[Paper Review] On the Boltzmann Equation for Weakly Nonlinear Wave Equations

Herbert Spohn|ArXiv.org|Jun 6, 2007

Gas Dynamics and Kinetic Theory29 references5 citations

TL;DR

This paper formulates a kinetic theory for weakly nonlinear wave equations using the Boltzmann equation framework, deriving a phonon Boltzmann equation for systems with cubic and quartic nonlinearities. It shows that entropy production is non-negative and identifies stationary solutions as those corresponding to thermal equilibrium when 3-phonon mergers or cubic on-site potentials are present, resolving long-standing questions about relaxation in systems like the Fermi-Pasta-Ulam chain.

ABSTRACT

We explain how the kinetic theory of L. Boltzmann is applied to weakly nonlinear wave equations.

Motivation & Objective

• To extend the Boltzmann kinetic theory to weakly nonlinear wave equations, particularly in the context of anharmonic lattices.
• To understand the relaxation dynamics and approach to thermal equilibrium in systems like the Fermi-Pasta-Ulam (FPU) chain.
• To clarify the role of different nonlinearities (cubic vs. quartic) in determining the rate of relaxation and the existence of collisional invariants.
• To establish conditions under which the kinetic equation yields thermal equilibrium as the unique stationary solution.
• To provide a rigorous kinetic description of energy transport and relaxation in weakly nonlinear wave systems using Wigner functions and collision integrals.

Proposed method

• Formulates the scalar wave equation with cubic nonlinearity on a discrete lattice, using a Fourier transform to express dynamics in momentum space.
• Introduces the Wigner function to represent the state of the system, enabling a phase-space description of wave dynamics.
• Derives the kinetic equation via time-dependent perturbation theory, leading to a collision integral involving four-wave interactions.
• Analyzes the collision integral using delta functions enforcing energy and momentum conservation: $\delta(\sum \sigma_j \omega_j)$ and $\delta(\sum \sigma_j k_j)$.
• Identifies collisional invariants as functions $\psi(k)$ satisfying $\psi(k) = a + c\omega(k)$, which characterize stationary solutions.
• Evaluates entropy production via the Boltzmann H-theorem, showing non-negativity and equality to zero only for collisional invariants.

Experimental results

Research questions

• RQ1 Under what conditions does the kinetic equation derived for weakly nonlinear wave equations yield thermal equilibrium as the unique stationary solution?
• RQ2 How do different types of nonlinearity—cubic (e.g., FPU-α) versus quartic (e.g., FPU-β or on-site potential)—affect the relaxation rate and the structure of the collision integral?
• RQ3 What is the role of 3-phonon and 2-phonon merger processes in determining the existence of non-equilibrium stationary states?
• RQ4 How does the kinetic theory explain the absence of thermalization in the original FPU experiment, particularly in the FPU-α chain?
• RQ5 In what function space are collisional invariants well-defined, and how does the absence of flat parts in the dispersion relation $\omega(k)$ affect the classification of these invariants?

Key findings

• The entropy production in the derived kinetic equation is non-negative, and vanishes if and only if $1/W(k)$ is a positive collisional invariant.
• For a dispersion relation without flat parts, the only collisional invariants are of the form $\psi(k) = a + c\omega(k)$, implying a two-parameter family of stationary solutions.
• The inclusion of 3-phonon mergers or a cubic on-site potential ensures that only thermal equilibrium (corresponding to $W(k) \propto \omega(k)^{-1}$) is a stationary solution.
• In the FPU-α chain with cubic nonlinearity, the collision term vanishes, indicating poor relaxation and explaining the lack of thermalization observed numerically.
• For the FPU-β chain with quartic nonlinearity, the energy current correlation decays as $t^{-3/5}$, consistent with slow relaxation predicted by kinetic theory.
• With a quartic on-site potential, the energy current correlation decays exponentially, and molecular dynamics simulations confirm rapid convergence to equilibrium, validating the kinetic theory in this regime.