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Steepest-entropy-ascent quantum thermodynamics (SEAQT) is a mathematical and theoretical framework for intrinsic quantum thermodynamics (IQT), a unified theory of quantum mechanics and thermodynamics. In the theoretical framework, entropy is viewed as a measure of energy load sharing among available energy eigenlevels, and a unique relaxation path of a system from an initial non-equilibrium state to a stable equilibrium is determined from the greatest entropy generation viewpoint. The SEAQT modeling has seen a great development recently. However, the applications have mainly focused on gas phases, where a simple energy eigenstructure (a set of energy eigenlevels) can be constructed from appropriate quantum models by assuming that gas-particles behave independently. The focus of this research is to extend the applicability to solid phases, where interactions between constituent particles play a definitive role in their properties so that an energy eigenstructure becomes quite complicated and intractable from quantum models. To cope with the problem, a highly simplified energy eigenstructure (so-called ``pseudo-eigenstructure") of a condensed matter is constructed using a reduced-order method, where quantum models are replaced by typical solid-state models. The details of the approach are given and the method is applied to make kinetic predictions in various solid-state phenomena: the thermal expansion of silver, the magnetization of iron, and the continuous/discontinuous phase separation and ordering in binary alloys where a pseudo-eigenstructure is constructed using atomic/spin coupled oscillators or a mean-field approximation. In each application, the reliability of the approach is confirmed and the time-evolution processes are tracked from different initial states under varying conditions (including interactions with a heat reservoir and external magnetic field) using the SEAQT equation of motion derived for each specific application. Specifically, the SEAQT framework with a pseudo-eigenstructure successfully predicts: (i) lattice relaxations in any temperature range while accounting explicitly for anharmonic effects, (ii) low-temperature spin relaxations with fundamental descriptions of non-equilibrium temperature and magnetic field strength, and (iii) continuous and discontinuous mechanisms as well as concurrent ordering and phase separation mechanisms during the decomposition of solid-solutions. Ph. D. Many engineering materials have physical and chemical properties that change with time. The tendency of materials to change is quantified by the field of thermodynamics. The first and second laws of thermodynamics establish conditions under which a material has no tendency to change; these conditions are called equilibrium states. When a material is not in an equilibrium state, it is able to change spontaneously. Classical thermodynamics reliably identifies whether a material is susceptible to change, but it is incapable of predicting how change will take place or how fast it will occur. These are kinetic questions that fall outside the purview of thermodynamics. A relatively new theoretical treatment developed by Hatsopoulos, Gyftopoulos, Beretta and others over the past forty years extends classical thermodynamics into the kinetic realm. This framework, called steepest-entropy-ascent quantum thermodynamics (SEAQT), combines the tools of thermodynamics with quantum mechanics through a postulated equation of motion. Solving the equation of motion provides a kinetic description of the path a material will take as it changes from a non-equilibrium state to stable equilibrium. To date, the SEAQT framework has been applied primarily to systems of gases. In this dissertation, solid-state models are employed to extend the SEAQT approach to solid materials. The SEAQT framework is used to predict the thermal expansion of silver, the magnetization of iron, and the kinetics of atomic clustering and ordering in binary solid-solutions as a function of time or temperature. The model makes it possible to predict a unique kinetic path from any arbitrary, non-equilibrium, initial state to a stable equilibrium state. In each application, the approach is tested against experimental data. In addition to reproducing the qualitative kinetic trends in the cases considered, the SEAQT framework shows promise for modeling the behavior of materials far from equilibrium.