Hydrodynamics and time correlation functions for cellular automata

Hydrodynamic excitations in lattice gas cellular automata are described in terms of equilibrium time correlation functions for the local conserved variables. For large space and time scales the linearized hydrodynamic equations are obtained to Navier-Stokes order. Exact expressions for the associated susceptibilities and transport coefficients are identified in terms of correlation functions. The general form of the time correlation functions for conserved densities in the hydrodynamic limit is given and illustrated by some examples suitable for comparison with computer simulation. The transport coefficients are related to time correlation functions for the conserved fluxes in a way analogous to the Green-Kubo expressions for continuous fluids. The general results are applied for a one-component fluid and several types of binary diffusion. Also discussed are the effects of unphysical slow modes such as staggered particle or momentum densities.     

Fokker-Planck equation approach to fluctuations about nonequilibrium steady states

We examine the properties of steady states in systems which interact at the boundary with a nonequilibrium environment. The examination is based on a nonlinear Fokker-Planck equation, the structure of which is determined by the fact that it also governs the time evolution of the equilibrium fluctuations of the system. The nonlinearities in the Fokker-Planck equation may have two origins: thermodynamic nonlinearities which arise if the thermodynamic potential is not a bilinear function of the state variables, and nonlinear mode coupling which arises if the transport coefficients depend on the state. While these nonlinearities have only a small effect on the equilibrium fluctuations of a system away from critical points, they are shown to be important for the determination of fluctuations about nonequilibrium steady states. In particular the state dependence of the transport coefficients may lead to deviations from local equilibrium and to a breakdown of detail balance. An explicit formula for the time correlations of fluctuations about the nonequilibrium steady state is obtained. The formula leads to long-range correlations in fluids in the presence of a temperature gradient. The result is compared with earlier approaches to the same problem. Finally, we study the linear response to external forces and obtain a generalization of the fluctuation-dissipation formula relating the response functions with the nonequilibrium correlation functions.     

Shear viscosity of strongly coupled Yukawa systems on finite length scales

The Yukawa shear viscosity has been calculated using nonequilibrium molecular dynamics. Near the viscosity minimum, we find exponential decay consistent with the Navier-Stokes equation, with significant deviations on finite length scales for larger viscosity values. The viscosity is determined to be nonlocal on a scale length consistent with the correlation length, revealing the length scales necessary for obtaining transport coefficients in the hydrodynamic limit by nonequilibrium molecular dynamics methods. Our results are quasiuniversal with respect to excess entropy for excess entropies well below unity.     

Nonlinear transport in the Boltzmann limit

Formal expressions for the irreversible fluxes of a simple fluid are obtained as functionals of the thermodynamic forces and local equilibrium time correlation functions. The Boltzmann limit of the correlation functions is shown to yield expressions for the irreversible fluxes equivalent to those obtained from the nonlinear Boltzmann kinetic equation. Specifically, for states near equilibrium, the fluxes may be formally expanded in powers of the thermodynamic gradients and the associated transport coefficients identified as integrals of time correlation functions. It is proved explicitly through nonlinear Burnett order that the time correlation function expressions for these transport coefficients agree with those of the Chapman-Enskog expansion of the nonlinear Boltzmann equation. For states far from equilibrium the local equilibrium time correlation functions are determined in the Boltzmann limit and a similar equivalence to the Boltzmann equation solution is established. Other formal representations of the fluxes are indicated; in particular, a projection operator form and its Boltzmann limit are discussed. As an example, the nonequilibrium correlation functions for steady shear flow are calculated exactly in the Boltzmann limit for Maxwell molecules.Research supported in part by NSF grant PHY 76-21453.     

A collisional approach to the calculation of time correlation functions. Transport coefficients of gases

A method of successive approximations is proposed for the evaluation of the time-correlation functions such as those that give the thermal transport coefficients of gases. The method is based on a calculation of the changes in correlations of appropriate functions of the molecular velocity which are a result of collisions in the gas. The decaying rates of the correlations are expressed as integrals of the differential collision cross section. When the first approximation is introduced in the expressions for thermal transport coefficients, results are obtained for the coefficient of binary diffusion and the viscosity and thermal conductivity of single-component systems which are identical with those of the first Chapman-Enskog solutions of the Boltzmann and Enskog equations. For the coefficients of viscosity and thermal conductivity in multicomponent systems, it is shown that the first approximation leads to expressions of the form of the Sutherland and Wassiljewa relations, respectively.     

Green–Kubo Expressions for a Granular Gas

The transport coefficients for a gas of smooth, inelastic hard spheres are obtained from the Boltzmann equation in the form of Green–Kubo relations. The associated time correlation functions are not simply those constructed from the fluxes of conserved densities. Instead, fluxes constructed from the reference local homogeneous distribution occur as well. The analysis exposes some complexities to be expected in the application of linear response methods to granular systems.     

On fluctuations in μ-space

A master equation is derived microscopically to describe the fluctuating motion of the particle density in . space. This equation accounts for the drift motion of particles and is valid for any inhomogeneous gas. The Boltzmann equation is obtained from the first moment of this equation by neglecting the second cumulant (the pair correlation function). The successive moments form coarse-grained BBGKY-like hierarchy equations, in which small spatial regions with r_ij < the force range are smeared out. These hierarchy equations are convenient for investigating the nonequilibrium long-range pair correlation function, which arises mainly from sequences of isolated binary collisions and gives rise to the much-discussed long-time tail and the logarithmic term in the density expansion of transport coefficients. It is shown to have a spatial long tail, like the Coulombic potential, in a steady laminar flow. The stochastic nature of the nonlinear Boltzmann-Langevin equation is also investigated; the random source term is found to be expressed as a linear superposition of Poisson random variables and to become Gaussian in special cases.     

Quantum thermal transport in nanostructures

In this colloquia review we discuss methods for thermal transport calculations for nanojunctions connected to two semi-infinite leads served as heat-baths. Our emphases are on fundamental quantum theory and atomistic models. We begin with an introduction of the Landauer formula for ballistic thermal transport and give its derivation from scattering wave point of view. Several methods (scattering boundary condition, mode-matching, Piccard and Caroli formulas) of calculating the phonon transmission coefficients are given. The nonequilibrium Green's function (NEGF) method is reviewed and the Caroli formula is derived. We also give iterative methods and an algorithm based on a generalized eigenvalue problem for the calculation of surface Green's functions, which are starting point for an NEGF calculation. A systematic exposition for the NEGF method is presented, starting from the fundamental definitions of the Green's functions, and ending with equations of motion for the contour ordered Green's functions and Feynman diagrammatic expansion. In the later part, we discuss the treatments of nonlinear effects in heat conduction, including a phenomenological expression for the transmission, NEGF for phonon-phonon interactions, molecular dynamics (generalized Langevin) with quantum heat-baths, and electron-phonon interactions. Some new results are also shown. We briefly review the experimental status of the thermal transport measurements in nanostructures.     

HNC-type approximation for transport processes in electrolytic solutions

On the basis of the diffusion approach in the theory of transport processes of electrolytic solutions we introduce a direct correlation force as a generalization of the direct correlation function in equilibrium. Starting from an approximation for the three-particle distribution function we derive a HNC (hypernetted chain)-type equation for calculation of binary distribution functions in nonequilibrium. The derivation is consistent with equilibrium theory.     

Statistical transport theory of reacting two-component fluid: Light scattering from an A⇄B liquid

A theory of light scattering from a simple A?B reacting fluid in which molecular anisotropy has been neglected is presented. The theory is a molecular-statistical one based on Mori's linear response formalism. The time-dependent correlation functions are associated with transport coefficients and the zero time correlation functions are associated with thermodynamic derivatives. The effects of the reaction are observed from both density and concentration fluctuations as well as from cross correlations between density and concentration fluctuations.     

Response function theory for far-from-equilibrium statistical systems

A formalism to determine the response function of a sample in conditions far from thermal equilibrium is presented. It consists in a generalization of scattering theory coupled to the statistical theory of irreversible processes, the nonequilibrium statistical operator method, developed by Zubarev. The scattering cross section is expressed in terms of double-time correlation functions, which are related to appropriate nonequilibrium thermodynamic Green's functions. The latter are also used to treat generalized transport equations, and, as an illustration, the method is applied to the study of the time-resolved Raman spectroscopy of a photoexcited semiconductor plasma.     

First-principles investigation on transport properties of NiO monowire-based molecular device

The electronic transport properties of novel NiO monowire connected to the gold electrodes are investigated using density functional theory combined with nonequilibrium Green's functions formalism. The densities of states of the monowire under various bias conditions are discussed. The transport properties are discussed in terms of the transmission spectrum and current–voltage characteristics of NiO monowire. The transmission pathways provide the insight to the transmission of electrons along the monowire. With different bias voltages, current in the order of few microampere flows across the monowire. The applied voltage controls the flow of current through the monowire, which can be used to control the current efficiently in the low order of magnitude in the molecular device.     

Non Existence of a Density Expansion of the Transport Coefficients in a Moderately Dense Gas

Two examples of developments in Statistical Mechanics, on whose success the experts agreed, are discussed. First and foremost, a systematic generalization of the Boltzmann equation for a dilute gas from binary to higher order particle collisions, expected to lead to a density expansion of the transport coefficients. This would imitate, in nonequilibrium, the density expansion of the thermodynamic quantities in equilibrium. However, by studying concrete models, it became clear that because of the neglect of collective collision effects, not relevant in equilibrium, the purely formal expressions for the transport coefficients obtained this way exhibited divergences. Secondly, a related purely mechanical problem, concerned the question of the maximum number of collisions between three identical hard balls in infinite space. The answer, on intuitively convincing grounds, was generally accepted to be three, as in one dimension. Also this was incorrect, as a computer simulation first showed. The message here is: beware for common opinions and purely formal or intuitive arguments: test them on concrete models first.     

Microscopic theory of the long-wavelength modes of two-component plasmas and ionic liquids: III. The space-time correlation functions

The identity between the exact screening length obtained from the static charge density correlation function and the one which appears in the Einstein relation between the transport coefficients of electrical conductivity and mass diffusion is demonstrated from first principles. For the space-time correlation functions of the number densities we show that their long-wavelength behaviour is completely determined by the four hydrodynamical modes of the two-component system of neutral particles. For charged particle systems there are only three hydrodynamical modes while we have moreover to add the two charge relaxation modes in order to exhaust the long-wavelength limit of the first sum-rule. The strengths with which the various modes appear in the space-time correlation functions have been computed exactly in the limit of long wavelengths.     

The coupling between translational and rotational motions

The coupling between molecular reorientation and translational displacement is studied in a fluid of rough spheres. The correlation functions frequently encountered in thermal neutron scattering and laser light scattering are computed using a binary collision approximation. These are compared with the corresponding uncoupled results. In the diffusion limit coupled and uncoupled diffusion coefficients are found. The Hubbard relation is generalized. The maximum deviation between the coupled and uncoupled results occur for wavenumbers commonly found in thermal neutron scattering. The ratio of uncoupled and coupled correlation times displays regions where translation-rotation coupling is clearly important. In these regions there are important differences in the computed coupled and uncoupled correlation functions.     

Transport coefficients of liquid N2 computed by molecular dynamics

Summary With three potential functions including quadrupole interaction, we computed the shear and the bulk viscosity coefficients as well as the thermal conductivity by molecular dynamics for a number of states of liquid nitrogen. The results indicated that the transport coefficients are determinable with a statistical error of about 10 per cent, even for the angledependent molecular potentials in the rigid-rotor approximation. An exception represents the bulk viscosity, for which systems of large particle numbers are necessary to reach the few perent accuracy for states near the triple point. The quadrupole interaction affects weakly the time correlation functions and thus the transport coefficients. In several cases, it improves, however, the agreement with experimental data. Paper presented at the workshop ?Highlights on Simple Liquids?, held in Turin at ISI on 1–3 May, 1989.     

Hydrodynamic origin of diffusion in nanopores

We study the transport of a subcritical Lennard-Jones fluid in a cylindrical nanopore, using a combination of equilibrium and nonequilibrium as well as dual control volume grand canonical molecular dynamics methods. We show that all three techniques yield the same value of the transport coefficient for diffusely reflecting pore walls, even in the presence of viscous transport. We also demonstrate that the classical Knudsen mechanism is not manifested, and that a combination of viscous flow and momentum exchange at the pore wall governs the transport over a wide range of densities.