Thermodynamically admissible form for discrete hydrodynamics

We construct a discrete model of fluid particles according to the GENERIC formalism. The model has the form of Smoothed Particle Hydrodynamics including correct thermal fluctuations. A slight variation of the model reproduces the Dissipative Particle Dynamics model with any desired thermodynamic behavior. The resulting algorithm has the following properties: mass, momentum and energy are conserved, entropy is a non-decreasing function of time and the thermal fluctuations produce the correct Einstein distribution function at equilibrium.Comment: 4 page

Generalized Green-Kubo formulas for fluids with impulsive, dissipative, stochastic and conservative interactions

We present a generalization of the Green-Kubo expressions for thermal transport coefficients $\mu$ in complex fluids of the generic form, $\mu= \mu_\infty +\int^\infty_0 dt V^{-1} _0$, i.e. a sum of an instantaneous transport coefficient $\mu_\infty$, and a time integral over a time correlation function in a state of thermal equilibrium between a current $J$ and a transformed current $J_\epsilon$. The streaming operator $\exp(t{\cal L})$ generates the trajectory of a dynamical variable $J(t) =\exp(t{\cal L}) J$ when used inside the thermal average $_0$. These formulas are valid for conservative, impulsive (hard spheres), stochastic and dissipative forces (Langevin fluids), provided the system approaches a thermal equilibrium state. In general $\mu_\infty \neq 0$ and $J_\epsilon \neq J$, except for the case of conservative forces, where the equality signs apply. The most important application in the present paper is the hard sphere fluid.Comment: 14 pages, no figures. Version 2: expanded Introduction and section II specifying the classes of fluids covered by this theory. Some references added and typos correcte

Lebesgue perturbation of a quasi-definite Hermitian functional. The positive definite case

16 pages, no figures.-- MSC2000 codes: 33C47; 42C05.MR#: MR1988489 (2004d:42042)Zbl#: Zbl 1032.42030In this work we study the problem of orthogonality with respect to a sum of measures or functionals. First we consider the case where one of the functionals is arbitrary and quasi-definite and the other one is the Lebesgue normalized functional. Next we study the sum of two positive measures. The first one is arbitrary and the second one is the Lebesgue normalized measure and we obtain some relevant properties concerning the new measure. Finally we consider the sum of a Bernstein–Szegö measure and the Lebesgue measure. In this case we obtain more simple explicit algebraic relations as well as the relation between the corresponding Szegö’s functions.First (A.C.) and third (C.P.) author's research was supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant number BFM2000-0015 and by Universidad de Vigo and Xunta de Galicia. Second author (F.M.)'s research was supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM2000-0206-C04-01 and INTAS project INTAS2000-272.Publicad

Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein-Szegö measure

24 pages, no figures.-- MSC2000 codes: 33C47, 42C05.-- Dedicated to Professor Dr. Mariano Gasca with occasion of his 60th anniversary.MR#: MR2350346 (2008m:33032)Zbl#: Zbl 1109.33010In the present paper we study the orthogonal polynomials with respect to a measure which is the sum of a finite positive Borel measure on [0,2π] and a Bernstein–Szegö measure. We prove that the measure sum belongs to the Szegö class and we obtain several properties about the norms of the orthogonal polynomials, as well as, about the coefficients of the expression which relates the new orthogonal polynomials with the Bernstein–Szegö polynomials. When the Bernstein–Szegö measure corresponds to a polynomial of degree one, we give a nice explicit algebraic expression for the new orthogonal polynomials.The research was supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant number BFM2000-0015, as well as BFM2003-06335-C03-C02.Publicad

On the microscopic foundation of dissipative particle dynamics

Mesoscopic particle based fluid models, such as dissipative particle dynamics, are usually assumed to be coarse-grained representations of an underlying microscopic fluid. A fundamental question is whether there exists a map from microscopic particles in these systems to the corresponding coarse-grained particles, such that the coarse-grained system has the same bulk and transport properties as the underlying system. In this letter, we investigate the coarse-graining of microscopic fluids using a Voronoi type projection that has been suggested in several studies. The simulations show that the projection fails in defining coarse-grained particles that have a physically meaningful connection to the microscopic fluid. In particular, the Voronoi projection produces identical coarse-grained equilibrium properties when applied to systems with different microscopic interactions and different bulk properties.Comment: First revisio

Dissipative Particle Dynamics with energy conservation

Dissipative particle dynamics (DPD) does not conserve energy and this precludes its use in the study of thermal processes in complex fluids. We present here a generalization of DPD that incorporates an internal energy and a temperature variable for each particle. The dissipation induced by the dissipative forces between particles is invested in raising the internal energy of the particles. Thermal conduction occurs by means of (inverse) temperature differences. The model can be viewed as a simplified solver of the fluctuating hydrodynamic equations and opens up the possibility of studying thermal processes in complex fluids with a mesoscopic simulation technique.Comment: 5 page

Static and Dynamic Properties of Dissipative Particle Dynamics

The algorithm for the DPD fluid, the dynamics of which is conceptually a combination of molecular dynamics, Brownian dynamics and lattice gas automata, is designed for simulating rheological properties of complex fluids on hydrodynamic time scales. This paper calculates the equilibrium and transport properties (viscosity, self-diffusion) of the thermostated DPD fluid explicitly in terms of the system parameters. It is demonstrated that temperature gradients cannot exist, and that there is therefore no heat conductivity. Starting from the N-particle Fokker-Planck, or Kramers' equation, we prove an H-theorem for the free energy, obtain hydrodynamic equations, and derive a non-linear kinetic equation (the Fokker-Planck-Boltzmann equation) for the single particle distribution function. This kinetic equation is solved by the Chapman-Enskog method. The analytic results are compared with numerical simulations.Comment: 22 pages, LaTeX, 3 Postscript figure

Using force covariance to derive effective stochastic interactions in dissipative particle dynamics

There exist methods for determining effective conservative interactions in coarse grained particle based mesoscopic simulations. The resulting models can be used to capture thermal equilibrium behavior, but in the model system we study do not correctly represent transport properties. In this article we suggest the use of force covariance to determine the full functional form of dissipative and stochastic interactions. We show that a combination of the radial distribution function and a force covariance function can be used to determine all interactions in dissipative particle dynamics. Furthermore we use the method to test if the effective interactions in dissipative particle dynamics (DPD) can be adjusted to produce a force covariance consistent with a projection of a microscopic Lennard-Jones simulation. The results indicate that the DPD ansatz may not be consistent with the underlying microscopic dynamics. We discuss how this result relates to theoretical studies reported in the literature.Comment: 10 pages, 10 figure

A discretized integral hydrodynamics

Using an interpolant form for the gradient of a function of position, we write an integral version of the conservation equations for a fluid. In the appropriate limit, these become the usual conservation laws of mass, momentum and energy. We also discuss the special cases of the Navier-Stokes equations for viscous flow and the Fourier law for thermal conduction in the presence of hydrodynamic fluctuations. By means of a discretization procedure, we show how these equations can give rise to the so-called "particle dynamics" of Smoothed Particle Hydrodynamics and Dissipative Particle Dynamics.Comment: 10 pages, RevTex, submitted to Phys. Rev.