Martingale properties of entropy production and a generalized work theorem with decoupled forward and backward processes

TL;DR

This work develops a forward-backward decoupled, stochastic-calculus framework to study entropy production in nonequilibrium Langevin dynamics. By introducing a backward process with an arbitrary initial condition and constructing an exponential martingale, the authors derive a generalized work theorem that holds conditionally on the initial state, extending beyond classical ensemble averaging. The approach applies to both overdamped and underdamped dynamics and reveals the pivotal role of the fluctuation-dissipation relation: deviations from $D=1/(\beta\gamma)$ generally break the standard work theorem, while under certain velocity-distribution assumptions the generalized result can persist. The results offer new interpretations of entropy production, enable trajectory-specific thermodynamic bounds, and suggest practical implications for high-dimensional and biological systems, as well as directions for further exploration in discrete-state settings.

Abstract

By decoupling forward and backward stochastic trajectories, we construct a family of martingales and work theorems for both overdamped and underdamped Langevin dynamics. Our results are made possible by an alternative derivation of work theorems that uses tools from stochastic calculus instead of path-integration. We further strengthen the equality in work theorems by evaluating expectations conditioned on an arbitrary initial state value. These generalizations extend the applicability of work theorems and offer new interpretations of entropy production in stochastic systems. Lastly, we discuss the violation of work theorems in far-from-equilibrium systems.

Figures (5)

• Figure 1: Schematic of standard backward driving protocols and the associated probability densities. (a) The forward driving protocol $\lambda_t$ as a function of forward time $t$. (b) The backward driving protocol $\tilde{\lambda}_{s}$ is obtained by counting time backwards from the terminal time $t_1$. (c) Probability densities of the forward ($\rho$) and the standard backward ($\tilde{\rho}$) processes described in previous work Manzano2021February. With an initial distribution $\tilde{\rho}(x_{0},0) = \rho(t_1)$, $\tilde{\rho}(s)$ indexed by the backward time $s$ evolves under the time-reversed driving protocol $\tilde{\lambda}_{s}$.
• Figure 2: Decoupling of the forward and backward processes. (a) In our derivations, we employ the probability density $\psi(x,t)$ of a backward process indexed by forward time $t$. The initial condition is arbitrary so that in general $\psi(x,t_1) \neq \rho(x,t_1)$. (b) Trajectories of the forward processes $x_t$ sampled from the initial distribution $\rho(x_{0},0)$ are shown in grey, while trajectories with a specific initial value $x_0$ are shown in red. The original work theorem uses averages over the grey trajectories, while our generalized work theorem considers averages over the red trajectories. See Eqs. \ref{['eq:main']} and \ref{['eq:manzano2']}.
• Figure 3: (a) Schematic representation of the continuum kinetic proofreading process. The receptor state is defined by the activation level $x$ and the ligand-binding status $\alpha \in \{0,1\}$. When the ligand is absent ($\alpha = 0$), the system relaxes to the stable state at $x = 0$. Ligand binding ($\alpha = 1$) introduces an external force $f$ that biases the activation dynamics, enabling transitions to higher activation levels. Binding and unbinding events follow exponential waiting time distributions, $\tau_+ \sim \operatorname{Exp}(k_+)$ and $\tau_- \sim \operatorname{Exp}(k_-)$, respectively. Activation occurs when $x > x^{*}$, with the probability of activation depending exponentially on the unbinding rate $k_-$, facilitating high selectivity in ligand recognition. Ligand unbinding before ($\alpha = 0$) resets the activation level to $x = 0$, analogous to erasing memory as constrained by Landauer's principle. (b) A Simplified drift-diffusion representation of the process with bound ligand ($\alpha =1$). The dynamics reduce to an overdamped Langevin equation under a constant force $f$. Starting from $x_0 = 0$, the system evolves according to $\gamma \mathrm{d} x_t = f \mathrm{d} t + \sqrt{\frac{2 \gamma}{\beta}} \mathrm{d} B_t$. The work performed by the force is given by $W = f \Delta x_\tau$, where $\Delta x_\tau$ is the displacement over time $\tau$. This model captures the fundamental principles of energy expenditure and memory erasure in ligand-receptor systems.
• Figure 4: Numerical results for the quantities $\mathbb{E}[e^{\theta_{\tau}}]$ (red), $\mathbb{E}[\theta_{\tau}]$ (blue), and $\mathbb{E}[\Delta \Sigma_{\tau}]$ (green) for different choices of $t_2$. The red line is at unity, consistent with our work theorem. By contrast, $\mathbb{E}[\theta_{\tau}]$ and $\mathbb{E}[\Delta \Sigma_{\tau}]$ vary with $t_2$, shown by the blue and green curves, respectively.
• Figure 5: Numerical results of the exponentiated entropy production $\mathbb{E} \left[e^{\beta (W_t - \Delta F_t)}\right]$ (blue) and $\mathbb{E} \left[e^{\theta_t}\right]$ (red) for the OU process with different diffusion coefficients $D$. In both cases, when $D=1/(\beta \gamma)=1$, both quantities are equal to unity.