W. W. Wood

Monte Carlo Calculations for Hard Disks in the Isothermal‐Isobaric Ensemble

The N p T ensemble for hard disks is formulated as an equivalent N V T ensemble for a pseudopotential interaction in a reduced configuration space, with exact relations for small periodic systems emphasized. The Monte Carlo method originally devised by Metropolis et al. is then adapted to the calculation of the equation of state and radial distribution function. The present paper then describes some results obtained for a small system (12 hard disks), as an example of the method, and with emphasis upon the statistical reliability of the results and on the effect of different pseudorandom‐number‐generating procedures. At least for such small systems it is possible to obtain, as a byproduct, the N V T ‐ensemble equation of state over a range of densities from an N p T ‐ensemble calculation at a single pressure. Subsequent papers will describe results already in hand for larger systems, including reasonably reliable extrapolations to the “thermodynamic” limit.

Equation of State of Classical Hard Spheres at High Density

Under certain conditions, an asymptotic expression for the equation of state of a classical mechanical system of N ν‐dimensional (ν=1, 2, or 3) hard spheres confined in a volume V is obtained in the form pVNkT=ν(1–1/N)(V/V0)−1+O(1). This expression agrees with the leading term in V/V0—1 of the usual free‐volume approximation for N = ∞. The conditions under which this conclusion is established are a restriction to a finite number of molecules N with periodic boundary conditions, and the requirement that as V→V0 the accessible configuration states approach a close‐packed configuration whose coordination number c satisfies the requirement c≥2ν−2(ν−1)/N. Such a limiting configuration, from which only an infinitesimal region of configuration space is accessible under an infinitesimal expansion, is called a stable configuration; the above restriction on the coordination number is a necessary condition for stability. The difficulties which appear as N→∞ are indicated.

Application of the Monte Carlo Method to the Lattice‐Gas Model. I. Two‐Dimensional Triangular Lattice

The Monte Carlo numerical method for obtaining statistical mechanical averages in the petite canonical ensemble has been applied to the two‐dimensional triangular lattice‐gas model. The Monte Carlo procedure described by Rosenbluth et al. and by Wood and Parker has been extended slightly to include the calculation of the partition function. The results are compared with the exact solutions for the two‐dimensional lattice and the agreement is well within the fluctuation displayed by the numerical method. The calculations are found to be relatively insensitive to a change in the pseudo‐random number sequence and to display no exceptional behavior in the region of the first‐order phase transition for small lattices.

NpT‐Ensemble Monte Carlo Calculations for the Hard‐Disk Fluid

The NpT‐ensemble Monte Carlo method previously described, in which the hard‐disk interaction is represented by a soft pseudopotential in a reduced configuration space with a fixed periodic enclosure, is used to calculate the equation of state and radial distribution function in the high‐density fluid phase. The equation of state agrees within statistical error with the (3, 3) Pade approximant to the six‐term virial expansion and, for large enough systems, with Salsburgs analysis of the N dependence. The results also agree with recent Monte Carlo calculations by Chae, Ree, and Ree using the NVT ensemble. Some qualitative results for systems of 48 and 90 disks which lend further support to the now‐accepted presence of a solid–fluid phase transition are also presented.

Recent monte carlo calculations of the equation of state of Lenard-Jones and hard sphere molecules

The investigations repor ted at this sympos ium are in the process of publication elsewhere [1, 2]; since the la t te r papers are r a the r detailed, we will content ourselves here with a ra ther brief survey of the work, with only a few addit ional results. The t e rm (~ Monte Carlo ,) has come into use to designate numerical methods in which specifically stochastic elements are introduced, in contras t to the whole body of classical numerical techniques which consist of numerical evaluations of complete ly de te rminan t algebraic expressions. The par t icular Monte Carlo me thod used here was devised by METROPOHS et al. [3]; its essential fea ture is t ha t i t produces a Markov chain [4] in which the individual 5 Ia rkov s tates are points in the usual configuration space of s tat is t ical mechanics, for a sys tem of N molecules confined a t a t e m p e r a t u r e T in ~ volume Y. The t ransi t ion probabil i t ies character izing the Markov chain are de termined in such a way tha t the va lue of any funct ion of configuration state, averaged over all s tates developed in the Markov cha.in, approaches the pe t i t canonical ensemble average value of the same function. I t should be noticed t ha t the present me thod does no t consist in an evaluat ion of the configurational phase in tegral b y means of a random, selection of points in configuration space. A me thod of the l a t t e r sort, which m a y be compared with a sequence of independent throws of a die, is indeed

Comments on the closure problem in the statistical theory of fluids

It is shown that if the BBGKY hierarchy is truncated by the approximation for the quadruplet correlation function g1234=g123g124g134g234/ [g12g13g14g23g24g34], the resulting system of equations yields solutions for g12 and g123 which are path dependent. This result is consistent with an earlier argument by Raveche and Green that the closure leads to a nonconservative force on a particle in a fixed triplet of particles. The magnitude of the path dependence is investigated and its consequences are discussed.

Investigations of the Detonation Properties of Condensed Explosives with Equations of State Based on Intermolecular Potentials. I. RDX with Fixed Product Composition

An attempt is made to determine an average pair potential of intermolecular force for the mixture of product gases of the condensed explosive RDX (cyclotrimethylenetrinitramine) from measurements of the detonation velocity. The principal approximations are that the products of detonation consist of CO, H2O, and N2 in equimolar proportions and that the equation of state for this mixture can be calculated from theories of the free‐volume type with the use of an average intermolecular potential. It was found that the average intermolecular potential was fairly well determined at distances somewhat less than the crossover distance.This result was almost independent of the choice of equation of state, although the calculated values of detonation temperature and pressure were not.

Tables of the Lennard‐Jones and Devonshire Equation of State at High Temperatures and Densities

The Lennard‐Jones and Devonshire equation of state is computed for the following ranges of the reduced variables: temperature, kT/e*, 20 to 150 in steps of 10; volume, v/v*, 0.3 to 0.7 in steps of 0.05 and 0.7 to 1.3 in steps of 0.1.