Unified projection operator formalism in nonequilibrium statistical mechanics

Abstract

The method of projection operators, which plays an important role in the field of nonequilibrium statistical mechanics, has been established with the use of the Liouville-von Neumann equation for a density matrix to eliminate irrelevant information from a whole system. We formulate a unified and general projection operator method for dynamical variables. The main features of our formalism parallel those for the Liouville-von Neumann equation. (1) Two types of basic equations, time-convolution and time-convolutionless decompositions, are systematically obtained without specifying a projection operator. (2) Expansion formulas for both decompositions are also obtained. (3) Problems incorporating a time-dependent Liouville operator can be flexibly treated. We apply the formulas to problems in random frequency modulation and low field resonance. In conclusion, our formalism yields a more direct and easier means of determining the average time evolution of an operator than the one for the Liouville-von Neumann equation.

Nonadiabatic effect on the quantum heat flux control

“…For the factorized initial condition between the reduced system ρ χ (0) and Gibbs states of the environmental systems ρ E (= ν=L,R ρ ν ), the time evolution of ρ χ (τ ) is obtained as [13], which corresponds to a TCL (time-convolutionless) type of master equation [16][17][18][19][20].…”

Quantum Master Equation with Damping Operator Including Interaction Effect for the Raman-Coupled Model with Cavity Damping

“…The time-evolution of the reduced density matrix is governed by the quantum master equation which is derived by means of the projection operator method [16][17][18][19][20]. We assume that the whole system is initially in the quantum state ρ(0) ⊗ ρ E , where ρ E is the thermal equilibrium state of the environmental system,…”

“…Although the non-Markovian quantum master equation has recently taken much attention [7][8][9][10][11][12][13][14][15], the Markovian quantum master equation still has some problems. The explicit form of the quantum master equation can be derived from the Liouville von-Neumann equation of a whole system consisting of a relevant system and thermal reservoir by eliminating irrelevant variables by means of the projection operator method [16][17][18][19][20] and the path-integral method [21][22][23]. For example, we obtain the quantum master equation for a harmonic oscillator of angular frequency ω [5,24], where ρ(t) is a density operator of the oscillator system and a and a † are bosonic annihilation and creation operators.…”

Quantum Retrodiction in Non-Equilibrium Thermo Field Dynamics

“…The derivation of the retrodictive time-evolution generator for the quantum non-Markovian dynamics is difficult. Such time-evolution generator can, however, be derived by means of the time-convolutionless projection operator method (Shibata and Arimitsu, 1980;Arimitsu, 1982;Uchiyama and Shibata, 1999). This will be published elsewhere.…”