Molecular field theory, the Onsager reaction field and the spherical model

We have examined the physical and formal aspects of the Onsager method 1 of solution of the many dipole problem. Our principal conclusion is: the Onsager prescription of subtracting out the reaction field from the mean molecular field (e.g. Weiss field in ferromagnets, Lorentz field in dielectrics, van der Waals field in fluids) is the necessary modification of molecular field theory which guarantees the fluctuation theorem of statistical mechanics (equality of the mean fluctuation of a coordinate to its susceptibility in response to an external field to which it is coupled).We develop the theory in detail for the Ising model. In the paramagnetic phase, we show that the Onsager method reduces to the spherical model. This reduction is somewhat incidental, and does not occur in the magnetized phase. On the contrary, the Onsager method then leads to a system of simultaneous differential equations for the magnetization, which is in general intractable. We introduce a scheme of successive approximations, and recover the Weiss theory and the generalized spherical model after the first and second iterations, respectively.Finally, we indicate how to apply these ideas in the liquid-vapor system.