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

Henning Kirchberg*, Abraham Nitzan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We considered discrete and continuous representations of a thermodynamic process in which a random walker (e.g., a molecular motor on a molecular track) uses periodically pumped energy (work) to pass N sites and move energetically downhill while dissipating heat. Interestingly, we found that, starting from a discrete model, the limit in which the motion becomes continuous in space and time ((Formula presented.)) 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 also studied processes in which the entropy production is kept constant as (Formula presented.) at the cost of a modified speed or diffusion coefficient. Furthermore, we also combined this dynamics with work against an opposing force, which made it possible to study the effect of discretization of the process on the thermodynamic efficiency of transferring the power input to the power output. Interestingly, we found that the efficiency was increased in the limit of (Formula presented.). Finally, we investigated the same process when transitions between sites can only happen at finite time intervals and studied the impact of this time discretization on the thermodynamic variables as the continuous limit is approached.

Original languageEnglish
Article number1218
Issue number8
StatePublished - Aug 2023
Externally publishedYes


FundersFunder number
National Science FoundationCHE1953701
University of Pennsylvania


    • continuous state space
    • discrete state space
    • entropy production
    • stochastic thermodynamics
    • thermodynamic process


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