Anomalous Fluctuations for a Perturbed Hamiltonian System with Exponential Interactions

Abstract

A one-dimensional Hamiltonian system with exponential interactions perturbed by a conservative noise is considered. It is proved that energy superdiffuses and upper and lower bounds describing this anomalous diffusion are obtained.

On the Fibonacci Universality Classes in Nonlinear Fluctuating Hydrodynamics

“…However, in the diffusive case 1 one expects the Gaussian scaling function to be the true scaling limit, up to possible logarithmic corrections. For specific models there are numerical [17,18,19] and mathematically rigorous results [1,2] that suggest that the true scaling form in case 2 is indeed generally a Lévy distribution. However, the coefficients A α , E α arising from the mode-coupling equations are not believed to correspond to the true values.…”

“…A microscopic configuration at time t is thus given by η(t) = {η k (t) : k ∈ Λ }. 1 The generator of the dynamics is denoted by L . The translation operator is denoted by T and defined by the property T (η k ) = η k+1 , and similar for functions of the local state variables.…”

“…• Case 2: For modes α such that I α = / 0 and α / ∈ I α the dynamical exponents satisfy the nonlinear recursion z α = min β ∈I α 1 + 1 z β and the scaling function is given 1] given in [20] (Lévy universality class).…”

“…In Proposition 1 the parameters ρ i = n i k are the conserved stationary densities and one finds immediately the compressibility matrix K with matrix elements K αβ = ρ α δ αβ . From the definition (22) of the generator one computes the microscopic currents defined up to an irrelevant constant by the discrete continuity equation (1). The factorization of the invariant measure then yields the stationary currentdensity relation…”

“…Also the spatiotemporal fluctuations of the densities ρ α (x,t) of globally conserved quantities N α (such as mass, energy, etc.) are frequently characterized by non-diffusive scaling properties with dynamical exponents z α = 2, including the celebrated Kardar-Parisi-Zhang (KPZ) universality class where z = 3/2 [11,27], and another universality class with z = 3/2 [1,2,17,25].…”