Equilibrium of disordered systems : constructing the appropriate valleys in each sample via strong disorder renormalization in configuration space

TL;DR

This paper introduces a strong disorder renormalization method in configuration space to construct and analyze the longest-lived valleys in disordered systems, providing insights into their equilibrium properties and non-equilibrium dynamics.

Contribution

It develops a general RG procedure for disordered systems that constructs valleys and computes their thermodynamic properties, applicable to any master equation.

Findings

Successfully constructed valleys and computed their free energies, energies, and entropies.

Applied the method to a 2D directed polymer in a random medium, analyzing valley property distributions.

Abstract

To describe the equilibrium properties of disordered systems and the possible emergence of various 'phases' at low temperature, we adopt here the 'broken ergodicity' point of view advocated in particular by Palmer [Adv. Phys. 31, 669 (1982)] : the aim is then to construct the valleys of configurations that become separated by diverging barriers and to study their relative weights, as well as their internal properties. To characterize the slow non-equilibrium dynamics of disordered systems, we have recently introduced in [C. Monthus and T. Garel, J. Phys. A 41, 255002 (2008) and arxiv:0804.1847] a strong disorder renormalization procedure in configuration space, based on the iterative elimination of the smallest barrier remaining in the system. In the present paper, we show how this renormalization procedure allows to construct the longest-lived valleys in each disordered sample, and to obtain their free-energies, energies and entropies. This explicit RG formulation is very general since it can be defined for any master equation, and it gives new insights into the main ingredients of the droplet scaling picture. As an application, we have followed numerically the RG flow for the case of a directed polymer in a two-dimensional random medium to obtain histograms of the free-energy, entropy and energy differences between the two longest-lived valleys in each sample.