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Brownian motion surviving in the unstable cubic potential and the role of Maxwell's demon

Luca Ornigotti1,*, Artem Ryabov2,†, Viktor Holubec2,3,‡, and Radim Filip1,§

  • 1Department of Optics, Palacký University, 17. listopadu 1192/12, CZ-771 46 Olomouc, Czech Republic
  • 2Charles University, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, CZ-180 00 Praha 8, Czech Republic
  • 3Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany

  • *luca.ornigotti@gmail.com
  • †rjabov.a@gmail.com
  • ‡Viktor.holubec@mff.cuni.cz
  • §filip@optics.upol.cz
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Phys. Rev. E 97, 032127 – Published 23 March, 2018

DOI: https://doi.org/10.1103/PhysRevE.97.032127

Abstract

The trajectories of an overdamped particle in a highly unstable potential diverge so rapidly, that the variance of position grows much faster than its mean. A description of the dynamics by moments is therefore not informative. Instead, we propose and analyze local directly measurable characteristics, which overcome this limitation. We discuss the most probable particle position (position of the maximum of the probability density) and the local uncertainty in an unstable cubic potential, V(x)∼x3, both in the transient regime and in the long-time limit. The maximum shifts against the acting force as a function of time and temperature. Simultaneously, the local uncertainty does not increase faster than the observable shift. In the long-time limit, the probability density naturally attains a quasistationary form. We interpret this process as a stabilization via the measurement-feedback mechanism, the Maxwell demon, which works as an entropy pump. The rules for measurement and feedback naturally arise from the basic properties of the unstable dynamics. All reported effects are inherent in any unstable system. Their detailed understanding will stimulate the development of stochastic engines and amplifiers and, later, their quantum counterparts.

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Phys. Rev. E 97, 032127– Published 23 March, 2018
  • Received 22 August 2017
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