Abstract
The second lecture was devoted to one-dimensional models of insulating solids. The heat flux expression can be described as the average work of the force exerted on an atom by its left-hand neighbor. The difficulty is that (unlike equilibrium systems) there is no explicit expression for the stationary measure. Only in the case of the harmonic chain with thermostats represented by Langevin forces can this stationary measure, which is Gaussian, be calculated. The problem with the harmonic chain is that the different modes don't interact with each other, so for a periodic chain the dynamics don't allow for equilibration.
In the case of the anaharmonic chain, the question of achieving equilibrium and equipartition of energy between modes dates back to the mid-1950s with the Fermi-Pasta-Ulam chain. Some anaharmonic chains, such as the Toda chain, are integrable and thus possess an infinite number of conserved quantities, which, like the harmonic chain, prevents them from reaching equilibrium. Apart from these integrable cases, simulations of anaharmonic chains generally show an anomalous Fourier law with the following main characteristics: a current that decays like a non-integer power law of size, a non-linear temperature profile even when thermostat temperatures are close, algebraically decaying current correlations, and fluctuations that decay with a diffusion law where the Laplacian is replaced by a fractional Laplacian.
The lecture ended by mentioning certain models introduced in recent years, such as the HCME chain, which are harmonic chains to which a stochastic element is added by exchanging pulses from neighboring sites at random times. These oscillator chains, which are simpler than anaharmonic chains, conserve energy and impulse, and exhibit most of the characteristics of the anomalous Fourier law.
