2001/06/06 by Nicolas Martzel, N. Martzel, Claude Aslangul +1 · 5 citations
Materials Science · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Material Dynamics and Properties #Statistical Mechanics and Entropy #cond-mat.stat-mech
paper · pdf · doi:10.1088/0305-4470/34/50/305
13 pages, 8 figures
arxiv created 2001/06/06 · openalex publication_date 2001/12/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We review some properties of the stationary states of the Fokker–Planck equation for N interacting particles within a mean-field approximation, which yields a non-linear integrodifferential equation for the particle density. Analytical results show that for attractive long-range potentials the steady state is always a precipitate containing one or several clusters of small size. For arbitrary potential, linear stability analysis allows the statement of the conditions under which the uniform equilibrium state is unstable against small perturbations and, via the Einstein relation, definition of a critical temperature T c separating the two phases, uniform and precipitate. The corresponding phase diagram turns out to be strongly dependent on the pair-potential. In addition, numerical calculations reveal that the transition is hysteretic. We finally discuss the dynamics of relaxation for the uniform state suddenly cooled below T c .