Entropy production and non-Gaussianity of fast processes at weak damping

TL;DR

This paper develops a data‑driven framework to quantify entropy production in weakly damped, non‑equilibrium systems by decomposing the rate $oldsymbol{\sigma_t}$ into four positive contributions associated with a nonzero mean velocity, a non‑thermal velocity width, position–velocity correlations, and non‑Gaussian velocity statistics. It formulates a variational Fisher information approach to estimate the non‑Gaussian contribution from trajectory data and shows that three Gaussian terms—determined by first and second moments—capture most of the dissipation in levitated nanoparticle experiments with nonlinear driving. The authors apply the method to transient, nonlinear pulses in cubic/quartic/inverted potentials and demonstrate that Gaussian components dominate entropy production, while non‑Gaussian velocity statistics yield a distinct positive contribution that also influences Shannon entropy dynamics. The work provides a practical, data‑driven toolkit for diagnosing driving mechanisms and bounding dissipation in weakly damped systems, with potential extensions to multi‑particle, strongly damped, and quantum regimes.

Abstract

We present a method of estimating the rate of entropy production in underdamped dynamics by decomposing it into contributions originating in different non-equilibrium effects. Specifically, a non-zero average velocity, a non-thermal width of the velocity distribution, correlations between position and velocity and non-Gaussian velocity statistics represent different ways in which the system can be out of equilibrium and each give rise to a positive contribution to the overall entropy production rate. We demonstrate that each contribution can be separately estimated from experimental trajectory data of levitated nano-particles subject to non-linear forces. We find that the majority of the entropy production rate can be attributed to the first three contributions which can be estimated from the first and second moments of the position and velocity and therefore result in a useful \enquote{Gaussian} estimate for the entropy production rate.

Figures (7)

• Figure 1: Illustration of different types of non-equilibrium phase-space densities for and underdamped system. In contrast to the equilibrium phase-space density (blue, center), a non-equilibrium system non-zero average velocity (density displaced from $v = 0$, top right), a width that is different from the thermal velocity (density stretched or compressed in $v$-direction, top left), correlations between position and velocity (density rotated in the $x$-$v$ plane, bottom left) or a non-Gaussian velocity density (more weight in the tails or center of the density, bottom right). Our main result is that each of these features gives rise to a separate, positive contribution to the entropy production rate of the system, which sum to the total entropy production rate.
• Figure 2: (a) Schematic representation of the experimental setup to generate non-linear potentials. (b) Representations of the 3 kinds of potential generated in the pulsed protocols. (c) Motional spectra for the different initial states, at 2mBar without and with pre-cooling, and at 80mBar. (d) Example protocol with 9$\mu s$ evolution in a quartic potential.
• Figure 3: The non-Gaussian parameter Eq. (\ref{['non-gaussian-1d']}) as a function of the polynomial order $M$ of the basis functions for different sizes $N$ of the data set. The dashed line corresponds to a value of $\alpha^\text{nG} = 0.05$.
• Figure 4: a) The four positive contributions to the dimensionless entropy production rate Eq. (\ref{['entropy-decompositon-dimensionless']}) as a function of time for a trapped particle driven by a quartic-potential force pulse. b) The two contributions to the dimensionless rate of change of the Shannon entropy Eq. (\ref{['shannon-decompositon-dimensionless']}) and the total value as a function of time. c) The overall Gaussian and non-Gaussian contribution to the entropy production rate as a function of time. The force pulse starts at $t = 3000$ samples and lasts $375$ samples. The sampling rate is $4.17 \cdot 10^6$ samples/s.
• Figure 5: a) The four positive contributions to the dimensionless entropy production rate Eq. (\ref{['entropy-decompositon-dimensionless']}) as a function of time for a trapped particle driven by an inverted-potential force pulse. b) The two contributions to the dimensionless rate of change of the Shannon entropy Eq. (\ref{['shannon-decompositon-dimensionless']}) and the total value as a function of time. c) The overall Gaussian and non-Gaussian contribution to the entropy production rate as a function of time. The force pulse starts at $t = 3000$ samples and lasts $375$ samples. The sampling rate is $4.17 \cdot 10^6$ samples/s.