2007/02/28 by Haye Hinrichsen · 1 citation
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Critical exponent #Differential (mechanical device) #Distribution (mathematics) #Exponent #Limit (mathematics) #Lévy flight #Phase transition #Sigma #Theoretical and Computational Physics #cond-mat.stat-mech #stochastic dynamics and bifurcation
paper · pdf · doi:10.1088/1742-5468/2007/07/p07006
published as J. Stat. Mech.: Theor. Exp. P07066 (2007) · LaTeX, 39 pages, 13 figures, minor revisions
arxiv created 2007/03/19 · openalex publication_date 2007/07/04 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
This paper 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-neighbour couplings by unrestricted Lévy flights with a power-law distribution P ( r )∼ r − d −σ controlled by an exponent σ. Similarly, the temporal evolution can be modified by introducing waiting times Δ t between subsequent moves which are distributed algebraically as P (Δ t )∼(Δ t ) −1−κ . It turns out that such systems with Lévy-distributed long-range interactions still exhibit a continuous phase transition with critical exponents varying continuously with σ and/or κ 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 and with fractional derivatives and . 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 Lévy flights cut off at the actual distance to the nearest particle. In the continuum limit such truncated Lévy flights can be described to leading order by terms involving fractional powers of the density field while the differential operators remain short-ranged.