Designing the statistical mechanics of classical topological states in artificial nano- and micro-materials (Invited)
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Quantitative Predictive Theories through Integrating Quantum, Statistical, Equilibrium, and Nonequilibrium Thermodynamics
Today's thermodynamics is largely based on the combined law for equilibrium systems and statistical mechanics derived by Gibbs in 1873 and 1901, respectively, while irreversible thermodynamics for nonequilibrium systems resides essentially on the Onsager Theorem as a separate branch of thermodynamics developed in 1930s. Between them, quantum mechanics was invented and was quantitatively solved in terms of density functional theory (DFT) in 1960s. Furthermore, these three scientific domains operate based on different principles and are very much separated from each other. In analogy to the parable of the blind men and the elephant articulated by Perdew, they individually represent different portions of a complex system and thus are incomplete by themselves alone, resulting in the lack of quantitative agreement between their predictions and experimental observations. Over the last two decades, the author's group has developed a multiscale entropy approach (recently termed as zentropy theory) that integrates DFT-based quantum mechanics and Gibbs statistical mechanics and is capable of accurately predicting entropy and free energy of complex systems. Furthermore, in combination with the combined law for nonequilibrium systems developed by Hillert, the author developed the theory of cross phenomena beyond the phenomenological Onsager Theorem. The zentropy theory and theory of cross phenomena jointly provide quantitative predictive theories for systems from electronic to any observable scales as reviewed in the present work.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Capturing the fractocohesive length scale in elastomers through a statistical mechanics-based gradient enhanced damage model
Here, this study aims to examine modeling flaw sensitivity in elastomers. The direct incorporation of polymer chain statistical mechanics considerations into a continuum stretch-based gradient-enhanced damage formulation, in turn, allows a representation of diffuse chain damage and fracture events that align with known micromechanical mechanisms. Through a series of numerical experiments, we simulate crack propagation and extract the fracture energy as an output of the model, while keeping track of the micromechanical signatures of diffuse chain damage that accommodate fracture propagation and eventually influence flaw sensitivity. Finally, by combining the fracture toughness and the work to rupture, we identify a fractocohesive length of the material, corresponding to the full width of the damage process zone. As the damage-to-fracture cascade in the proposed GED model is influenced by the introduction of a length scale associated with network imperfection and long-range load transfer, the emerging relationship of the two length scales is discussed, providing a potential link between microscopic damage mechanisms and the observed macroscopic fracture response.
36 MATERIALS SCIENCE↗
Quantifying the degree of disorder and associated phenomena in materials through zentropy: Illustrated with Invar Fe 3 Pt
Disorder exists in all materials at finite temperatures, implicating that the degree of disorder (ƒ DoD ) is a key parameter to tailor macroscopic functionalities of materials. Here, in this work, we propose to quantify ƒ DoD as a function of temperature using properties such as configurational entropy predicted by zentropy – a theory to represent total entropy of a system via a nested formula through the integration of quantum mechanics and statistical mechanics. Taking Invar Fe 3 Pt as an example through first-principles based calculations from 0 K to finite temperatures, we demonstrate the capability of the present approach in predicting ƒ DoD and associated phenomena in Fe 3 Pt such as Curie temperature and negative thermal expansion, showing a good agreement with experimental data.
36 MATERIALS SCIENCE↗
2023 Roadmap on molecular modelling of electrochemical energy materials
New materials for electrochemical energy storage and conversion are the key to the electrification and sustainable development of our modern societies. Molecular modelling based on the principles of quantum mechanics and statistical mechanics as well as empowered by machine learning techniques can help us to understand, control and design electrochemical energy materials at atomistic precision. Therefore, this roadmap, which is a collection of authoritative opinions, serves as a gateway for both the experts and the beginners to have a quick overview of the current status and corresponding challenges in molecular modelling of electrochemical energy materials for batteries, supercapacitors, CO 2 reduction reaction, and fuel cell applications.
25 ENERGY STORAGE↗
Symposium MT02: Statistical Mechanics-Based Computational Tools for the Study of Phase Transformation in Complex Materials (Final Report)
Symposium MT02 brought together a diverse and interdisciplinary community of scientists specializing in Statistical Mechanics-based computational modeling to investigate phase transformations in materials exhibiting complex disordered structures. As the demand for materials with extreme performance metrics grows—from aerospace components to next-generation optical fibers—the ability to predict microstructural evolution under non-equilibrium conditions has become paramount. The primary goal of this symposium was to identify, evaluate, and discuss advanced computational tools capable of designing precise manufacturing conditions to tailor material properties efficiently. By fostering a dialogue between computational theorists and experimentalists, the symposium sought to establish new protocols for predicting how processing history—such as cooling rates or strain paths—dictates the final microstructure.
36 MATERIALS SCIENCE↗
An over-approximation of entropy for elemental multiconfigurational ground state electronic structures
The ability to predict the chemical and physical properties of a material is directly related to the structure and interactions of its electrons. For materials comprised of f-block elements (the lanthanides and actinides found in the last two rows of the periodic table), the complexity of electronic structure has presented great difficulty in understanding, modeling, and predicting material properties. The complexity of multiconfigurational ground state electronic structures is illustrated herein by the combinatorics of electron permutations within individual and cumulative occupancy configurations. A non-integer orbital occupancy representation of multiconfigurational ground states is described for superposition mixing between multiple near-energy degenerate occupancy configurations and generalized in such a way that established ground states are returned by approximation for elements with less-complex electronic structures. By considering the occupancy configurations as statistical mechanics macrostates, and the permutations of electrons as statistical mechanics microstates within those macrostates, an over-approximation of entropy for multiconfigurational elemental ground state electronic structures has been calculated.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Watch and learn—a generalized approach for transferrable learning in deep neural networks via physical principles
Transfer learning refers to the use of knowledge gained while solving a machine learning task and applying it to the solution of a closely related problem. Such an approach has enabled scientific breakthroughs in computer vision and natural language processing where the weights learned in state-of-the-art models can be used to initialize models for other tasks which dramatically improve their performance and save computational time. Here we demonstrate an unsupervised learning approach augmented with basic physical principles that achieves fully transferrable learning for problems in statistical physics across different physical regimes. By coupling a sequence model based on a recurrent neural network to an extensive deep neural network, we are able to learn the equilibrium probability distributions and inter-particle interaction models of classical statistical mechanical systems. Our approach, distribution-consistent learning, DCL, is a general strategy that works for a variety of canonical statistical mechanical models (Ising and Potts) as well as disordered interaction potentials. Using data collected from a single set of observation conditions, DCL successfully extrapolates across all temperatures, thermodynamic phases, and can be applied to different length-scales. This constitutes a fully transferrable physics-based learning in a generalizable approach.
97 MATHEMATICS AND COMPUTING↗
Relative State Counting for Semiclassical Black Holes
It has been shown that entropy differences between certain states of perturbative quantum gravity can be computed without specifying an ultraviolet completion. This is analogous to the situation in classical statistical mechanics, where entropy differences are defined but absolute entropy is not. Unlike in classical statistical mechanics, however, the entropy differences computed in perturbative quantum gravity do not have a clear physical interpretation. Here we construct a family of perturbative black hole states for which the entropy difference can be interpreted as a relative counting of states. Conceptually, this Letter begins with the algebra of mass fluctuations around a fixed black hole background, and points out that while this is a type I algebra, it is not a factor and therefore has no canonical definition of entropy. As in previous work, coupling the mass fluctuations to quantum matter embeds the mass algebra within a type II factor, in which entropy differences (but not absolute entropies) are well defined. It is then shown that for microcanonical wave functions of mass fluctuation, the type II entropy difference equals the logarithm of the dimension of the extra Hilbert space that is needed to map one microcanonical window to another using gauge-invariant unitaries. The Letter closes with comments on type II entropy difference in a more general class of states, where the von Neumann entropy difference does not have a physical interpretation, but “one-shot” entropy differences do. Published by the American Physical Society 2024
Akers, Chris (ORCID:0000000227929827)↗
Random insights into the complexity of two-dimensional tensor network calculations
Projected entangled pair states (PEPS) offer memory-efficient representations of some quantum many-body states that obey an entanglement area law and are the basis for classical simulations of ground states in two-dimensional (2d) condensed matter systems. However, rigorous results show that exactly computing observables from a 2d PEPS state is generically a computationally hard problem. Yet approximation schemes for computing properties of 2d PEPS are regularly used, and empirically seen to succeed, for a large subclass of (“not too entangled”) condensed matter ground states. Adopting the philosophy of random matrix theory, in this work, we analyze the complexity of approximately contracting a 2d random PEPS by exploiting an analytic mapping to an effective replicated statistical mechanics model that permits a controlled analysis at a large bond dimension. Through this statistical-mechanics lens, we argue that (i) although approximately sampling wave-function amplitudes of random PEPS faces a computational-complexity phase transition above a critical bond dimension, and (ii) one can generically efficiently estimate the norm and correlation functions for any finite bond dimension. Furthermore, these results are supported numerically for various bond-dimension regimes. It is an important open question whether the above results for random PEPS apply more generally also to PEPS representing physically relevant ground states.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Zentropy theory for accurate prediction of free energy, volume, and thermal expansion without fitting parameters
Based on statistical mechanics, a macroscopically homogeneous system, i.e., a single phase in the present context, is composed of many independent configurations that the system embraces. The macroscopical properties of the system are determined by the properties and statistical probabilities of those configurations with respect to external conditions. The volume of a single phase is thus the weighted sum of the volumes of all configurations. Consequently, the derivative of the volume to temperature of a single phase depends on both the derivatives of the volumes of every configuration to temperature and the derivatives of their statistical probabilities to temperature, with the latter introducing nonlinear emergent behaviors. It is shown that the derivative of the volume to the temperature of the single phase can be negative, i.e., negative thermal expansion, due to the symmetry-breaking non-ground-state configurations with smaller volumes than that of the ground-state configuration and the rapid increase of the statistical probabilities of the former, and negative thermal expansion can be predicted without fitting parameters from the zentropy theory that combines quantum mechanics and statistical mechanics with the free energy of each configuration predicted from quantum mechanics and the partition function of each configuration calculated from its free energy.
36 MATERIALS SCIENCE↗
Metastability of stratified magnetohydrostatic equilibria and their relaxation
Motivated by explosive releases of energy in fusion, space and astrophysical plasmas, we consider the nonlinear stability of stratified magnetohydrodynamic equilibria against two-dimensional interchanges of straight magnetic-flux tubes. We demonstrate that, even within this restricted class of dynamics, the linear stability of an equilibrium does not guarantee its nonlinear stability: equilibria can be metastable. We show that the minimum-energy state accessible to a metastable equilibrium under non-diffusive two-dimensional dynamics can be found by solving a combinatorial optimisation problem. These minimum-energy states are, to good approximation, the final states reached by our simulations of destabilised metastable equilibria for which turbulent mixing is suppressed by viscosity. To predict the result of fully turbulent relaxation, we construct a statistical mechanical theory based on the maximisation of Boltzmann's mixing entropy. This theory is analogous to the Lynden-Bell statistical mechanics of collisionless stellar systems and plasma, and to the Robert–Sommeria–Miller theory of two-dimensional vortex turbulence. Our theory reproduces well the results of our numerical simulations for sufficiently large perturbations to the metastable equilibrium.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Entanglement transitions from restricted Boltzmann machines
The search for novel entangled phases of matter has lead to the recent discovery of a new class of “en- tanglement transitions,” exemplified by random tensor networks and monitored quantum circuits. Most known examples can be understood as some classical ordering transitions in an underlying statistical mechanics model, where entanglement maps onto the free-energy cost of inserting a domain wall. In this paper we study the possibility of entanglement transitions driven by physics beyond such statistical mechanics mappings. Motivated by recent applications of neural-network-inspired variational Ansätze, we investigate under what conditions on the variational parameters these Ansätze can capture an entanglement transition. We study the entanglement scaling of short-range restricted Boltzmann machine (RBM) quantum states with random phases. For uncorrelated random phases, we analytically demonstrate the absence of an entanglement transition and reveal subtle finite-size effects in finite-size numerical simulations. Introducing phases with correlations decaying as 1/r α in real space, we observe three regions with a different scaling of entanglement entropy depending on the exponent α. We study the nature of the transition between these regions, finding numerical evidence for critical behavior. Furthermore, our work establishes the presence of long-range correlated phases in RBM-based wave functions as a required ingredient for entanglement transitions.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Automated identification and calculation of prompt effects in kinetic mechanisms using statistical models
The kinetics of prompt dissociation involves rovibrationally excited species (generally formed by exothermic reactions) which may dissociate or isomerize prior to thermalization via collisions with the bath gas. Treating such rovibrationally excited species (so-called "hot" species) with standard kinetic phenomenology may result in incorrect macroscopic representation of their reactivity. Here this work presents the first fully automated methodology for the calculation of prompt effects of a chosen species in a kinetic mechanism, including (i) reaction selection; (ii) theoretical calculation of rate constants and prompt branching fractions; and (iii) final rate constant fitting. The energy partition between hot fragments is estimated using a variety of statistical models, including a new physically sound microcanonical statistical model based on the rovibrational density of states of the fragments. The methodology is validated against literature data for the prompt dissociations of HCO and C 3 H 7 radicals. The microcanonical statistical model is in better agreement with trajectory simulations for larger species and is thus applicable for practical systems that typically involve large molecules, for which direct dynamics calculations are impractical. The automated workflow is applied to the evaluation of the effects of prompt dissociation for two isomeric radicals C 4 H 7 1-3 (1-methylallyl) and C 4 H 7 1-4 (3-buten-1-yl). Twelve H-atom abstraction reactions are selected and the corresponding rate constants are computed with first principles theory. The microcanonical statistical model predicts that prompt dissociations of C 4 H 7 1-3 and C 4 H 7 1-4 are already significant at 1000K, resulting in differences of up to an order of magnitude at 2000K with respect to the phenomenological thermal rate constants. To illustrate the effects of prompt dissociation on simulations of experimental data, the calculated prompt rate constants are implemented in both CRECK and C3MechV3.3 kinetic mechanisms. Simulations of experimental flame data illustrate the noticeable impact of prompt dissociation kinetics on the high-temperature combustion reactivity of C 4 H 8 -1 and C 4 H 8 -2.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Entanglement features of random neural network quantum states
Restricted Boltzmann machines (RBMs) are a class of neural networks that have been successfully employed as a variational ansatz for quantum many-body wave functions. Here, we develop an analytic method to study quantum many-body spin states encoded by random RBMs with independent and identically distributed complex Gaussian weights. By mapping the computation of ensemble-averaged quantities to statistical mechanics models, we are able to investigate the parameter space of the RBM ensemble in the thermodynamic limit. We discover qualitatively distinct wave functions by varying RBM parameters, which correspond to distinct phases in the equivalent statistical mechanics model. Notably, there is a regime in which the typical RBM states have near-maximal entanglement entropy in the thermodynamic limit, similar to that of Haar-random states. However, these states generically exhibit nonergodic behavior in the Ising basis, and do not form quantum state designs, making them distinguishable from Haar-random states.
36 MATERIALS SCIENCE↗
CASM Monte Carlo: Calculations of the thermodynamic and kinetic properties of complex multicomponent crystals
Monte Carlo techniques play a central role in statistical mechanics approaches that connect macroscopic thermodynamic and kinetic properties to the electronic structure of a material. This paper describes the implementation of Monte Carlo techniques for the study of multicomponent crystalline materials within the Clusters Approach to Statistical Mechanics (CASM) software suite, and demonstrates their use in model systems to calculate free energies and kinetic coefficients, study phase transitions, and construct phase diagrams from first principles. Many crystal structures are complex, with multiple sublattices occupied by differing sets of chemical species, along with the presence of vacancies or interstitial species. This imposes constraints on concentration variables, the form of thermodynamic potentials, and the values of kinetic transport coefficients. The framework used by CASM to formulate thermodynamic potentials and kinetic transport coefficients accounting for arbitrarily complex crystal structures is presented and demonstrated with examples of increasing complexity. Additionally, an overview of the capabilities of the CASM software specific to Monte Carlo methods is given, and a new CASM software package is introduced, casm-flow, which helps automate the setup, submission, management, and analysis of Monte Carlo simulations.
Cluster expansion↗
Computing Thermodynamic Properties of Fluids Augmented by Nanoconfinement: Application to Pressurized Methane
Nanoconfined fluids exhibit remarkably different thermodynamic behavior compared to the bulk phase. These confinement effects render predictions of thermodynamic quantities of nanoconfined fluids challenging. In particular, confinement creates a spatially varying density profile near the wall that is primarily responsible for adsorption and capillary condensation behavior. Significant fluctuations in thermodynamic quantities, inherent in such nanoscale systems, coupled to strong fluid–wall interactions give rise to this near-wall density profile. Empirical models have been proposed to explain and model these effects, yet no first-principles based formulation has been developed. We present a statistical mechanics framework that embeds such a coupling to describe the effect of the fluid–wall interaction in amplifying the near-wall density behavior for compressible gases at elevated pressures such as pressurized methane in confinement. We show that the proposed theory predicts accurately the adsorbed layer thickness as obtained with small-angle neutron scattering measurements. Furthermore, the predictions of density under confinement from the proposed theory are shown to be in excellent agreement with available experimental and atomistic simulations data for a range of temperatures for nanoconfined methane. While the framework is presented for evaluating the near-wall density, owing to its rigorous foundation in statistical mechanics, the proposed theory can also be generalized for predicting phase-transition and nonequilibrium transport of nanoconfined fluids.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗