## 5 Harmonic, Translation-Invariant Open Dynamics  
Let us consider the time dependence of the elementary open components of a translation-invariant open dynamics in more detail. We simplify the discussion by restricting our attention to a simple dynamics where the equation of motion for the reduced system density matrix is local in time and contains only the first time derivative. A further simplification is that we chose harmonic dynamics where the equation of motion contains quadratic terms in the canonical operators. The most general harmonic translation-invariant master equation of the Lindblad form [10] is:  
\[\partial_{t}\rho = \left[\frac{i\hbar}{m}\mathbf{\
abla}\mathbf{\
abla}_{d}+\frac{i}{\hbar}f \mathbf{x}_{d}-\frac{d_{0}+d_{2}\
u^{2}-2 mv\xi}{2\hbar}\mathbf{x}_{d}^{2}\right. \tag{9}\] \[\left.+i\bigg{(}\frac{d_{2}\
u}{m}-\xi\bigg{)}\mathbf{x}_{d}\mathbf{\
abla }-\
u\mathbf{x}_{d}\mathbf{\
abla}_{d}+\frac{\hbar d_{2}}{2m^{2}}\mathbf{\
abla}^{2}\right]\rho,\]  
with \(\mathbf{\
abla}=\partial/\partial\mathbf{x}\) and \(\mathbf{\
abla}_{d}=\partial/\partial\mathbf{x}_{d}\). The first two terms correspond to the Neumann equation of a closed system under the influence of a dragging force \(\mathbf{f}\). The parameter \(\
u\) generates [PERSON]'s friction force; \(d_{0}>0\) and \(d_{2}>0\) control the decoherence; \(\xi\) is the coefficient of a total time derivative term in the effective Lagrangian and represents no dynamical process [11]. The positivity of the density matrix is preserved for weak friction as long as \(m^{2}(\
u^{2}+4\xi^{2})\leq 2d_{0}d_{2}\). The solution of this master equation or one of its simplified forms has been studied extensively [12; 13], and here, we confine ourselves only to some simple remarks about the time dependence of the elementary open components. The elementary state components of the translation-invariant dynamics are labeled by the wave vector \(\mathbf{q}\), which might be complex. It is easy to see that the master Equation (9) preserves such an \(\mathbf{x}\)-dependence; hence, the time evolution keeps the subspaces \(\mathcal{A}_{q}\) of the form \(\rho(\mathbf{x},\mathbf{x}_{d})=e^{i\mathbf{q}\mathbf{x}}\chi(\mathbf{x}_{d})\) closed. This expression might be used in the full coordinate space or only in a restricted region, depending on the actual problem. The master equation for the factor \(\chi(\mathbf{x}_{d})\),  
\[\partial_{t}\chi = \bigg{(}i\frac{\hbar\mathbf{q}}{m}-\mathbf{x}_{d}\
u\bigg{)}\mathbf{\
abla}_{ d}\chi \tag{10}\] \[+\bigg{[}\frac{\hbar\mathbf{q}^{2}d_{2}}{2m^{2}}+\bigg{(}\frac{f}{ \hbar}+\frac{\mathbf{q}d_{2}}{m}\
u-\mathbf{q}\xi\bigg{)}ix_{d}-\frac{d_{0}+d_{2}\
u^ {2}-2 mv\xi}{2\hbar}\mathbf{x}_{d}^{2}\bigg{]}\chi,\]  
can easily be solved,  
\[\chi(t,\mathbf{x}_{d})=\chi_{i}\bigg{(}x_{d}e^{-vt}-\frac{\hbar\mathbf{q}}{m\
u}(1-e^ {-vt})\bigg{)}e^{a_{q}(t)+i\mathbf{k}_{q}(t)x_{d}-\frac{d}{2}(1-e^{-2 vt})x_{d}^{2}}, \tag{11}\]  
where the initial condition \(\chi_{i}(\mathbf{x}_{d})=\chi(0,\mathbf{x}_{d})\) is used and:  
\[a_{q}(t) = -\frac{1}{2 mv^{2}}[2i\mathbf{q}\mathbf{f}(vt-1+e^{-vt})+\hbar\mathbf{q}^{2} \xi(1+2 ie^{-vt}+e^{-2 vt})]\] \[-\frac{\hbar\mathbf{q}^{2}}{4m^{2}\
u^{3}}[d_{0}(-e^{-2 vt}+4e^{-vt}-3+ 2 vt)+d_{2}\
u^{2}(1-e^{-2 vt})],\] \[\mathbf{k}_{q}(t) = \frac{1-e^{-vt}}{\hbar\
u}(f-i\hbar q\xi e^{-vt})-i\frac{\mathbf{q}}{ 2 mv^{2}}[d_{0}(1-e^{-vt})^{2}-d_{2}\
u^{2}(1-e^{-2 vt})],\] \[d = \frac{d_{0}+d_{2}\
u^{2}-2 mv\xi}{2\hbar\
u}. \tag{12}\]  
We make here only a few general comments about these elementary components, and their application to simple problems was discussed in [14]. The solution gives the elementary components as the product of two factors: one is the initial condition, \(\chi_{i}(\mathbf{x}_{0})\), considered at the initial coordinate of the characteristic passing \(\mathbf{x}_{d}\) at \(t\). This factor displays the presence of the diffusive forces: the information of the initial state is reduced to a single number, \(\chi_{i}(i\hbar\mathbf{q}/mv)\), as \(t\to\infty\).