Abstract It is shown that the relevant part of an equilibrium two-time correlation function of operators related to a subsystem of S < N particles, selected from the whole system of N ≫ 1 quantum particles, obeys the exact completely closed (homogeneous) evolution equation. This equation differs from the conventional Nakajima–Zwanzig equation by the absence of the undesirable inhomogeneous term comprising all interparticle correlations at the initial time moment. The equation defines the subdynamics in a subspace of S selected particles and is obtained without any use of approximation like the Boltzmann ‘molecular chaos’, factorized initial state or Bogoliubov’s principle of weakening of initial correlations. In the second order in a small inter-particle interaction, this equation reduces to the Markovian evolution equation, which for a one-particle correlation function ( S = 1) takes the form of the linear generalized quantum Boltzmann-like equation for a relevant part of a correlation function (one particle momentum distribution function) with the additional terms conditioned by initial correlations taken into account. This equation makes it possible to obtain the solution for the corresponding one-particle correlation function, which is valid for all timescales and reveals the influence of initial correlations on the evolution in time process. On a large timescale t ≫ t cor ∼ ℏ / k B T (decoherence time), the initial correlations cease to influence the damping process and the evolution enters a kinetic stage. However, the factor related to initial correlations survives on the large timescale, and can contribute to the kinetic coefficients.
For a system of N≫1 spinless quantum particles, the projection operator is introduced, which, surprisingly, exactly transforms the inhomogeneous (with irrelevant initial correlations term) Nakajima-Zwanzig generalized master equation (GME) into completely closed (homogeneous) GME with initial correlations included into the kernel governing its evolution. The obtained equation is equivalent to the closed linear evolution equation for an S-particle (S<N) statistical operator, which holds for any initial condition and all timescales. Thus, the introduced projector implements the idea of the subdynamics in the subspace of S-particle complex. It also allows for precise treatment of the initial correlations on an equal footing with collisions. No "molecular chaos" -type approximation is used. The obtained homogeneous equation is further specialized in the linear approximation in the particles density n for the kernel and considered in detail for the one-particle statistical operator. The influence of both the quantum correlations, caused by the particles' statistics, and initial correlations, related to the interparticle interaction, is considered in detail and at different timescales. Connection of the linear equation for a one-particle statistical operator to the nonlinear quantum Boltzmann equation is shown.
A projection operator is introduced, which exactly transforms the inhomogeneous Nakajima--Zwanzig generalized master equation for the relevant part of a system +bath statistical operator, containing the inhomogeneous irrelevant term comprising the initial corrrelations, into the homogeneous equation accounting for initial correlations in the kernel governing its evolution. No "molecular chaos"-like approximation has been used. The obtained equation is equivalent to completely closed (homogeneous) equation for the statistical operator of a system of interest interacting with a bath. In the Born approximation (weak system-bath interaction) this equation can be presented as the time-local Redfield-like equation with additional terms caused by initial correlations. As an application, a quantum oscillator, interacting with a Boson field and driven from a Gibbs initial equilibrium system+bath state by an external force, is considered. All terms determining the oscillator evolution over time are explicitly calculated at all timescales. It is shown, how the initial correlations influence the evolution process. It is also demonstrated, that at the large timescale this influence vanishes, and the evolution equation for the quantum oscillator statistical operator acquires the Lindblad form..
Abstract A system of <?CDATA
N≫1?> N ≫ 1 interacting spinless quantum particles, described by a statistical operator F(t), is considered. A time-dependent projection operator formalism for a family of projectors, which select a statistical operator <?CDATA
FS(t)?> F S ( t ) for a group of S < N relevant particles by integration of the variables of the irrelevant N − S ‘environment’ particles, is presented. This formalism results in a nonlinear version of the inhomogeneous Nakajima–Zwanzig generalized master equation (GME) for the relevant part of the statistical operator F(t), which contains an undesirable initial condition term related to the irrelevant part of the statistical operator depending on N variables. By introducing an additional operator identity, the obtained nonlinear inhomogeneous Nakajima–Zwanzig equation is exactly converted into a homogeneous nonlinear equation, accounting for initial correlations in the kernel governing its evolution. Both of these equations are equivalent to the corresponding evolution equations for <?CDATA
FS(t)?> F S ( t ) and take into account the dynamics of the environment particles. The equations for <?CDATA
FS(t)?> F S ( t ) are specialized in the linear approximation for the particle density n and analyzed for a one-particle statistical operator <?CDATA
F1(t)?> F 1 ( t ) . It turns out that, in addition to the correlations caused by interparticle interaction, the irrelevant initial condition term in the inhomogeneous equation for <?CDATA
F1(t)?> F 1 ( t ) , defined by the two-particle correlations, also contains quantum correlations conditioned by the quantum particle statistics. The latter do not vanish with time and cannot be disregarded in the conventional way by applying e.g. Bogoliubov’s principle of weakening of initial correlations. In order to take initial quantum and other correlations (commonly discarded in an unconvincing way) into consideration, the obtained nonlinear homogeneous GME has been applied. Exact in the first approximation in n, we obtain a new homogeneous equation for <?CDATA
F1(t)?> F 1 ( t ) , which accounts for correlations and is valid for any time moment t. It is shown how this reversible-in-time equation changes at different timescales and converts into the irreversible quantum Boltzmann equation on the macroscopic timescale without the use of the conventional approximations.
A novel approach to accounting for the influence of initial system-bath correlations on the dynamics of an open quantum system, based on the conventional projection operator technique, is suggested. To avoid the difficulties of treating the initial correlations, the conventional Nakajima-Zwanzig inhomogeneous generalized master equations (GMEs) for a system's reduced statistical operator and correlation function are exactly converted into the homogeneous GMEs (HGMEs), which take into account the initial correlations in the kernel governing the evolution of these HGMEs. In the second order (Born) approximation in the system-bath interaction, the obtained HGMEs are local in time and valid at all timescales. They are further specialized for a realistic equilibrium Gibbs initial (at
t=t0) system+bath state (for a system reduced statistical operator an external force at
t>t0 is applied) and then for a bath of oscillators (Boson field). As an example, the evolution of a selected quantum oscillator (a localized mode) interacting with a Boson field (Fano-like model) is considered at different timescales. It is shown explicitly how the initial correlations influence the oscillator evolution process. In particular, it is shown that the equilibrium system's correlation function acquires at the large timescale the additional constant phase factor conditioned by survived initial system-bath correlations.
A regular approach to accounting for initial correlations, which allows to go beyond the unrealistic random phase (initial product state) approximation in deriving the evolution equations, is suggested. An exact homogeneous equation for a two-time equilibrium correlation function for the dynamical variables of a subsystem interacting with a boson field (heat bath) is obtained. No conventional approximation like RPA or Bogoliubov's principle of weakening of initial correlations is used. The obtained equation takes into account the initial correlations in the kernel governing its evolution. The solution to this equation is found in the second order of the kernel expansion in the electron-phonon interaction, which demonstrates that generally the initial correlations influence the correlation function's evolution in time. It is shown that this influence vanishes on a large timescale. The developed approach is applied to the Fröhlich polaron and the low-temperature polaron mobility (which was under a long-time debate) is found with a correction due to initial correlations.
Using the method of non-equilibrium statistical Zubarev operator an approach to the description of kinetics taking into account the nonlinear hydrodynamic fluctuations for quantum Bose system is proposed. The non-equilibrium statistical operator that consistently describes both the kinetic and nonlinear hydrodynamic fluctuations in quantum liquid is calculated. A kinetic equation for the non-equilibrium one-particle distribution function and generalized Fokker-Planck equation for non-equilibrium distribution function of hydrodynamic variables: densities of momentum, energy and particle numbers are obtained. A structural function of hydrodynamic fluctuations in cumulant representation is calculated. It provides a possibility to analyzing the generalized Fokker-Planck equation in Gaussian and higher approximations for dynamic correlations of hydrodynamic variables that is important to describe the quantum turbulent processes. The generalized Gross-Pitaevkii equation for the order parameter which takes into account nonlinear hydrodynamic fluctuations is obtained.