Correlation Functions and Finite–Size Effects in Granular Media

A model is considered, where the local order parameter is an n–component vector. This model allows us to calculate correlation functions, describing the correlations between local order parameter at different spatial coordinates. The longitudinal and transverse Fourier–transformed two–point correlation functions G ∥ ( k ) $$G_{\parallel }(\mathbf{k})$$ and G ⊥ ( k ) $$G_{\perp }(\mathbf{k})$$ in presence of an external field h are considered in some detail. In the thermodynamic limit, these correlation functions exhibit the so-called Goldstone mode singularities below certain critical temperature at an infinitesimal external field h = + 0 $$h = +0$$ . The actual model can be applied to granular media, in which case it describes a small particle and, therefore, the finite–size effects have to be taken into account. Based on Monte Carlo simulation data for different system (lattice) sizes, we have found that the correlation functions are reasonably well described by certain analytic approximation formulas.