Abstract
Diffusive systems, such as lattice gas models, are a class of systems that verify Fourier's law. In the last fifteen years or so, we have developed a fairly complete theory, the macroscopic theory of fluctuations, which allows us to understand the fluctuations and large deviations in current and density of these diffusive systems. After describing a few examples of diffusive systems: exclusion models, independent walkers and the KMP (Kipnis, Presutti, Marchioro) model, the lecture focused on determining the transport coefficients, which are the only characteristics of a given system that need to be known within the framework of this macroscopic fluctuation theory.
Three methods can be envisaged for determining these transport coefficients: (i) measuring the current in an open system in contact at both ends with thermostats at similar temperatures, (ii) measuring current fluctuations for a system at equilibrium, (iii) measuring the current induced by a field (gravity or electric in the case of particle transport). For some systems, gradient models, the energy current is written as a gradient, making it easy to determine the transport coefficients. For non-gradient systems, however, it is generally impossible to determine the transport coefficients exactly. Using the principle of entropy minimization, however, we have a variational method for determining these coefficients approximately.
