dc.contributor.author Menegaki, Angeliki dc.date.accessioned 2021-11-10T16:58:33Z dc.date.available 2021-11-10T16:58:33Z dc.date.submitted 2021-06 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/330536 dc.description.abstract The main results of my work contribute to the mathematical study of microscopic non-equilibrium systems that were first introduced in order to derive macroscopic physical laws such as Fourier's law. In particular the main objective is to determine the scaling of the spectral gap, i.e. the relaxation rate, in terms of the number of the particles for a paradigmatic model describing heat transport, the chain of oscillators. The mathematical study of this model started at the end of $90$s and it is challenging due to the degeneracy of the dynamics as the noise is not assumed to act to all the degrees of freedom, leading to lack of ellipticity and coercivity. We give bounds on the spectral gap for weak nonlinearities of the chain, i.e. perturbations around linear homogeneous chains and also a complete answer for the linear, homogeneous and disordered, chain of oscillators as well as $d$-dimensional grids of oscillators. The methods range from hypocoercivity inspired techniques, in the sense of Villani, to spectral analysis of discrete Schrödinger operators. Moreover we study heat conduction in gases addressing, with both analytic and probabilistic techniques, the question of the existence, and properties, of a non-equilibrium steady state for the nonlinear BGK model, introduced by Bhatnagar, Gross and Krook, with diffusive boundary conditions. The case that we address concerns large boundary temperatures away from the equilibrium case. Furthermore, besides non-equilibrium phenomena in many particle systems, this thesis deals with the question of deriving nonlinear diffusion equations from microscopic stochastic processes. We present a new, quantitative, unified method to show that the particle densities of one-dimensional processes on a periodic lattice, including the zero-range and simple exclusion jump processes as well as diffusion processes of Ginzburg-Landau type, converge to the solution of a nonlinear diffusion equation with an explicit, uniform in time, convergence rate. We discuss how we can extend the result to all the dimensions. Finally a study of the scaling of the spectral gap for all the mean field $\mathcal{O}(n)$ models of Ginzburg-Landau type using semiclassical tools, is included in this thesis. This concerns the spectral gap as a function of the number of particles, spins, for the dynamics below and at the critical temperature, with and without an external magnetic field. dc.description.sponsorship EPSRC dc.rights All Rights Reserved dc.rights.uri https://www.rioxx.net/licenses/all-rights-reserved/ dc.subject non-equilibrium statistical mechanics dc.subject chain of oscillators dc.subject spectral gap dc.subject hypoellipticity dc.subject hypocoercivity dc.subject non-equilibrium steady states dc.subject kinetic equations dc.subject quantitative hydrodynamical limits dc.subject zero-range process dc.subject Ginzburg-Landau process dc.title Quantitative studies and Hydrodynamical limits for interacting particle systems dc.type Thesis dc.type.qualificationlevel Doctoral dc.type.qualificationname Doctor of Philosophy (PhD) dc.publisher.institution University of Cambridge dc.identifier.doi 10.17863/CAM.77980 rioxxterms.licenseref.uri https://www.rioxx.net/licenses/all-rights-reserved/ dc.contributor.orcid Menegaki, Angeliki [0000-0003-1192-3567] rioxxterms.type Thesis dc.publisher.college Christs dc.type.qualificationtitle Doctor of Philosophy in Mathematics pubs.funder-project-id EPSRC (1946599) cam.supervisor Mouhot, Clément
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