Comments on the Experimental and Theoretical Study of Transport Phenomena in Simple Liquids (original) (raw)

As we shall see, the exact kinetic equations for a dense fluid can be displayed only in the most formal way at the present time. Consequently, their asymptotic Markovian form is unknown, and the forms of the equations derived to describe a dense fluid are based on an intuitive analysis of the nature of random processes. Indeed, at present the only kinetic equations applicable to the description of phenomena in the liquid state are those derived using the time-smoothing technique introduced by Kirkwood [5], or its equivalent. The method of obtaining equations satisfied by the one-and twomolecule distribution functions / (1) (Γ' 1 ; t)> / {2) (Γ 2 ; t), respectively, is essentially that of integrating the iV-molecule distribution function / {Ν) (Γ Ν \t) over the subphase space of all the other molecules in the system. Now, f iN) satisfies the Liouville equation, and is not known explicitly. Therefore, one may only obtain differential equations for / ( 1 ) and / ( 2 ) by integrating the Liouville equation term by term. The result is a coupled hierarchy of equations; i.e., the equation for/ (1) also involves / ( 2 ) , the equation for / ( 2 ) also involves / ( 3 ) , and so on. It is necessary to truncate this hierarchy at some point in order to obtain closed equations for/« 1 ) and/< 2 >. For a classical fluid we describe the system of iV-structureless molecules in the volume V by use of the Hamiltonian equations of motion. These equations have some interesting general implications. Since there is one equation for each degree of freedom of the system, it follows that the phase of the system at any instant is uniquely determined by the phase at any other instant. In accordance with the definition of a Markovian random process, it follows that the phase of the system Γ Ν may be regarded as a Markovian process of a simple kind. (The transition probability is a δ-function, since the increment of the variable Γ Ν has only one possible value for each time instant.) The kinetic equations for the reduced distribution functions/ (1) ,/ (2) ,..., are concerned with the random variables ΓΊ(1), Γ 2 (1,2),..., which are of smaller dimensionality. Now, it is well known that the projection of a Markovian process of 6iV dimensions onto a space of smaller dimensionality (6, 12,..., dimensions) generally yields a random process of higher order. Thus, /\(1), Γ 2 (1, 2),..., will be non-Markovian processes of high order. This general feature has been obtained in the analysis of Prigogine and co-workers. They find that the stochastic interaction term has the form of a time convolution over the history of the variable. The important result is that when the system has reached a stationary state, the kinetic equations reduce to Markovian form. At this point it is legitimate to raise the question: What is the connection, if any, between the equations of hydrodynamics in microscopic or * The successive collisions suffered by a particular molecule are the result of its motion through an environment which is assumed to be (statistically) unaffected by the rebounding molecules. + The dynamical events are, of course, the interaction of the molecule, pair of molecules, etc., under consideration, with their environment. Clearly, the phase of the molecule, pair, etc., is not independent of the phase during a previous interval; it is the phase of the environment which is (assumed) independent of the phase during a previous interval. ]. * There is no kinetic contribution to the bulk viscosity; in other words: k = 0. * The lifetime of the correlations is, in fact, of the order of a relaxation time. Nevertheless, the destruction term vanishes in an interaction time. * In the present context the friction coefficient should be regarded as a diffusion (i.e., transport) coefficient in momentum space. + Expressions for transport coefficients of both types are often referred to as Kubo relations. * Referred to hereafter as PNM. * Hereafter referred to as BBR (see Berne et al. [51], Boon et al. [52]). * See Helfand [54]. + See Boon et al. [52]. * This proof is slightly more general because it is valid regardless of the time dependence of the phenomena considered, while the use of the autocorrelation formalism introduces an implicit restriction to the study of quasi stationary processes. * See Rice and Allnatt [68]. * See diagram on page 316. * Referred to hereafter as CR. See Sedgwick and Collins [69], and Collins and Raffel [70]. * Note added in proof: Data for τ/asa function of pressure in Ar and 0 2 have been reported by De Bock, Gravendock, and Awouters [7a]. * Note added in proof: Measurements have recently been reported by Naugle, Lunsford, and Singer [4-112], and by Madigorsky [24a]. + A complete bibliography of publications dealing with the transport properties of simple liquids is given in Section 4.1. * For instance, a similar development for the neon-deuterium pair, which can be considered in a certain sense to be "isotopic species" (see Table ), leads to the correct qualitative result, but with a discrepancy of about 30 % between theory and experiment. + The.same kind of calculation performed for the case of the thermal conductivity would lead to equivalent conclusions. * This system was examined from a classical point of view in Section 3.3.