A New Principle of Equilibrium by Lewis G.N.

By Lewis G.N.

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For the moment being we assume that the aik ’s are given either from phenomenological considerations or, else, from calculations and continue exploring the general structure of master equations. 2 Positive-semidefiniteness of the Relaxation Matrix All inequalities among relaxation parameters can be deduced from this property of A, and it is certainly worthwhile to recall some useful theorems. Note again that writing down master equations by introducing relaxation on purely phenomenological grounds, as is frequently done for interpretations of experiments, would require a careful discussion of their exact solutions in order to obtain the same information as from the analysis of positive-semidefiniteness of A.

The energy trapping due to a quantum interference between atoms. The generator (147) is of a mean-field type as studied in the Subsubsect. 5. The associated nonlinear Schr¨ odinger equation (127) is of the following form, i δN d ψt = ωS 3 ψt − i dt 2 ψt | S + | ψt S − ψt − ψt | S − | ψt S + ψt with δ = δ 1 − e−βω . Introducing the parameterization √ iΘt pt e ψt = √ , pt ∈ [0, 1] , 1 − pt eiΘ t Θt , Θ t ∈ R , , (149) (150) one obtains from (149) d d Θt = Θ t = 0 , dt dt dpt = −δ N (1 − pt )pt , dt (151) with the solution Θt = Θ0 , Θt=Θ pt = eδ N (t−tD ) + 1 tD = The radiated energy 1 ln δN p0 1 − p0 0 −1 .

145) Then, the dissipative part of the generator defined by (143) may be written as LD = LD + LD , (146) where LD is given by (143) with amn replaced by amn = amn − δmn and LD is obtained putting amn = δmn . According to the results of Subsubsect. 4 - LD , - L D and - LD are positive operators on the Hilbert space B(H(N ) ) with a scalar product (N ) (A, B)β = Tr(ρβ A B). Hence if LD X = 0 then LD X = 0, too. Because, as stated above, LD X = 0 if and only if X = c11, then (145) implies that our system returns to the equilibrium.

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