Abstract
Under the description of quantum mechanics, the dynamics of an isolated system is fully determined by Hamiltonian, which preserves the probability as a result. However, when one tries to understand how a quantum system approaches thermal equilibrium, entropy, as a mean to characterize the level of disorder, would stay constant in time. Such a paradox can be attributed to the assumption of isolated system at the beginning, hence, it is necessary to consider open system, where the dynamics is no longer determined by Hamiltonian. The thesis discusses how the thermal equilibrium entropy relates to mode counting in modern physics perspective, here we would compare three different ways of counting and look into thermodynamics quantities derived from each method, especially those expected to be extensive. Next we’d study the dynamics in the open quantum system, it is rare to see the existence of exact solvable example. Here we propose hedgehog model that quantitatively describes the thermalized process in general. Numerous number of spin units and central spin comprise the global system but we only focus on central spin and how environment influences the central spin. This model features the analytic solution in time domain, in addition, systems with different parameters can be categorized into different groups, each group is corresponding to a unique thermodynamics limit curve, this indicates the universal property of solution. In the end, this model implies we have to take a deeper thought about the meaning of thermal equilibrium in view of quantum mechanics.