Abstract
After recalling Fourier's and Fick's laws which, combined with the laws of conservation of energy or particle number, lead to the equations of heat or diffusion, this first lecture described the oldest models, such as those based on the idea of a mean free path in a gas, enabling these laws to be deduced. It then showed how Kubo's formula can be used to deduce transport coefficients from temporal current correlations measured at equilibrium. The simplest models of fluids or solids (such as the perfect gas or the harmonic solid) do not verify Fourier's law: indeed, Fourier's law predicts an energy current inversely proportional to the length of the system, whereas for these too-simple models the transport (of atoms for the perfect gas and of phonons for the harmonic solid) is ballistic and the current does not depend on the length of the system. Surprisingly, numerous simulations have shown that in low dimensions (dimensions 1 and 2) Fourier's law is not verified either, even for interacting particle gases or anharmonic solids. For these systems, we observe an anomalous Fourier law, i.e. an energy current proportional to a non-integer negative power of the system length.
