In this thesis, we investigate variational structures for ﬂuctuations in Markov processes, with a particular focus on interacting particle systems (such as the simple exclusion process and the zero-range process). A great part of this thesis is devoted to time-reversal symmetry. We discuss the acceleration of convergence to the steady state for dissipative systems, where we revisit the fact that ‘breaking detailed balance’ accelerates the convergence to equilibrium and extend known results to the case of interacting particle systems and their hydrodynamic scaling limits. The theoretical ﬁndings are supported by simulations of independent particles and the zero-range process in one and two space dimensions. We further investigate a general Ψ-Ψ? structure for the Onsager-Machlup functional Φ, which can be used to represent several large-deviation rate functions for particle diﬀusions, Markov chains and Macroscopic Fluctuation Theory. We discuss a splitting of the thermodynamic force acting on the system in time-reversal symmetric and antisymmetric parts, for which we prove a ‘generalised Hamilton-Jacobi orthogonality’. Finally, we apply this structure to a special class of interacting particle systems (which includes the simple-exclusion process and a large class of zero-range processes) and show how the individual terms of the Ψ-Ψ? structure converge to their hydrodynamic counterparts (as known from Macroscopic Fluctuation Theory).
|Date of Award
|22 Nov 2018
|Johannes Zimmer (Supervisor) & Robert Jack (Supervisor)