Stochastic Energetics for Non-Gaussian Processes

By introducing a new stochastic integral, we investigate the energetics of classical stochastic systems driven by non-Gaussian white noises. In particular, we introduce a decomposition of the total energy difference into the work and the heat for each trajectory, and derive a formula to calculate the heat from experimental data on the dynamics. We apply our formulation and results to a Langevin system driven by a Poisson noise.

Figure 1

A schematic of the $ϵ$ smoothing.

Figure 2

Numerical results on the consistency between Eq. (7) and $\widehat{Y}$. We fix the variance of the Poisson noise $\lambda I^2=4.0$ and simulate the equation until $t=1.0$ with $ϵ=0.001$. As $I$ increases, the difference between the $*$ integral and the Stratonovich integral becomes significant. In the Gaussian limit of $I\to 0$, $Z$ is expected to equal unity in both calculuses as denoted by an open circle.

Figure 3

The check of the first law of thermodynamics through the numerical solution of Eq. (16) with the intensity $I=\pm 2.0$ where the Poisson noises are added at $\tau =1.0$ and $\tau =2.0$. The result of ${\tilde{U}}^{*}$ based on the $*$ calculus agrees with the total energy $\tilde{U}$, while ${\tilde{U}}^S$ based on the Stratonovich calculus is different from $\tilde{U}$.