Entropy Production of Nanosystems with Time Scale Separation

Energy flows in biomolecular motors and machines are vital to their function. Yet experimental observations are often limited to a small subset of variables that participate in energy transport and dissipation. Here we show, through a solvable Langevin model, that the seemingly hidden entropy production is measurable through the violation spectrum of the fluctuation-response relation of a slow observable. For general Markov systems with time scale separation, we prove that the violation spectrum exhibits a characteristic plateau in the intermediate frequency region. Despite its vanishing height, the plateau can account for energy dissipation over a broad time scale. Our findings suggest a general possibility to probe hidden entropy production in nanosystems without direct observation of fast variables.

Figure 1

(a) The potential switching model with a fast switching time constant ${\tau }_f=1/r$ and a slow relaxation time constant ${\tau }_s=\gamma /k$. The effective potential $U_e(x)$ in the fast switching limit $\epsilon \ll 1$ is shown. (b) Rate of energy input against $\epsilon$. (c) Frequency spectra of the velocity correlation and response functions at various values of $\epsilon$. (d) Frequency spectra of the FRR violation whose integral yields the hidden entropy production that balances energy input in (b). Parameters: $k=\gamma =1$ and $L=5$. Open circles and stars in (c) and (d) give results from a simulated trajectory of length $T_{sp}=10^4{\tau }_s$ and sampling rate $2{\tau }_f^{-1}$. The response function is reconstructed from data using a perturbation strength $h=0.5$. (See Supplemental Material [29].)

Figure 2

(a) Illustration of a general nonequilibrium Markov process with a dissipative cycle formed by fast and slow processes. (b) Effective system at large time scale separation $\epsilon \ll 1$, where the circulating probability flux is hidden, along with the associated entropy production. (c) Eigenvalue spectrum of the master equation for a time scale separated system. Fast modes are well separated in their decay rates ${\lambda }_j$ from the slow ones. The latter form a nearly degenerate band at the bottom that defines the slow dynamics of the effective system.