The effect of a constant number of particles on the pair correlation function and the density profile in a one-dimensional lattice gas

Abstract

A one-dimensional lattice gas (Ising model) of lengthL and with nearest-neighbor couplingJ is considered in a canonical ensemble with fixed number of particlesN=L/2. Exact expressions and asymptotic forms for largeL are derived for the density-density correlation function, using periodic boundary conditions, and for the density (magnetization) profile, using antisymmetric boundary conditions. The density-density correlation function,g, assumes for temperaturesT> T, withT = 2J(κ_BlnL)⁻¹ and forL large, the formg(x) =g ^gc(x) +BL ⁻¹ +a(x)L ⁻¹ +O(L⁻²) wherex is a distance between considered lattice sites,B is known from earlier work of Lebowitz and Percus,(1b) anda(x) decays exponentially forx → ∞. For T⩽T′, the correlation function and the density profile behave differently, the latter exhibiting a step in the middle of the interface.