Quantum relaxation to equilibrium?

Question

Source and context: Im reading “The Theory of Quantum Open Systems” by Breuer and Petruccione. As an application of the just derived Lindblad equation for the dynamics of the reduced density matrix $\rho_S(t)$ of subsystem $S$: $$ \frac{d}{dt} \rho_S(t) = -i [H_S, \rho_S(t)] +\mathcal{D}(\rho_S(t)), \tag{1} $$ the authors “show” that for any initial state $\rho_S(0)$ in contact with an stationary heat bath $\rho_B = \rho_{th} \equiv \exp (-\beta H_b)/\mathcal{Z}$ $$ \rho_S(t) \longrightarrow \rho_{th} \quad \text{as} \quad t \rightarrow \infty, $$ as one naturally expects. They “show” this by proving that $\rho_{th}$ is stationary, i.e. $\frac{d}{dt}\rho_{th}=0$ (I can follow that proof okay).

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Questions:

1. Why do they just prove that $\rho_{th}$ is stationary? (I don’t find this very impressive as $\rho_{th}$ doesn’t depend on time by definition). In fact, I would have thought that according to what they showed, $\rho_{th}$ is a stationary solution of subsystem $S$ as I expected $\frac{d}{dt}\rho_{th}=0$ to be true by definition and hence $\rho_{th}$ to satisfy (1).

2. Wouldn’t one want to explicitly show that no matter what $\rho_S(0)$ was, in the limit $t\rightarrow \infty$ then $\rho_S\rightarrow \rho_{th}$?

Answer 1

Uhm I don't exactly know which page of Breuer&Petruccione you are referring to, but maybe the following remarks can help:

• In the context of GKLS (Lindblad) master equations, a stationary state is defined as an eigenvector of the Liouvillian $\mathcal{L}$ with zero eigenvalue. The Liouvillian is defined as the generator of the dynamical semigroup (Eq.(3.47) of my edition of Breuer&Petruccione): $\rho(t)=\exp\mathcal{L}t[\rho(0)]$. Then, given a generic (time-independent) density matrix $\rho$, we say that it is a stationary state if $\mathcal{L}[\rho]=0$. The meaning of this can be easily deduced from the definition of Liouvillian.
• The characterization of stationary states for GKLS master equation is a difficult task, with still a lot of ongoing research. There are some remarkable results (not really discussed in Breuer&Petruccione), such as:

1. If the Hilbert space of the system is finite, there is always at least one stationary state of the dynamics [1,2].
2. If the stationary state is unique (say $\rho_{th}$), then the semigroup is relaxing, i.e. $\rho_S\rightarrow \rho_{th}$ for $t\rightarrow \infty$ and for any $\rho_S$ in the state space of the system (the condition in your second point) [1].
3. There are some sufficient and necessary conditions about the uniqueness of stationary states. See the nice review below [3].

• Under certain assumptions on the microscopic model (not always satisfied in physical systems), it can be shown that there is a unique stationary state of the Markovian master equation derived from this model, which is the Gibbs state (section 3.3.2 of your textbook).

Further references:

[1] Rivas and Huelga. "Open Quantum Systems. An Introduction.", Springer Berlin (2012)

[2] Baumgartner and Narnhofer. "Analysis of quantum semigroups with GKS–Lindblad generators: II. General." Journal of Physics A: Mathematical and Theoretical 41, 395303 (2008).

[3] Nigro. "On the uniqueness of the steady-state solution of the Lindblad–Gorini–Kossakowski–Sudarshan equation." Journal of Statistical Mechanics: Theory and Experiment 2019, 043202 (2019).