E. G. D. Cohen

Dynamical ensembles in stationary states

We propose, as a generalization of an idea of Ruelles to describe turbulent fluid flow, a chaotic hypothesis for reversible dissipative many-particle systems in nonequilibrium stationary states in general. This implies an extension of the zeroth law of thermodynamics to nonequilibrium states and it leads to the identification of a unique distribution μ describing the asymptotic properties of the time evolution of the system for initial data randomly chosen with respect to a uniform distribution on phase space. For conservative systems in thermal equilibrium the chaotic hypothesis implies the ergodic hypothesis. We outline a procedure to obtain the distribution μ: it leads to a new unifying point of view for the phase space behavior of dissipative and conservative systems. The chaotic hypothesis is confirmed in a nontrivial, parameter-free, way by a recent computer experiment on the entropy production fluctuations in a shearing fluid far from equilibrium. Similar applications to other models are proposed, in particular to a model for the Kolmogorov-Obuchov theory for turbulent flow.

Analysis of the transport coefficients for simple dense fluid: Application of the modified Enskog theory

Abstract The viscosity (η) and thermal-conductivity (λ) coefficients for argon, oxygen, and p-hydrogen have been calculated from the modified Enskog theory (i.e., the hard-sphere Enskog theory adapted to include experimental PVT data) and compared with experiment over a wide range of experimental conditions. Specifically, experimental data and theoretical predictions for the first density corrections, η1 and λ1, were examined and the temperature and density dependences of the experimental and theoretical transport coefficients in the liquid were studied. A brief comparison of the modified Enskog theory with some other theories is included. Overall, the modified Enskog theory, with the exception of the critical region for the thermal conductivity, is found to give reasonable agreement with experiment (to within about 10–15%) for densities generally not exceeding twice the critical density. Qualitatively the theory does not distinguish between the viscosity and thermal conductivity in the liquid. This is discussed by comparing the experimental and theoretical derivatives ( ∂Δη ∂T ) Q and ( ∂Δλ ∂T ) Q , where Δη and Δλ are excess functions. The qualitative features of the theory are discussed in some detail leading to a method by which the predictive capability of the theory can be improved.

Extension of the fluctuation theorem.

Heat fluctuations are studied in a dissipative system with both deterministic and stochastic components for a simple model: a Brownian particle dragged through water by a moving potential. An extension of the stationary state fluctuation theorem is derived. For infinite time, this reduces to the conventional fluctuation theorem only for small fluctuations; for large fluctuations, it gives a much larger ratio of the probabilities of the particle to absorb rather than supply heat. This persists for finite times and should be observable in experiments similar to a recent one carried out by Wang et al.

Difficulties in the Kinetic Theory of Dense Gases

For the determination of the transport coefficients of a dense gas, the long‐time behavior of the pair distribution function F2 for small intermolecular distances is obtained from a density expansion in terms of the first distribution function F1. On the basis of the dynamics of small groups of particles, it is shown that this expansion contains divergences so that it cannot be used for (a) the computation of the long‐time behavior of F2 beyond O(n); (b) the demonstration of the decay of the initial state beyond O(n2). Similar divergences are encountered in the computation of the transport coefficients from time‐correlation functions. The nature of the divergences suggests (a) there is no kinetic stage in the approach of a dense gas to equilibrium, in the sense of Bogoliubov; (b) a weak logarithmic density dependence of the transport coefficients.

On the generalization of the boltzmann equation to general order in the density

Abstract An expansion of the pair distribution function as a functional of the first distribution function in powers of the density is derived for a system not in equilibrium. This is achieved, using cluster expansions that are analogous to the Ursell-Uhlenbeck-Kahn- and the Husimi-expansions known from equilibrium statistical mechanics. On the basis of this procedure it is shown, that the density expansions of the non-equilibrium- and the equilibrium pair distribution function can be characterized by the same diagrams, where in the non-equilibrium case the first distribution function takes the role of the density in the equilibrium case. The general term in a systematic generalization of the Boltzmann equation to higher densities follows immediately from the expansion.

Fifty years of kinetic theory

While the kinetic theory of dilute gases based on Boltzmanns 1872 equation was essentially completed by Chapman and Enskog around 1915, a concentrated effort to generalize the Boltzmann equation systematically to higher densities to obtain expansions of the transport coefficients in powers of the density was only started about 1945. Around 1965 it was realized, however, that this was impossible and that non-analytic terms in the density would occur. This ended the classical period of kinetic theory and inaugurated the modern era at about the same time as this occurred in the theory of phase transitions. New phenomena for which a kinetic, i.e., molecular, theory were given, include: the long time (tail) correlations in equilibrium time correlation functions related to the transport coefficients and later the long spatial correlations of density fluctuations in non-equilibrium stationary states. In addition a kinetic theory has been developed for the transport properties of dense gases and liquids. The theory is based on identifying the relevant collision processes as suggested by computer simulations of dense hard sphere fluids and by neutron scattering experiments of atomic liquids: ring and cage collisions. The theory has been applied to dense simple liquids as well as to concentrated colloidal suspensions. Finally, some open problems and current investigations are discussed.

Statistics and dynamics

The classical statistics of Boltzmann and Gibbs will be discussed. As pointed out long ago by Einstein, there is a connection of the statistics applicable to a system and its underlying dynamics. This follows, for example, from the existence of quantum (Bose–Einstein or Fermi–Dirac) statistics, where the underlying dynamics is quite different from classical mechanics. In 1988 Tsallis proposed a very successful new statistics. Its connection with an underlying dynamics as well as with the nonextensivity of systems are still matters of active investigation.

On the Kinetic Theory of Dense Gases

A statistical mechanical theory of a dense gas that is not in equilibrium is presented, which is completely analogous to the well known theory of a dense gas in equilibrium. In particular, an expansion of the pair distribution function in powers of the density for a gas not in equilibrium is given, corresponding with that in equilibrium to all orders in the density, that can be represented by the same diagrams. The expansion can be reduced to that derived by Bogolubov, Uhlenbeck, and Choh from a solution of the B‐B‐G‐K‐Y hierarchy. The conditions for the validity of the expansion are, for an infinite system at not too high density, and after the lapse of some time after t = 0: (1) a statistical assumption at t = 0; (2) some conditions on the interaction potential; (3) coarse‐grained distribution functions. A simple generalization of the Boltzmann equation to general order in the density is included. Also, the connection with a Master equation for a spatially homogeneous system is discussed.

Transport coefficients in dense gases I: The dilute and moderately dense gas

Abstract Explicit expressions for the transport coefficients in a classical dense gas in terms of the intermolecular forces are derived by a systematic evaluation in powers of the density of general formulae for transport coefficients in terms of time correlation functions. The evaluation can be reduced to the computation of the very long time behaviour of certain time correlation functions in a dense gas in equilibrium. This can be accomplished by similar methods as are used in the theory of transport coefficients in dense gases from non-equilibrium distribution functions. In this paper the general expressions for the viscosity and heatconductivity are calculated to lowest and first order in the density, and the results are shown to be equivalent with the results of Chapman and Enskog, and of Choh and Uhlenbeck respectively.

The transport properties and the equation of state of gaseous para- and ortho-hydrogen and their mixtures below 40°K

Synopsis Using a spherically symmetrical Lennard-Jones potential field, the following properties of para-H2, ortho-H2 and their mixtures were quantum-mechanically calculated: 1) viscosity and heat conductivity, with special attention paid to the dependence of viscosity on the para-ortho concentration ratio; 2) coefficients of self-diffusion and of mutual diffusion; 3) thermal diffusion ratio; 4) second virial coefficient. The agreement between experiment and the calculated viscosity and heat conductivity coefficients is good. Our calculations on the concentration dependence of viscosity show that pH2 has a greater viscosity than nH2, contrary to previous theoretical predictions, but in accordance with the measurements of Becker and Stehl and Van Itterbeek and Coremans. A similar effect in the second virial coefficient must be negligibly small, according to both experiment and theory. The second virial coefficient itself is about 10% smaller than the experimental value, which is probably due to the non-spherical character of the real potential field.