Generalized Langevin equations and memory effects in non-equilibrium statistical physics

The dynamics of many-body complex processes is a challenge that many scientists
from various fields have to face. Reducing the complexity of systems involving a
large number of bodies in order to reach a simple description for observables captur-
ing the main features of the process is a difficult task for which different approaches
have been proposed over the past decades. In this thesis we introduce new tools
to describe the coarse-grained dynamics of arbitrary observables in non-equilibrium
processes. Following the projection operator formalisms introduced first by Mori
and Zwanzig, and later on by Grabert, we first derive a non-stationary Generalized
Langevin Equation that we prove to be valid in a wide spectrum of cases. This
includes in particular driven processes as well as explicitly time-dependent observ-
ables. The equation exhibits a priori memory effects, controlled by a so-called non-
stationary memory kernel. Because the formalism does not provide extensive infor-
mation about the memory kernel in general, we introduce a set of numerical meth-
ods aimed at evaluating it from Molecular Dynamics simulation data. These proce-
dures range from simple dimensionless estimations of the strength of the memory
to the determination of the entire kernel. Again, the methods introduced are very
general and require as input a small number of quantities directly computable from
numerical of experimental timeseries. We finally conclude this thesis by using the
projection operator formalisms to derive an equation of motion for work and heat in
dissipative processes. This is done in two different ways, either by using well-known
integral fluctuation theorems, or by explicitly splitting the dynamics into adiabatic
and dissipative parts.