Statistical Mechanics of Phase Transitions with a Hierarchy of Structures

Abstract

Rescaling (renormalization-group) [1] and restructuring [2] transformations are used in the statistical mechanics of systems with different complexities at different length scales, starting with the microscopic scale. These transformations are effected by a partial trace of the partition function, yielding effective coupling constants that incorporate the free energy contribution of the structures and length scales that are summed out of the problem. An example of this approach is a quantitative treatment [3] of the phase diagram of krypton adsorbed onto graphite. At the smallest length scales, adatom adsorption (first and second layer) and vibrations at adsorption sites are considered. At the next scale, sublattice formation and vacancy occurrence on the hexagonal graphite surface are considered. At a larger scale, heavy and superheavy domain wall formation, wall crossings and annihilations (dislocations) [4] are considered. This problem is treated by a sequence of 5 different transformations. Starting with the microscopic potentials, phase diagrams are obtained in temperature versus pressure or coverage, featuring disordered, commensurate and incommensurate solid phases, in good agreement with experiments [5]. The mechanism for reentrant phase transitions is divulged. Such solutions can be considered approximate treatments of physical systems or, alternatively, exact solutions of “hierarchical models” [6]. This “realizability” on hierarchical lattices insures the robustness of the approximation.