Since every invertible square matrix can be decomposed as At=e(Z+iH)t, you don’t actually need a unitary assumption. You can just say that after billions of years, all but the largest Z-matrices have died out.
There’s another tie between statistics and quantum evolution called the Wick rotation. If you set t=iβ, then E[e(Z+iH)t]=E[e−Hβ] so the inverse-temperature is literally imaginary time! You can recover the Boltzmann distribution by looking at the expected number of particles in each state: E[⟨n|e(Z+iH)t|n⟩]=e−βEn where En is the nth eigenvalue (energy in the nth state).
A couple things to add:
Since every invertible square matrix can be decomposed as At=e(Z+iH)t, you don’t actually need a unitary assumption. You can just say that after billions of years, all but the largest Z-matrices have died out.
There’s another tie between statistics and quantum evolution called the Wick rotation. If you set t=iβ, then E[e(Z+iH)t]=E[e−Hβ] so the inverse-temperature is literally imaginary time! You can recover the Boltzmann distribution by looking at the expected number of particles in each state: E[⟨n|e(Z+iH)t|n⟩]=e−βEn where En is the nth eigenvalue (energy in the nth state).