Anomalous diffusion for a class of systems with two conserved quantities

Abstract

We introduce a class of one dimensional deterministic models of energyvolume conserving interfaces. Numerical simulations show that these dynamics are genuinely super-diffusive. We then modify the dynamics by adding a conservative stochastic noise so that it becomes ergodic. System of conservation laws are derived as hydrodynamic limits of the modified dynamics. Numerical evidence shows these models are still super-diffusive. This is proven rigorously for harmonic potentials.

Anomalous Heat Transport in One Dimensional Systems: A Description Using Non-local Fractional-Type Diffusion Equation

“…The above results suggest that, in the open set-up where the system is connected to two reservoirs at different temperatures, this model would exhibit anomalous scaling of the steady state current j with system size N . In [30], it has been numerically shown that indeed j ∼ 1/ √ N . Recently, an understanding of the open system was achieved using the fractional equation description, which we now discuss [34].…”

“…For quadratic V (r), i.e harmonic chains, there are a macroscopic number of conserved quantities and transport becomes ballistic. In this case a number of studies have considered augmenting the Hamiltonian dynamics with a stochastic component such that the system again has only three conserved quantities [9,[29][30][31]. In this case one again recovers the typical features of anomalous transport and several exact results are possible.…”

“…The probability Q(x) can now be expressed in terms of H(x) as Q(x) = (Q l − Q r )H(x) + Q r , which can be checked easily to satisfy Eq. (30).…”

“…We refer to this model as Harmonic chain with volume exchange (HCVE). This model was introduced in [30] where it was shown that the energy current auto-correlation decays as ∼ 1/ √ t, implying super-diffusive transport. It is easy to see that this system has only two conserved quantities namely, the total "volume" i η i and the total energy i η 2 i .…”

Computational Statistical Physics

“…In addition, the study by G. Stoltz and C. Bernardin on thermal transport in onedimensional chains of oscillators whose kinetic and potential energy functions are the same, has been accepted and is now published [13].…”