[Paper Review] Toward the Fourier law for a weakly interacting anharmonic crystal

Carlangelo Liverani, Stefano Olla|arXiv (Cornell University)|Jun 15, 2010

Advanced Thermodynamics and Statistical Mechanics7 references4 citations

TL;DR

This paper establishes the first step toward deriving the Fourier law of heat conduction in weakly coupled anharmonic crystals by proving that, in the diffusive time scale $\varepsilon^{-2}t$, the energies of oscillators evolve according to a conservative, non-gradient Ginzburg-Landau stochastic differential equation. The result relies on hypocoercivity and hypoellipticity of the noise-perturbed dynamics, showing that energy currents emerge via a central limit theorem in the weak coupling limit.

ABSTRACT

For a system of weakly interacting anharmonic oscillators, perturbed by an energy preserving stochastic dynamics, we prove an autonomous (stochastic) evolution for the energies at large time scale (with respect to the coupling parameter). It turn out that this macroscopic evolution is given by the so called conservative (non-gradient) Ginzburg-Landau system of stochastic differential equations. The proof exploits hypocoercivity and hypoellipticity properties of the uncoupled dynamics.

Motivation & Objective

• To rigorously derive the macroscopic evolution of energy in a weakly interacting anharmonic crystal under stochastic energy-conserving dynamics.
• To bridge the gap between microscopic Hamiltonian dynamics and the diffusive scaling leading to the heat equation.
• To establish that the energy dynamics at the $\varepsilon^{-2}$ time scale converges to a conservative, non-gradient stochastic process.
• To validate the use of stochastic perturbations as a tool to induce ergodicity and mixing without altering macroscopic energy transport.

Proposed method

• Introduce a weak coupling parameter $\varepsilon$ between anharmonic oscillators with energy-conserving noise on each particle’s kinetic energy.
• Apply a diffusive time rescaling $t \to \varepsilon^{-2}t$ to observe energy current fluctuations.
• Use the generator $L_\varepsilon = \{H_\varepsilon, \cdot\} + \sigma^2 \sum X_i^2$ to model the stochastic dynamics, where $X_i$ generates rotations on momentum spheres.
• Establish convergence of the energy process by analyzing the martingale component of the energy evolution and controlling the quadratic variation via hypoellipticity.
• Prove that the limiting energy process satisfies a degenerate SDE of the form $d\mathcal{E}_i = \sigma^{-1}\sqrt{2\mathcal{E}_1\mathcal{E}_2}\,dw_t - 2\sigma^{-2}(\mathcal{E}_1 - \mathcal{E}_2)\,dt$, representing a non-gradient Ginzburg-Landau system.
• Leverage hypocoercivity and hypoellipticity to control the convergence of the energy process and ensure the validity of the central limit theorem for currents.

Experimental results

Research questions

• RQ1 Can the energy dynamics of weakly coupled anharmonic oscillators converge to a macroscopic stochastic process in the diffusive limit $\varepsilon^{-2}t$?
• RQ2 Does the presence of energy-conserving noise induce sufficient mixing to justify a central limit theorem for energy currents?
• RQ3 Is the limiting macroscopic energy evolution a conservative, non-gradient Ginzburg-Landau SDE, as expected from hydrodynamic theory?
• RQ4 Can the non-gradient nature of the current be rigorously derived from microscopic dynamics using weak coupling and stochastic perturbations?
• RQ5 Does the limiting SDE correspond to a non-linear heat equation under further space-time diffusive scaling?

Key findings

• In the limit $\varepsilon \to 0$, the energies $\mathcal{E}_{1,\varepsilon}(t)$ and $\mathcal{E}_{2,\varepsilon}(t)$ converge to a diffusion process on $\mathbb{R}_+^2$ governed by the generator $\mathcal{L} = 2\sigma^{-2}(\partial_{\mathcal{E}_1} - \partial_{\mathcal{E}_2})\mathcal{E}_1\mathcal{E}_2(\partial_{\mathcal{E}_1} - \partial_{\mathcal{E}_2})$.
• The limiting dynamics is described by the SDE $d\mathcal{E}_1 = \sigma^{-1}\sqrt{2\mathcal{E}_1\mathcal{E}_2}\,dw_t - 2\sigma^{-2}(\mathcal{E}_1 - \mathcal{E}_2)\,dt$, which is conservative and non-gradient.
• The martingale component $\varepsilon M^u_{\varepsilon^{-2}t}$ converges to a Wiener process, with quadratic variation controlled by the $L^2$-norm of the vector fields $X_i$.
• The error terms from higher-order corrections vanish in the limit $\varepsilon \to 0$, due to the use of hypocoercivity and the Schwartz inequality on the $S\phi$ term.
• The non-gradient nature of the current arises from the fact that the current $j$ is not the gradient of a local function of the energies, unlike in the harmonic case.
• The result reduces the derivation of the heat equation to a two-step program: first, the convergence to the Ginzburg-Landau SDE is rigorously established; second, the diffusive scaling of space and time must be analyzed to recover the non-linear heat equation.