Ordering in statistical systems on the way to the thermodynamic limit

TL;DR

This paper introduces the concept of order indices to quantitatively describe how ordering phenomena develop in finite statistical systems as they approach the thermodynamic limit, using mean-field models.

Contribution

It presents a new approach to quantify preordering in finite systems through order indices, illustrating their growth towards phase transitions in various models.

Findings

Order indices increase with system size, indicating growing order.

Preordering phenomena can be characterized before reaching the thermodynamic limit.

Demonstrates the approach for Bose-Einstein condensation, superconductivity, magnetization, and crystallization.

Abstract

It is well known that the mathematically accurate description of ordering and related symmetry breaking in statistical systems requires to consider the thermodynamic limit. But the order does not appear from nowhere, and yet before the thermodynamic limit is reached, there should exist some kind of preordering that appears and grows in the process of increasing the system size. The quantitative description of growing order, under the growing system size, is developed by introducing the notion of {\it order indices}. The rigorous proof of the phase transition existence is a separate difficult problem that is not the topic of the present paper. We illustrate the approach resorting to several models in the mean-field approximation, which makes it possible to demonstrate the notion of order indices for finite systems in a clear way. We show how the order grows on the way to the thermodynamic limit for Bose-Einstein condensation, arising superconductivity, magnetization, and crystallization phenomena.