Energy Conversion and Entropy Production in Biased Random Walk Processes -- from Discrete Modeling to the Continuous Limit

TL;DR

This paper explores how different assumptions in the limiting process from discrete to continuous models of thermodynamic random walks affect entropy production, efficiency, and other thermodynamic variables, revealing non-uniqueness and potential efficiency gains.

Contribution

It introduces a detailed analysis of the non-uniqueness in the continuous limit of discrete thermodynamic models and examines the impact on entropy production and efficiency.

Findings

Entropy production depends on the chosen limiting observables.

Efficiency increases as the model approaches the continuous limit.

Different constraints lead to different thermodynamic behaviors in the limit.

Abstract

We consider discrete and continuous representations of a thermodynamic process in which a random walker (e.g. a molecular motor on a molecular track) uses a periodically pumped energy (work) to pass $N$ sites and move energetically downhill while dissipating heat. Interestingly, we find that, starting from a discrete model, the limit in which the motion becomes continuous in space and time ($N\to \infty$) is not unique and depends on what physical observables are assumed to be unchanged in the process. In particular, one may (as usually done) choose to keep the speed and diffusion coefficient fixed during this limiting process, in which case the entropy production is affected. In addition, we study also processes in which the entropy production is kept constant as $N\to \infty$ at the cost of modified speed or diffusion coefficient. Furthermore, we also combine this dynamics with work against an opposing force, which makes it possible to study the effect of discretization of the process on the thermodynamic efficiency of transferring power input to power output. Interestingly, we find that the efficiency is increased in the limit of $N\to\infty$. Finally, we investigate the same process when transitions between sites can only happen at finite time intervals and study the impact of this time discretization on the thermodynamic variables as the continuous limit is approached.