The question whether and how closed quantum systems equilibrate is still debated today. In this thesis a generic spin system is analysed and criteria to classify unique equilibration dynamics are developed. Furthermore, the eigenstate thermalization hypothesis is investigated as a possible cause for the unique equilibrium. For both problems novel numerical methods for solving the time-dependent Schroedinger equation based on series expansions and typicality are developed. Furthermore, the problem of markovian dynamics on the level of single measurements is discussed.

Our everyday experiences teach us that any imbalance like temperature gradients, non-uniform particle-densities etc. will approach some equilibrium state if not subjected to any external force. Phenomenological descriptions of these empirical findings reach back to the 19th century where Fourier and Fick presented descriptions of relaxation for macroscopic systems by stochastic approaches. However, one of the main goals of thermodynamics remained the derivation of these phenomenological description from basic microscopic principles. This task has gained much attraction since the foundation of quantum mechanics about 100 years ago. However, up to now no such conclusive derivation is presented. In this dissertation we will investigate whether closed quantum systems may show equilibration, and if so, to what extend such dynamics are in accordance with standard thermodynamic behavior as described by stochastic approaches. To this end we consider i.a. Markovian dynamics, Fokker-Planck and diffusion equations. Furthermore, we consider fluctuation theorems as given e.g. by the Jarzynski relation beyond strict Gibbsian initial states. After all we find indeed good agreement for selected quantum systems.

The first part of the present Phd thesis is devoted to transport investigations in disordered quantum systems. We aim at quantitatively determining transport parameters like conductivity, mean free path, etc., for simple models of spatially disordered and/or percolated quantum systems in the limit of high temperatures and low fillings using linear response theory. We find the transport behavior for some models to be in accord with a Boltzmann equation, i.e., long mean free paths, exponentially decaying currents although there are no band-structures to start from, while this does not apply to other models even though they are also almost completely delocalized. The second part of the present PhD thesis addresses the issue of initial state independence (ISI) in closed quantum system. The relevance of the eigenstate thermalization hypothesis (ETH) for the emergence of ISI equilibration is to some extent addressed. To this end, we investigate the Heisenberg spin-ladder and check the validity of the ETH for the energy difference operator by examining the scaling behavior of the corresponding ETH-fluctuations, which we compute using an innovative numerical method based on typicality related arguments. While, the ETH turns out to hold for the generic non-integrable models and may therefore serve as the key mechanism for ISI for this cases, it does not hold for the integrable Heisenberg-chain. However, close analysis on the dynamic of substantially out-of-equilibrium initial states indicates the occurrence of ISI equillibration in the thermodynamic limit regardless of whether the ETH is violated. Thus, we introduce a new parameter $v$, which we propose as an alternative of the ETH to indicate ISI equillibration in cases, in which the ETH does not strictly apply.

Based on the publications [P1–P6], the cumulative dissertation at hand addresses quite diverse aspects of non-equilibrium behavior in isolated quantum systems. The works presented in publications [P1, P2] concern the issue of finding generally valid upper bounds on equilibration times, which ensure the eventual occurrence of equilibration in isolated quantum systems. Recently, a particularly compelling bound for physically relevant observables has been proposed. Said bound is examined analytically as well as numerically. It is found that the bound fails to give meaningful results in a number of standard physical scenarios. Continuing, publication [P4] examines a particular integral fluctuation theorem (IFT) for the total entropy production of a small system coupled to a substantially larger but finite bath. While said IFT is known to hold for canonical states, it is shown to be valid for microcanonical and even pure energy eigenstates as well by invoking the physically natural conditions of “stiffness” and “smoothness” of transition probabilities. The validity of the IFT and the existence of stiffness and smoothness are numerically investigated for various lattice models. Furthermore, this dissertation puts emphasis on the issue of the route to equilibrium, i.e., to explain the omnipresence of certain relaxation dynamics in nature, while other, more exotic relaxation patterns are practically never observed, even though they are a priori not disfavored by the microscopic laws of motion. Regarding this question, the existence of stability in a larger class of dynamics consisting of exponentially damped oscillations is corroborated in publication [P6]. In the same vein, existing theories on the ubiquity of certain dynamics are numerically scrutinized in publication [P3]. Finally, in publication [P5], the recently proposed “universal operator growth hypothesis”, which characterizes the complexity growth of operators during unitary time evolution, is numerically probed for various spin-based systems in the thermodynamic limit. The hypothesis is found to be valid within the limits of the numerical approach.