Density Expansions of Transport Coefficients for Gases

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The determination of density expansions for transport coefficients for dilute and moderately dense gases has been an active field of research in the kinetic theory of gases for at least a century, beginning, perhaps, with the fundamental work of David Enskog in 1911 (S. Chapman and T. G. Cowling, 1971). A density expansion of a transport or thermodynamic property of a dilute and moderately dense gas composed of particles with central, short-ranged forces, and obeying classical mechanics, is usually thought to be a power series expansion of such quantities in powers of the reduced density of the gas, ~n = na^d, where n=N/V is the number density of N particles in a contained of volume V, a is a characteristic diameter of a molecule of the gas, and d is the number of spatial dimensions of the system. The thermodynamic properties of dilute and moderately dense gases can indeed be expressed to a high degree of approximation by means of such series expansions, called ‘‘virial expansions’’, in powers of the gas density where the lowest order term is the ideal gas limit. The coefficients of the powers are determined by the potential energies of interactions among small groups of particles considered in isolation from the other particles in the gas (J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, 1964; L. Reichl, 1998). The ideal gas term is obtained if one ignores all interactions between the particles, the first correction to it involves two particle interactions, the next term, three particle interactions, and so on. The same situation does not obtain if one attempts to apply similar methods to obtain useful expansions for transport properties, such as the coefficients of shear and bulk viscosity, thermal conductivity, diffusion and so on, in series of powers of the gas density, with lowest order term the values obtained by use of the Boltzmann transport equation. Instead, due to the presence of long-range and long-time dynamical correlations between the particles produced by correlated sequences of collisions, the coefficients in any power series expansion diverge after a few terms (J. R. Dorfman and H. van Beijeren, 1977). The order in density of the first divergent term in the series depends upon the spatial dimensions of the system. The physical reason for these divergences has its origin in the attempt to express transport properties as power series expansions in the density of the gas. Each term in this expansion is determined by the dynamics of small groups of particles, considered in isolation from the rest of the particles in the gas. An important dynamical property of particles in the gas, the mean free path, is treated in an inappropriate way in this expansion. In a gas, each particle travels only a mean free path between collisions, on the average. However, dynamical processes in which particles are treated as if they can travel freely for arbitrarily large distances determine the terms in the power series expansion. This situation suggests that the terms in the density expansion should be combined in such a way as to correctly incorporate the mean free path damping of the likelihood that a particle would travel a large distance between successive collisions. This can be accomplished by summing the most divergent terms in each order of density in the expansion (K. Kawasaki and I. Oppenheim, 1965; J. R. Dorfman and H. van Beijeren, 1977).

Although the divergences in the virial coefficients can be removed by resummations of the diverging terms in the viral expansion, the resulting structure of the density expansion of transport coefficients is much more complicated than a simple power series, and little is known about all but the first few terms in this expansion. Combinations of powers and logarithmic terms in the density certainly appear in the expansion. Moreover, the same kind of resummations that lead to the known finite results for three dimensional systems still lead to divergent transport coefficients for two dimensional systems, via a different, but closely related mechanism, an effect of the presence of long-range dynamical correlations that exist in all non-equilibrium fluids. The structure of the ‘‘Navier-Stokes’’ hydrodynamics for two-dimensional systems is apparently inherently non-linear and non-analytic, due to these correlations. For the sake of brevity, we shall restrict this discussion to classical systems, but quantum systems are similar in most essential features, except for systems at low temperatures or high densities.


The Boltzmann and Enskog theories

A systematic way of calculating the transport coefficients for a dilute gas is provided by the ‘‘Boltzmann transport equation’’. This is an equation for the single-particle distribution function, \(f (\mathbf{r}, \mathbf{v}, t) \) for particles in the gas, defined so that, for a one-component gas, \(f (\mathbf{r}, \mathbf{v}, t)\delta\mathbf{r}\delta\mathbf{v}\) is the number of gas particles in a small region of space and velocity about the point \((\mathbf{r}, \mathbf{v})\) at time t. For mixtures one defines a separate function for each species. By means of probabilistic and mechanical arguments, Boltzmann derived an equation for this distribution function, which provides the foundation for the kinetic theory of dilute gases and allows for a derivation of the Navier-Stokes equations together with expressions for all of the transport coefficients in terms of the gas density, temperature, and the forces between the particles. The derivation of the Navier-Stokes equations requires an assumption that the gas is close to a ‘‘local equilibrium state’’, that is a ‘‘Maxwell-Boltzmann’’ equilibrium distribution with a local temperature, density, and mean velocity that depend upon position and time. It is also assumed that the local variables vary on a scale that is long compared to the mean free path of the particles between collisions. These assumptions lead to the ‘‘normal solution’’ of the Boltzmann equation that, in turn, leads to the Navier-Stokes, and higher order hydrodynamic equations together with explicit expressions for the associated transport coefficients (S. Chapman and T. G. Cowling, 1971).

For many years it was assumed that one could expand the transport coefficients for moderately dense gases in a virial expansion in much the same way as one does for equilibrium properties of such gases. This supposition was supported by the ‘‘Enskog theory’’ for transport coefficients for a ‘‘hard sphere gas’’ that does lead to a virial expansion of the transport coefficients. The Enskog theory is constructed from Boltzmann's method by: (1) modifying the expression for the rate at which binary collisions take place in the gas by a factor of the local equilibrium two-particle correlation function at contact which accounts for the fact that two particles cannot collide at points in space that are already occupied by other particles; (2) for hard spheres there are additional mechanisms of momentum and energy transport, not included in the Boltzmann equation, due to the instantaneous, “collisional transfer” transport of energy and momentum across a distance of a molecular diameter whenever two hard sphere particles collide. The Enskog theory takes these effects into account and leads to density expansions of transport coefficients about the Boltzmann result in the form of a virial series in the density. However Enskog's method does not provide a systematic expansion of transport coefficients in powers of the density. It only applies to hard sphere particles, for which simultaneous collisions of more than two particles have zero measure in phase space, and for which collisional transfer of momentum and energy are clear and well defined. Further, the Enskog method approximates the dynamics of hard sphere collisions in terms of a simple excluded volume approximation

Time correlation functions

In addition to methods based upon a fundamental kinetic equation for distribution functions in a fluid, another approach to a microscopic theory for hydrodynamics is based upon ‘‘time correlation functions’’ This method, first proposed by M. S. Green, R. Kubo, and others, is based upon the determination of solutions to the ‘‘Liouville equation’’ for many-particle systems that are close to a state of local equilibrium, similar to the solution method used in the normal solutions of the Boltzmann equation, but adjusted to an N-particle distribution function (W. A. Steele, 1971). This method also leads to the Navier-Stokes equations but with expressions for the associated transport coefficients that should be applicable to fluids at any density. These expressions are integrals of time-correlation functions and, for a transport coefficient, \(\tau\ ,\) take the form \[ \tau = \int_{0}^{\infty} dt < J_{\tau}(\Gamma,t) J_{\tau}(\Gamma,0)>, \] where the angular brackets denote an average over an equilibrium ensemble distribution function, where the positions and momenta of the \(N\) particles are denoted by \(\Gamma\ ,\) and \(J_{\tau}(\Gamma,0)\) is the initial value of a microscopic current whose form depends on the phase space variables and upon the particular transport coefficient, \(\tau\ ,\) being considered, while \(J_{\tau}(\Gamma,t)\) is the value of this current at a time \(t\) later as obtained by solving the equations of motion for the system and following the phase space trajectory for a time interval \(t\ ,\) starting at the phase point \(\Gamma\ .\)

Formal, non-equilibrium virial expansions

The problem of obtaining formal density expansions of the kinetic equation and time correlation functions was solved through the work of N. N. Bogoliubov, M. S. Green, E. G. D. Cohen and others (E. G. D. Cohen, 1993, M. H. Ernst, 1998). Bogoliubov’s method was difficult to apply and not very transparent but it did stimulate other workers to look at the problem. Green and Cohen separately realized that the same kind of cluster expansion methods used to obtain equilibrium virial expansions could be adapted and applied to non-equilibrium systems, as well. The procedure for obtaining a density expansion for the corrections to the Boltzmann equation follows a few well-defined steps that can also be applied to a formal evaluation of the time correlation functions as a density expansion about their low-density limit. One begins by integration the Liouville equation over the phase variables of all but one of the particles in order to obtain an exact equation for the one-particle distribution function, \(F_{1}({\mathbf{r}}_{1}, {\mathbf {p}}_{1},t)\ .\) This leads to the so-called ‘‘first BBGKY hierarchy equation’’ that is an equation for the single-particle distribution function, but there is an interaction term that necessarily involves the two-particle distribution function, \(F_{2} ({\mathbf{r}}_{1}, {\mathbf{p}}_{1}, {\mathbf{r}}_{2},{\mathbf{p}}_{2},t)\ ,\) when the two particles are interacting with each other. Similarly one can obtain an equation for the two-particle distribution function, but it contains an interaction term that requires knowledge of the three-particle distribution function, and so on.

This chain of equations can be made useful by means of non--equilibrium cluster expansion methods. The simplest cluster expansions, corresponding to the “Ursell expansions” for equilibrium systems, are not useful for times long compared to the mean free time between collisions, since they each contain secular terms, that is, terms growing as powers of the time \(t\ .\) The secular terms reflect the fact that the volumes of phase space where two particles collide at any time in the interval \([0,t]\) grow as powers of the time, \(t\ .\) Since the hydrodynamic equations are valid on time scales long compared to the mean free time between collisions, these expansions, as they stand, are not appropriate for a derivation of a useful kinetic equation, valid on long time scales, thus not useful for a derivation of the Navier-Stokes equations. This difficulty can be overcome by assuming that there are only short ranged correlations between the particles in the initial state of the gas (cf. J. R. Dorfman and H. van Beijeren, 1977). The simplest assumption is that at \(t=0\ ,\) all of the \(n\)-particle distribution functions have a factorized form as products of one particle distribution functions, or \(F_n(1,2,...,n,t=0) = \prod_{i=1}^{n}F_1(i,t=0)\ .\) One removes the secular terms in the cluster expansions by inverting the cluster expansion for \(F_1(i,t)\) in order to express \(F_{1}(i,t=0)\) as an expansion in products of \(F_1(t)\ ,\) in much the same way that the fugacity is expressed as a series in the density in the equilibrium case. Then this expansion can be inserted in the expansion for \(F_2(t)\) in a series involving products of one-particle functions, \(F_1(t)\ ,\) structurally similar to the ‘‘Husimi expansion’’ for equilibrium functions. When this expansion is inserted in the first BBGKY hierarchy equation, one obtains the ‘‘generalized Boltzmann equation’’, a closed equation for \(F_1(t)\) where the interaction term is an expansion in a power series in the density. The first term is the Boltzmann equation and the higher terms provide the sought after density corrections to it.

Using the resulting kinetic equation as a starting point for a derivation of the Navier-Stokes equations, one obtains explicit expressions for all the coefficients in the virial expansions of the transport coefficients in terms of the dynamical events taking place among an isolated group of particles over some time \(t\) long compared to a typical microscopic time scale. One is then left with the problem of determining what the relevant dynamical events are, and determining their contributions to the non-equilibrium virial coefficients. A very similar procedure can be used to evaluate the time correlation function expressions for transport coefficients in the form of a virial expansion, with results that are identical with those resulting from the generalized Boltzmann equation.

Divergences in the virial expansion of the transport coefficients

The procedure just outlined solves the problem of generalizing the Boltzmann equation results for the transport coefficients as a virial expansion in the density, provided one can evaluate expressions for the virial coefficients. As this involves the determination of the possible dynamical events taking place in a group, or cluster, of a given number of particles, the next problem to be solved is to determine the dynamical events that contribute to each of the virial coefficients and to evaluate their contributions by carrying out the relevant integrals. It is here that a feature of non-equilibrium virial expansions appears that is not present in equilibrium virial expansions. Unlike the equilibrium virial coefficients where the configurations of small groups of particles contribute only when the particles are close to each other, the dynamical events that contribute to the non-equilibrium virial coefficients can take place over large distances and over large time intervals. In Figure 1 we illustrate one of

Figure 1: A three=body correlated collision sequence

the dynamical events that contributes to the three-body contribution to the non-equilibrium virial coefficients. In addition to simultaneous encounters of three or four particles that take place over relatively short times, there are also contributions from correlated collision sequences that can take place over arbitrarily long time intervals. Both phase space estimates and explicit calculations of the contributions of such correlated collision sequences to the non-equilibrium virial coefficients for transport coefficients reveal a new and deep problem. For two dimensional systems of particles interacting with short ranged, repulsive forces the contributions to the three- and higher- body virial coefficients from correlated collision sequences ‘‘diverge’’ as the time \(t\) approaches infinity, starting as \(\ln t \)for the three-body term, and growing as successively higher powers of \(t\) for successive virial coefficients. For systems in three dimensions, a \(\ln t\) divergence appears in the four-body term, and the higher virial coefficients grow as successively higher powers of t. In other words, non-equilibrium virial expansions, unlike their equilibrium counterparts do not provide a useful representation of the density dependence of the transport coefficients for a moderately dense gas (E. D. G. Cohen, 1993, M. H. Ernst, 1998). The source of the problem can be easily identified. The non-equilibrium virial coefficients contain contributions from sequences of collisions among small groups of particles in which the particles can travel arbitrarily large distances between its collisions in the sequence. In actuality such long paths occur with very small probability. Instead, a typical trajectory is interrupted by collisions with particles in the gas. This means that the typical distance between collisions is a mean free path length. Virial expansions force us to ignore this ever-present collision damping of the probability for a long, free trajectory between successive collisions. Nevertheless, virial expansions of the transport coefficients are directly related to virial expansions of time correlation functions, via the Green-Kubo relations, and these expansions can be used to determine the short time behavior of the time correlation functions (de Schepper, Ernst and Cohen, 1981).

In order to obtain a well-behaved density expansion, not necessarily a virial expansion, of transport coefficients for a moderately dense gas, one must renormalize the virial expansion by summing the most divergent contributions in each order of the density (K. Kawasaki and I. Oppenheim, 1965). This resummation, known as the ‘‘ring sum’’, introduces the mean free path damping of the trajectory probabilities into now modified expressions for transport coefficients as functions of *The resummation has two important consequences:

  • There are now logarithmic terms in the density expansion of the transport coefficients, of relative order \(\tilde{n}\ln{\tilde{n}}\) in two dimensions and \({\tilde{n}}^2\ln{\tilde{n}}\) in three dimensions. Therefore the density expansion for transport coefficients, denoted by \(\tau\ ,\) for three dimensional gases is given by

\[ \frac{\tau(\tilde{n},T)}{\tau_0(\tilde{n},T)} = 1 + a_{1,\tau}\tilde{n} + a_{2,\tau}{\tilde{n}}^2 + a_{2,\tau}^{\prime}{\tilde{n}}^2\ln{\tilde{n}} + \cdots . \] where \(\tau_0\) is the Boltzmann value. Terms beyond those explicitly above include higher powers and combinations of powers and logarithms of the reduced density \(\tilde{n}\ ,\) however little more than that is known. For hard spheres particles the coefficients \(a_{1,\tau}\) and \(a_{2,\tau}^{\prime}\) are known exactly. There are reasonable estimates available for \(a_{2,\tau}\ ,\) also for hard spheres. The appearance of the logarithmic terms can easily be understood as an effect of the mean free path damping, since it provides a cut-off on the order of a few mean free times on time integrals that would otherwise be logarithmically divergent.

  • The ring sum provides a microscopic explanation of the \(t^{-d/2}\ ,\) or ‘‘long time tails’’ in the time correlation functions appearing in the integrands of the Green-Kubo expressions for transport coefficients, discovered by means of computer simulations by B. J. Alder and T. Wainwright (B. J. Alder and T, Wainwright, 1970). Kinetic theory provides a quantitative description of this decay as resulting from the effects of the dynamical events included in the ring summation. For three-dimensional systems the long-time tail contribution to transport coefficients can be large and are the dominant contribution for fluids that are near a critical points of a phase transition. The long time tail contributions to two-dimensional systems are divergent. The long time tail decays of the Green-Kubo integrands and the resulting divergence of the Green-Kubo integrals in two dimensions requires a re-examination of the microscopic derivations of the Navier-Stokes and higher order hydrodynamic equations. The conclusion reached from such an examination is that the familiar linear laws relating currents of mass, momentum and energy to gradients in temperature, mean velocity, and pressure must be modified, at the Navier-Stokes equations for two dimensional fluids and at the higher order gradient terms beyond the Navier-Stokes order for three dimensional fluids (M. H. Ernst and J. R. Dorfman, 1975).

Conclusion: Can this picture be checked?

In view of the many complications that arise when attempting to develop density expansions for a fundamental kinetic equation beyond the Boltzmann, dilute gas limit, or for transport coefficients beyond their dilute gas limit, it is imperative to check the results obtained so far with experimental results or with the results of computer simulated molecular dynamics. Comparisons have been carried out for three kinds of observations:

  • Efforts to detect the presence of terms of order \({\tilde{n}}^2\ln {\tilde{n}}\) in the density expansions of transport coefficients in experiments on moderately dense gases at room temperature. have been inconclusive. However comparisons of the theoretical results for hard-sphere systems, with exact values for \(a_{1,\tau}\),\(a_{2,\tau}\prime\) and an estimate of the coefficient \( a_{2,\tau}\ ,\) in the expression above, with computer results of Alder, Gass, and Wainwright suggest good agreement over a range of densities beyond the dilute gas limit (J. R. Dorfman, T. R Kirkpatrick and J. V. Sengers, 1994). This is illustrated in Figure 2, where the dashed line is the result obtained by keeping only the linear order term in the density.
Figure 2: Theoretical and computer results for the density dependence of the coefficient of shear viscosity for hard sphere gases.

The presence of logarithmic terms in the density expansion of the coefficient of diffusion has been unambiguously demonstrated by Bruin and van Leeuwen for a simplified model gas, the ‘‘Lorentz gas’’, where a particle moves in a system of randomly placed, fixed scatterers with which it makes elastic, specular collisions (C. Bruin, 1974). For this simplified model it is possible to calculate several in the expansion of the diffusion coefficient in powers, and products of powers and logarithms, of the scatterer density. The diffusion of electrons in helium gas at temperatures below \(4 K \)has been studied using a quantum version of the theory described here. The diffusion coefficient for the electrons as a function of the helium density also has logarithmic terms, and the theoretical results and experimental data are in good agreement

  • As mentioned above, the long time tails were discovered by means of computer experiments (B. J. Alder and T. Wainwright, 1967). Theoretical values for the coefficients of these exponential decays can be obtained by means of kinetic theory using the ring summation that was used to renormalize the divergent terms in the virial expansions of the transport coefficients. This resummation leads both to the logarithmic terms in density expansions discussed above, as well as to the long time tails (Y. Pomeau, 1971; J. R. Dorfman and E. G. D. Cohen, 1970; see also Y. Pomeau and P. Resibois,1975) The excellent agreement between theoretical and computer simulation values for the coefficients of the long time tail decays, over a range of densities, provides confirmation of the basic soundness of the theory. One should also mention that the same theory can be generalized to cover the properties of transport coefficients near a critical point of a phase transition. This generalization, known as ‘‘mode-coupling theory’’ also leads to results that are in good agreement with experiments.
  • The ring resummation can be applied to calculation of the spectrum of light scattered by a fluid maintained in a non-equilibrium stationary state. Kirkpatrick, and co-workers have shown that this leads to a significant enhancement of the central, or Rayleigh peak, of the scattered light for a fluid that has a stationary temperature gradient (T. R. Kirkpatrick, E. G. D. Cohen, and J. R. Dorfman, 1982). Sengers and co-workers have measured this enhancement and demonstrated the excellent agreement of the theoretical and experimental results (Ortiz de Zarate and J. V. Sengers, 2006).


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