Order--disorder separation: Geometric revision

TL;DR

This paper explores the relationship between order and disorder in multiparticle systems, showing how measure concentration leads to phase space separation and influences system dynamics and the arrow of time.

Contribution

It introduces a geometric framework connecting measure concentration to order--disorder separation, providing new insights into system evolution and symmetry effects.

Findings

Phase space divides into small-diameter disordered states and large-measure ordered states.

System dynamics involve high acceleration, interpreted as collision-driven evolution.

The time arrow from order to disorder is explained by measure concentration effects.

Abstract

After Boltzmann and Gibbs, the notion of disorder in statistical physics relates to ensembles, not to individual states. This disorder is measured by the logarithm of ensemble volume, the entropy. But recent results about measure concentration effects in analysis and geometry allow us to return from the ensemble--based point of view to a state--based one, at least, partially. In this paper, the order--disorder problem is represented as a problem of relation between distance and measure. The effect of strong order--disorder separation for multiparticle systems is described: the phase space could be divided into two subsets, one of them (set of disordered states) has almost zero diameter, the second one has almost zero measure. The symmetry with respect to permutations of particles is responsible for this type of concentration. Dynamics of systems with strong order--disorder separation has high average acceleration squared, which can be interpreted as evolution through a series of collisions (acceleration--dominated dynamics). The time arrow direction from order to disorder follows from the strong order--disorder separation. But, inverse, for systems in space of symmetric configurations with ``sticky boundaries" the way back from disorder to order is typical (Natural selection). Recommendations for mining of molecular dynamics results are presented also.