Linking Time and Length Scale in Complex Solids with Statistical Mechanics
The principles of statistical mechanics have in the past been successfully used to interpret the thermodynamic behavior of fluids from the atomic standpoint. The formulations of statistical mechanics for solids, however, are limited to certain topics and areas. The intrinsic multi-scale nature of complex solids poses issues that are not present in fluids. There exists no comprehensive theoretical framework for solids, derived from atomistic theories, that includes defect dynamics and links to macroscopic deformation. Here we begin with an atomistic description of a solid and apply the principles of statistical mechanics to derive the dynamics of defects and ultimately describe the macroscopic behavior such as plastic deformation with different scales naturally coupled together. The current formulation capitalizes on the essential difference between a crystal and a fluid. In a fluid, atoms execute long-range motions and are described by their coordinates ri measured from a single origin. In crystals, on the other hand, nuclei are associated with a lattice. To capture the presence of a lattice and ultimately the deformation, we associate with each nucleus two classes of variables: the displacement si of the corresponding lattice point from its nominal position and the relative atomic displacement qi. Because there are only three absolute degrees of freedom per atom, these six variables are related by constraints. We then introduce a hierarchical development across time/length scales by identifying fast and slow variables. First, the familiar Born-Oppenheimer (BO) approximation allows us to integrate out the “fast” electron degrees of freedom. Then, we invoke a BO-like Ansatz to construct a constrained distribution function for lattice points, integrating out the “fast” vibrational (phonon) degrees of freedom. Next we consider point defects (vacancies and interstitials) and dislocations, which modify overall constraints. Thus, we derive a successive series of constrained distribution functions and ultimately obtain a hierarchy of dynamical equations for each scale that exists in a complex crystal, from the quantum world to the macroscopic. These equations, such as the transport of defects and lattice deformation, are naturally coupled with dynamics on other scales in a way that can be calculated from statistical mechanics. Consequently, we demonstrate that by including the lattice points into the distribution function and applying BO-like approximations, we are able to start from atomistic dynamics and construct a comprehensive theory for complex crystals.