# Geometry of quantum observables and thermodynamics of small systems.

@article{Olshanii2015GeometryOQ, title={Geometry of quantum observables and thermodynamics of small systems.}, author={Maxim Olshanii}, journal={Physical review letters}, year={2015}, volume={114 6}, pages={ 060401 } }

We show that for classical and quantum observables, the integrability-to-ergodicity transition leaves constant the sum of (a) the ensemble variance of the temporal average and (b) the ensemble average of temporal variance. The induced Frobenius (Hilbert-Schmidt) geometry of quantum observables encodes how eigenstate thermalization appears, the inverse participation ratio decreases, and the integrals of motion disappear during the transition. We use it to optimize the set of conserved quantities…

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