Random transitions described by the stochastic Smoluchowski-Poisson system and by the stochastic Keller-Segel model

Area of Science:

• Statistical Physics
• Complex Systems
• Non-equilibrium Thermodynamics

Background:

• Phase transitions from homogeneous to inhomogeneous states occur in systems with long-range interactions below a critical temperature.
• Metastable states in such systems can exhibit bistability, switching between configurations.
• Finite particle number (N) effects and fluctuations become significant near critical points, challenging mean-field approximations.

Purpose of the Study:

• To numerically investigate random transitions between two metastable states in a one-dimensional self-gravitating Brownian gas.
• To analyze the role of fluctuations (finite N effects) in these transitions.
• To compare the observed phenomenology with the Kramers problem and the Arrhenius law for metastable state lifetimes.

Main Methods:

• Numerical solution of N-body Langevin equations.
• Numerical solution of the stochastic Smoluchowski-Poisson system, incorporating fluctuations.
• Analysis of residence time distributions and average lifetimes.

Main Results:

• Observed random transitions between two metastable states, exhibiting bistability.
• Demonstrated that these transitions follow the Kramers problem phenomenology for a double-well potential.
• Found Poissonian residence time distributions and Arrhenius law scaling for average lifetimes, proportional to exp(ΔF/kBT).

Conclusions:

• Metastable state lifetimes in systems with long-range interactions scale exponentially with particle number (e^N), potentially appearing stable for large N.
• Near critical points or for moderate N, fluctuations reduce the free energy barrier, decreasing lifetime and invalidating mean-field theory.
• The findings are applicable to bacterial chemotaxis dynamics described by stochastic Keller-Segel models.