Entropic hypocoercivity via the Polchinski flow: an application to the lattice sine-Gordon model

Abstract

We develop a multiscale criterion for exponential decay of relative entropy in the canonical stochastic quantization of lattice field theories, the underdamped Langevin dynamics of the field and its momentum. For smooth bounded perturbations of Gaussian fields, the criterion requires the uniform operator-norm bound on the smoothed interaction Hessian along the Polchinski flow of Bauerschmidt and Bodineau to have a small $L^2$ norm in the scale variable. The decay constants depend only on this $L^2$ norm and the friction. The proof measures the transfer of momentum dissipation to position on the same scales that decompose entropy. A phase-space extension of the Polchinski flow gives Gaussian coarse-graining that commutes with free kinetic evolution. Integrating position--momentum cross terms over these scales yields a corrector bounded by entropy. A force defect accounts for changes in the averaged interaction force away from equilibrium; its evolution in scale controls the errors in the kinetic estimate. For the massive two-dimensional lattice sine--Gordon model with $0<β<4π$ and small dimensionless local coupling, the criterion gives entropy decay for all finite-entropy initial laws, uniformly in the lattice spacing and the volume. For friction proportional to the mass $m$, the rate has optimal order $m$.