Gaussian Versus Mean-Field Models: Contradictory Predictions for the Casimir Force Under Dirichlet–Neumann Boundary Conditions

The mean-field model (MFM) is the workhorse of statistical mechanics: one normally accepts that it yields results which, despite differing numerically from correct ones, are not “very wrong”, in that they resemble the actual behavior of the system as eventually obtained by more advanced treatments. This, for example, turns out to be the case for the Casimir force under, say, Dirichlet–Dirichlet, (+,+) and (+,−) boundary conditions (BC) for which, according to the general expectations, the MFM is attractive for similar BC or repulsive for dissimilar BC force, with the principally correct position of the maximum strength of the force below or above the critical point Tc. It turns out, however, that this is not the case with Dirichlet–Neumann (DN) BC. In this case, the mean-field approach leads to an attractive Casimir force. This contradiction with the “boundary condition rule” is cured in the case of the Gaussian model under DN BC. Our results, which are mathematically exact, demonstrate that the Casimir force within the MFM is attractive as a function of temperature T and external magnetic field h, while for the Gaussian model, it is repulsive for h=0 and can be, surprisingly, both repulsive and attractive for h≠0. The treatment of the MFM is based on the exact solution of one non-homogeneous, nonlinear differential equation of second order. The Gaussian model is analyzed in terms of both its continuum and lattice realization. The obtained outcome teaches us that the mean-field results should be accepted with caution in the case of fluctuation-induced forces and ought to be checked against the more precise treatment of fluctuations within the envisaged system.