• Open Access

Stochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound

Sosuke Ito and Andreas Dechant
Phys. Rev. X 10, 021056 – Published 15 June 2020

Abstract

We investigate the connection between the time evolution of averages of stochastic quantities and the Fisher information and its induced statistical length. As a consequence of the Cramér-Rao bound, we find that the rate of change of the average of any observable is bounded from above by its variance times the temporal Fisher information. As a consequence of this bound, we obtain a speed limit on the evolution of stochastic observables: Changing the average of an observable requires a minimum amount of time given by the change in the average squared, divided by the fluctuations of the observable times the thermodynamic cost of the transformation. In particular, for relaxation dynamics, which do not depend on time explicitly, we show that the Fisher information is a monotonically decreasing function of time and that the minimal required time is determined by the initial preparation of the system. We further show that the monotonicity of the Fisher information can be used to detect hidden variables in the system and demonstrate our findings for simple examples of continuous and discrete random processes.

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  • Received 30 October 2018
  • Revised 24 February 2020
  • Accepted 20 April 2020

DOI:https://doi.org/10.1103/PhysRevX.10.021056

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & ThermodynamicsGeneral Physics

Authors & Affiliations

Sosuke Ito1,2 and Andreas Dechant3,4

  • 1Universal Biology Institute, the University of Tokyo, 7-3-1 Hongo Bunkyo-ku Hongo, Tokyo 113-0033, Japan
  • 2JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan
  • 3WPI-Advanced Institute of Materials Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan
  • 4Department of Physics 1, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

Popular Summary

To arrive at a consistent description of any physical system, we need to account for the information it contains. The most famous example of this is Maxwell’s demon, a hypothetical imp that permits fast-moving particles to accumulate on one side of a barrier and slow-moving particles on the other. The resolution of this apparent violation of the second law of thermodynamics is to accept information as a physical quantity, just like mass or energy, but it is not clear how this abstract quantity relates to tangible observables such as heat and work. We establish such a relation for stochastic systems, which can be described in terms of time-dependent probabilities. We show that the information encoded in the evolution of these probabilities provides a limit to how fast any thermodynamic observable can change with time.

Measuring the time evolution of some observable quantity yields information about the system. We relate this information to the so-called Fisher information, which describes the amount of information carried by a random variable. This then leads to a speed limit for the time evolution of observables, determined by its fluctuations and its Fisher information. This relation connects thermodynamic observables to their stochastic fluctuations and the information contained in the probabilistic description of the system.

We show that, in the absence of time-dependent forces, the Fisher information always decreases with time and use this to identify hidden degrees of freedom in the system. This may be useful for biological systems like molecular motors, where the microscopic biochemical processes are often not directly accessible in experiments. It will be interesting to see whether the presence of these processes can be inferred from a measurement of, for example, the motion of the motor. Another important open question is whether and under what conditions nonlinear dynamics, which often arise as effective models of interacting systems, also have a decreasing Fisher information.

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Vol. 10, Iss. 2 — April - June 2020

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