[Paper Review] Macroscopic parameters of Fokker-Planck flows

Igor A. Tanski|ArXiv.org|Jun 6, 2006

Statistical Mechanics and Entropy2 references3 citations

TL;DR

This paper derives macroscopic hydrodynamic equations for Fokker-Planck systems by averaging over velocity distributions, showing that the resulting continuum model behaves like an isothermal, viscous, compressible fluid. Using Fourier transforms of the distribution function, it derives conservation laws for mass, momentum, and energy, and establishes a state equation linking the Fokker-Planck coefficients (damping α and diffusion k) to fluid viscosity and temperature via k/α = kBT.

ABSTRACT

The aim of this work is to investigate properties of solutions of Fokker - Planck equation in the context of continuum mechanics. We show that average quantities, calculated for these solutions approximately satisfy equations of isothermal motion of viscous ideal gas.

Motivation & Objective

• To derive macroscopic hydrodynamic equations from the Fokker-Planck equation using velocity-space averaging.
• To identify conservation laws (mass, momentum, energy) satisfied by averaged quantities in non-interacting particle systems.
• To establish a state equation for the Fokker-Planck continuum, relating stress and pressure to kinetic parameters.
• To connect Fokker-Planck coefficients (α, k) to fluid properties like viscosity and temperature via k/α = kBT.
• To analyze the fundamental solution of the Fokker-Planck equation and verify hydrodynamic consistency in the long-time limit.

Proposed method

• Define macroscopic quantities (density ρ, velocity ui, stress σij, energy e) via velocity-space averages of the distribution function n(xi, vi).
• Use the Fourier transform M(t, xi, qj) of the distribution function to express macroscopic parameters as derivatives of M at qj = 0.
• Derive evolution equations for M by multiplying the Fokker-Planck equation by exp(−ivkqk) and integrating over velocities.
• Substitute qj = 0 into transformed equations to obtain mass, momentum, and energy conservation laws.
• Assume a Gaussian form for the Fourier transform M to close the system and derive a state equation resembling the ideal gas law.
• Verify consistency of derived equations with the fundamental solution of the Fokker-Planck equation in the long-time limit.

Experimental results

Research questions

• RQ1 Do the averaged macroscopic quantities of Fokker-Planck solutions satisfy standard continuum mechanics conservation laws?
• RQ2 Can the Fokker-Planck system be interpreted as a viscous, compressible fluid with an equation of state?
• RQ3 What is the physical interpretation of the Fokker-Planck coefficients α and k in terms of fluid viscosity and temperature?
• RQ4 How do the macroscopic parameters (density, velocity, stress) evolve in the fundamental solution of the Fokker-Planck equation?
• RQ5 Is the momentum conservation law satisfied by the derived hydrodynamic equations in the long-time limit?

Key findings

• The averaged density ρ and velocity ui satisfy the continuity equation ∂ρ/∂t + ∂(ρui)/∂xi = 0.
• The momentum conservation law ∂(ρui)/∂t + ∂(ρuiuj)/∂xj − ∂σij/∂xj + αρui = 0 is derived and verified for the fundamental solution.
• The energy conservation law ∂e/∂t + ∂Fi/∂xi − 2αe = −3kρ holds, with Fi being the energy flux.
• The stress tensor σij is related to the velocity variance, and the equation of state p = 2/3 E is recovered, consistent with kinetic theory.
• The relation k/α = kBT is established, linking the Fokker-Planck coefficients to temperature via Boltzmann’s constant.
• In the long-time limit, the density ρ approximates the solution of a diffusion equation, and the system exhibits viscous, compressible fluid behavior with strain rate εij = (α/θ)(1−e−αt)²δij.