Superdiffusivity of Two Dimensional Lattice Gas Models

Abstract

Abstract. It was proved [4,13] that stochastic lattice gas dynamics converge to the Navier-Stokes equations in dimension d = 3 in the incompressible limits. In particular, the viscosity is finite. We proved that, on the other hand, the viscosity for a two dimensional lattice gas model diverges faster than log log t. Our argument indicates that the correct divergence rate is (log t) 1/2 . This problem is closely related to the logarithmic correction of the time decay rate for the velocity auto-correlation function of a tagged particle.

Superdiffusive Bounds on Self-repellent Brownian Polymers and Diffusion in the Curl of the Gaussian Free Field in d=2

“…Given the explicit expressions (11), (12) and (13) for J 1 (v) and J 2 (v), in order to prove the superdiffusive lower bound in (5) we need to prove efficient upper bounds on J 3 (v).…”

“…(See [12] for another example.) In the Appendix we present a non-rigorous, nevertheless very instructive scaling argument which explains this conjecture.…”

Diffusivity of Lattice Gases

“…Remark 2. The d = 1, j (ρ) = 0, j (ρ) = 0 case bears many similarities in scaling of the diffusivity to the d = 2 lattice gas models for Navier-Stokes studied in [20] and certain random diffusions in random environment in d = 2 [33]. This is not completely coincidental as there are some structural similarities between models in d = 1 at inflection points and models in d = 2 without preferred direction which make the both superdiffusivity and the estimates based on degree 3 test functions coincide in the two cases.…”

“…Note that the full predictions for the superdiffusivities can be obtained formally from the variational method if one assumes certain scaling properties of the optimal test function, and (more severely) that all off diagonal terms in the computations vanish. The formal computations are described in [20,27,33]. Since they are fairly analogous in our case, we do not repeat them here.…”

Superdiffusive bounds on self-repellent precesses in d=2 — extended abstract

“…The proof of Theorem 1 follows the main lines of [5]. See also [4]. However on the computational level there are notable differences.…”