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Irreversibility and randomness

Nino Dekkers, Klaas Landsman

TL;DR

The paper investigates how irreversible macroscopic evolution, such as Boltzmann-type equations, can emerge from reversible microscopic dynamics by grounding molecular chaos in algorithmic randomness. It analyzes three settings—a rigorous Boltzmann-Grad-based derivation (Lanford–Deng–Hani–Ma) and two toy models, the Ehrenfest urn and the Kac ring—and shows that when microstates or trajectories are $P$-random, the macroscopic distribution $f(t,ullet)$ satisfies a Boltzmann-like equation and exhibits time-asymmetric behavior. Time-reversal of $P$-random paths is exponentially unlikely, illustrating the arrow of time, while explicit $P$-random trajectories are largely non-constructible due to Chaitin incompleteness. The work highlights the role of i.i.d. initial data, the distinction between averages and pointwise behavior, and how randomness-based derivations may extend Boltzmann reasoning beyond traditional proofs.

Abstract

We make precise sense of the idea of "molecular chaos" through algorithmic randomness of microscopic trajectories, and ground macroscopic irreversibility in the lack of symmetry under time reversal of this property. This concept of randomness is defined relative to an underlying probability measure P on the space of trajectories. In deterministic models like Newtonian N-particle flow in dilute gases of hard spheres (as considered by Boltzmann) or the Kac ring model these may be reduced to their initial conditions, in which case P makes the particles i.i.d. at t=0. In the (stochastic) Ehrenfest urn model, on the other hand, the importance of trajectories as the decisive random objects comes out more clearly. We consider each of these models from this point of view, including a conceptual analysis of the recent (post-Lanford) microscopic derivation of the full Boltzmann equation for long times. We also show to which extent algorithmic randomness is stronger than necessary for the derivation of Boltzmann-like equations, in giving rise to an infinite number of other macroscopic properties. In the light of Chaitin's incompleteness theorems for algorithmic randomness, the price for this scenario is the impossibility of explicitly displaying algorithmically random microscopic trajectories.

Irreversibility and randomness

TL;DR

The paper investigates how irreversible macroscopic evolution, such as Boltzmann-type equations, can emerge from reversible microscopic dynamics by grounding molecular chaos in algorithmic randomness. It analyzes three settings—a rigorous Boltzmann-Grad-based derivation (Lanford–Deng–Hani–Ma) and two toy models, the Ehrenfest urn and the Kac ring—and shows that when microstates or trajectories are -random, the macroscopic distribution satisfies a Boltzmann-like equation and exhibits time-asymmetric behavior. Time-reversal of -random paths is exponentially unlikely, illustrating the arrow of time, while explicit -random trajectories are largely non-constructible due to Chaitin incompleteness. The work highlights the role of i.i.d. initial data, the distinction between averages and pointwise behavior, and how randomness-based derivations may extend Boltzmann reasoning beyond traditional proofs.

Abstract

We make precise sense of the idea of "molecular chaos" through algorithmic randomness of microscopic trajectories, and ground macroscopic irreversibility in the lack of symmetry under time reversal of this property. This concept of randomness is defined relative to an underlying probability measure P on the space of trajectories. In deterministic models like Newtonian N-particle flow in dilute gases of hard spheres (as considered by Boltzmann) or the Kac ring model these may be reduced to their initial conditions, in which case P makes the particles i.i.d. at t=0. In the (stochastic) Ehrenfest urn model, on the other hand, the importance of trajectories as the decisive random objects comes out more clearly. We consider each of these models from this point of view, including a conceptual analysis of the recent (post-Lanford) microscopic derivation of the full Boltzmann equation for long times. We also show to which extent algorithmic randomness is stronger than necessary for the derivation of Boltzmann-like equations, in giving rise to an infinite number of other macroscopic properties. In the light of Chaitin's incompleteness theorems for algorithmic randomness, the price for this scenario is the impossibility of explicitly displaying algorithmically random microscopic trajectories.
Paper Structure (6 sections, 3 theorems, 90 equations)

This paper contains 6 sections, 3 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Irreversibility of the original Boltzmann equation
  3. Irreversibility in the Ehrenfest model
  4. Irreversibility in the Kac ring model
  5. Algorithmic randomness
  6. Time reversal

Key Result

Theorem 3.1

Let the initial condition $\alpha \in [0,1]$, which parametrizes $\mathbb{P}_1$ and hence $\mathbb{P}$, be a computable real. For each initial segment $\mathbb{T}\subset{\mathbb N}$ and each $\mathbb{P}$-random path $\omega\in(2^{{\mathbb N}})^\mathbb{T}$, we have in norm, cf. 3.39, where $f(t)$ is a solution of the toy Boltzmann equation for $t \in {\mathbb T}$ with initial condition $f(0) = \a

Theorems & Definitions (4)

  • Theorem 3.1
  • Theorem 4.1
  • Definition A.1
  • Theorem A.2: Levin, Gács