Search

It is shown that the grand partition function for a system of interacting particles can be obtained by a method which entirely bypasses tbe second quantization formalism. Equations were determined for the coordinate representation of the density matrix with Born-von Karman conditions for particles with Boltzmann statistics. The equations for the canonical and grand canonical partition functions were derived for particles with Boltzmann, Bose, and Fermi statiatics. (M.C.G.)

Van Kampen's cluster expansion method for a classical real gas - in which the canonical (configurational) partition function is expanded into a infinite product - is generalized in order to make it applicable to more general systems with weak correlations. Two applications are considered: a simplified derivation of the formal expressions for the virial coefficients of a quantum gas and a deriviation of the necessary condition for the validity of Curie's law for an isolated spin system with dipole-dipole and exchange interactions.

Synopsis From Saha's function and from the electrical conductivity, determined both theoretically and experimentally, mean electron densities and mean arc temperatures for three cross-sections (1 mm, 2 mm and 3 mm from cathode) were derived for 5 arcs of 48, 97, 148, 246 and 287 amp struck between two vertical tungsten electrodes approximately 4 mm apart in the atmosphere of argon maintained at atmospheric pressure. Various physical data calculated or derived include: partition functions for argon, intensity of the electric field, mean free path, mobility and drift velocity of electrons, current density and jet velocity. The computation of the partition functions was extended over the wide range of temperature from 5,000°K to 50,000°K for neutral, singly, doubly and trebly ionized atoms. The approximate distribution of the electric field and the value of its intensity across the arc were obtained from a field map. The mean free path and the mobility of electrons were calculated from theoretical equations and some published data.

Synopsis Kramers' variational method is used to derive the partition functions per spin for the cubic ferromagnetic structures. These partition functions are compared with the partition functions derived by the approximate methods of Bethe, Kirkwood and Zernike. The case of the two dimensional square ferromagnet with both nearest and next nearest neighbour interactions is also briefly discussed.

Some relations for the partition functions of two-dimensional infinite triangular and honeycomb lattices with Ising interaction between neighbours without magnetic field are given in the general case of different interactions in the three directions. Exact evaluation of these partition functions is obtained with the aid of the method of Bruria Kaufman. The theory of this method is simplified by avoiding the use of abstract group-properties. Some thermodynamic properties of the isotropic hexagonal lattices are given. These are compared with the corresponding properties of the quadratic lattice.

The singularities observable at second-order phase transitions ( e.g. , the divergence of the specific heat) can occur only if the size of the considered system is very large (strictly speaking. infinite). Therefore some pronounced modifications of critical phenomena have to be expected in small systems. We investigate these modifications by numerical calculations for the case of the three-dimensional Ising model (the total number of spins in the considered systems being in the range from N ≈ 10 N ≈ 10 3 ; for N ⪕ 54 the exact zero-field partition functions are given, while we derive the properties of the larger systems using the Monte-Carlo technique). Assuming systems with free surfaces we find also a dependence on the shape of the system. Considering the behaviour of the energy, the specific heat, and the magnetization, one observes both a rounding of the critical anomalies and a shift to lower temperatures. This behaviour is in good quantitative agreement with the conclusions which can be drawn from the exactly soluble two-dimensional Ising model.

In view of recent investigations on the viscosities of 1-2-dichloroethylene, carried out at this laboratory, some aspects of Eyring's theory of viscous flow are considered. It is concluded that the difference between the experimental quantity E vis = R . ′d (1n ƞ) d( 1 T ) and Eyring's “energy of activation for viscous flow” (U∗) cannot be neglected, as has been done in previous work by several authors. For many compounds, Evis has been found to be independent of temperature over the ranges measured. It appears to be impossible to explain this behavior on the basis of Eyring's theory. Until a definite solution of the problems concerning the liquid state, such as the concepts of the “free volume” and of the partition functions of the molecules, has been given, in connection with a well-established theory of viscous flow, the independence of temperature of Evis has to be regarded as an unexplained empirical fact.

Work on graph theoretical methods in classical statistics is summarized. The application of these methods to the calculation of the classical pantition function for an Nparticle system with additive two-body forces is described. The relation of the two basic theorems of Mayer's theory to the general theorems in the theory of graphs is indicated. The two Mayer theorems reduce the problem of calculating the partition function to the calculation of the star integrals. The different types of stars are classified according to their complexity or connectivity. Applications to calculations of partition functions in plasma are also discussed. (M.C.G.)