# Quantum relaxation to equilibrium?

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).

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 $$\rho_S\rightarrow \rho_{th}$$?

• On what page of Breuer's book does this claim come from? Commented Mar 2, 2021 at 18:09
• Uh on page number 132, I believe there is only one edition? Commented Mar 2, 2021 at 18:19

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).

• Thank you so much for your answer and references! For future learners like me, ref [1] above contains the answer to my question in section 5.4 Steady states of homogeneous Markov processes. In particular the proof of @Goffredo_Gretzky 's points 1. and 2. are explained in great detail in theorems 5.3 and 5.4 with accesible proofs offered. If the manuscript doesn’t change noticeably this corresponds to pgs. 45-48. This review would make a great book on open quantum systems! Commented Mar 5, 2021 at 10:11
• It actually is: springer.com/gp/book/9783642233531 Commented Mar 5, 2021 at 17:20