Optimal stochastic transport in inhomogeneous thermal environments

¹ Cancer Cell Biophysics, Institute for Cancer Research and Treatment at Candiolo (IRC@C) Str. Prov. 142 km 3.95, 10060 Candiolo, Torino, Italy, EU
² Department of Computational Biology, AlbaNova University Centre, KTH Royal Institute of Technology SE-106 91 Stockholm, Sweden, EU
³ Nordita, KTH Royal Institute of Technology and Stockholm University - Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden, EU
⁴ INFN, Sezione di Torino - via P. Giuria 1, 10125 Torino, Italy, EU
⁵ ACCESS Linnaeus Centre, KTH Royal Institute of Technology - SE-100 44 Stockholm, Sweden, EU
⁶ Department of Information and Computer Science, Aalto University - PO Box 15400, FI-00076 Aalto, Finland, EU
⁷ Physics of Biological Systems, Institut Pasteur and CNRS UMR 3525 - 28 rue du docteur Roux, 75015 Paris, France, EU

Received: 13 March 2013
Accepted: 2 July 2013

Abstract

We consider the optimization of the average entropy production in inhomogeneous temperature environments within the framework of stochastic thermodynamics. For systems modeled by Langevin equations (e.g. a colloidal particle in a heat bath) it has been recently shown that a space-dependent temperature breaks the time reversal symmetry of the fast velocity degrees of freedom resulting in an anomalous contribution to the entropy production of the overdamped dynamics. We show that optimization of entropy production is determined by an auxiliary deterministic problem formally analogous to motion on a curved manifold in a potential. The “anomalous contribution” to entropy plays the role of the potential and the inverse of the diffusion tensor is the metric. We also find that entropy production is not minimized by adiabatically slow, quasi-static protocols but there is a finite optimal duration for the transport process. As an example we discuss the case of a linearly space-dependent diffusion coefficient.