Static equilibrium for isolated quantum systems

[@GeoffryGifari]

hi jon, there are some things i found interesting: 1. what are the assumptions we need for the equilibriation bound? 2. Can we say that matrix elements Aₘₙ become more diagonal over time? 3. Does the theorem depend on interaction Hamiltonian? 4. Can we prove that the system goes to equilibrium monotonically (no out-of-equilibrium "spikes" along the way) 5. Does equilibriation become less predictable as system size gets smaller (more "quantum", hence time-reversal invariant) 6. What is the physical interpretation of tr(ω²) again? 7. Is the equilibrium state unique? i know its a lot, no pressure

[@JonathonRiddell]

Hey! 1. The assumptions are we are in an isolated quantum system, and our system satisfies the no resonance condition on the spectrum. 2. The magnitude of the entries are independent on time (we are mostly in the Schrodinger picture here). 3. The no resonance condition guarantees that we have interactions / the Hamiltonian isn't trivial. 4. No, in fact, this can't be true unless we are in the thermodynamic limit, by the Poincare Recurrence theorem:) I have a video on it. 5. Yes, in fact, for sufficiently small systems you might not even expect to see something that looks like equilibration. 6. If you take the reciprocal, you can treat that as the effective dimension of the dynamics, i.e the effective size of the Hilbert space. 7. Yes, well, up to phases on c_m's, in theory you could construct slightly different states that have the same omega. No worries! I'm glad to have these questions :)

[@GeoffryGifari]

is it right to say that as long as there is interaction, the equilibriation bound holds, but the form of interaction itself only contributes to relaxation time?

[@JonathonRiddell]

That plus the intersection needs to be sufficiently complicated along with the initial condition being sufficiently complicated (i.e we need a lot of eigenstates of the Hamiltonian in the dynamics).

[@benburdick9834]

Could you elaborate on how one should think about purity? Would it be correct to think of it as a measure of how 'quantum' your state is? Also, how many inequalities did we use to get to this result? It seems like we used two, in which case, shouldn't we be worried that we are getting further from sigma than we have to be?

[@JonathonRiddell]

The purity should be regarded as a measure of how pure our state is. If Omega was a pure state then the purity would be 1. In our example this would only happen if our initial state was an eigenstate of the Hamiltonian. So some people call the inverse of the purity in this case "the effective dimension". So we want omega to be very mixed (we expect it). For the inequality you are exactly right, we used 3 bounds in the end, which took us away from the real quantity. In fact this might make our bound an order of magnitude bigger than the true value( arxiv.org/abs/2112.09475 figure 6 shows this). But the important idea here is instead, we get an upper bound that goes to zero exponentially fast, it is the scaling that matters here.

[@GeoffryGifari]

Oh and one thing i've thought about after reading about nonequilibrium systems: if a system does not equilibriate, is it necessary for work to be supplied for it to remain that way?

[@JonathonRiddell]

In this context no, but it could definitely depend on the system in question. If the system / conditions without interference would equilibrate you would need to do work / interfere to stop it. But equilibration isn't always guaranteed :)

[@GeoffryGifari]

@@JonathonRiddell i'm getting into the thermodynamics of life (jeremy england's work is one of them). fascinating how organism's far-from-equilibrium state can be maintaned through energy flux