Irreversibility and randomness

Abstract

We make the idea of "molecular chaos" precise through algorithmic randomness of microscopic trajectories. Apart from making this idea concrete, this has three advantages. First, it sharpens "typicality" approaches to irreversible macroscopic evolution equations like the Boltzmann equation in giving a criterion for individual (as opposed to typical) trajectories to induce irreversible macroscopic behaviour. Second, algorithmic randomness theory comes with an effective ergodic theorem, which in our toy models yields the autonomy of macroscopic equations induced by microscopic data. Third, it relates macroscopic irreversibility of random trajectories to a lack of symmetry under time reversal of the (algorithmic) criterion for randomness. This criterion is defined via an underlying probability measure P on the space of microscopic trajectories. In deterministic models these reduce to their initial conditions. Following most literature, including the recent derivation of the Boltzmann equation for long times, our P makes all particles i.i.d. at t=0. After some qualitative comments on the Boltzmann equation in historical perspective, we realize our scenario in two toy models, viz. the (stochastic) Ehrenfest urn model and the (deterministic) Kac ring model. We finally discuss the relevance of Chaitin's incompleteness theorems, which (here) state the impossibility of explicitly displaying algorithmically random microscopic trajectories, despite their ubiquity.

40 pages. Final version