A linear integral equation derived by Mayer for the radial distribution function of fluids has been investigated numerically. To compare with other recent investigations of integral equations, a Lennard‐Jones 6–12 p...
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A linear integral equation derived by Mayer for the radial distribution function of fluids has been investigated numerically. To compare with other recent investigations of integral equations, a Lennard‐Jones 6–12 potential was used and the 55°C isotherm of argon investigated. Results for the pressure and the radial distribution function are less satisfactory than are obtained with other integral equations. An estimate of the validity of Kirkwood's superposition approximation in the range of the calculation was also made.
The general structure for the distributionfunctions (reduced density matrices) for systems composed of a number of elements is given by taking the variation with respect to the distributionfunctions in the formalism...
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The general structure for the distributionfunctions (reduced density matrices) for systems composed of a number of elements is given by taking the variation with respect to the distributionfunctions in the formalism of the cluster variation method. The parameters or the Lagrange multipliers occurring in the distributionfunctions must be determined by the reducibility condition of the distributionfunctions or by the stationariness condition of the free energy.
Statistical mechanical averages of vectors and tensors characterizing the configurations of polymethylene chains with n = 5–200 bonds have been calculated, these quantities being expressed in the internal reference f...
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Statistical mechanical averages of vectors and tensors characterizing the configurations of polymethylene chains with n = 5–200 bonds have been calculated, these quantities being expressed in the internal reference frame attached to the first two bonds. Specifically, the persistence vector a ≡ 〈r〉 and the averaged center‐of‐mass vector 〈g〉 have been obtained as functions of chain length. They are compared to the corresponding vectors for various model chains. Configurational averages of the Cartesian tensors formed from ρ = r ? a have been computed up to and including the tensor of seventh rank. Density distributionfunctionsWa (ρ), evaluated from these moment tensors using the three‐dimensional Hermite series elaborated in the preceding paper, are approximately cyclindrically symmetric about a principal axis that lies close to the direction of a. The second moment (and, hence, the dispersion) along this axis is smaller than the moments perpendicular thereto. Acentricity and skewness of the distribution Wa (ρ) is large for n < 40 bonds; they are appreciable even at n = 100 bonds. For n ≥ 50, moments of the normalized density distribution can be represented in good approximation by a model chain consisting of freely jointed segments originating at the terminus of a, with each segment taken to be equivalent to 20 bonds of the real chain. The traditional approximation of the unnormalized density distribution about r = 0 by a freely jointed equivalent chain is subject to much greater error. The second moment tensor of gyration, averaged over all configurations, is approximately cylindrically symmetric, but its component (and its dispersion) along the symmetry axis is greater than in the transverse directions. This tensor is contrasted with that obtained by summing moments along the principal axes for individual configurations.
A new method (a topological moment method) is presented for treating statistical problems of ring polymer chains of which the topological state (Gauss linking coefficient) is prescribed. It is shown that the topologic...
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A new method (a topological moment method) is presented for treating statistical problems of ring polymer chains of which the topological state (Gauss linking coefficient) is prescribed. It is shown that the topological interaction among ring polymers formally resembles the excluded volume effect among linear polymers. A ’’topological parameter γ’’ which corresponds to the binary cluster integral β in the excluded volume problem is newly introduced for representing the strength of the topological interaction among submolecules. The parameter γ is given as a function of bond length, bond angle, internal‐rotational potential, and of interaction potentials among chain elements. Analytical expressions for the distribution function of the intermolecular distance and the second virial coefficient are derived and they are computed numerically.
A set of molecular distributionfunctions for molecules confined to a small volume is developed which makes explicit use of the short‐range nature of most intermolecular forces. The excess chemical potential for syst...
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A set of molecular distributionfunctions for molecules confined to a small volume is developed which makes explicit use of the short‐range nature of most intermolecular forces. The excess chemical potential for systems of molecules with rigid impenetrable cores is expressed in terms of this specialized set or directly related probability functions. The general theory for rigid‐sphere systems is formulated and is used in the analysis of the kernel appearing in the set of integral equations developed by Kirkwood and Salsburg.
Exact expressions are obtained for the distributionfunctions, in configuration and momentum space, for a system of bosons or spinless fermions interacting via pairwise harmonic forces.
Exact expressions are obtained for the distributionfunctions, in configuration and momentum space, for a system of bosons or spinless fermions interacting via pairwise harmonic forces.
Radial distributionfunctions are computed for the Gaussian model of a fluid by approximating the density expansion with a Padé approximant . Several choices ofandare studied by comparing these calculatedwith tho...
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Radial distributionfunctions are computed for the Gaussian model of a fluid by approximating the density expansion with a Padé approximant . Several choices ofandare studied by comparing these calculatedwith those previously obtained using the Percus–Yevick and hypernetted chain integral equations, a Monte Carlo method, and simple truncation of the density expansion. It is found that, if a few terms in the density expansion are known, the range of densities for which this information is useful may be extended by the proper choice ofandin the Padé approximant. A form ofis presented which approximatesquite well for a large range of densities.
The relaxationfunction of stress can be interpreted in terms of the distribution function of relaxation times. The retardation function of strain can be interpreted in terms of the distribution function of strain. The...
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The relaxationfunction of stress can be interpreted in terms of the distribution function of relaxation times. The retardation function of strain can be interpreted in terms of the distribution function of strain. The two distributionfunctions, or spectra, belonging to the same physical system are related by integral transforms. These transforms have been discussed in earlier papers. However, the discussion was not entirely satisfactory because it missed the appearance of single lines associated with truncated spectra, and because it employed integrals over improper and ill‐defined functions. This paper gives an alternative treatment which is believed to be more satisfactory mathematically. The distributionfunctions are found to be the imaginary components of two complex, analytical functions which are the generatingfunctions of all other functions of viscoelastic theory.
It is demonstrated that the relationship between pair potentials and distributionfunctions is experimentally ambiguous for dense fluids. Calculations in the Percus—Yevic approximation show that qualitatively differe...
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It is demonstrated that the relationship between pair potentials and distributionfunctions is experimentally ambiguous for dense fluids. Calculations in the Percus—Yevic approximation show that qualitatively different potentials can be chosen which predict pair distributions which are virtually indistinguishable, and vice versa.
A linearized version of the nonequilibrium BBGYK hierarchy applicable to simple fluid systems near equilibrium is derived. A sequence of successive extended dynamical superposition approximations is defined to truncat...
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A linearized version of the nonequilibrium BBGYK hierarchy applicable to simple fluid systems near equilibrium is derived. A sequence of successive extended dynamical superposition approximations is defined to truncate the hierarchy at any arbitrary level. A formal solution, applicable either to the truncated or the complete hierarchy, is obtained by iteration of the Laplace transformed hierarchy followed by inversion of the transform. For the lowest truncation, which produces a single kinetic equation for the one‐body reduced distribution function, the Fourier–Laplace transform of the solution is summed to a closed form.
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