So-called Mean Field Research Articles

Influence of additive noise on noise-induced phase transitions in nonlinear chains

The influence of additive noise on phase transitions in nonlinear chains induced by multiplicative noise is studied. These chains can be considered as finite difference approximations of nonlinear partial differential equations of diffusion type. The phase transitions considered manifest themselves in the disruption of the symmetry and the birth of new equilibrium states. In the so-called mean field approximation, it is shown that additive noise significantly shifts the boundaries of the phase transition.

Propagation effects in a collective atomic recoil laser

We present a theoretical investigation of propagation effects in a collective atomic recoil laser (CARL) operating in the FEL limit. We consider the cases where the system evolves while in free space and while enclosed in a ring cavity. In the case where no cavity is present, we show that the scattered radiation consists of soliton-like superfluorescent pulses. In the case of a ‘good’ cavity we arrive analytically at a condition to neglect propagation effects. This condition implies that in order to use the so-called mean field approximation, the condition (ΔL)l32c→0, T → 0 must be satisfied with (Tlc(ΔL)23 finite where lc is the cooperation length of the system, T is the transmission coefficient of the mirrors, L and Λ are the sample length and cavity length respectively. We confirm the validity of this condition using a numerical analysis and provide a simple physical interpretation. In the mean field limit, we show that if the cavity linewidth is greater than the spectral width of the pulse emitted by the sample, the emission remains superfluorescent and is not sensitive to the presence of the cavity. We also show that in the opposite case the emission is sensitive to the cavity parameters and no longer superfluorescent.

Competition effects in steady-state diffusion-limited reactions: Renormalization group approach

We consider the steady-state competition between perfectly absorbing sinks distributed uniformly in a spherical region. An equation for the coarse-grained concentration of diffusing particles inside the region is derived following the renormalization group approach. The total flux of diffusing particles, their coarse-grained concentration, and the characteristic penetration length are found. We then compare our results with so-called mean field approximation, and show that the latter approach cannot correctly describe the coarse-grained concentration in the system under consideration.

Ohm's law for mean magnetic fields

The magnetic fields associated with plasmas frequently exhibit small-amplitude MHD fluctuations. It is useful to have equations for the magnetic field averaged over these fluctuations, the so-called mean field equations. Under very general assumptions, it is shown that the effect of MHD fluctuations on a force-free plasma can be represented by one parameter in Ohm's law, which is effectively the coefficient of electric current viscosity.

Note on non-local mean field approximations for amorphous Heisenberg spin system

A non-local mean field approximation for amorphous Heisenberg ferromagnets is derived. It is shown that the so-called homogeneous mean field approximation is an inconsistent approximation and that their results have to be modified.

Spin glasses with long-range interactions

A spin-glass model with a general form of long-range interactions is introduced. The replica method and the renormalization group theory are used and the results are explained from the viewpoint of frustration effects compared with the non- random case. The so-called mean field approximation is shown to be invalid for 3-d RKKY interaction systems.