Equation of motion for the reduced density matrix

Question

The equation of motion for the density matrix of a many body isolated quantum system is the von Neumann's equation: $\dot{\rho }(t)=i[\rho (t),H]$. How about the equation of motion for the reduced density matrix of one particle, is there a general procedure to obtain it?

Answer 1

There are a number of schemes that have been developed over the years to describe the dynamics of the reduced density matrix. The problem that you encounter is that you have to make a choice between computability versus generality.

Let me make an overview of some of the approaches you can find. We typically think of system consisting of a some particle in contact with a heat bath (a reservoir). Starting with the Liouville equation $$ \frac{d W}{dt} = -i[H,W(t)] \equiv \mathcal{L}(t)W(t)$$ where $W(t)$ is the density matrix of the full system, $H$ the full Hamiltonian and $\mathcal{L}(t)$ the Liouville operator.

The reduced density matrix is obtained by projecting to the relevant subspace,

$$\rho(t) = PW(t)$$ where $P^2=1$ is a projection operator. This operator takes the partial trace of $W$, as is standard with reduced density matrices, and the reduced density matrix is still an operator on the total Hilbert space.

We now substitute $W(t) = \rho(t) + (1-P)W(t)$ in the Liouville equation, and project with $P$ and $(1-P)$ from the left. This gives two coupled equations of motion, one for $\rho$ and one for $(1-P)W$. Solving formally for $(1-P)W$ and substituting this back into $\dot{\rho}$ gives the Nakajima-Zwanzig equation,

$$\dot{\rho}(t) = P\mathcal{L}\rho(t) + \int_0^t dt'P\mathcal{L}e^{(1-P)\mathcal{L}t'}(1-P)\mathcal{L}\rho(t-t') + P\mathcal{L} e^{(1-P)\mathcal{L}t}(1-P)W(0) $$

It is a mixed integral and differential equation (integro-differential). It shows that the time evolution of the reduced density matrix not only depends on the state of $\rho$ at time $t$, but also on the history of the system leading up to that time, including the initial entanglement between the subsystem and the reservoir. This retarded effect comes in through the second term, the integral. It has a natural interpretation: in the strictest sense you have to keep track of what the external reservoir is doing if you want to describe the dynamics of the subsystem. This "information" has to be stored somewhere, hence the integral over some memory kernel.

Simplifications have to be made. The most straightforward one disregards the initial term $W(0)$ (such that you do not worry about the initial conditions, which definitely does not apply to all systems) and assumes a Markovian time evolution, i.e. where the time evolution only depends on the state at time $t$. This leads to the Born-Markov approximation, which is an example of a Lindblad equation.

The Lindblad equation assumes some features like positivity of the density matrix at all times, trace preserving and no external time dependence. But do note that it is not unitary (which is ok!). Assuming the Hamiltonian can be written as $H=H_S+ H_R + H_I$ (subsystem, reservoir and interaction) we can write down

$$\dot{\rho}(t) = -i[H_s,\rho] +\frac{1}{2} \sum_j \left\lbrace -\frac{1}{2}\hat{V}_n^\dagger \hat{V}_n\rho(t) - \frac{1}{2}\rho \hat{V}_n^\dagger\hat{V}_n + \hat{V}_n\rho \hat{V}_n^\dagger\right\rbrace $$

Here the sum runs over a specific basis of the subsystem, and the operator $\hat{V}$ describes all the effective dynamics of the subsystem due to coupling with the reservoir. This is probably the best known equation of motion for the reduced density matrix, although it is not as general as the NZ-equation (which is not really useful though).

To give some insight to this equation: the last term represents quantum jumps or (quantum) fluctuations between states of the subsystem - transitions between states of the subsystem due to the external reservoir. The first two terms are, mathematically speaking, there to ensure positivity and preserve the trace of the reduced density matrix. They are dissipative terms.

Answer 2

The more general and direct procedure is to take the partial trace over the rest of particles at both sides of the whole equation. One obtains the exact equation $\dot{\sigma_1 }(t)=iTr_{2,...N}[\rho (t),H]$ for the one-particle density matrix $\sigma_1 =Tr_{2,...N}\rho$.

Different physical approximations can be introduced in this exact equation. For instance, taking the non-correlated state $\rho = \sigma_1 \otimes \sigma_2 \cdots \otimes \sigma_N$ we obtain the mean field master equation.

NOTES: The Nakajima-Zwanzig equation given above by Olaf is dimensionally incorrect. I have not checked all the equation but in a first look the integral term is lacking a $P\mathcal{L}$ term because this term is obtained from the $P\mathcal{L}(1-P)$ subspace. Even corrected this equation is problematic both physically and computationally. The Lindblad equation can give violation of translation invariance and when this is corrected by adding a counter-term the improved equation fails to give the correct approach to equilibrium.