Executive Summary : | The proposal explores the meaning and character of thermodynamic equilibrium, a crucial aspect of statistical mechanics that has not been thoroughly explored in the literature. The equilibrium state is what remains after a system has been left in contact with a heat bath at temperature T for a long time. The equilibrium properties can be obtained from the partition function without knowing the relaxation dynamics that brought the system to equilibrium. The study aims to modify the relaxation dynamics to reach an equilibrium state that fluctuates closer to the global minimum. It is not widely appreciated that relaxation dynamics plays a key role in equilibrium. Fluctuations in equilibrium are present on all time and energy scales, and the research focuses on hysteresis, a history-dependent effect and nonequilibrium phenomena. The system's evolution depends on its initial state, and the system evolves to a steady state independent of the initial state. Glauber dynamics are run on two separate initial states: all spins up and all spins down. As the dynamics progress, the diminishing difference between the magnetization of the two states is a measure of the closeness to the equilibrium state. The study focuses on analytic and numerical work on the Bethe lattice, with a critical temperature Tc known analytically from the partition function and a critical temperature sensed by the dynamics. Exploratory studies show that the difference between simulations and partition function values gradually reduces as we go from single spin to multispin flip dynamics. |