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Claude Cohen Tannoudji

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Accueil du site > Séminaires > LKB > Geometry of quantum observables and thermodynamics of small systems

Geometry of quantum observables and thermodynamics of small systems

Exposé de Maxim Olshanii (Univ. of Mass. Boston)

Vendredi 23 novembre 2012, 9h15, salle de réunion du siège de l’IFRAF, au 4e étage du bâtiment Rataud à l’ENS au 45 rue d’Ulm, 75005 Paris

Résumé

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 talk 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, near-integrable, or mesoscopic systems.

In conclusion, we propose to use the proposed geometric structures to construct a « Maxwell demon of nonergodicity » : a measurement that allows to select useful beyond-ergodicity fluctuations of the temporal averages around the ensemble mean. We regard the cooling of nano-sized opto-mechanical devices as the ultimate field of application for these ideas.

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