Phase Transition for Discrete Non Linear Schrödinger Equation in Three and Higher Dimensions

2022/12/01 by Dey, Partha S., Kirkpatrick, Kay, Krishnan, Kesav
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2212.00276

Abstract

We analyze the thermodynamics of the focusing discrete nonlinear Schrödinger equation in dimensions d≥ 3 with general nonlinearity p>1 and under a model with two parameters, representing inverse temperature and strength of the nonlinearity, respectively. We prove the existence of limiting free energy and analyze the phase diagram for general d,p. We also prove the existence of a continuous phase transition curve that divides the parametric plane into two regions involving the appearance or non-appearance of solitons. Appropriate upper and lower bounds for the curve are constructed. We also look at the typical behavior of a function chosen from the Gibbs measure for certain parts of the phase diagram.

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