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Showing papers in "Physica A-statistical Mechanics and Its Applications in 1978"


Journal ArticleDOI
TL;DR: In this article, it is shown that there exist simple formulae for splitting correlations, which arise when dealing with widely used models of random processes such as Poisson, gaussian and Markov jump processes with an exponential form of the covariance.
Abstract: New tools for the solution of stochastic differential equations are presented. It is shown that there exist simple formulae for splitting correlations, which arise when dealing with widely used models of random processes such as Poisson, gaussian and Markov jump processes with an exponential form of the covariance. The use of the formulae is demonstrated.

312 citations


Journal ArticleDOI
TL;DR: A general treatment of the problem of diagonalizing a hamiltonian which is a homogeneous quadratic expression in a finite number of boson construction operators has been given in this article.
Abstract: A general treatment has been given of the problem of diagonalizing a hamiltonian which is a homogeneous quadratic expression in a finite number of boson construction operators. The treatment provides us with several systematic procedures to diagonalize such hamiltonians in practice; each algorithm in principle requires only a single unitary diagonalization of a hermitian matrix.

280 citations


Journal ArticleDOI
TL;DR: The transition from dynamical (regular) to stochastical behaviour in nonlinear quantum systems is considered in this paper, based on the determination of a discrete time mapping of the creation and annihilation operators in the Heisenberg picture and the projection of this mapping on the phase space of coherent states.
Abstract: The transition from dynamical (regular) to stochastical behaviour in nonlinear quantum systems is considered. A method of describing the stochastical instability is proposed for quantum systems. The method is based on the determination of a discrete time mapping of the creation and annihilation operators in the Heisenberg picture and the projection of this mapping on the phase space of coherent states. The condition that the phase correlation vanishes is found for a nonlinear quantum oscillator perturbed by the external periodical force, which is a linear combination of delta pulses. The kinetic equation describing the relaxation of the system in the space of c-numbers formed by the projections of the operators is derived.

204 citations


Journal ArticleDOI
TL;DR: In this article, the authors derived the second description from the first by means of a systematic expansion procedure of the Chapman-Enskog type in terms of the inverse friction coefficient, and also derived equations describing the initial period, when the Smoluchowski description is not yet valid.
Abstract: The motion of a Brownian particle in an external field can be described on two levels: by a Fokker-Planck equation for the joint probability distribution of position and velocity, and by a Smoluchowski equation for the distribution in position space only. We derive the second description, with corrections, from the first by means of a systematic expansion procedure of the Chapman-Enskog type in terms of the inverse friction coefficient. We also derive equations describing the initial period, when the Smoluchowski description is not yet valid; in particular we find formulae connecting the initial value to be used for the Smoluchowski equation with that of the full Fokker-Planck equation. The special case of an harmonically bound Brownian particle can be solved exactly; the results are used to check and to illustrate our expressions for general potential.

196 citations


Journal ArticleDOI
TL;DR: In this article, the authors present rigorous working equations for an automatic Wheatstone bridge employed in fluid thermal conductivity measurements by the transient hot-wire technique, focusing on the inevitable small differences between the two wires employed for the cancellation of the end effects in the experimental arrangement.
Abstract: The paper presents rigorous working equations for an automatic Wheatstone bridge employed in fluid thermal conductivity measurements by the transient hot-wire technique. Particular attention is concentrated upon the inevitable small differences between the two wires employed for the cancellation of the end effects in the experimental arrangement. It is shown that from experimental measurements of the change in the difference of resistance of the two wires as a function of time during their transient heating, it is possible to determine the temperature rise of a wire acting as a finite section of an infinite wire. The correct expressions for the calculation of the heat flux in the wire are also given.

130 citations


Journal ArticleDOI
TL;DR: In this article, the diffusion tensor for a Brownian particle in a periodic field of force is studied in the strong damping limit, in which the Smoluchowski equation is valid.
Abstract: The diffusion tensor for a Brownian particle in a periodic field of force is studied in the strong damping limit, in which the Smoluchowski equation is valid. A general relation between the diffusion tensor and the Smoluchowski “relaxation operator” is derived; the effect of the periodic force, at least in the simplest situation of diagonal and uniform friction, appears as a dressing of the bare particle mass to an effective tensor mass. From this the explicit form of the diffusion coefficient as a functional of the potential energy in the one-dimensional case is obtained, showing a temperature dependence which deviates at high temperatures from a simple Arrhenius behaviour. Finally, the expression for the mobility of the Brownian particle is derived, and by comparison with the expression for the diffusion coefficient the Einstein relation between diffusion and mobility is proved to be satisfied.

120 citations


Journal ArticleDOI
TL;DR: In this article, the authors derived the kinetic equation for a density matrix, which describes the relaxation of the internal states of encountering particles dissolved in an inert medium, under the following assumptions: a) the random motion of reacting particles in a liquid is considered to be a classical Markoffian process; b) the concentration of reacted particles is small enough.
Abstract: The kinetic equation for a density matrix, which describes the relaxation of the internal states of encountering particles dissolved in an inert medium, has been derived under the following assumptions: a) the random motion of reacting particles in a liquid is considered to be a classical Markoffian process; b) the concentration of reacting particles is small enough. The equation obtained is shown to be the generalization of that of the familiar impact theory of pressure broadening for the case of any type of encountering particle motion. Our general formulae are concretized in accordance with the physical situations of rectilinear, diffusion, and stochastic jump motion of the encountering particles.

81 citations


Journal ArticleDOI
TL;DR: In this article, self-diffusion data for methane obtained by the NMR spin-echo technique are reported at −50°, 25° and 50°C over the density range 2 to 27 mol dm−3.
Abstract: Self-diffusion data for methane obtained by the NMR spin-echo technique are reported at −50°, 25° and 50°C over the density range 2 to 27 mol dm−3. The apparatus was calibrated using literature atmospheric pressure tracer data for water and benzene and the calibration was checked using high pressure tracer data for benzene obtained by a separate technique. The methane results are found to be somewhat lower than those of earlier workers, but are consistent with limiting values calculated using the recent numerical intermolecular potential of Matthews and Smith. The data suggest that the first “virial” coefficient in the density expansion changes sign just above −50°C, though measurements at lower densities than are accessible with the present apparatus would be needed to confirm this conclusion. Chandler's rough hard sphere model is applied to the high density data and these are consistent with there being little, if any, translational-rotational coupling for diffusive motion of CH4. A relation is established between the hard sphere diameter and temperature allowing the calculation of D for liquid and dense gaseous methane in the region 90 20 mol dm−3. A comparison is made between recent molecular dynamics calculations of D and the experimental measurements.

73 citations


Journal ArticleDOI
TL;DR: In this article, it is shown that an explicit form for the dynamical equation of state may be obtained if in the neighbourhood of some state of thermodynamic equilibrium it is permissible to consider the equilibrium pressure and the thermodynamic affinities (conjugate to the internal degrees of freedom) as linear functions of ν, T and the internal variables and if, moreover, the phenomenological coefficients which occur in the equations for the irreversible processes, may be considered as constants.
Abstract: In this paper we investigate the relation (dynamical equation of state) among the (mechanical hydrostatic) pressure P , the volume ν and the temperature T of an isotropic medium with an arbitrary number, say n , of scalar internal degrees of freedom. It is assumed that the irreversible processes in the medium are due to volume viscosity and to changes in the internal variables and that these processes can be described with the aid of non-equilibrium thermodynamics. It is shown that an explicit form for the dynamical equation of state may be obtained if in the neighbourhood of some state of thermodynamic equilibrium it is permissible to consider the equilibrium pressure and the thermodynamic affinities (conjugate to the internal degrees of freedom) as linear functions of ν, T and the internal variables and if, moreover, the phenomenological coefficients, which occur in the equations for the irreversible processes, may be considered as constants. This dynamical equation of state has the form of a linear relation among P , ν, T , the first n derivatives with respect to time of P and of T and the first n +1 derivatives with respect to time of ν.

72 citations


Journal ArticleDOI
TL;DR: In this article, the transverse time-dependent spin-spin correlation functions (Sx1(0)SxR+1(t) ) and Sy1( 0)SyR+ 1(t ) were computed at zero temperature for the one-dimensional XY model with the hamiltonian H N = - Σ N i = 1 [(1 + γ)S x i S x i R i+1 + (1 − 1 + hS z i ].
Abstract: We compute exactly the transverse time-dependent spin-spin correlation functions 〈Sx1(0)SxR+1(t)〉 and 〈Sy1(0)SyR+1(t)〉 at zero temperature for the one-dimensional XY model that is defined by the hamiltonian H N = - Σ N i=1 [(1 + γ)S x i S x i+1 + (1 − γ)S y i S y i+1 + hS z i ] . We then analyze these correlation functions in two scaling limits: (a) γ fixed, h → 1, R → ∞, t → ∞ such that ‖(h − 1)/γ‖[R 2 − γ 2 t 2 ] 1 2 is fixed, and (b) h fixed less than one, γ → 0+, R → ∞, t → ∞ such that γ[R 2 − (1 − h 2 )t 2 ] 1 2 is fixed. In these scaling regions we give both a perturbation expansion representation of the various scaling functions and we express these scaling functions in terms of a certain Painleve transcendent of the third kind. From these representations we study both the small and large scaling variable limits in both the space-like and time-like regions.

69 citations


Journal ArticleDOI
TL;DR: In this paper, the authors report on new, accurate (± 0.3%) measurements of the viscosity of binary and multicomponent mixtures involving the four saturated hydrocarbons: CH 4, C 2 H 6, C 3 H 8 and C 4 H 10.
Abstract: The paper reports on new, accurate (±0.3%) measurements of the viscosity of binary and multicomponent mixtures involving the four saturated hydrocarbons: CH 4 , C 2 H 6 , C 3 H 8 and C 4 H 10 . Measurements were restricted to atmospheric pressure and the temperature range 25–195°C. The measurements confirm that the pure components conform to the extended law of corresponding states of Kestin, Ro and Wakeham and prove that the same is true about their mixtures. Measurements on the binary mixtures serve to determine the scaling factors σ 12 and ϵ 12 for binary interactions, and measurements on multicomponent mixtures confirm the continued validity of the extended law of corresponding states for these complex molecules. The measurements yield values of the binary diffusion coefficients of the six pairs. The binary diffusion coefficients for C 2 H 6 - n ·C 4 H 10 and C 3 H 8 - n ·C 4 H 10 do not seem to have been measured before.

Journal ArticleDOI
TL;DR: In this article, the complete diffusion matrix describing the translation and rotation of two spherically symmetric polymers in hydrodynamic interaction in a viscous fluid is evaluated in terms of a series expansion in inverse powers of the distance l between centers.
Abstract: We evaluate the complete diffusion matrix describing the translation and rotation of two spherically symmetric polymers in hydrodynamic interaction in a viscous fluid. Explicit results in terms of a series expansion in inverse powers of the distance l between centers is obtained by a method of reflections. Recently derived Faxen theorems allow direct calculation of the diffusion matrix, rather than its inverse the friction matrix. The results can be easily compared with those for hard spheres with mixed slip-stick boundary conditions.

Journal ArticleDOI
TL;DR: In this paper, the authors restrict themselves to finite and infinite one-dimensional systems of equal particles interacting with two-body potentials of simple type, among others to convexrepulsive interactions (at a given number density of the system) and to Lennard-Jones-type potentials.
Abstract: From the empirical fact that at sufficiently low temperatures most substances crystallize one may be led to the assumption that for (infinitely) large systems of interacting particles the configuration with minimum potential energy per particle will be a periodic structure. Plausible as this assumption may seem, as far as we are aware rigorous proof has not yet been given. In this paper we restrict ourselves to finite and infinite one-dimensional systems of equal particles interacting with two-body potentials of simple type, among others to convexrepulsive interactions (at a given number density of the system) and to Lennard-Jones-type potentials. For these cases a number of theorems have been proved rigorously. The theorems are stated in the Introduction and the proofs are given in subsequent sections.

Journal ArticleDOI
TL;DR: In this paper, a general solution of the linear Navier-Stokes equation for incompressible viscous flow about a sphere with mixed slip-stick boundary conditions for arbitrary incident flow is presented.
Abstract: We present the general solution of the linear Navier-Stokes equation for incompressible viscous flow about a sphere with mixed slip-stick boundary conditions. Exphclt results are obtained in cartesian coordinates. The relation wRh Lamb's method is indicated. We derive expressions for a set of fundamental tensors, introduced by Brenner, describing the perturbed flow. Finally the force density induced on the sphere is evaluated. In this paper we give the solution of the linear Navier-Stokes equation for incompressible viscous flow about a sphere with mixed slip-stick boundary conditions for arbitrary incident flow. A method of solution was presented long ago by Lamb l) in terms of solid spherical harmonics, but his solution left something to be desired from a practical point of view. Lamb's solution is described in some detail in the monograph by Happel and Brenner2). The difficulty with Lamb's solution is that application to particular problems often requires writing the harmonics in terms of spherical coordinates, whereby a specific choice of axes, alien to the problem at hand, is made. For many problems an approach using cartesian coordinates is more convenient. In particular for the problem of hydrodynamic interaction of two spheres this approach has recently proved effective3"4). In the following we persistently use the cartesian point of view and solve the problem by the so-called method of cartesian ansatz. The solution is a direct generalization of earlier results obtained for constant, linear or quadratic incident flow3-7). At the end we show the relation to Lamb's solution. As a benefit we obtain explicit results for a set of tensors introduced by Brenner 8) in his fundamental paper on Stokes resistance. Brenner only gave explicit

Journal ArticleDOI
TL;DR: In this article, the authors compared eight possible definitions of the surface thickness for two density profiles, and then the 10-90 thickness t was shown to be in satisfactory agreement with the other six.
Abstract: We compare eight possible definitions of the surface thickness for two density profiles. One definition is eliminated, and then the 10–90 thickness t is shown to be in satisfactory agreement with the other six. Our earlier work on the liquid-vapour interface is taken further, with a physical interpretation of the slowly varying density limit for the surface energy ∈; an improved estimate of t using experimental values of ∈ and the bulk energy; and an estimate of t using experimental values of ∈ and the surface tension. We obtain t ≈ 1.3d (d is the atomic diameter) for Ar, Kr and Xe near their triple points. A comparison is made with other estimates for t/d. These vary from 1 to above 3. Our results are in best agreement with Monte-Carlo simulations of the interface. The results for liquid 4He at zero temperature are discussed briefly.

Journal ArticleDOI
TL;DR: The theory of the ground state of the two-dimensional non-ideal Bose gas is presented in this paper, and conditions for the validity of the ladder and the Bogolubov approximations are derived.
Abstract: The theory of the ground state of the two-dimensional non-ideal Bose gas is presented. The conditions for the validity of the ladder and the Bogolubov approximations are derived. These conditions ensure the existence of a Bose condensate in the ground state of two-dimensional systems. These conditions are different from the corresponding conditions for the three-dimensional case. The connection between the effective interaction and the two-dimensional scattering amplitude at some characteristic energy κ2/2m (≠0) is obtained (f(κ = 0) = ∞ in the two-dimensional case).

Journal ArticleDOI
TL;DR: In this article, the authors derived Faxen theorems for the force, the torque and the symmetric force dipole moment acting on a spherically symmetric polymer suspended in arbitrary flow.
Abstract: We derive Faxen theorems for the force, the torque and the symmetric force dipole moment acting on a spherically symmetric polymer suspended in arbitrary flow. The results are immediate generalizations of corresponding theorems for a hard sphere with mixed slip-stick boundary conditions.

Journal ArticleDOI
TL;DR: In this paper, the authors describe absolute measurements of the thermal conductivity of four pure monatomic gases, He, Ne, Ar and Kr, performed at room temperature (≈300 K) in a high-precision instrument based on the transient hotwire principle.
Abstract: The paper describes absolute measurements of the thermal conductivity of four pure monatomic gases, He, Ne, Ar and Kr. The measurements were performed at room temperature (≈300 K) in a high-precision instrument based on the transient hot-wire principle. In the cases of He, Ne and Ar the measurements were performed in the pressure range 0.8 MPa to 35 MPa, whereas for Kr the pressure range covered was 1.1 to 16 MPa. The precision and reproducibility of the experimental data are estimated to be 0.2%, whereas their accuracy is judged to be one of ±1.5%. In addition to the absolute values of the thermal conductivity, we have evaluated our results on a relative basis to take full advantage of their high precision. This has been made possible through the known Eucken factors for the monatomic gases at low density in combination with accurate viscosity data. The uncertainty in the reported thermal conductivity evaluated on the relative basis is one of ±0.4%. The thermal conductivity as a function of density for Ne, Ar and Kr has been subjected to a statistical analysis in order to ascertain whether a logarithmic term exists in its density expansion. It is concluded in all cases that a polynomial density expansion represents the data as well as a polynomial of second order plus a logarithmic term. This supports the view that a logarithmic term, if it exists, must be multiplied by a very small coefficient.

Journal ArticleDOI
TL;DR: In this article, the authors present new experimental data for the viscosity of binary mixtures of xenon with the remaining monatomic gases, helium, neon, argon, and krypton.
Abstract: This paper presents new experimental data for the viscosity of binary mixtures of xenon with the remaining monatomic gases, helium, neon, argon, and krypton. The measurements have been performed in a high precision oscillating-disk viscometer at atmospheric pressure and within the temperature range 25–500°C. The data have an estimated uncertainty of ±0.1% at 25°C increasing to ±0.3% at 500°C. The collision integrals for the interactions of xenon with the other monatomic species conform to the extended law of corresponding states formulated by Kestin, Ro and Wakeham. For each binary interaction the scaling parameters σij and ∈ij have been obtained. The ensemble of experimental results can be correlated by means of the appropriate kinetic theory expressions reinforced by the extended law of corresponding states. The deviations do not exceed 0.5%. The binary diffusion coefficients were calculated from the measured mixture viscosity and compared with the available experimental results. The standard deviation was estimated as ±2% which is within the mutual uncertainty of the two sets.

Journal ArticleDOI
TL;DR: In this paper, the self-diffusion coefficient of chlorotrifluoromethane at 30°, 50° and 75°C over the density range 3 to 15 mol dm -3 was reported.
Abstract: In this paper are reported NMR spin-echo measurements of the self-diffusion coefficient of chlorotrifluoromethane at 30°, 50° and 75°C over the density range 3 to 15 mol dm -3 . No evidence was found for the type of critical region anomaly recently reported by Duffield and Harris for carbon dioxide. The high density data are correlated with molecular dynamics computer simulated data for hard spheres. The translational-rotational coupling factor for Chandler's rough hard sphere model is found to be 0.90±0.02. For the viscosity the coupling factor is 0.77±0.02. These values are found to be consistent with limits set by classical hydrodynamic theory.

Journal ArticleDOI
TL;DR: In this article, a rigorous closed set of differential equations is derived for the correlations between x and y components of spins for the inhomogeneous XY-model with spin 12 in the high-temperature limit.
Abstract: A rigorous closed set of differential equations is derived for the correlations between x- and y-components of spins for the inhomogeneous XY-model with spin 12 in the high-temperature limit. The correlations can also be derived from a Lagrangian which describes the motion in a classical lattice problem.

Journal ArticleDOI
TL;DR: In this article, a generalized Smoluchowski equation was derived for the momenta in terms of the reciprocal of the friction coefficient, and a mode-mode coupling equation for the local concentration, which still depends on the random forces of the solvent.
Abstract: To describe dynamical properties of a system of interacting Brownian particles stochastic transport equations are derived for the positions of the particles and their concentration fluctuations. This is achieved by an expansion of the Langevin equation for the momenta in terms of the reciprocal of the friction coefficient. As a by-product this procedure gives a new derivation of the generalized Smoluchowski equation. Using a local equilibrium approximation for the configurational distribution function a mode-mode coupling equation is derived for the local concentration, which still depends on the random forces of the solvent. For the interaction free case the relation to the ordinary diffusion approach is established.

Journal ArticleDOI
TL;DR: In this paper, a method for predicting the composition dependence of the thermal conductivity of dense gas mixtures is presented, where the Thorne-Enskog hard-sphere theory is used as an interpolating formula between the end points.
Abstract: In this paper we present a method for predicting the composition dependence of the thermal conductivity of dense gas mixtures. The method requires the knowledge of the thermal conductivities of the pure component gases at high density, of the zero-density values of the thermal conductivities both of the pure components and of one binary mixture, and of the virial coefficients Bij and their derivatives dB ij dT . The Thorne-Enskog hard-sphere theory, after a minor correction for consistency with the Onsager reciprocal relations, is then used as an interpolating formula between the end points. An extension of the method to mixtures of dense polyatomic gases is also provided. For these gases, the transport of internal energy is assumed to be entirely kinetic (diffusion mechanism). The method has been applied to the three binary monatomic systems He-Ne, Ne-Ar and He-Ar at 30°C and up to a pressure of 20 MPa. The calculated values of the thermal conductivity of these systems agree with the best available experimental data to within the uncertainty of the latter (about 2%). No comparison was performed for the polyatomic species owing to the unavailability of reliable experimental data.

Journal ArticleDOI
TL;DR: In this paper, the self-diffusion coefficient for Lennard-Jones molecules has been determined by molecular dynamics for densities up to the critical one and for temperatures ranging from T = 1.3∈/k to T = 5.56∈ /k.
Abstract: The self-diffusion coefficient for Lennard-Jones molecules has been determined by molecular dynamics for densities up to the critical one and for temperatures ranging from T = 1.3∈/ k to T = 5.56∈/ k . At low density, the results are in a good agreement with the theoretical predictions; at elevated densities agreement with the available experimental results is found, although the scarcity of experimental data prevents a general comparison between real systems and the molecular dynamical calculations. Additional computations with different pair-potentials lead to results which throw some doubts on the reliability of the so-called Modified Enskog Theory.

Journal ArticleDOI
TL;DR: In this article, a system of exact algebraic relations for spin correlation functions on any so-called "boundary set" of an Ising model on an arbitrary planar graph is derived.
Abstract: A system of exact algebraic relations is derived for the spin correlation functions on any so-called “boundary set” of an Ising model on an arbitrary planar graph. One way of expressing these relations is to say that every higher-order correlation function is equal to a Pfaffian of the pair correlation function on the same set of boundary spins.

Journal ArticleDOI
TL;DR: In this paper, a new recursion method was proposed for the calculation of the thermodynamics of Ising-spin models with short range interaction, which is exact in one dimensions, while for higher dimensions a truncation scheme is developed.
Abstract: A new recursion method is proposed for the calculation of the thermodynamics of Ising-spin models with short range interaction. The method is exact in one dimensions, while for higher dimensions a truncation scheme is developed. The approach is tested in the two-dimensional Ising model. The results are within 0.7% of the exacr reduced free energy per spin.

Journal ArticleDOI
TL;DR: In this article, the equivalence between the n-state trapping model and the continuous-time random walk model is established, where a distinct first-waiting-time distribution is introduced.
Abstract: We compare two model descriptions of transport in a material with traps. We establish the equivalence between the n -state trapping model where equilibrium initial conditions are used, and the continuous-time random walk description where a distinct first-waiting-time distribution is introduced. Tunaley's result on the frequency independence of the ac conductivity is corroborated by this equivalence.

Journal ArticleDOI
TL;DR: In this paper, the frequency dependent friction coefficient of a spherical Brownian particle of radius a in a compressible fluid close to its critical point is evaluated taking into account the large correlation length ζ of the density fluctuations in the fluid.
Abstract: The frequency dependent friction coefficient of a spherical Brownian particle of radius a in a compressible fluid close to its critical point is evaluated taking into account the large correlation length ζ of the density fluctuations in the fluid. The expression found depends for finite frequencies on the ratio a /Gx. It is furthermore shown that a negative contribution to the amplitude of the usual t −3/2 long time tail of the velocity autocorrelation function of the particle appears very close to the critical point when ζ ⪢ a .

Journal ArticleDOI
TL;DR: In this article, a comprehensive axiomatic system of the non-equilibrium thermodynamics of a surface of discontinuity is formulated, and the general formulation of the axioms is applied to the problem of an n -component visco-elastic isotropic non-polar mixture.
Abstract: A comprehensive axiomatic system of the non-equilibrium thermodynamics of a surface of discontinuity is formulated. The general formulation of the axioms is applied to the problem of an n -component visco-elastic isotropic non-polar mixture. The effect of mass currents from one phase to the other on the formulation of the hypothesis of local equilibrium is discussed.

Journal ArticleDOI
TL;DR: In this article, direct inversion methods are used to analyze the extended principle of corresponding states of Kestin and co-workers in terms of the pair interaction potential, and reference to the potential makes possible a number of extensions and improvements to the corresponding-states correlations.
Abstract: Direct inversion methods are used to analyze the extended principle of corresponding states of Kestin and co-workers in terms of the pair interaction potential. It is shown how reference to the potential makes possible a number of extensions and improvements to the corresponding-states correlations. Specifically, the range of validity of the principle is made more precise, the consistency between gas viscosities and second virial coefficients is checked, and it is shown how to improve the available temperature range over which predictions can be made and the precision with which properties can be predicted.