Jani Lukkarinen

Fourier's law for a harmonic crystal with self-consistent stochastic reservoirs

We consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the “exterior” left and right heat baths are at specified values TLand TR, respectively, while the temperatures of the “interior” baths are chosen self-consistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fouriers law with a finite positive thermal conductivity which can also be computed using the Green–Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich, and Visscher in 1970 who first studied this model. In the other limit, d>>1, the stationary infinite volume heat conductivity behaves as (ldd)−1where ldis the coupling to the intermediate reservoirs. We also analyze the effect of having a non-uniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit.

Kinetic Limit for Wave Propagation in a Random Medium

We study crystal dynamics in the harmonic approximation. The atomic masses are weakly disordered, in the sense that their deviation from uniformity is of the order

Energy Transport in Weakly Anharmonic Chains

\sqrt{\epsilon}

Not to Normal Order—Notes on the Kinetic Limit for Weakly Interacting Quantum Fluids

. The dispersion relation is assumed to be a Morse function and to suppress crossed recollisions. We then prove that in the limit

Paradoxical reflection in quantum mechanics

\epsilon\to 0

Nonequilibrium stationary states of harmonic chains with bulk noises

, the disorder-averaged Wigner function on the kinetic scale, time and space of order