Thermal Transport in Low Dimensions by Stefano Lepri

Author:Stefano Lepri
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(5.3)

where , with {T x, x+1( j)} the random times when N x, x+1 jumps, and p x (t −) is the velocity of the particle x just before time t, i.e. .

The evolution of the probability density on the configurations then follows the Fokker-Planck equations:

(5.4)

where A is the Hamiltonian operator

(5.5)

while S is the generator of the random exchanges

(5.6)

where p x, x+1 is the configuration obtained exchanging p x and p x+1.

This stochastic perturbation of the Hamiltonian dynamics has the property to conserve the total energy, and in the unpinned case the resulting dynamics conserves also the total momentum (), and the volume or strain of the chain (). It has also the property that these are the only conserved quantities. In this sense it gives the necessary ergodicity to the dynamics [13, 23].

We have also considered other type of conservative random dynamics, like a continuous random exchange of the momenta of each triplets {p x−1, p x , p x+1}. The intersection of the kinetic energy sphere p x−1 2 + p x 2 + p x+1 2 = C with the plane p x−1 + p x + p x+1 = C′ gives a one dimensional circle. Then we define a dynamics on this circle by a standard Wiener process on the corresponding angle. This perturbation is locally more mixing, but it gives the same results for the macroscopic transport.

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