Irreversibility and randomness

TL;DR

This paper explores how the concept of algorithmic randomness in microscopic trajectories underpins macroscopic irreversibility, analyzing various models and discussing the implications of randomness for deriving Boltzmann-like equations.

Contribution

It provides a formal framework linking microscopic trajectory randomness to macroscopic irreversibility and examines the strength of randomness needed for kinetic equations.

Findings

Algorithmic randomness explains irreversibility in microscopic models.

Randomness is stronger than necessary for Boltzmann equation derivation.

Incomputability of trajectories due to incompleteness theorems.

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.