Exclusion statistics for structured particles on topologically correlated states. I. Single species lattice gases

A statistical thermodynamics description of particles having a set of spatially correlated states with statistical exclusion is developed. A general approximation for the density of states is presented from a state-counting ansatz recently introduced accounting for the multiple state exclusion statistical phenomena as a consequence of state spatial correlations. The multiple exclusion statistics is characterized by an exclusion correlation constant g_{c} which is consistently determined within the formalism from proper thermodynamic limits. The analytical form of g_{c} is given in terms of the Lambert function from the particle-lattice geometry. A generalized statistical distribution is obtained reducing to Haldane's statistics and Wu's distribution in the limiting case of particles on a set of spatially uncorrelated states. The problem of hard rods (k-mers) on a square lattice is studied with this formalism. From the entropy density dependence of the isotropic (I) and fully oriented nematic (N) phases, the approximation predicts two transitions, I→N and high-coverage N→I (disordered), only for k≥7 with the entropy at saturation matching to the known value from a Monte Carlo (MC) simulation. Critical coverage of both transitions is given for k=7 to k=20 in the first and second orders of approximations, in qualitative and quantitative agreement with results from MC simulations. State exclusion frequency e(n) and exclusion average G(n) functions are introduced and given in terms of the chemical potential to obtain a thermodynamic characterization of the state exclusion evolution on density. Results of chemical potential and state exclusion are shown for ideal lattice gases of k-mers, squares, and rectangles on a square lattice. Analytical results are compared with fast-relaxation grand canonical MC simulations.