Among the many non-equilibrium processes encountered in nature we deal with two different but related aspects. One is the non-equilibrium relaxation process that is at the origin of \'aging phenomena••, and the other one is a non-equilibrium phase transition, called ••dynamic phase transition••. One of the main purposes of our research is to explore more realistic situations than studied previously. Indeed, in the study of aging phenomena certain kinds of disorder effects are considered, and we introduce the ••surface•• as a spatial boundary to the system undergoing the dynamic phase transition. In order to observe these processes as clearly as possible, we study in both cases simple spin systems. Using Monte Carlo simulations we first investigate aging in three-dimensional Ising spin glasses as well as in two-dimensional Ising models with disorder quenched to low temperatures. The time-dependent dynamical correlation length L(t) is determined numerically and the scaling behavior of various two-time quantities as a function of L(t)/L(s) is discussed where t and s are two different times. For disordered Ising models deviations of L(t) from algebraic growth law show up. The generalized scaling forms as a function of L(t)/L(s) reveal a generic simple aging scenario for Ising spin glasses as well as for disordered Ising ferromagnets. We also study the local critical phenomena at a dynamic phase transition by means of numerical simulations of kinetic Ising models with surfaces subjected to a periodic oscillating field. We examine layer-dependent quantities, such as the period-averaged magnetization per layer Q(z) and the layer susceptibility ¥ö(z), and determine local critical exponents through finite size scaling. Both for two and three dimensions, we find that the values of the surface exponents differ from those of the equilibrium critical surface. It is revealed that the surface phase diagram of the non-equilibrium system is not identical to that of the equilibrium system in three dimensions. / Ph. D.

Dynamical properties of non-equilibrium systems, similar to equilibrium ones, have been shown to obey robust time scaling laws which have enriched the concept of physical universality classes. In the first part of this Dissertation, we present the results of our investigations of some of the critical dynamical properties of systems belonging to the Voter or the Directed Percolation (DP) universality class. To be more precise, we focus on the aging properties of two-state and three-state Potts models with absorbing states and we determine temporal scaling of autocorrelation and autoresponse functions. We propose a novel microscopic model which exhibits non-equilibrium critical points belonging to the Voter, DP and Ising Universality classes. We argue that our model has properties similar to the Generalized Voter Model (GVM) in its Langevin description. Finally, we study the time evolution of the width of interfaces separating different absorbing states. The second part of this Dissertation is devoted to the applications of Machine Learning models in physical systems. First, we show that a trained Convolutional Neural Network (CNN) using configurations from the Ising model with conserved magnetization is able to find the location of the critical point. Second, using as our training dataset configurations of Ising models with conserved or non-conserved magnetization obtained in importance sampling Monte Carlo simulations, we investigate the physical properties of configurations generated by the Restricted Boltzmann Machine (RBM) model. The first part of this research was sponsored by the US Army Research Office and was accomplished under Grant Number W911NF-17-1-0156. The second part of this work was supported by the United States National Science Foundation through grant DMR-1606814. / Doctor of Philosophy / Physical systems with equilibrium states contain common properties with which they are categorized in different universality classes. Similar to these equilibrium systems, non-equilibrium systems may obey robust scaling laws and lie in different dynamic universality classes. In the first part of this Dissertation, we investigate the dynamical properties of two important dynamic universality classes, the Directed Percolation universality class and the Generalized Voter universality class. These two universality classes include models with absorbing states. A good example of an absorbing state is found in the contact process for epidemic spreading when all individuals are infected. We also propose a microscopic model with tunable parameters which exhibits phase transitions belonging to the Voter, Directed Percolation and Ising universality classes. To identify these universality classes, we measure specific dynamic and static quantities, such as interface density at different values of the tunable parameters and show that the physical properties of these quantities are identical to what is expected for the different universal classes. The second part of this Dissertation is devoted to the application of Machine Learning models in physical systems. Considering physical system configurations as input dataset for our machine learning pipeline, we extract properties of the input data through our machine learning models. As a supervised learning model, we use a deep neural network model and train it using configurations from the Ising model with conserved dynamics. Finally, we address the question whether generative models in machine learning (models that output objects that are similar to inputs) are able to produce new configurations with properties similar to those obtained from given physical models. To this end we train a well known generative model, the Restricted Boltzmann Machine (RBM), on Ising configurations with either conserved or non-conserved magnetization at different temperatures and study the properties of configurations generated by RBM. The first part of this research was sponsored by the US Army Research Office and was accomplished under Grant Number W911NF-17-1-0156. The second part of this work was supported by the United States National Science Foundation through grant DMR-1606814.

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