Abstract
Fluctuating hydrodynamics describes the large-scale properties of diffusive systems through the noisy evolution of two fields: density and current, which are linked by a conservation law. It is not always easy to deduce the equations of fluctuating hydrodynamics from a microscopic model. However, this is possible for sufficiently simple models such as symmetrical exclusion, and relatively easy to do for independent walkers.
After introducing the various properties of diffusive systems that can be determined using fluctuating hydrodynamics (quasi-potential, onset and relaxation trajectories of fluctuations, large current deviation functions, those of empirical density profiles), the fourth lecture showed how mean density profiles or correlations of density fluctuations can be obtained. The limit of a large system translates into a low-noise limit for fluctuating hydrodynamics. In this limit, as can easily be seen in the case of a Langevin equation in the low-noise limit, the problem most often reduces to finding the trajectory that minimizes a certain action. The tools of analytical mechanics and the principle of least action can thus be used.
The special cases of a double well where the idea of an optimal trajectory is not sufficient, or that of an out-of-equilibrium system where the quasi-potential is non-analytical, have also been discussed.
