Classical Many-body Problem Research Articles

Exact solution of the classical and quantal one-dimensional many-body problems with the two-body potential {ie383-01}

Exact solution of the classical and quantal one-dimensional many-body problems with the two-body potential {ie383-01}

The second moment of the complexity of a graph

A graph consists of a set of vertices some pairs of which are joined by a single edge. A tree is a graph with the property that each pair of vertices is connected by precisely one path, i.e. , a sequence of distinct vertices joined consecutively by edges. The complexity c of a graph G(n, k) with n vertices and k edges is the number of trees with n vertices which are subgraphs of G(n, k) . The distribution of c over the class of all graphs G(n, k) is of physical interest because it throws light on the classical many-body problem. (See, e.g. [9].) Ford and Uhlenbeck [3] gave numerical data which suggested that the distribution of c tends to normality for increasing n if k is near No moments higher than the first were known in general and they remarked in [4] that even “the second would be worth knowing”. The main object in this paper is to derive a formula for the second moment of c .

Indirect Exponential Coupling in the Classical Many-Body Problems

The usual coupling procedure consists of multiplying the interparticle potential ${U}_{N}({\mathrm{r}}^{N})$, by a coupling parameter $\ensuremath{\lambda}$ and then expanding thermodynamic functions in powers of $\ensuremath{\lambda}$. The Kirkwood variation of this procedure couples only one particle of the system, resulting in an integrodifferential equation for distribution functions, which also can be expanded in powers of the coupling parameter. These expansions converge and are valid only for weakly coupled systems. If the Ursell $f$ bonds are coupled instead of the direct interaction potentials, we can expand certain thermodynamic functions in powers of the exponential coupling parameters; for actual physical systems these expansions are practically finite low-order polynomials in the coupling parameters. Integrodifferential equations for distribution functions are derived, and it is seen that distribution functions are given by ratios of two practically finite polynomials in the exponential coupling parameters. The coefficients in these polynomials are finite even for strongly singular (e.g., hard sphere) potentials. The method provides a well-defined expansion parameter for the Kirkwood-Salzburg hierarchy and appears related to the $f$-bond chain summation and nodal expansion methods. Present and possible future applications include: theory of fused salts and electrolytes, theory of ferroelectricity, ion pairing in semiconductors, equation of state of the high-temperature electron gas, and problems of phase transitions. The possibility of applying exponential coupling to quantum-mechanical systems is noted.

A Monte Carlo Equation of State for Mixtures of Hard-Sphere Molecules

PROGRESS in the theory of mixtures is limited by the lack of adequate approximate solutions of the classical many-body problem. Thus at densities greater than those appropriate to the virial expansion no satisfactory estimate of the properties of mixtures is possible. However, the Monte Carlo method enables the evaluation of the many-dimensional configurational integral to be performed by a sampling procedure. The method is applicable to any form of intermolecular potential.