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In deriving the master equation, I am coming across the Markov Approximation which says:

Suppose environment $E$ and system $S$ interact and exchange some energy with each other. Then $E$ would recover back to thermal equilibrium faster than $S$ because $E$ is much larger than $S$. Due to this, from the point of view of the $S$, the state of the environment seems to be constant in time i.e. $$\rho_{SE}(t) = \rho_{S}(t) \otimes\rho_E(0)$$ where $\rho_{SE}(t)$ is the combined system-environment state as a function of time and $\rho_E(0)$ is the initial state of the environment. Moreover, since environment is quick to recover from any energy exchange than the system, the interaction or the coupling between the system and the environment is deemed "weak".

I have 2 questions about this approximation:

  1. Why does $E$ being large mean that its correlation time is very short? It makes sense that if environment loses or gains energy, then the effect of that on the total energy of environment will be little. But its not clear why that entails that the relaxation time of environment is smaller than that of the system.
  2. Why does the quick recovery of environment make the interaction "weak"? If this is true, then realistically we can never design systems that are strongly coupled to the environment. This is because we would then need a system as large as environment and designing such system is impossible.
Eesh Starryn
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1 Answers1

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  1. If the environment consists of an continuum of frequencies then the environment's correlations decay. For a specific example see Breuer's book, on chapter 3, High temperature master equation section, where these correlation times are explicitly calculated, and behave like $D(t) \propto e^{-t/\Omega_C}$, where $\Omega_c$ is the cutoff frequency. Note, however, that the higher the cutoff frequency, the faster the decay, and therefore it is NOT true that just having a continuum is enough to have fast correlation decay time.
  2. A weak interaction with the environment is a separate approximation. A priori you could have strong coupling to the environment while at the same time having a fast bath. An extreme example is polaron transformed master equation, where the coupling to the environment is strong, and the approximation instead assumes high cutoff frequency.
peep
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