Quantifying the validity and breakdown of the overdamped approximation in stochastic thermodynamics: Theory and experiment

Abstract

Stochastic thermodynamics provides an important framework to explore small physical systems where thermal fluctuations are inevitable. In the studies of stochastic thermodynamics, some thermodynamic quantities, such as the trajectory work, associated with the complete Langevin equation (the Kramers equation) are often assumed to converge to those associated with the overdamped Langevin equation (the Smoluchowski equation) in the overdamped limit under the overdamped approximation. Nevertheless, a rigorous mathematical proof of the convergence of the work distributions to our knowledge has not been reported so far. Here we study the convergence of the work distributions explicitly. In the overdamped limit, we rigorously prove the convergence of the extended Fokker-Planck equations including work using a multiple timescale expansion approach. By taking the linearly dragged harmonic oscillator as an exactly solvable example, we analytically calculate the work distribution associated with the Kramers equation, and verify its convergence to that associated with the Smoluchowski equation in the overdamped limit. We quantify the accuracy of the overdamped approximation as a function of the damping coefficient. In addition, we experimentally demonstrate that the data of the work distribution of a levitated silica nanosphere agrees with the overdamped approximation in the overdamped limit, but deviates from the overdamped approximation in the low-damping case. Our work fills a gap between the stochastic thermodynamics based on the complete Langevin equation (the Kramers equation) and the overdamped Langevin equation (the Smoluchowski equation), and deepens our understanding of the overdamped approximation in stochastic thermodynamics.

Functionals in stochastic thermodynamics: how to interpret stochastic integrals

“…Much has been written about the noise interpretation related to the whitenoise limit and the small-mass limit in stochastic differential equations (SDEs), with the two limits usually studied separately [12-17, 19, 27-31], and, more recently, about the behavior of stochastic thermodynamic quantities in the small-mass limit [18,20,22,23,[32][33][34][35][36][37][38][39][40][41]. As mentioned above, we will recover several of these previous results along our analysis, and present them in a single framework.…”

Stochastic thermodynamics of holonomic systems

“…In the overdamped conditions, the system is analyzed in the so-called high friction limit hypothesis with a negligible system inertia. The validity and the accuracy of the overdamped approximation have been recently discussed [70], and the overdamped stochastic thermodynamics has been further generalized to systems with multiple reservoirs [71]. The first step of our analysis concerns the elaboration of the overdamped Langevin equation and the corresponding Smoluchowski equation.…”

Kovacs memory effect with an optically levitated nanoparticle

“…1, cannot be applied. This warns against the adoption of such an approximation under certain circumstances, because it may return erroneous conclusions and predictions, as has been previously observed in similar situations of low damping [41] or when the temperature of the system is allowed to evolve with time [42,43]. The accurate quantification of time scales in thermalization processes of Brownian systems is also required for the correct evaluation of performances and efficiencies in the implementation of thermodynamics processes and thermal machines at the microscale, the so called stochastic thermodynamics [44][45][46].…”