equality elegantly connects the energy dissipation rate of a moving object with its measurable violation of the Fluctuation-Dissipation Theorem (FDT). Although proven for Langevin processes, its validity remains unclear for discrete Markov systems whose forward and backward transition rates respond asymmetrically to external perturbation. A typical example is a motor protein called kinesin. Here we show generally that the FDT violation persists surprisingly in the high-frequency limit due to the asymmetry, resulting in a divergent FDT violation integral and thus a complete breakdown of the Harada-Sasa equality. A renormalized FDT violation integral still well predicts the dissipation rate when each discrete transition produces a small entropy in the environment. Our study also suggests a new way to infer this perturbation asymmetry based on the measurable high-frequency-limit FDT violation.

Motivation & Objective

• To clarify the connection between energy dissipation and FDT violation in discrete Markov processes with asymmetric perturbation responses.
• To address the breakdown of the Harada-Sasa equality in such systems due to divergent FDT violation integrals at high frequencies.
• To develop renormalization schemes that restore the validity of the Harada-Sasa equality when entropy production per jump is small.
• To provide a protocol for experimentally inferring dissipation rates and perturbation asymmetry from measurable correlation and response spectra.

Proposed method

• Derives the FDT violation integral for a general discrete Markov process with asymmetric forward and backward transition rates under external perturbation.
• Identifies that high-frequency FDT violation persists due to perturbation asymmetry, leading to a divergent integral and breakdown of the Harada-Sasa equality.
• Introduces two renormalization schemes: one using a renormalized temperature $ T_{re} $ derived from the high-frequency limit of the correlation-to-response ratio, and another using an effective friction coefficient $ \gamma_{re} $.
• Defines a renormalized FDT violation integral $ I_{re} $ that remains finite and well-predicts the dissipation rate $ \dot{q} $ when entropy change per jump is small.
• Validates the approach numerically using a 1D hopping model for kinesin, showing $ \gamma_{re}I_{re} \approx \dot{q} $ under small $ \Delta U/T $.
• Establishes a linear relationship between high-frequency FDT violation and the asymmetry factor $ \theta $, enabling experimental inference of $ \theta $.

Experimental results

Research questions

• RQ1 Does the Fluctuation-Dissipation Theorem remain valid at high frequencies in discrete Markov systems with asymmetric perturbation responses?
• RQ2 Why does the Harada-Sasa equality fail in such systems despite finite dissipation rates?
• RQ3 Can a renormalized FDT violation integral restore the predictive power of the Harada-Sasa equality when entropy production per jump is small?
• RQ4 Can the high-frequency FDT violation be used to experimentally infer the perturbation asymmetry factor $ \theta $?

Key findings

• The FDT violation persists at high frequencies due to perturbation asymmetry, leading to a divergent FDT violation integral that invalidates the original Harada-Sasa equality.
• A renormalized FDT violation integral $ I_{re} $, using a renormalized temperature $ T_{re} $, remains finite and accurately predicts the dissipation rate $ \dot{q} $ when $ \Delta U/T \ll 1 $.
• The effective friction coefficient $ \gamma_{re} $, derived from the high-frequency limit of the response function, converges to $ \gamma_{*} $ under small entropy production, ensuring $ \gamma_{re}I_{re} \approx \dot{q} $.
• The high-frequency FDT violation $ \mathcal{V}_{\infty} $ is linearly proportional to the asymmetry factor $ \theta $, enabling experimental inference of $ \theta $ from measurable spectra.
• Numerical validation using a 1D kinesin model confirms that $ \gamma_{re}I_{re} $ well estimates $ \dot{q} $ when $ \Delta U/T $ is small.
• The renormalized approach restores the predictive power of the Harada-Sasa equality in discrete Markov systems where the original formulation fails due to divergence.