Problems in nonequilibrium fluctuations across scales: A path integral approach

In this thesis we study stochastic systems evolving with Markov jump processes. In a first work we discuss different representations of the stochastic evolution: the master equation, the generalized Langevin equation, and
their path integrals. The description is used to derive the generating functions for out of equilibrium observables,
together with the typical approximation techniques. In a second work the path integral is used to enforce
thermodynamic consistency across scales. The description of identical units with all-to-al interactions is reduced
from a micro- to a meso- to a macroscopic level. A suitable scaling of the dynamics and of the thermodynamic
observables allows to preserve the thermodynamical structure at the different levels. In a third work we focus
on the large deviation properties of chemical networks. The path integral allows to compute the dominant
trajectories that constitute macroscopic fluctuations. For bi-stable systems the existence of multiple macroscopic
contributions results in a phase transition for the macroscopic current. In a fourth work we study the response of
such chemical currents to external perturbations. Out of equilibrium the system can display negative differential
response, a feature that offers different strategies to minimize external or internal disturbances. Finally, in a
fifth work, we start from a quantum system where part of the system can be traced out to act as multiple
reservoirs at different temperatures. Using the Schwinger-Keldysh contour and Green's functions we can obtain
the generating function for the different parts of the hamiltonian. The statistics of thermodynamic observables
is accessible even in the strong coupling regime, while the semi-classical approximation is in agreement with the
classical counterpart.