Kinetic theory of dense fluids

A kinetic theory of dense fluids is presented in this series of papers. The theory is based on a kinetic equation for subsystems which represents a subset of equations structurally invariant to the sizes of the subsystem that includes the Boltzmann equation as an element at the low density limit. There exists a H-function for the kinetic equation and the equilibrium solution is uniquely given by the canonical distribution functions for the subsystems comprising the entire system. The cluster expansion is discussed for the N-body collision operator appearing in the kinetic equation. The kinetic parts of transport coefficients are obtained by means of a moment method and their density expansions are formally obtained. The Chapman-Enskog method is discussed in the subsequent paper.     

Kinetic theory of dense fluids. III. Density dependence of transport coefficients for dense gases

The viscosity coefficient obtained in a previous paper of this series is calculated as a function of density by developing the N-particle collision operator into a dynamic cluster expansion. The excess transport coefficient Δη is given in an exponential form,

$\text{Δν=ν}{}_0\text{exp}\text{∑}\text{l=1}\text{∞}\text{β}{}_{\mathrm{l}}\text{n}{}^{\mathrm{l}}\text{?1}$

where η₀ is the two-body Chapman-Enskog result for the transport coefficient, n is the density, and β_l is a density-independent quantity consisting of connected cluster contributions of (l + 2) particles. Therefore, the leading term β₁ consists of connected three-body cluster contributions. The excess shear viscosity coefficient is calculated for a monatomic hard-sphere fluid by computing β_l up to the three-body contributions and the result is compared with the molecular dynamics result by Ashurst and Hoover and also with the experimental data on Ar at 75°C. In spite of the crudity of the potential model used and the approximations made the agreement is good. The result can be improved if l-body clusters (l ? 4) are included in the calculation. The thermal conductivity coefficient can be obtained in a similar form by using exactly the same procedure used for the viscosity coefficient.     

Kinetic theory of dense fluids. II. The generalized Chapman-Enskog solution

In the second paper of this series we solve the kinetic equation proposed in the previous paper by a method following the spirit of Chapman and Enskog (generalized Chapman-Enskog method). The zeroth-order solution to the kinetic equation leads to the Euler equations in hydrodynamics for real fluids, and the first-order solution to the Navier-Stokes equations for real fluids. General formulas for transport coefficients such as viscosity and heat-conductivity coefficients are obtained for dense fluids, which are given in terms of time-correlation functions of fluxes conjugate to the thermodynamic forces. The results have the same formal structures as the time-correlation functions in linear response theory except for the collision operator appearing in place of the Liouville operator in the evolution operator for the system.     

Kinetic theory of dense fluids. IV. Entropy balance equation and irreversible thermodynamics

The viscosity coefficient obtained in a previous paper of this series is calculated as a function of density by developing the N-particle collision operator into a dynamic cluster expansion. The excess transport coefficient Δη is given in an exponential form, where η₀ is the two-body Chapman-Enskog result for the transport coefficient, n is the density, and β_l is a density-independent quantity consisting of connected cluster contributions of (l + 2) particles. Therefore, the leading term β₁ consists of connected three-body cluster contributions. The excess shear viscosity coefficient is calculated for a monatomic hard-sphere fluid by computing β_l up to the three-body contributions and the result is compared with the molecular dynamics result by Ashurst and Hoover and also with the experimental data on Ar at 75°C. In spite of the crudity of the potential model used and the approximations made the agreement is good. The result can be improved if l-body clusters (l 4) are included in the calculation. The thermal conductivity coefficient can be obtained in a similar form by using exactly the same procedure used for the viscosity coefficient.     

A new sum rule for dense fluids

A new high frequency kinetic sum rule for classical simple fluids is obtained. It involves only the static pair- and triplet-correlation functions; its self part and the local limit depend only on the pair correlation function.     

Kinetic equation for a dense soliton gas

We propose a general method to derive kinetic equations for dense soliton gases in physical systems described by integrable nonlinear wave equations. The kinetic equation describes evolution of the spectral distribution function of solitons due to soliton-soliton collisions. Owing to complete integrability of the soliton equations, only pairwise soliton interactions contribute to the solution, and the evolution reduces to a transport of the eigenvalues of the associated spectral problem with the corresponding soliton velocities modified by the collisions. The proposed general procedure of the derivation of the kinetic equation is illustrated by the examples of the Korteweg-de Vries and nonlinear Schr?dinger (NLS) equations. As a simple physical example, we construct an explicit solution for the case of interaction of two cold NLS soliton gases.     

Phenomenological equations for multicomponent fluids

Ahydrodynamic equation of motion for each component of a multicomponent fluid is derived on the basis of nonequilibrium thermodynamics. Special care has been directed to the choice of state variables. In some limiting cases, this equation leads to customary phenomenological equations, such as the equation for diffusion and the Navier-Stokes equation. The viscosity is a consequence of nonlocal coupling of forces and fluxes. The reciprocity between the linear coefficients is examined closely.     

Anisometric molecules in dense fluids

The Kerr constant of anisometric non-polar molecules in solvents of spherical molecules is calculated using the model of an ellipsoidal cavity in an isotropic dielectric continuum. It is shown that the Kerr constants has the same form as for dilute gases:

provided that effective polarizability elements α _xx ^l * (optical) and α _xx ^s * (static) replace the vacuum values α _xx ^l and α _xx ^s in Γ*, which then reads Γ* = 1/2[(α _xx ^l * - α _yy ^l *)(α _xx ^s *) - α _yy ^s *) + circular permutation].

Each effective polarizability element α _xx * (either optical or static) is composed of two contributions, one pertaining to the anisometric solute itself and another arising from the anisotropic distortion of the polarization in the neighbouring continuum. These two contributions are of opposite signs, so that large variations of the magnitude of the Kerr effect are predicted, depending essentially upon the mean refractivity and permittivity of the solute compared with those of the solvent.     

On wave equations for viscous fluids

This paper deals with wave equations describing small-amplitude disturbances in horizontally stratified, continuously varying, viscous fluids; gradients of the static pressure and of the coefficient of viscosity are neglected. A set of equations in first-order matrix form, which describes coupled longitudinal and transverse disturbances, is treated by the methods ofClemmow andHeading and ofHeading.The work of this paper could be extended in a number of ways; for example, the effect of a gravitational field could be included, and the coefficient of viscosity could be allowed to vary with position.     

Mutual diffusion in dense supercritical fluids

An expression for the mutual diffusion coefficient of a dense binary system of hard spheres, derived by using the Percus-Yevick approximation for the contact radial distribution function, together with Thorne's extension of the Enskog theory, is used to study the variation of the mutual diffusion coefficient with pressure, composition and ratio of the molecular diameters. Applications are made to real systems.     

Boltzmann entropy for dense fluids not in local equilibrium

Using computer simulations, we investigate the time evolution of the (Boltzmann) entropy of a dense fluid not in local equilibrium. The macrovariables M describing the system are the (empirical) particle density f=[f(x,v)] and the total energy E. We find that S(f(t),E) is a monotone increasing in time even when its kinetic part is decreasing. We argue that for isolated Hamiltonian systems monotonicity of S(M(t))=S(M(X(t))) should hold generally for "typical" (the overwhelming majority of) initial microstates (phase points) X0 belonging to the initial macrostate M0, satisfying M(X0)=M(0). This is a consequence of Liouville's theorem when M(t) evolves according to an autonomous deterministic law.     

Structural relaxation in dense hard-sphere fluids

The long-time decay of the shear-stress autocorrelation function is shown to be quantitatively related to the decay of correlations between the orientation of bonds connecting colliding pairs of particles. Within computational uncertainties, we find that orientational correlations in high-density fluids decay as a stretched exponential in time, with an exponent that is independent of density. However, at low densities the decay is exponential. In two-dimensional systems the decay is exponential, even at high density.     

A kinetic theory of dense fluids

A new kinetic equation is developed which incorporates the desirable features of the Enskog, the Rice-Allnatt, and the Prigogine-Nicolis-Misguich kinetic theories of dense fluids. Advantages of the present theory over the latter three theories are (1) it yields the correct local equilibrium hydrodynamic equations, (2) unlike the Rice-Allnatt theory, it determines the singlet and doublet distribution functions from the same equation, and (3) unlike the Prigogine-Nicolis-Misguich theory, it predicts the kinetic and kinetic-potential transport coefficients. The kinetic equation is solved by the Chapman-Enskog method and the coefficients of shear viscosity, bulk viscosity, thermal conductivity, and self-diffusion are obtained. The predicted bulk viscosity and thermal conductivity coefficients are singular at the critical point, while the shear viscosity and self-diffusion coefficients are not.     

Theoretical and analytical radial distribution function for dense fluids

The ‘identical particles in quasi-mean potential energy field’ assumption was used to derive the approximate theoretical and analytical radial distribution function (RDF) for dense fluids through solving the two-body Schrödinger equation and using the first-order perturbation method. The theoretical and analytical expressions of RDF can save much computation time in calculating the thermodynamics properties of fluids and may make the statistical mechanics theories comparable with the equation of state method that is currently widely used in physics, chemistry and technology. The calculated properties for argon by this RDF fit well with the experimental data of reference over a very wide range of conditions, including dense fluids (liquid phase and dense gas), critical point, and dilute gas, in which the pair potential and the Axilrod-Teller three body interaction were considered. The extensive practical application of this model for science and technology needs further investigation.     

Kinetic equations with soluble moment equations

If the moments of the kernel in a kinetic equation are polynomials, then the moment equations have a symmetry property, which permits the general solution in the form of a Laguerre series, and its coefficients (Laguerre moments) satisfy the same equations as the ordinary moments.     

Validity of some regularities of dense fluids for ionic liquids

Five of the best known regularities have been investigated for different classes of ionic liquids. The regularities are near linearity of the isothermal bulk modulus as function of pressure, common intersection point of isotherms of the bulk modulus as a function of density, common intersection point of isotherms of isobaric expansion coefficient or isothermal compressibility versus pressure, and common intersection point for the isobars of internal pressure as a function of temperature. These regularities were evaluated by Tait and GMA equation of states.     

Time correlation functions in classical dense fluids

The velocity autocorrelation functions and memory functions of dense classical fluids may be directly obtained from the static radial distribution function g(r) in an approximate way. Following the Mori projection operator formalism, the memory functions may be related to the fluctuating force correlation. At low densities, these functions may be evaluated by following the trajectories of particle pairs in the interatomic potential. At higher densities, the force correlation functions can be evaluated approximately from particle pair trajectories via the potential of the mean force. The contributions to the memory function come mainly from particle pairs with rather specific and rather short interatomic distances. At higher temperatures, this specific distance is even shorter, hence the memory function decays quickly with time. At lower temperatures, a negative region of the memory function may develop. On the other hand, there is relatively little density dependence of the normalized memory function. The results for argon fluids at various densities and temperatures agree satisfactorily with the molecular dynamics and the Enskog values. The decrease of the diffusion coefficient with density is partly due to the nature of g(r) which reflects the stronger clustering of atoms at higher densities.