the quantitative details listed in the section on magnetism. Applications Magnetism In a spin system, the equal-time correlation function is especially well-studied. It describes the canonical ensemble (thermal) average of the scalar product of the spins at two lattice points over all possible orderings: Here the brackets mean the above-mentioned thermal average. Schematic plots of this function are shown for a ferromagnetic material below, at, and above its Curie temperature on the left. Even in a magnetically disordered phase, spins at different positions are correlated, i.e., if the distance r is very small (compared to some length scale ), the interaction between the spins will cause them to be correlated. The alignment that would naturally arise as a result of the interaction between spins is destroyed by thermal effects. At high temperatures exponentially-decaying correlations are observed with increasing distance, with the correlation function being given asymptotically by where r is the distance between spins, and d is the dimension of the system, and is an exponent, whose value depends on whether the system is in the disordered phase (i.e. above the critical point), or in the ordered phase (i.e. below the critical point). At high temperatures, the correlation decays to zero exponentially with the distance between the spins. The same exponential decay as a function of radial distance is also observed below , but with the limit at large distances being the mean magnetization . Precisely at the critical point, an algebraic behavior is seen where is a critical exponent, which does not have any simple relation with the non-critical exponent introduced above. For example, the exact solution of the two-dimensional Ising model (with short-ranged ferromagnetic interactions) gives precisely at criticality , but above criticality and below criticality . As the temperature is lowered, thermal disordering is lowered, and in a continuous phase transition the correlation length diverges, as the correlation length must transition continuously from a finite value above the phase transition, to infinite below the phase transition: with another critical exponent . This power law correlation is responsible for the scaling, seen in these transitions. All exponents mentioned are independent of temperature. They are in fact universal, i.e. found to be the same in a wide variety of systems. Radial distribution functions One common correlation function is the radial distribution function which is seen often in statistical mechanics and fluid mechanics. The correlation function can be calculated in