Non-equilibrium Phase Transitions with Long-Range Interactions

TL;DR

This review explores non-equilibrium phase transitions with long-range interactions, focusing on Levy flights and fractional derivatives, revealing continuous phase transitions with critical exponents depending on interaction parameters.

Contribution

It provides a comprehensive overview of recent theoretical advances in modeling long-range interactions in non-equilibrium phase transitions using fractional calculus.

Findings

Systems with Levy-distributed interactions show continuous phase transitions.

Critical exponents vary with interaction parameters sigma and kappa.

Fractional derivatives effectively describe long-range effects in these systems.

Abstract

This review article gives an overview of recent progress in the field of non-equilibrium phase transitions into absorbing states with long-range interactions. It focuses on two possible types of long-range interactions. The first one is to replace nearest-neighbor couplings by unrestricted Levy flights with a power-law distribution P(r) ~ r^(-d-sigma) controlled by an exponent sigma. Similarly, the temporal evolution can be modified by introducing waiting times Dt between subsequent moves which are distributed algebraically as P(Dt)~ (Dt)^(-1-kappa). It turns out that such systems with Levy-distributed long-range interactions still exhibit a continuous phase transition with critical exponents varying continuously with sigma and/or kappa in certain ranges of the parameter space. In a field-theoretical framework such algebraically distributed long-range interactions can be accounted for by replacing the differential operators nabla^2 and d/dt with fractional derivatives nabla^sigma and (d/dt)^kappa. As another possibility, one may introduce algebraically decaying long-range interactions which cannot exceed the actual distance to the nearest particle. Such interactions are motivated by studies of non-equilibrium growth processes and may be interpreted as Levy flights cut off at the actual distance to the nearest particle. In the continuum limit such truncated Levy flights can be described to leading order by terms involving fractional powers of the density field while the differential operators remain short-ranged.