Statistical mechanical theory for steady state systems. II. Reciprocal relations and the second entropy

The concept of second entropy is introduced for the dynamic transitions between macrostates. It is used to develop a theory for fluctuations in velocity, and is exemplified by deriving Onsager reciprocal relations for Brownian motion. The cases of free, driven, and pinned Brownian particles are treated in turn, and Stokes' law is derived. The second entropy analysis is applied to the general case of thermodynamic fluctuations, and the Onsager reciprocal relations for these are derived using the method. The Green-Kubo formulas for the transport coefficients emerge from the analysis, as do Langevin dynamics.     

Relationships between the single particle barrier hopping theory and thermodynamic, disordered media, elastic, and jamming models of glassy systems

The predictions of the ultralocal limit of the activated hopping theory of highly viscous simple fluids and colloidal suspensions [K. S. Schweizer and G. Yatsenko, J. Chem. Phys. 127, 164505 (2007), preceding paper] for the relaxation time and effective activation barrier are compared with those of diverse alternative theoretical approaches and computer simulation. A nonlinear connection between the barrier height and excess pressure as empirically suggested by simulations of polydisperse repulsive force fluids is identified. In the dense normal and weakly dynamical precursor regime, where entropic barriers of hard spheres are nonexistent or of order the thermal energy, agreement with an excess entropy ansatz is found. In the random close packing or jamming limit, the barrier hopping theory predicts an essential singularity stronger than the free volume model, which is in agreement with the simplest entropic droplet nucleation and replica field theoretic approaches. Upon further technical simplification of the theory, close connections with renormalization group and nonperturbative memory function based studies of activated transport of a Brownian particle in a disordered medium can been identified. Several analytic arguments suggest a qualitative consistency between the barrier hopping theory and solid-state elastic models based on the high frequency shear modulus and a molecular-sized apparent activation volume. Implications of the analysis for the often high degeneracy of conflicting explanations of glassy dynamics are discussed.     

Statistical mechanical theory for steady state systems. VII. Nonlinear theory

The second entropy theory for nonequilibrium thermodynamics is extended to the nonlinear regime and to systems of mixed parity (even and odd functions of molecular velocities). The steady state phase space probability density is given for systems of mixed parity. The nonlinear transport matrix is obtained and it is shown to yield the analog of the linear Onsager-Casimir reciprocal relations. Its asymmetric part contributes to the flux and to the production of second entropy. The nonlinear transport matrix is not simply expressible as a Green-Kubo fluctuation equilibrium time correlation function. However, here the first nonlinear correction to the transport coefficient is given explicitly as a type of the Green-Kubo equilibrium time correlation function. The theory is illustrated by application to chemical kinetics.     

Scaling of the memory function and Brownian motion

It has been recently shown that the velocity autocorrelation function of a tracer particle immersed in a simple liquid scales approximately with the inverse of its mass. With increasing mass the amplitude is systematically reduced and the velocity autocorrelation function tends to a slowly decaying exponential, which is characteristic for Brownian motion. We give here an analytical proof for this behavior and comment on the usual explanation for Brownian dynamics which is based on the assumption that the memory function is proportional to a Dirac distribution. We also derive conditions for Brownian dynamics of a tracer particle which are entirely based on properties of its memory function.     

Statistical mechanical theory for steady-state systems. III. Heat flow in a Lennard-Jones fluid

A statistical mechanical theory for heat flow is developed based upon the second entropy for dynamical transitions between energy moment macrostates. The thermal conductivity, as obtained from a Green-Kubo integral of a time correlation function, is derived as an approximation from these more fundamental theories, and its short-time dependence is explored. A new expression for the thermal conductivity is derived and shown to converge to its asymptotic value faster than the traditional Green-Kubo expression. An ansatz for the steady-state probability distribution for heat flow down an imposed thermal gradient is tested with simulations of a Lennard-Jones fluid. It is found to be accurate in the high-density regime at not too short times, but not more generally. The probability distribution is implemented in Monte Carlo simulations, and a method for extracting the thermal conductivity is given.     

The van Hove distribution function for brownian hard spheres: dynamical test particle theory and computer simulations for bulk dynamics

We describe a test particle approach based on dynamical density functional theory (DDFT) for studying the correlated time evolution of the particles that constitute a fluid. Our theory provides a means of calculating the van Hove distribution function by treating its self and distinct parts as the two components of a binary fluid mixture, with the "self?" component having only one particle, the "distinct" component consisting of all the other particles, and using DDFT to calculate the time evolution of the density profiles for the two components. We apply this approach to a bulk fluid of Brownian hard spheres and compare to results for the van Hove function and the intermediate scattering function from Brownian dynamics computer simulations. We find good agreement at low and intermediate densities using the very simple Ramakrishnan-Yussouff [Phys. Rev. B 19, 2775 (1979)] approximation for the excess free energy functional. Since the DDFT is based on the equilibrium Helmholtz free energy functional, we can probe a free energy landscape that underlies the dynamics. Within the mean-field approximation we find that as the particle density increases, this landscape develops a minimum, while an exact treatment of a model confined situation shows that for an ergodic fluid this landscape should be monotonic. We discuss possible implications for slow, glassy, and arrested dynamics at high densities.     

Specific low-shear viscosity of a liquid dispersion of solid particles from variational theory for the Fuchs stability ratio

The contribution of a solid phase to the low-shear viscosity of a solid–liquid dispersion, i.e., the specific viscosity, η_SP, is investigated theoretically by applying a variational procedure. The Fuchs stability ratio has been interpreted as a general steady-state equilibrium constant for aggregation and has been extended to a functional form which describes the motion of two Brownian units in a dispersion. Application of the Euler–Lagrange equation under the validity of an adiabatic-like approximation for the Hamiltonian (approximately Brownian kinetic energy and inertial potential field) yields a constraint that involves specific viscosity, solid volume fraction, φ, interparticle energy and correlation functions of the dispersed phase. The Einstein formula is found as the limit of the Saito equation when an infinitely dilute hard-sphere suspension is considered, while a general closed form expression, η_SP=η_SP(φ), is proposed for a concentrated suspension. It depends on the particle coordination number and affinity, returns the low density expansion predicted by effective-medium-type theories for the viscosity, and can be represented as the sum of two dominant contributions, associated respectively with the first peak of the radial distribution function and the second peak of the total correlation function. Application to experimental data, concerning latex particles in cis-decalin and interacting silica–water systems, is presented and discussed. Received: 31 March 2000/Accepted: 13 July 2000     

Effect of correlated noises on directed motion

The Langevin equation for a Brownian particle, in contact with a heat bath which offers state dependent friction, is considered to study the directed motion in presence of two external correlated noises. The effects of correlation on transport of the Brownian particle in a symmetric periodic potential is studied and it has been found that the steady state current increases with increase in the degree of correlation. This property suggests that by controlling the degree of correlation one can enhance the current in a properly designed experiment.     

Friction and diffusion of a Brownian particle in a mesoscopic solvent

The friction and diffusion coefficients of a massive Brownian particle in a mesoscopic solvent are computed from the force and the velocity autocorrelation functions. The mesoscopic solvent is described in terms of free streaming of the solvent molecules, interrupted at discrete time intervals by multiparticle collisions that conserve mass, momentum, and energy. The Brownian particle interacts with the solvent molecules through repulsive Lennard-Jones forces. The decays of the force and velocity autocorrelation functions are analyzed in the microcanonical ensemble as a function of the number N of solvent molecules and Brownian particle mass and diameter. The simulations are carried out for large system sizes and long times to assess the N dependence of the friction coefficient. The decay rates of these correlations are confirmed to vary as N(-1) in accord with earlier predictions. Hydrodynamic effects on the velocity autocorrelation function and diffusion coefficient are studied as a function of Brownian particle mass and diameter.     

Diagrammatic kinetic theory for a lattice model of a liquid. I. Theory

We present a diagrammatic formalism for the time correlation functions of density fluctuations for an excluded volume lattice gas on a simple d-dimensional hypercubic lattice. We consider a multicomponent system in which particles of different species can have different transition rates. Our theoretical approach uses a Hilbert space formalism for the time dependent dynamical variables of a stochastic process that satisfies the detailed balance condition. We construct a Liouville matrix consistent with the dynamics of the model to calculate both the equation of motion for multipoint densities in configuration space and the interactions in the diagrammatic theory. A Boley basis of fluctuation vectors for the Hilbert space is used to develop two formally exact diagrammatic series for the time correlation functions. These theoretical techniques are generalizations of methods previously used for spin systems and atomic liquids, and they are generalizable to more complex lattice models of liquids such as a lattice gas with attractive interactions or polymer models. We use our formalism to construct approximate kinetic theories for the van Hove correlation and self-correlation function. The most simple approximation is the mean field approximation, which is exact for the van Hove correlation function of a one component system but an approximation for the self-correlation function. We use our first diagrammatic series to derive a two site multiple scattering approximation that gives a simple analytic expression for the spatial Fourier transform of the self-correlation function. We employ our second diagrammatic series to derive a simple mode coupling type approximation that provides a system of equations that can be solved for the self-correlation function.     

Analysis of particle-wall interactions during particle free fall

In this study, the vertical motion of a particle in a quiescent fluid falling toward a horizontal plane wall is analyzed, based on simplified models. Using the distance between the particle and wall as a parameter, the effects of various forces acting on the particle and the particle motion are examined. Without the colloidal and Brownian forces being included, the velocity of small particles is found to be approximately equal to the inverse of the drag force correction function used in this study as the particle approaches the near-wall region. Colloidal force is added to the particle equation of motion as the particle moves a distance comparable to its size. It is found that the particle might become suspended above or deposited onto the wall, depending on the Hamaker constant, the surface potentials of the particle and wall, and the thickness of the electrical double layer (EDL). For strong EDL repulsive force and weaker van der Waals (VDW) attractive force, the particle will become suspended above the wall at a distance at which the particle velocity is zero. This location is referred to as the equilibrium distance. The equilibrium distance is found to increase with increased in EDL thickness when a repulsive force barrier appears in the colloidal force interaction. For the weak EDL repulsive force and strong VDW attractive force case, the particle can become deposited onto the wall without the Brownian motion effect. The Brownian jump length was found to be very small. Many Brownian jumps would be required in a direction toward the wall for a suspended particle to become deposited.     

Statistical mechanical theory for steady state systems. VI. Variational principles

Several variational principles that have been proposed for nonequilibrium systems are analyzed. These include the principle of minimum rate of entropy production due to Prigogine [Introduction to Thermodynamics of Irreversible Processes (Interscience, New York, 1967)], the principle of maximum rate of entropy production, which is common on the internet and in the natural sciences, two principles of minimum dissipation due to Onsager [Phys. Rev. 37, 405 (1931)] and to Onsager and Machlup [Phys. Rev. 91, 1505 (1953)], and the principle of maximum second entropy due to Attard [J. Chem.. Phys. 122, 154101 (2005); Phys. Chem. Chem. Phys. 8, 3585 (2006)]. The approaches of Onsager and Attard are argued to be the only viable theories. These two are related, although their physical interpretation and mathematical approximations differ. A numerical comparison with computer simulation results indicates that Attard's expression is the only accurate theory. The implications for the Langevin and other stochastic differential equations are discussed.     

Thermally activated escape rate for a Brownian particle in a double-well potential for all values of the dissipation

The translational Brownian motion in a (2-4) double-well potential is considered. The escape rate, the position correlation function and correlation time, and the generalized susceptibility are evaluated from the solution of the underlying Langevin equation by using the matrix-continued fraction method. The escape rate and the correlation time are compared with the Kramers theory of the escape rate of a Brownian particle from a potential well as extended by Mel'nikov and Meshkov [J. Chem. Phys. 85, 1018 (1986)]. It is shown that in the low-temperature limit, the universal Mel'nikov and Meshkov expression for the escape rate provides a good estimate of both escape rate and inverse position correlation time for all values of the dissipation including the very low damping (VLD), very high damping (VHD), and turnover regimes. Moreover, for low barriers, where the Mel'nikov and Meshkov method is not applicable, analytic equations for the correlation time in the VLD and VHD limits are derived.     

Memory effects in collective dynamics of Brownian suspensions

We obtain macroscopic equations for average suspension velocity and particle current in a Brownian suspension valid on long time scales for which the memory effects are important. The coefficients in these equations depend solely on local properties of the medium. This formalism allows one to obtain well-defined theoretical expressions for transport coefficients, free of the integrals diverging with the size of the system. As an example, the expression for long-time collective diffusion coefficient is derived and the memory contribution to this coefficient is estimated.     

Molecular theory of irreversibility

A generalization of the Gibbs entropy postulate is proposed based on the Bogolyubov-Born-Green-Kirkwood-Yvon hierarchy of equations as the nonequilibrium entropy for a system of N interacting particles. This entropy satisfies the basic principles of thermodynamics in the sense that it reaches its maximum at equilibrium and is coherent with the second law. By using a generalization of the Liouville equation describing the evolution of the distribution vector, it is demonstrated that the entropy production is a non-negative quantity. Moreover, following the procedure of nonequilibrium thermodynamics a transport matrix is introduced and a microscopic expression for this is derived. This framework allows one to perform the thermodynamic analysis of nonequilibrium steady states with smooth phase-space distribution functions which, as proven here, constitute the states of minimum entropy production when one considers small departures from stationarity.     

Critical examination of the supermolecule density functional theory calculations of intermolecular interactions

The results of calculations employing twelve different combinations of exchange and correlation functionals are compared with results of ab initio calculations for two different configurations of the water dimer and three different configurations of the thymine-adenine complex. None of the density functional theory (DFT) treatments could properly reproduce the results of coupled-cluster calculations for all configurations examined. The DFT approaches perform well when the interaction energy is dominated by the electrostatic component and the dispersion energy is less important. Two mechanisms that compensate for the missing dispersion component were identified. The first one is the decrease of the magnitude of the intermolecular exchange-repulsion and the second one is the increase of the magnitude of the attractive deformation energy. For some functionals both effects are observed together, but for some other ones only the second effect occurs. The three correlation functionals that were examined were found to make only very small contributions to the deformation energy. The examination of angular and distance dependence of the interactions shows that the currently available DFT approaches are not suitable for developing intermolecular potential energy surfaces. They could however be used to find global minima on potential energy surfaces governed by intermolecular electrostatic interactions. Additional single point ab initio calculations are recommended as the means of validating optimized structures.     

Statistical mechanical theory of liquid entropy

The multiparticle correlation expansion for the entropy of a classical monatomic liquid is presented. This entropy expresses the physical picture in which there is no free particle motion, but, rather, each atom moves within a cage formed by its neighbors. The liquid expansion, including only pair correlations, gives an excellent account of the experimental entropy of most liquid metals, of liquid argon, and of the hard-sphere liquid. The pair correlation entropy is well approximated by a universal function of temperature. Higher-order correlation entropy, due to n-particle irreducible correlations for n ≥ 3, is significant in only a few liquid metals, and its occurrence suggests the presence of n-body forces. When the liquid theory is applied to the study of melting, we discover the important classification of normal and anomalous melting, according to whether there is not or is a significant change in the electronic structure upon melting, and we discover the universal disordering entropy for melting of a monatomic crystal. Interesting directions for future research are extension to include orientational correlations of molecules, theoretical calculation of the entropy of water, application to the entropy of the amorphous state, and correlational entropy of compressed argon. We clarify the relation among different entropy expansions in the recent literature. © 1994 John Wiley & Sons, Inc.