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,

$\mathrm {B} _{t}(\rho _{0})=\rho _{t}$

This evolution is said to be dissipative if it satisfies these five properties:

(1)$\mathrm {B} _{t}$ is linear.

(2)$\mathrm {B} _{t}$ is positive.

(3)$\mathrm {B} _{t}$ is trace preserving.

(4)$\|\mathrm {B} _{t}(\rho )\|_{1}\leq \|\rho \|_{1}$

(4)Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \Beta_t(\rho)=\sum_n V_{nt}\rhoV^*_{nt}}

External links

• example.com