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Will a Large Complex System be a Maxwell Demon?

Matthew P Leighton

TL;DR

The results show the probability of a finding a demon decreases at least exponentially, and in some cases double-exponentially, with the number of degrees of freedom, ultimately suggesting that large complex demons can only arise through a process of selection.

Abstract

Emerging evidence suggests that physical systems operating as Maxwell demons, in which some subsystem of a larger system extracts heat energy from its environment in an apparent local violation of the second law, are commonplace throughout biology. Should these findings surprise us, or is Maxwell demon behavior inevitable in sufficiently large complex systems? In this Letter we pose the question of how likely it is that a random stochastic system with many degrees of freedom will operate as a Maxwell demon, considering null models for both continuous and discrete random dynamics. Our results show the probability of a finding a demon decreases at least exponentially, and in some cases double-exponentially, with the number of degrees of freedom, ultimately suggesting that large complex demons can only arise through a process of selection.

Will a Large Complex System be a Maxwell Demon?

TL;DR

The results show the probability of a finding a demon decreases at least exponentially, and in some cases double-exponentially, with the number of degrees of freedom, ultimately suggesting that large complex demons can only arise through a process of selection.

Abstract

Emerging evidence suggests that physical systems operating as Maxwell demons, in which some subsystem of a larger system extracts heat energy from its environment in an apparent local violation of the second law, are commonplace throughout biology. Should these findings surprise us, or is Maxwell demon behavior inevitable in sufficiently large complex systems? In this Letter we pose the question of how likely it is that a random stochastic system with many degrees of freedom will operate as a Maxwell demon, considering null models for both continuous and discrete random dynamics. Our results show the probability of a finding a demon decreases at least exponentially, and in some cases double-exponentially, with the number of degrees of freedom, ultimately suggesting that large complex demons can only arise through a process of selection.
Paper Structure (5 sections, 33 equations, 2 figures)

This paper contains 5 sections, 33 equations, 2 figures.

Table of Contents

  1. Langevin Dynamics with $N=2$
  2. Langevin Dynamics with $N\geq 2$
  3. Discrete Dynamics with $N=2$
  4. Discrete Dynamics with $N\geq2$
  5. Details of Numerical Sampling

Figures (2)

  • Figure 1: a) Probability that a random stochastic system obeying linear overdamped Langevin dynamics operates as a Maxwell demon. Theoretical lower (red) and upper (green) bounds, as well as the independence approximation (blue). Black points show numerical results from sampling random $\bm{A}$ matrices subject to the constraint of all positive eigenvalues (uncertainties are smaller than the displayed points), with $\epsilon=0.1$, while grey points show a more general ensemble (see Appendix E for details). Inset: logarithmic y-axis. b) Distribution of largest observed heat flows in random systems that are Maxwell demons for different numbers $N$ of degrees of freedom. Inset: mean and left/right standard deviations as functions of $N$.
  • Figure 2: a) Probability that a random stochastic system obeying discrete master equation dynamics operates as a Maxwell demon. Theoretical lower (red) and upper (green) bounds, as well as the independence approximation (blue). Black (dark gray, light gray) points show numerical results from sampling random transition rate matrices with $\sigma_\mathrm{discrete}=0.1$ ($\sigma_\mathrm{discrete}=1,\, 10$) (uncertainties are smaller than the displayed points). Inset: logarithmic y-axis. b) Distribution of largest observed heat flows in random systems that are Maxwell demons for different numbers $N$ of degrees of freedom. Inset: logarithmic y-axis to illustrate tails of the distributions.