John D. Ramshawhttps://works.bepress.com/john_ramshaw/Recent works by John D. Ramshawen-usCopyright (c) 2019 All rights reserved.Fri, 01 Mar 2019 08:00:00 +00003600Thermodynamic Derivation of Classical Density Functional Theoryhttps://works.bepress.com/john_ramshaw/106/<p>Classical density functional theory has evolved into a major branch of statistical and condensed matter physics. The fundamental equation of the equilibrium theory is $\delta {A}_{t}/\delta n({\bf{r}})+\phi ({\bf{r}})=\mu ,$ where ${A}_{t}[n({\bf{r}})]$ is the thermal Helmholtz free energy of the system as a functional of its non-uniform local number density $n({\bf{r}})$, $\phi ({\bf{r}})$ is the external potential, μ is the uniform chemical potential, and $\delta /\delta n({\bf{r}})$ denotes an isothermal functional derivative. This equation implicitly determines $n({\bf{r}})$, and is ordinarily derived from the grand canonical ensemble (GCE) of statistical mechanics. Here we show that it can also be simply derived from thermodynamics alone, and is therefore not inherently statistical in character or specific to the GCE. This derivation further shows that the isothermal functional derivative of A t is identically equal to the isentropic functional derivative of the internal energy E t , which is not immediately obvious in the statistical theory. This development shows that certain aspects of density functional theory are essentially macroscopic rather than statistical in character, thereby clarifying the physical content of the theory and making it more accessible to students and nonspecialists.</p>
Fri, 01 Mar 2019 08:00:00 +0000https://works.bepress.com/john_ramshaw/106/ArticlesSupercanonical Probability Distributionshttps://works.bepress.com/john_ramshaw/103/<p>The canonical probability distribution describes a system in thermal equilibrium with an infinite heat bath. When the bath is finite the distribution is modified. These modifications can be derived by truncating a Taylor-series expansion of the entropy of the heat bath, but their form depends on the expansion parameter chosen. We consider two such expansions, which yield supercanonical (i.e., higher-order canonical) distributions of exponential and power-law form. The latter is identical in form to the "Tsallis distribution," which is therefore a valid asymptotic approximation for an arbitrary finite heat bath, but bears no intrinsic relation to Tsallis entropy.</p>
Wed, 08 Aug 2018 07:00:00 +0000https://works.bepress.com/john_ramshaw/103/ArticlesStatistical Analogues of Thermodynamic Extremum Principleshttps://works.bepress.com/john_ramshaw/102/<p>As shown by Jaynes, the canonical and grand canonical probability distributions of equilibrium statistical mechanics can be simply derived from the principle of maximum entropy, in which the statistical entropy $S=-\,{k}_{{\rm{B}}}{\sum }_{i}{p}_{i}\mathrm{log}{p}_{i}$ is maximised subject to constraints on the mean values of the energy E and/or number of particles N in a system of fixed volume V. The Lagrange multipliers associated with those constraints are then found to be simply related to the temperature T and chemical potential μ. Here we show that the constrained maximisation of S is equivalent to, and can therefore be replaced by, the essentially unconstrained minimisation of the obvious statistical analogues of the Helmholtz free energy F = E − TS and the grand potential J = F − μN. Those minimisations are more easily performed than the maximisation of S because they formally eliminate the constraints on the mean values of E and N and their associated Lagrange multipliers. This procedure significantly simplifies the derivation of the canonical and grand canonical probability distributions, and shows that the well known extremum principles for the various thermodynamic potentials possess natural statistical analogues which are equivalent to the constrained maximisation of S.</p>
Thu, 01 Mar 2018 08:00:00 +0000https://works.bepress.com/john_ramshaw/102/ArticlesEntropy Production and Volume Contraction in Thermostated Hamiltonian Dynamicshttps://works.bepress.com/john_ramshaw/101/<p>Patra et al. [<a href="https://doi.org/10.1142/S0218127416500899" target="_blank">Int. J. Bifurcat. Chaos 26, 1650089 (2016)</a>] recently showed that the time-averaged rates of entropy production and phase-space volume contraction are equal for several different molecular dynamics methods used to simulate nonequilibrium steady states in Hamiltonian systems with thermostated temperature gradients. This equality is a plausible statistical analog of the second law of thermodynamics. Here we show that those two rates are identically equal in a wide class of methods in which the thermostat variables z are determined by ordinary differential equations of motion (i.e., methods of the Nosé-Hoover or integral feedback control type). This class of methods is defined by three relatively innocuous restrictions which are typically satisfied in methods of this type</p>
Wed, 15 Nov 2017 08:00:00 +0000https://works.bepress.com/john_ramshaw/101/ArticlesGeneral Formalism for Singly-Thermostated Hamiltonian Dynamicshttps://works.bepress.com/john_ramshaw/42/A general formalism is developed for constructing modified Hamiltonian dynamical systems which preserve a canonical equilibrium distribution by adding a time evolution equation for a single additional thermostat variable. When such systems are ergodic, canonical ensemble averages can be computed as dynamical time averages over a single trajectory. Systems of this type were unknown until their recent discovery by Hoover and colleagues. The present formalism should facilitate the discovery, construction, and classification of other such systems by encompassing a wide class of them within a single unified framework. This formalism includes both canonical and generalized Hamiltonian systems in a state space of arbitrary dimensionality (either even or odd), and therefore encompasses both few- and many-particle systems. Particular attention is devoted to the physical motivation and interpretation of the formalism, which largely determine its structure. An analogy to stochastic thermostats and fluctuation-dissipation theorems is briefly discussed.Sun, 01 Nov 2015 07:00:00 +0000https://works.bepress.com/john_ramshaw/42/ArticlesNumerical stability in multifluid gas dynamics with implicit drag forces, Computer Physics Communicationshttps://works.bepress.com/john_ramshaw/99/<div class="line" id="line-29"><span style="color: rgb(46, 46, 46); font-family: Arial, Helvetica, "Lucida Sans Unicode", "Microsoft Sans Serif", "Segoe UI Symbol", STIXGeneral, "Cambria Math", "Arial Unicode MS", sans-serif; font-size: 16px;">The numerical stability of a conventional explicit numerical scheme for solving the inviscid multifluid dynamical equations describing a multicomponent gas mixture is investigated both analytically and computationally. Although these equations do not explicitly contain diffusion terms, it is well known that they reduce to a single-fluid diffusional description when the drag coefficients in the species momentum equations are large. The question then arises as to whether their numerical solution is subject to a diffusional stability restriction on the time step in addition to the usual Courant sound-speed stability condition. An analytical stability analysis is performed for the special case of a quiescent binary gas mixture with equal sound speeds and temperatures. It is found that the Courant condition is always sufficient to ensure stability, so that no additional diffusional stability restriction arises for any value of the drag coefficient, however large. This result is confirmed by one-dimensional computational results for binary and ternary mixtures with unequal sound speeds, which remain stable even when the time step exceeds the usual diffusional limit by factors of order 100.</span></div>John D. Ramshaw et al.Thu, 01 Oct 2015 00:00:00 +0000https://works.bepress.com/john_ramshaw/99/ArticlesApproximate Equations of State in Two-Temperature Plasma Mixtureshttps://works.bepress.com/john_ramshaw/100/<p>Approximate thermodynamic state relations for multicomponent atomic and molecular gas mixtures are often constructed by artificially partitioning the mixture into its constituent materials and requiring the separated materials to be in temperature and pressure equilibrium. Iterative numerical algorithms have been employed to enforce this equilibration and compute the resulting approximate state relations in single-temperature mixtures. In partially ionized gas mixtures, there is both theoretical and empirical evidence that equilibrating the chemical potentials, number densities, or partial pressures of the free electrons is likely to produce more accurate results than equilibrating the total pressures. Moreover, in many situations of practical interest the free electrons and heavy particles have different temperatures. In this paper, we present a generalized algorithm for equilibrating the heavy-particle and electron temperaturesand a third user-specified independent thermodynamic variable in a two-temperature plasmamixture. Test calculations based on the equilibration of total pressure vs. electron pressure are presented for three different mixtures.</p>
Sat, 01 Feb 2014 08:00:00 +0000https://works.bepress.com/john_ramshaw/100/ArticlesTurbulent Concentration Diffusion in Multiphase Flowhttps://works.bepress.com/john_ramshaw/23/In multifluid multiphase flow models, the velocity of each phase is determined by its own momentum equation, which is coupled to the other phases by pairwise interphase drag forces proportional to velocity differences. When the drag coefficients are large, the phase velocities become nearly equal and the relative motion of the phases becomes diffusional rather than inertial. The multifluid momentum equations then reduce to a single momentum equation for the mixture and a system of linear relations that determine the small residual velocity differences between the phases. We derive such diffusional relations in a very general form that applies to nearly all multiphase flow models of this type. The simplest such models then reduce to 'drift flux' models that exhibit pressure diffusion but not ordinary (Fick's law) diffusion driven by concentration gradients. The question then arises of how to consistently account for turbulent concentration diffusion in models of this type. This question is resolved by specializing the general diffusional relations to a rather typical single-pressure multiphase flow model which includes turbulent momentum transport due to Reynolds stresses. This special case serves as a paradigm which illustrates that (a) turbulent concentration diffusion in multiphase flow is a direct consequence of the isotropic (turbulent pressure) part of the Reynolds stresses, provided the latter are expressed in their proper divergence form; and (b) the order of magnitude of the resulting turbulent diffusivities is just what one would anticipate on dimensional grounds. Properly formulated models of this type therefore automatically include turbulent concentration diffusion, and consequently require no further additions or modifications to introduce it.John D. RamshawSat, 01 Sep 2012 07:00:00 +0000https://works.bepress.com/john_ramshaw/23/ArticlesNonlinear Ordinary Differential Equations in Fluid Dynamicshttps://works.bepress.com/john_ramshaw/17/The equivalence between nonlinear ordinary differential equations (ODEs) and linear partial differential equations (PDEs) was recently revisited by Smith, who used the equivalence to transform the ODEs of Newtonian dynamics into equivalent PDEs, from which analytical solutions to several simple dynamical problems were derived. We show how this equivalence can be used to derive a variety of exact solutions to the PDEs describing advection in fluid dynamics in terms of solutions to the equivalent ODEs for the trajectories of Lagrangian fluid particles. The PDEs that we consider describe the time evolution of non-diffusive scalars, conserved densities, and Lagrangian surfaces advected by an arbitrary compressible fluid velocity field u(x, t). By virtue of their arbitrary initial conditions, the analytical solutions are asymmetric and three-dimensional even when the velocity field is one-dimensional or symmetrical. Such solutions are useful for verifying multidimensional numerical algorithms and computer codes for simulating advection and interfacial dynamics in fluids. Illustrative examples are discussed.John D. RamshawThu, 01 Dec 2011 08:00:00 +0000https://works.bepress.com/john_ramshaw/17/ArticlesA Discrete Impulsive Model for Random Heating and Brownian Motionhttps://works.bepress.com/john_ramshaw/21/<p>The energy of a mechanical system subjected to a random force with zero mean increases irreversibly and diverges with time in the absence of friction or dissipation. This random heating effect is usually encountered in phenomenological theories formulated in terms of stochastic differential equations, the epitome of which is the Langevin equation of Brownian motion. We discuss a simple discrete impulsive model that captures the essence of random heating and Brownian motion. The model may be regarded as a discrete analog of the Langevin equation, although it is developed ab initio. Its analysis requires only simple algebraic manipulations and elementary averaging concepts, but no stochastic differential equations (or even calculus). The irreversibility in the model is shown to be a consequence of a natural causal stochastic condition that is closely analogous to Boltzmann's molecular chaos hypothesis in the kinetic theory of gases. The model provides a simple introduction to several ostensibly more advanced topics, including random heating, molecular chaos, irreversibility, Brownian motion, the Langevin equation, and fluctuation-dissipation theorems.</p>
Sat, 01 Aug 2009 07:00:00 +0000https://works.bepress.com/john_ramshaw/21/Articles