Adiabatic elimination and reduced probability distribution functions in spatially extended systems with a fluctuating...

Area of Science:

• Nonlinear Dynamics
• Statistical Physics
• Computational Physics

Background:

• Understanding phase transitions and critical phenomena in systems with fluctuating parameters is crucial.
• The Ginzburg-Landau and Swift-Hohenberg equations model diverse physical phenomena, including pattern formation and superconductivity.

Purpose of the Study:

• To determine the stationary probability distribution functions of the order parameter near onset for 1D Ginzburg-Landau and Swift-Hohenberg equations.
• To investigate the impact of a fluctuating control parameter on these systems.
• To analytically solve for the probability distribution of the slowest evolving mode.

Main Methods:

• Perturbative expansion of the fluctuation intensity.
• Derivation of a hierarchy of Fokker-Planck equations for conditional probability distributions.
• Successive integration to obtain a Fokker-Planck equation for the slowest mode.
• Analytical solution of the resulting Fokker-Planck equation.

Main Results:

• The probability distribution function above onset takes the form P(A0) ~ A(delta)(0)e(-gammaA20).
• The exponents delta and gamma depend explicitly on the fluctuation intensity.
• An effective bifurcation threshold and moments of the order parameter above threshold can be calculated.

Conclusions:

• The study provides an analytical framework for understanding the influence of fluctuations on critical behavior.
• The derived probability distribution offers insights into the statistical properties of order parameters in noisy systems.
• The method is applicable to various models exhibiting similar dynamical behaviors.