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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.
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.
A broad program of research in theoretical chemistry Semiannual progress report, 1 Jul. - 31 Dec. 1967
Quantum mechanics and statistical mechanics applied to solution of biological and chemical problems - chemical bonding, resonance, spin, and transport phenomena
Mysteries of the ringed planets
An assessment is presented of the recent progress in the theory of planetary rings which was in evidence at the IAU's recent, 75th Colloquium. Observational material was dominated by spacecraft data, and theoretical consideration of the problems posed comes predominantly from gravitational mechanics. An understanding of collective effects, in light of both fluid mechanical and statistical mechanical methodologies, is being approached, and the importance of electromagnetic phenomena studies is noted. Voyager observations of Saturn's rings, and accumulating data from stellar occultations by the rings of Uranus, provided most of the observational material. Jupiter's faint ring was closely examined by the 1979 Voyager flight. These three known ring systems are found to exhibit such family resemblances as their proximity to the parent planet and magnetospheric environment.
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.
Measurement-induced criticality in random quantum circuits
We investigate the critical behavior of the entanglement transition induced by projective measurements in (Haar) random unitary quantum circuits. Using a replica approach, we map the calculation of the entanglement entropies in such circuits onto a two-dimensional statistical-mechanics model. In this language, the area- to volume-law entanglement transition can be interpreted as an ordering transition in the statistical-mechanics model. We derive the general scaling properties of the entanglement entropies and mutual information near the transition using conformal invariance. We analyze in detail the limit of infinite on-site Hilbert space dimension in which the statistical-mechanics model maps onto percolation. In particular, we compute the exact value of the universal coefficient of the logarithm of subsystem size in the n th Rényi entropies for n ≥ 1 in this limit using relatively recent results for the conformal field theory describing the critical theory of two-dimensional (2D) percolation, and we discuss how to access the generic transition at finite on-site Hilbert space dimension from this limit, which is in a universality class different from 2D percolation. Here, we also comment on the relation to the entanglement transition in random tensor networks, studied previously in Vasseur et al. [Phys. Rev. B 100, 134203 (2019)].
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.
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.
Strongly magnetized classical plasma models
Discrete particle processes in the presence of a strong external magnetic field were investigated. These processes include equations of state and other equilibrium thermodynamic relations, thermal relaxation phenomena, transport properties, and microscopic statistical fluctuations in such quantities as the electric field and the charge density. Results from the equilibrium statistical mechanics of two-dimensional plasmas are discussed, along with nonequilibrium statistical mechanics of the electrostatic guiding-center plasma (a two-dimensional plasma model).
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
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.
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.
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.
Strongly magnetized classical plasma models
The class of plasma processes for which the so-called Vlasov approximation is inadequate is investigated. Results from the equilibrium statistical mechanics of two-dimensional plasmas are derived. These results are independent of the presence of an external dc magnetic field. The nonequilibrium statistical mechanics of the electrostatic guiding-center plasma, a two-dimensional plasma model, is discussed. This model is then generalized to three dimensions. The guiding-center model is relaxed to include finite Larmor radius effects for a two-dimensional plasma.
Test particle propagation in magnetostatic turbulence. 2: The local approximation method
An approximation method for statistical mechanics is presented and applied to a class of problems which contains a test particle propagation problem. All of the available basic equations used in statistical mechanics are cast in the form of a single equation which is integrodifferential in time and which is then used as the starting point for the construction of the local approximation method. Simplification of the integrodifferential equation is achieved through approximation to the Laplace transform of its kernel. The approximation is valid near the origin in the Laplace space and is based on the assumption of small Laplace variable. No other small parameter is necessary for the construction of this approximation method. The n'th level of approximation is constructed formally, and the first five levels of approximation are calculated explicitly. It is shown that each level of approximation is governed by an inhomogeneous partial differential equation in time with time independent operator coefficients. The order in time of these partial differential equations is found to increase as n does. At n = 0 the most local first order partial differential equation which governs the Markovian limit is regained.
Evolution of cosmic string networks
A discussion of the evolution and observable consequences of a network of cosmic strings is given. A simple model for the evolution of the string network is presented, and related to the statistical mechanics of string networks. The model predicts the long string density throughout the history of the universe from a single parameter, which researchers calculate in radiation era simulations. The statistical mechanics arguments indicate a particular thermal form for the spectrum of loops chopped off the network. Detailed numerical simulations of string networks in expanding backgrounds are performed to test the model. Consequences for large scale structure, the microwave and gravity wave backgrounds, nucleosynthesis and gravitational lensing are calculated.
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.