The self consistent expansion applied to the factorial function

• We study the long time mean square displacement (MSD) of a particle swept by a random flow.
• We obtain the MSD for a family of random flows.
• For ordinary turbulence, which is a member of that family we find that the swept particle super diffuses.
• The exponent characterizing the super diffusion is 6/5.
• We explain why it is difficult to distinguish experimentally the slight super diffusion from diffusion.

Most of the interesting systems in statistical physics can be described as nonlinear stochastic field theories. A common feature in the theoretical study of such systems is that ordinary perturbation theory seldom works. On the other hand, there exists a useful tool for the study of systems of that generic nature. That tool, the Self Consistent Expansion (SCE) is technically similar to the ordinary perturbation expansion, in the sense that it is an expansion around a solvable problem. The key point which distinguishes the SCE from an ordinary perturbation expansion, is that the small parameter of the expansion is adjustable and determined inherently by optimization of the expansion. Therefore, it allows the adaptive SCE to remain accurate relative to the inflexible ordinary expansion.The goal of the present paper is to present the SCE by applying it to a well-known zero dimensional problem. We choose the evaluation of the factorial function, x!x!, as the test case for the SCE, because the Stirling approximation for that function is one of the best known asymptotic expansions, with a very wide use in statistical physics. We show that the SCE approximation holds for small and even negative arguments of the factorial function, where the Stirling expansion fails miserably. It does so without paying any penalty at high values of the argument, where the Stirling formula is excellent. We present numerical as well as analytic SCE approximations of the factorial function.