Inferring energy dissipation from violation of the fluctuation-dissipation theorem

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 way to infer this perturbation asymmetry based on the measurable high-frequency-limit FDT violation.

Figure 1

(a) One-dimensional (1D) hopping process. (b),(c) Multi-dimensional hopping models. The corresponding observable ${\mathcal{Q}}_n$, which is $x$ here, does not distinguish microscopic states within each colored block. As with the 1D hopping model, we assume that $\theta$ is the same for all red transitions. These models may describe molecular motors that hop along a discrete lattice with several internal chemical states. They also resemble sensory adaptation model in E. coli [35].

Figure 2

(a) A simplified Markov model for kinesin. (b) The experimentally suggested relations between transition rates $w_{\pm }$ and the external force $h$ [30], corresponding to the case with $\theta =0.5$. (c) The energy landscape connecting neighboring states. The external force only varies the energy barrier for the forward transition.

Figure 3

(a) The correlation spectrum ${\tilde{C}}_{\dot{x}}\left(\omega \right)$ and the (real part of) response spectrum ${\tilde{R}}_{\dot{x}}^{\prime }\left(\omega \right)$ for the velocity ${\dot{x}}_t$, obtained at $\mathrm{\Delta }U=1$ and $\theta =0.5$. A finite violation of FDT, ${\mathcal{V}}_{\infty }$, persists even in the high frequency limit. (b) The relative violation of the FDT in the high-frequency limit $[{\mathcal{V}}_{\infty }/{\tilde{C}}_{\dot{x}}\left(\infty \right)]$, the average drifting velocity ${\langle \dot{x}\rangle }_{ss}$, and the predicted dissipation rate ${\gamma }_{*}{I}_{*}$ based on the renormalized FDT violation integral against the actual dissipation rate $\dot{q}$. Here, the control parameter is the entropic change per jump, i.e., $\mathrm{\Delta }U/T$. (c) Verification of Eq. (11). The slope of each curve gives the corresponding $\theta$. Other parameters: $w_0=1$, $d=1$, and $h=0$.

Figure 4

(a) The energy landscape $U_n$ for the 1D hopping model, constructed from $U\left(x\right)=-sin(2\pi x/L)-\mathrm{\Delta }\mu x/L$ at a given lattice constant $d$. The landscapes for different $d$'s are shifted vertically for illustration. (b) The correlation and response spectrum for the velocity ${\dot{x}}_t\equiv {\dot{n}}_{t}d$, obtained at $d=0.1$ and $\theta =-0.2$. The FDT is restored in the high frequency limit with the renormalized temperature. (c) The relative high-frequency-limit violation of FDT against the average entropic change in the environment per jump: $\frac{1}{N}{\sum }_n|\mathrm{\Delta }{U}_n/T|$. (d),(e) Emergence of the renormalized HS equality (from different schemes) at small medium entropy production per jump. Other parameters: $w_0=1/d^2$, $T=1$, $L=1$, $\mathrm{\Delta }\mu =2.5$, and $h=0$.