Thermalization

Azimuth 2020-06-12

I’m wondering if people talk about this. Maybe you know?

Given a self-adjoint operator $H$ that’s bounded below and a density matrix $D$ on some Hilbert space, we can define for any $\beta > 0$ a new density matrix

$\displaystyle{ D_\beta = \frac{e^{-\beta H/2} \, D \, e^{-\beta H/2}}{\mathrm{tr}(e^{-\beta H/2} \, D \, e^{-\beta H/2})} }$

I would like to call this the thermalization of $D$ when $H$ is a Hamiltonian and $\beta = 1/kT$ where $T$ is the temperature and $k$ is Boltzmann’s constant.

For example, in the finite-dimensional case we can take $D$ to be the identity matrix, normalized to have trace 1. Then $D_\beta$ is the Gibbs state at temperature $T$ : that is, the state of thermal equilibrium at temperature $T.$

But I want to know if you’ve seen people do this thermalization trick starting from some other density matrix $D.$