Probabilistic approach to pattern selection

Abstract

The problem of pattern selection arises when the evolution equations have many solutions, whereas observed patterns constitute a much more restricted set. An approach is advanced for treating the problem of pattern selection by defining the probability distribution of patterns. Then the most probable pattern naturally corresponds to the largest probability weight. This approach provides the ordering principle for the multiplicity of solutions explaining why some of them are more preferable than other. The approach is applied to solving the problem of turbulent photon filamentation in resonant media.

Atomic squeezing under collective emission

“…As can be checked by straightforward calculations [46][47][48][49], for λ ≫ a, one has g ≫ 1, where the coupling g is defined in Eqs. (36).…”

“…The effective Fresnel number for each filament is less than one. Therefore the problem again can be reduced to the consideration of separate filaments, for each of which the single-mode description is applicable [46][47][48][49]. In the single-mode picture, we have u(r, t) = u(t) e ikz , w(r, t) = w(t) , s(r, t) = s(t) .…”

Absence of spin superradiance in resonatorless magnets

“…The retardation effects, due to the unit-step function entering Eqs. (26) and (27), manifest themselves at the time scale L c /c. Taking for L c the maximal value L c ∼ 1/k, the retardation time can be approximated by ω −1 .…”

“…Using inequality (25), the collective radiation rate (26) and the collective frequency shift (27) can be simplified. The retardation effects, due to the unit-step function entering Eqs.…”

“…Here we have presented a fully quantum-mechanical derivation of these expressions and, in addition, have obtained their generalizations (26), (27), and (29). In studying spin dynamics, it is necessary to take account of the longitudinal and transverse relaxation rates.…”