1208.0582 (Maxim Olshanii)

Geometry of quantum observables and thermodynamics of small systems    [PDF]

Maxim Olshanii

The concept of ergodicity---the convergence of the temporal averages of observables to their ensemble averages---is the cornerstone of thermodynamics. The transition from a predictable, integrable behavior to ergodicity is one of the most difficult physical phenomena to treat; the celebrated KAM theorem is the prime example. This Letter is founded on the observation that for many classical and quantum observables, the sum of the ensemble variance of the temporal average and the ensemble average of temporal variance remains approximately constant across the integrability-ergodicity transition. We show that this property induces a particular geometry of quantum observables---Frobenius (also known as Hilbert-Schmidt) one---that naturally encodes all the phenomena associated with the emergence of ergodicity: the Eigenstate Thermalization effect, the decrease in the inverse participation ratio, and the disappearance of the integrals of motion. As an application, we use this geometry to solve a known problem of optimization of the set of conserved quantities---coming from symmetries or from finite-size effects, regardless---to be incorporated in an extended thermodynamical theory of integrable and near-integrable systems. We regard the cooling of nano-sized opto-mechanical devices as the ultimate field of application for the "optimized" thermodynamics we developed.

View original: http://arxiv.org/abs/1208.0582