STATISTICAL THERMODYNAMICS OF NONUNIFORM FLUIDS
Statistical thermodynamic properties of nonuniform fluids
THERMODYNAMIC PROPERTY↗
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An asymptotic approach for the statistical thermodynamics of certain model systems
In classical statistical thermodynamics, calculating the configuration integral is both vital and elusive. Analytic relations for configuration integrals are desirable for modeling purposes, but it is typically impossible to obtain them. Certain systems become analytically tractable after replacing steep potential energies with harmonic potentials or athermal rigid constraints, but these approximations are often inadequate, especially when modeling the stretching of molecules. It is therefore necessary to develop a systematic approach to improve upon the approximations provided by these reference systems. Here, a general asymptotic approach is introduced, where the configuration integral for the full system is obtained in terms of that of the reference system and several corrections. This asymptotic approach is first demonstrated using the simple example of a classical three-dimensional oscillator. Next, the approach is applied to modeling the stretching of single polymer chains and to modeling thermally assisted crack growth, where results are verified with respect to numerical calculations. Overall, this asymptotic approach is a valid and effective tool for statistical thermodynamics in general.
Buche, Michael Robert↗
Nonequilibrium statistical thermodynamics of multicomponent interfaces
Nonequilibrium interfacial thermodynamics has important implications for crucial biological, physical, and industrial-scale transport processes. Here, we discuss a theory of local equilibrium for multiphase multicomponent interfaces that builds upon the “sharp” interface concept first introduced by Gibbs, allowing for a description of nonequilibrium interfacial processes such as those arising in evaporation, condensation, adsorption, etc. By requiring that the thermodynamics be insensitive to the precise location of the dividing surface, one can identify conditions for local equilibrium and develop methods for measuring the values of intensive variables at the interface. We then use extensive, high-precision nonequilibrium molecular dynamics (NEMD) simulations to verify the theory and establish the validity of the local equilibrium hypothesis. In particular, we demonstrate that equilibrium equations of state are also valid out of equilibrium, and can be used to determine interfacial temperature and chemical potential(s) that are consistent with nonequilibrium generalizations of the Clapeyron and Gibbs adsorption equations. We also show, for example, that, far from equilibrium, temperature or chemical potential differences need not be uniform across an interface and may instead exhibit pronounced discontinuities. However, even in these circumstances, we demonstrate that the local equilibrium hypothesis and its implications remain valid. These results provide a thermodynamic foundation and computational tools for studying or revisiting a wide variety of interfacial transport phenomena.
36 MATERIALS SCIENCE↗
Modeling single-molecule stretching experiments using statistical thermodynamics
Single-molecule stretching experiments are widely utilized within the fields of physics and chemistry to characterize the mechanics of individual bonds or molecules, as well as chemical reactions. Analytic relations describing these experiments are valuable, and these relations can be obtained through the statistical thermodynamics of idealized model systems representing the experiments. Since the specific thermodynamic ensembles manifested by the experiments affect the outcome, primarily for small molecules, the stretching device must be included in the idealized model system. Though the model for the stretched molecule might be exactly solvable, including the device in the model often prevents analytic solutions. In the limit of large or small device stiffness, the isometric or isotensional ensembles can provide effective approximations, but the device effects are missing. Here a dual set of asymptotically correct statistical thermodynamic theories are applied to develop accurate approximations for the full model system that includes both the molecule and the device. In conclusion, the asymptotic theories are first demonstrated to be accurate using the freely jointed chain model and then using molecular dynamics calculations of a single polyethylene chain.
74 ATOMIC AND MOLECULAR PHYSICS↗
STATISTICAL THERMODYNAMICS OF NONUNIFORM FLUIDS
Statistical evaluation of low order distribution functions and thermodynamic properties of nonuniform equilibrium flow
DISTRIBUTION FUNCTION↗
Statistical thermodynamics of A-B-A block copolymers. I
A-B-A block copolymers statistical thermodynamics, describing postulated microstructure theoretical model for phase transition prediction
Leary, D. F.↗
An integrated statistical-thermodynamic model for fission gas release and swelling in nuclear fuels
Here, we propose a new model for burst fission gas release induced by microcracking in ceramic nuclear fuels such as uranium dioxide. The model stipulates that the densities of defects in the fuel material, such as microcracks and fission gas bubbles on grain boundaries, evolve in accordance with the second law of thermodynamics. Central to the model is the notion of an effective temperature, conjugate to the configurational entropy of the fuel material, and directly linked to the burnup. The model predicts that microcracking, driven by the internal stress state of the fuel material, reduces the bubble storage capacity of grain boundaries, and accounts for burst fission gas release during rapid temperature transients that simulate power transients, reactor startup, and loss-of-coolant accident conditions.
11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
Circuit complexity and functionality: A statistical thermodynamics perspective
Circuit complexity, defined as the minimum circuit size required for implementing a particular Boolean computation, is a foundational concept in computer science. Determining circuit complexity is believed to be a hard computational problem. Recently, in the context of black holes, circuit complexity has been promoted to a physical property, wherein the growth of complexity is reflected in the time evolution of the Einstein-Rosen bridge (“wormhole”) connecting the two sides of an anti-de Sitter “eternal” black hole. Here, we are motivated by an independent set of considerations and explore links between complexity and thermodynamics for functionally equivalent circuits, making the physics-inspired approach relevant to real computational problems, for which functionality is the key element of interest. In particular, our thermodynamic framework provides an alternative perspective on the obfuscation of programs of arbitrary length—an important problem in cryptography—as thermalization through recursive mixing of neighboring sections of a circuit, which can be viewed as the mixing of two containers with “gases of gates.” This recursive process equilibrates the average complexity and leads to the saturation of the circuit entropy, while preserving functionality of the overall circuit. The thermodynamic arguments hinge on ergodicity in the space of circuits which we conjecture is limited to disconnected ergodic sectors due to fragmentation. The notion of fragmentation has important implications for the problem of circuit obfuscation as it implies that there are circuits of same size and functionality that cannot be connected via a polynomial number of local moves. Furthermore, we argue that fragmentation is unavoidable unless the complexity classes NP and coNP coincide, a statement that implies the collapse of the polynomial hierarchy of computational complexity theory to its first level.
Science & Technology - Other Topics↗
Gibbs' third variational principle in statistical thermodynamics
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Brown, W. B.↗
Equilibrium Statistical Thermodynamics of a Many-Particle System Coupled to an External Scalar Field
The equilibrium thermodynamics of a many-particle assembly in the presence of an external scalar field is examined. Two types of scalar coupling are considered: an external field coupled to the particle density and an external scalar field coupled to the energy density. It is shown that the broken translational and rotational invariance of the system due to the external field is reflected in the macroscopic physics by loss of the usual extensivity property of the system and by means of anisotropy in the response of the system to changes in the system lengths or to the system shape. In addition, the assumptions used in local equilibrium analyses are shown to be incorrect in principle. Nonlocal effects due to the external field must be included in the determination of the equation of state. Simple model calculations for a system in an external gravitational field and an externally imposed temperature field are presented as illustrations.
Salvino, R. E.↗
Thermodynamics and statistical mechanics
The basic thermodynamic properties of gases are reviewed and the relations between them are derived from the first and second laws. The elements of statistical mechanics are then formulated and the partition function is derived. The classical form of the partition function is used to obtain the Maxwell-Boltzmann distribution of kinetic energies in the gas phase and the equipartition of energy theorem is given in its most general form. The thermodynamic properties are all derived as functions of the partition function. Quantum statistics are reviewed briefly and the differences between the Boltzmann distribution function for classical particles and the Fermi-Dirac and Bose-Einstein distributions for quantum particles are discussed.
Source record↗
Thermodynamics and its prediction and CALPHAD modeling: Review, state of the art, and perspectives
Thermodynamics is a science concerning the state of a system, whether it is stable, metastable, or unstable, when interacting with its surroundings. The combined law of thermodynamics derived by Gibbs about 150 years ago laid the foundation of thermodynamics. In Gibbs combined law, the entropy production due to internal processes was not included, and the 2nd law was thus practically removed from the Gibbs combined law, so it is only applicable to systems under equilibrium, thus commonly termed as equilibrium or Gibbs thermodynamics. Gibbs further derived the classical statistical thermodynamics in terms of the probability of configurations in a system in the later 1800's and early 1900's. With the quantum mechanics (QM) developed in 1920's, the QM-based statistical thermodynamics was established and connected to classical statistical thermodynamics at the classical limit as shown by Landau in the 1940's. In 1960's the development of density functional theory (DFT) by Kohn and co-workers enabled the QM prediction of properties of the ground state of a system. On the other hand, the entropy production due to internal processes in non-equilibrium systems was studied separately by Onsager in 1930's and Prigogine and co-workers in the 1950's. In 1960's to 1970's the digitization of thermodynamics was developed by Kaufman in the framework of the CALculation of PHAse Diagrams (CALPHAD) modeling of individual phases with internal degrees of freedom. CALPHAD modeling of thermodynamics and atomic transport properties has enabled computational design of complex materials in the last 50 years. Our recently termed zentropy theory integrates DFT and statistical mechanics through the replacement of the internal energy of each individual configuration by its DFT-predicted free energy. The zentropy theory is capable of accurately predicting the free energy of individual phases, transition temperatures and properties of magnetic and ferroelectric materials with free energies of individual configurations solely from DFT-based calculations and without fitting parameters, and is being tested for other phenomena including superconductivity, quantum criticality, and black holes. Those predictions include the singularity at critical points with divergence of physical properties, negative thermal expansion, and the strongly correlated physics. Furthermore, those individual configurations may thus be considered as the genomic building blocks of individual phases in the spirit of the materials genome®. This has the potential to shift the paradigm of CALPHAD modeling from being heavily dependent on experimental inputs to becoming fully predictive with inputs solely from DFT-based calculations and machine learning models built on those calculations and existing experimental data through newly developed and future open-source tools. Furthermore, through the combined law of thermodynamics including the internal entropy production, it is shown that the kinetic coefficient matrix of independent internal processes is diagonal with respect to the conjugate potentials in the combined law, and the cross phenomena that the phenomenological Onsager flux and reciprocal relationships are due to the dependence of the conjugate potential of a molar quantity on nonconjugate molar quantities and other potentials, which can be predicted by the zentropy theory and CALPHAD modeling.
42 ENGINEERING↗
Automated reaction kinetics and network exploration (Arkane): A statistical mechanics, thermodynamics, transition state theory, and master equation software
The open-source statistical mechanics software described here, Arkane–Automated Reaction Kinetics and Network Exploration–facilitates computations of thermodynamic properties of chemical species, high-pressure limit reaction rate coefficients, and pressure-dependent rate coefficient over multi-well molecular potential energy surfaces (PES) including the effects of collisional energy transfer on phenomenological kinetics. Arkane can use estimates to fill in information for molecules or reactions where quantum chemistry information is missing. The software solves the internal energy master equation for complex unimolecular reaction systems. Inputs to the software include converged electronic structure computations performed by the user using a variety of supported software packages (Gaussian, Molpro, Orca, TeraChem, Q-Chem, Psi4). The software outputs high-pressure limit rate coefficients and pressure-dependent phenomenological rate coefficients, as well as computed thermodynamic properties (enthalpy, entropy, and constant pressure heat capacity) with added energy corrections. Some of the key features of Arkane include treatment of 1D, 2D or ND hindered internal rotation modes, treatment of free internal rotation modes, quantum tunneling effect consideration, transition state theory (TST) and Rice-Ramsperger-Kassel-Marcus (RRKM) rate coefficient computations, master equation solution with four implemented methods, inverse-Laplace transform of high-pressure limit rate coefficients into the energy domain, energy corrections based on bond-additivity or isodesmic reactions, automated and efficient PES exploration, and PES sensitivity analysis. The present work describes the design of Arkane, how it should be used, and refers to the theory that it employs. Arkane is distributed via the RMG-Py software suite (https://github.com/ReactionMechanismGenerator/RMG-Py).
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Statistics and thermodynamics of fracture
A probabilistic model of the fracture processes unifying the phenomenological study of long term strength of materials, fracture mechanics and statistical approaches to fracture is briefly outlined. The general framework of irreversible thermodynamics is employed to model the deterministic side of the failure phenomenon. The stochastic calculus is used to account for thg failure mechanisms controlled by chance; particularly, the random roughness of fracture surfaces.
Chudnovsky, A.↗
The photon gas formulation of thermal radiation
A statistical consideration of the energy, the linear momentum, and the angular momentum of the photons that make up a thermal radiation field was presented. A general nonequilibrium statistical thermodynamics approach toward a macroscopic description of thermal radiation transport was developed and then applied to the restricted equilibrium statistical thermostatics derivation of the energy, linear momentum, and intrinsic angular momentum equations for an isotropic photon gas. A brief treatment of a nonisotropic photon gas, as an example of the results produced by the nonequilibrium statistical thermodynamics approach, was given. The relativistic variation of temperature and the invariance of entropy were illustrated.
Ried, R. C., Jr.↗
Freely jointed chain models with extensible links
We report analytical relations for the mechanical response of single polymer chains are valuable for modeling purposes, on both the molecular and the continuum scale. These relations can be obtained using statistical thermodynamics and an idealized single-chain model, such as the freely jointed chain model. To include bond stretching, the rigid links in the freely jointed chain model can be made extensible, but this almost always renders the model analytically intractable. Here, an asymptotically correct statistical thermodynamic theory is used to develop analytic approximations for the single-chain mechanical response of this model. The accuracy of these approximations is demonstrated using several link potential energy functions. This approach can be applied to other single-chain models, and to molecular stretching in general.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Physical concepts in the development of constitutive equations
Proposed viscoplastic material models include in their formulation observed material response but do not generally incorporate principles from thermodynamics, statistical mechanics, and quantum mechanics. Numerous hypotheses were made for material response based on first principles. Many of these hypotheses were tested experimentally. The proposed viscoplastic theories and the experimental basis of these hypotheses must be checked against the hypotheses. The physics of thermodynamics, statistical mechanics and quantum mechanics, and the effects of defects, are reviewed for their application to the development of constitutive laws.
Cassenti, B. N.↗
Laplace's method applied to functional integrals
In statistical thermodynamics, partition functions are required to evaluate the current state. It is typically impossible to analytically calculate the resulting configuration integrals, but it is sometimes possible to obtain approximations in the limit of steep potential energies. This approach directly leverages Laplace’s method for approximating integrals and has been quite successful in recent applications. Motivated by models with an infinite number of degrees of freedom, this approach should be extended to the statistical thermodynamics of fields. To that end, Laplace’s method is developed for functional integrals.
97 MATHEMATICS AND COMPUTING↗