Lèvy noise induced steady states

Abstract

Systems far from equilibrium organize themselves to accommodate energy throughput. It is also in these nonequilibrium systems where noise has often been found to follow alpha-stable distributions, commonly called Lévy noise, rather than Gaussian distributions. There is no general theory that links these alpha-stable distributions to the resultant thermodynamic behavior of a system as a whole. Here two different model systems are investigated for which the assumption of Lévy noise leads to behavior that deviates from that seen at equilibrium. We begin by examining trajectories of overdamped noise-driven particles in a harmonic potential. These trajectories display broken time reversal symmetry due to the large displacements inherent to Lévy noise. A parameter to measure this symmetry breaking and estimate the stability parameter, [alpha], of the underlying noise is proposed. This parameter is applied to a time series of solar x-ray irradiance and compared to previous methods. Next, we study the same overdamped particles in a 2D system with simple semi-circular cavities. Lévy noise in such a system will lead to a preferential accumulation of particles in one cavity. The nonhomogeneous steady-state represents a lower entropy configuration in comparison to equilibrium. The chosen system leads to concise expressions for the distribution of particles within the cavities as well as the concomitant entropy reduction. Such structures maintained in nonequilibrium have been referred to as dissipative structures because they may aid the system in transporting or dissipating energy.