Inverse and Forward Problems of Statistical Mechanics: Classical and Quantum Systems

Abstract

Statistical mechanics connects the microscopic properties of systems of interacting entities (e.g., particles and spins) to their macroscopic properties.The ``forward'' approach of statistical mechanics aims to determine thermodynamics and kinetic features of a system with known interactions. In the ``inverse'' approach, one attempts to determine the interactions or configurations that realize a desired ``target'' microstructure descriptor. For the inverse problems, we first introduce sensitivity metrics that measure the variation in pair statistics associated with any given variation in the corresponding pair potentials. We identify illustrative cases in which distinctly different potential functions give very similar pair statistics, demonstrating the need for more precise inverse techniques. Motivated by this and other challenges, we introduce a new methodology that yields much more precise interactions than previous procedures and is able to treat challenging nonequilibrium pair statistics as well as exotic ``hyperuniform'' states. This methodology enables one to study nontrivial attributes of systems equilibrated under the optimized effective potentials. We also use the methodology to tackle the realizability problem of pair statistics and identify precise density ranges on which the unit-step function $g_2$ is realizable. Furthermore, we determine classical states that mimic pair statistics of quantum fermi gases, which could facilitate quantum-mechanical simulations. Finally, we study the structural degeneracy problem of whether one can ``hear the shape of a crystal'', i.e., whether a crystal is uniquely determined, up to isometry, by its radial distribution function. The identification of isospectral crystals enables one to study the degeneracy of the ground-state manifold. For the forward problems, we develop efficient methods to extract microstructural information from statistical descriptors. We provide further evidence that the time-dependent diffusion spreadability is a robust probe of the microstructure of complex media across length scales. To accurately compute the probability distribution of hole sizes in equilibrium crystals, we introduce a simulation technique based on biased sampling that achieves significantly higher efficiency and accuracy of hole probabilities than the standard unbiased method. In summary, our work provides significant insights and techniques to inverse and forward problems and is expected to facilitate the design of novel tunable materials.