Hard-Sphere Lattice Gases. I. Plane-Square Lattice

Abstract

A plane-square lattice gas of hard ``squares'' which exclude the occupation of nearest-neighbor sites is studied by deriving 13 terms of the activity and the virial series and nine terms of appropriate high-density expansions. Using the ratio and Padé approximant extrapolation techniques it is found that the gas undergoes a continuous (or ``second-order'') transition to an ordered state at an activity zt=3.80±2 and a density ρt=(0.740±0.008)ρmax. The ordered state is characterized by a difference of the sublattice occupation probabilities, R(z), which vanishes at the |transition as (z—zt)β with β≈⅛. The pressure at the transition is given by pa2/kBT=0.792±5. The compressibility exhibits a maximum at or near the transition point but probably remains finite and continuous through the transition. A suitably defined ``staggered compressibility,'' which measures the tendency towards sublattice ordering, diverges sharply as the transition is approached from either side. A double expansion which converges at all densities is derived and examined numerically and the exact behavior of finite lattices of N=4, 16, 20, and 24 sites is discussed.

Lattice gas with nearest- and next-to-nearest-neighbor exclusion

“…The model is thus a hard-square model. It should not be confused with other hard-square models, such as Fisher's [1] and Baxter's [2,3], and the nearest-neighbor exclusion hard-square model without further interactions [4][5][6][7][8][9]. The latter model, as well as that of Eq.…”

Monte Carlo simulations of two-dimensional hard core lattice gases

“…This system, which can also interpreted either as 45 o tilted hard-squares of linear size λ = √ 2 or as hard disks of radius √ 2/2, has been extensively studied and here we present some results for the sake of both completeness and comparison. Many different approaches have been used to describe its properties on a square lattice: series expansions [3,5,11,12,13], cluster variational and transfer matrix methods [5,14,15,16,17,18,19,20,21,22,23], renormalization group [24,25], Monte Carlo simulations [26,27,28,29,30,31,32,33], Bethe lattice [5,34,35,36,37,38], and more recently density functional theory [39]. Moreover, this model has also been considered because of its interesting mathematical [40,41,42] and dynamical [43,44,45,46,47,48,49,50,51,…”

“…The difficulty and the scarcity of exact solutions has stimulated the development of various approximate theories aimed at treating more complex interaction potentials. Examples of these are the high and the low temperature (or density) expansions [3], generalization of Bethe's methods for Ising model (which became known as cluster variational methods) [4], as well as the approximation schemes such as Rushebrook and Scoins [5]. Since melting is dominated by strong short ranged repulsive forces, a lattice gas in which particles interact exclusively through an extended hard core -a particle on one site prevents the neighboring sites from being occupiedhas attracted a particular attention [6].…”

Hard squares with negative activity

“…The hard-square model is a well-known model of two-dimensional statistical mechanics [1,2]. It describes a classical gas of particles on the square lattice, with the restriction that particles may not be on adjacent sites.…”