User:Sckavassalis/Markov dissipative evolution

Markov dissipative evolution is

Definition
For an open system, where the total system is defined by a density operator $$R$$, we assume it evolves according to the equation,

\mathrm{i}\frac{\partial R_t}{\partial t}=\frac{1}{\hbar}[H_{tot},R_t] $$ Where $$H_{tot}=H_{syst}+H_{envir} + \lambda v$$, acting on $$L^2(dxdy)$$, and $$H_{syst}$$ and $$H_{envir}$$ are Schrödinger operators.

If we define a reduced density matrix for our system at time $$t$$ as,
 * $$\rho_t=Tr_{envir}R_t$$,

which is given in terms of the partial trace $$Tr_{envir}$$ of $$R$$ over the environment variables. If initially $$R_0=\rho_0 \otimes \rho_{envir0}$$, we can define an evolution as,
 * $$\Beta_t(\rho_0)=\rho_t$$

This evolution is said to be dissipative if it satisfies these five properties:
 * (1)$$\Beta_t$$ is linear.
 * (2)$$\Beta_t$$ is positive.
 * (3)$$\Beta_t$$ is trace preserving.
 * (4)$$\|\Beta_t(\rho)\|_1\leq\|\rho\|_1$$
 * (4)$$\Beta_t(\rho)=\sum_n V_{nt}\rhoV^*_{nt}$$