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Foundations of algorithmic thermodynamics

Aram Ebtekar, Marcus Hutter

TL;DR

The paper develops an ensemble-free thermodynamic framework based on Gács' coarse-grained algorithmic entropy to describe individual coarse-grained states in Markovian systems. By combining stochastic thermodynamics with algorithmic information theory, it derives detailed and integral fluctuation relations, including algorithmic analogs of Jarzynski's equality and Landauer's principle, and formulates a comprehensive nonequilibrium second law. It clarifies when the algorithmic entropy $S_\pi(x|z) = K(x|z) + \log\pi(x)$ bounds actual work from a single state, as opposed to the Gibbs–Shannon entropy which describes ensemble averages, and extends the theory to open systems with reservoirs via heat and work decompositions. The work also provides applications to Maxwell's demon and information engines, offering a principled, state-level accounting of information processing and its thermodynamic costs, with potential extensions to quantum information and complex causal structures.

Abstract

Gács' coarse-grained algorithmic entropy leverages universal computation to quantify the information content of any given physical state. Unlike the Boltzmann and Gibbs-Shannon entropies, it requires no prior commitment to macrovariables or probabilistic ensembles, rendering it applicable to settings arbitrarily far from equilibrium. For measure-preserving dynamical systems equipped with a Markovian coarse-graining, we prove a number of fluctuation inequalities. These include algorithmic versions of Jarzynski's equality, Landauer's principle, and the second law of thermodynamics. In general, the algorithmic entropy determines a system's actual capacity to do work from an individual state, whereas the Gibbs-Shannon entropy only gives the mean capacity to do work from a state ensemble that is known a priori.

Foundations of algorithmic thermodynamics

TL;DR

The paper develops an ensemble-free thermodynamic framework based on Gács' coarse-grained algorithmic entropy to describe individual coarse-grained states in Markovian systems. By combining stochastic thermodynamics with algorithmic information theory, it derives detailed and integral fluctuation relations, including algorithmic analogs of Jarzynski's equality and Landauer's principle, and formulates a comprehensive nonequilibrium second law. It clarifies when the algorithmic entropy bounds actual work from a single state, as opposed to the Gibbs–Shannon entropy which describes ensemble averages, and extends the theory to open systems with reservoirs via heat and work decompositions. The work also provides applications to Maxwell's demon and information engines, offering a principled, state-level accounting of information processing and its thermodynamic costs, with potential extensions to quantum information and complex causal structures.

Abstract

Gács' coarse-grained algorithmic entropy leverages universal computation to quantify the information content of any given physical state. Unlike the Boltzmann and Gibbs-Shannon entropies, it requires no prior commitment to macrovariables or probabilistic ensembles, rendering it applicable to settings arbitrarily far from equilibrium. For measure-preserving dynamical systems equipped with a Markovian coarse-graining, we prove a number of fluctuation inequalities. These include algorithmic versions of Jarzynski's equality, Landauer's principle, and the second law of thermodynamics. In general, the algorithmic entropy determines a system's actual capacity to do work from an individual state, whereas the Gibbs-Shannon entropy only gives the mean capacity to do work from a state ensemble that is known a priori.
Paper Structure (21 sections, 8 theorems, 155 equations, 1 table)

This paper contains 21 sections, 8 theorems, 155 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Stochastic matrices
  5. Markov processes
  6. Stochastic thermodynamics
  7. Algorithmic information theory
  8. Theoretical analysis
  9. Coarse-grained algorithmic entropy
  10. Comparison against Gibbs-Shannon entropy
  11. Interactions with general reservoirs
  12. Constant temperature reservoirs
  13. Refining the second law
  14. Applications
  15. Inclusive dynamics and open systems
  16. ...and 6 more sections

Key Result

Theorem 1

Let $X,Y$ be $\mathcal{X},\mathcal{Y}$-valued random variables, and $P(y,\,x) := \Pr(Y=y \mid X=x)$. If the measures $\pi,\mu:\mathcal{X}\rightarrow\mathbb R^+$, $\nu:\mathcal{Y}\rightarrow\mathbb R^+$, and $P\pi$ are nonzero everywhere, then,

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Corollary 4: Algorithmic second law of thermodynamics
  • proof
  • Lemma 5
  • proof
  • ...and 6 more