Abstract
Taking up the case of the Langevin equation in the low-noise limit, the fifth lecture began by showing how to write the Hamilton-Jacobi equations for the quasi-potential. In the framework of macroscopic fluctuation theory, the non-locality of this quasi-potential is directly linked to the presence of long-range correlations. For systems in equilibrium, we know how to express the quasi-potential from the free energy. In non-equilibrium systems, it is generally impossible to calculate the quasi-potential, with the exception of a few models such as symmetric exclusion. Nevertheless, we can write the Hamilton-Jacobi equations satisfied by this quasi-potential. The lecture ended by showing how, in the case of symmetrical exclusion, the quasi-potential can be calculated from the expression of the weights given by the matrix ansatz, making it possible to check that the Hamilton-Jacobi equations are indeed satisfied.
