2.2 The Zubarev solution of the initial value problem

The solution of the problem how to find the missing signatures of ρð Þt not already described by ρrelð Þt was found by Zubarev [1, 2] generalizing the Bogoliubov principle of weakening of initial correlations [8]. He proposed to use the relevant statistical operator ρrelð Þ t<sup>0</sup> at some initial time t<sup>0</sup> as initial condition to construct ρð Þt ,

$$
\rho\_{t\_0}(t) = \mathbf{U}(t, t\_0) \rho\_{\rm rel}(t\_0) \mathbf{U}^\dagger(t, t\_0). \tag{22}
$$

The unitary time evolution operator Uð Þ t; t<sup>0</sup> is the solution of the differential equation

$$i\hbar\frac{\partial}{\partial t}\mathbf{U}(t, t\_0) = \mathbf{H}^t\mathbf{U}(t, t\_0),\tag{23}$$

with the initial condition Uð Þ¼ t0; t<sup>0</sup> 1. This unitary operator is known from the solution of the Schrödinger equation. If the Hamiltonian is not time dependent, we have

$$\mathbf{U}(t, t\_0) = \mathbf{e}^{-\frac{\mathbf{H}}{\hbar}\mathbf{H}(t - t\_0)}.\tag{24}$$

If the Hamiltonian is time dependent, the solution is given by a time-ordered exponent.

Now, it is easily shown that ρ<sup>t</sup><sup>0</sup> ð Þt is a solution of the von Neumann equation. All missing correlations not contained in ρrelð Þ t<sup>0</sup> are formed dynamically during the time evolution of the system. However, incorrect initial correlations contained in ρrelð Þ t<sup>0</sup> may survive for a finite time interval t � t0, and the self-consistency conditions (18) valid at t<sup>0</sup> are not automatically valid also at t.

To get rid of these incorrect initial correlations, according to the Bogoliubov principle of weakening of initial correlations, one can consider the limit t<sup>0</sup> ! �∞. According to Zubarev, it is more efficient to average over the initial time so that no special time instant t<sup>0</sup> is singled out. This is of importance, for instance, if there are long-living oscillations determined by the initial state. According to Abel's theorem, see [1–4], the limit t<sup>0</sup> ! �∞ can be replaced by the limit ϵ ! þ0 in the expression

$$\rho\_{\epsilon}(t) = \epsilon \int\_{-\infty}^{t} \mathbf{e}^{c(t\_1 - t)} \mathbf{U}(t, t\_1) \rho\_{\text{rel}}(t\_1) \mathbf{U}^{\dagger}(t, t\_1) \, \text{d}t\_1. \tag{25}$$

This averaging over different initial time instants means a mixing of phases so that long-living oscillations are damped out. Finally, we obtain the nonequilibrium statistical operator as

$$\rho\_{\text{NSO}}(t) = \lim\_{\epsilon \to 0} \rho\_{\epsilon}(t). \tag{26}$$

This way, ρrelð Þ t<sup>1</sup> for all times �∞ < t<sup>1</sup> < t serves as initial condition to solve the Liouville-von Neumann equation, according to the Bogoliubov principle of weakening of initial correlations. The missing correlations are formed dynamically during the time evolution of the system. The more information about the

nonequilibrium state are used to construct the relevant statistical operator, the less dynamical formation of the correct correlations in ρð Þt is needed. The limit t<sup>0</sup> ! �∞ is less active to produce the remaining missing correlating. The past that is of relevance, given by the relaxation time τ, becomes shorter, if the relevant (longliving) correlations are already correctly implemented. The limit ε ! þ0 has to be performed after the thermodynamic limit, see below.
