Particle fluctuations within sub-regions of an N-particle, two-dimensional fluid: Finite-size effects and compressibility

We derive an expression which relates fluctuations in particle number, in the thermodynamic limit with fluctuations within a sub-region of an N-particle, two-dimensional, simple fluid. The accuracy of our result is tested against simulation results for a two-dimensional, hard-sphere fluid at a low, moderate, and high density state and we find agreement to within the uncertainties of the simulation when restrictions associated with the result are enforced. Also, we demonstrate the bulk isothermal compressibility can be accurately determined directly from fluctuation data obtained from simulations. Our expression yields a quantitative description of the (explicit) size effects associated with fluctuations within sub-regions of an N-particle system, and in particular explicitly shows a relation between size effects and particle correlations.