Stochastic Calculus for Pathwise Observables of Markov-Jump Processes: Unification of Diffusion and Jump Dynamics

Abstract

Path-wise observables--functionals of stochastic trajectories--are at the heart of time-average statistical mechanics and are central to thermodynamic inequalities such as uncertainty relations, speed limits, and correlation-bounds. They provide a means of thermodynamic inference in the typical situation, when not all dissipative degrees of freedom in a system are experimentally accessible. So far, theories focusing on path-wise observables have been developing in two major directions, diffusion processes and Markov-jump dynamics, in a virtually disjoint manner. Moreover, even the respective results for diffusion and jump dynamics were derived with a patchwork of different approaches that are predominantly indirect. Stochastic calculus was recently shown to provide a direct approach to path-wise observables of diffusion processes, while a corresponding framework for jump dynamics remained elusive. In our work we develop, in an exact parallelism with continuous-space diffusion, a complete stochastic calculus for path-wise observables of Markov-jump processes. We formulate a "Langevin equation" for jump processes, define general path-wise observables, and establish their covariation structure, whereby we fully account for transients and time-inhomogeneous dynamics. We prove the known kinds of thermodynamic inequalities in their most general form and discus saturation conditions. We determine the response of path-wise observables to general (incl. thermal) perturbations and introduce a corresponding response-function formalism. We carry out the continuum limit to achieve the complete unification of diffusion and jump dynamics. In addition, we connect the framework to quantum unraveling and the Belavkin equation for open quantum systems, associating quantum and classical descriptions of thermal systems.

Figures (14)

• Figure 1: Comparison of key quantities in continuous and discrete space, namely the equations of motion, noise statistics, fundamental path observables, thermodynamic bounds, fluctuation-scale function, and response to perturbations (incl. temperature changes).
• Figure 2: Schematic comparison of discrete and continuous trajectories. In (a), a continuous trajectory is shown. (b) shows a discrete trajectory in a three-state MJP. The transitions $2\to 3$ (blue), $3\to 1$ (orange), $1\to 2$ (green), and $3\to 2$ (red) are marked with the respective colors. The displacement and jump increments $\mathrm{d} x_\tau$ and $\mathrm{d} n_{xy}(\tau)$ are presented in (c) and (d) for the continuous and discrete trajectories. The colors in (d) correspond to the different transitions in (b).
• Figure 3: Example trajectory and examples of an arising current and density. The part of the trajectory in (a) which contributes to the density $\rho_\tau$ with state function, $\textcolor{black}{V}_i(\tau) = \delta_{i2}$, is marked in blue. Similarly, the transitions contributing to the current $J_\tau$ with transition weights, $\kappa_{ij} = \delta_{i2}\delta_{j3} - \delta_{i3}\delta_{j2}$, are marked by orange arrows with numbers representing the weight of the transition. The resulting density and current can be seen in (b) and (c), respectively.
• Figure 4: Two densities, ${\rho}_t^\alpha$ and ${\rho}_t^\beta$, are used to visualize density (co)variances in the SAT model, see Sec. \ref{['SAT']}. These correspond to the state functions $V_i^\alpha = \delta_{i1} + \delta_{i2}$ and $V_i^\beta = \delta_{i5} + \delta_{i6}$ measuring if molecules of type $A$ and $B$ are in the channel, respectively. In (a), the (scaled) variances $\mathrm{var}({\rho}^{\alpha/\beta}_t)/t$ are shown together with the (scaled) covariance between them $\mathrm{cov}({\rho}^{\alpha}_t,{\rho}^{\beta}_t)/t$ from numerical simulations (colored) and analytical solutions Eq. \ref{['eq:DensityCov']} (dashed lines). The analytical variance of ${\rho}_t^\alpha$ (solid lines) and covariance (black dashed lines), both modulated by $1/t$, are shown in (b) for various values of $e^\mathrm{out}_B$. The inset shows $\mathrm{var}({\rho}_t^\beta)/t$. The numerics in (a) are evaluated using $N=10^4$ trajectories sampled using the celebrated Gillespie algorithm CelebratedGillespieCelebratedGillespie2. The initial condition is $p_i=(\delta_{i1}+\delta_{i3}+\delta_{i5})/3$ and the values of the parameters can be found in Tab. \ref{['tab:ParameterSAT']}. In (b), $e^\mathrm{out}_B = 40$ is used.
• Figure 5: Statistics of transitions and waiting times in the stationary SAT model. The average number of transition $\langle n_{ij}\rangle$ are shown in $(a)$ for a selection of edges. In (b), the average time spent $\langle \tau_i\rangle$ in each state is presented. The steady-state currents resulting from the transitions in (a) are presented in (c). The red errorbars are the respective standard deviations $\sqrt{\mathrm{var}(J_t)}$. In (c), the lower bound $|\langle J_t\rangle|\sqrt{2/\Delta S_\mathrm{tot}}\leq\sqrt{\mathrm{var}(J_t)}$ resulting from the steady-state TUR is added as purple errorbars. We use $t=10$ in every panel.