%0 Book Section %1 statphys23_0638 %A Mastrodonato, C. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K canonical ensemble function gap gaussian group haar measure measures mechanics on quantum statphys23 topic-1 typical unitary wave %T Typicality of the GAP Measure %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=638 %X Associated with any density matrix $\rho$ on a Hilbert space, there is a GAP measure, a natural probability measure $GAP(\rho)$ on the unit sphere of the Hilbert space, having covariance $\rho$. A system whose wave function is random with distribution $GAP(\rho)$ is a system which, according to quantum mechanics, is in the state $\rho$. For a system in thermal equilibrium with canonical density matrix $\rho_\exp(-H)$, its wave function should be regarded as random, with distribution $GAP(\rho_\beta)$ on the unit sphere of the system's Hilbert space. The proof of this claim is our main result. Crucial ingredients are: (i) the general claim that for a system 1 in interaction with a much larger system 2 and such that the composite is in a pure state $\psi$ for which the reduced density matrix of system 1 is fixed to be $\rho_1$, then a typical such $\psi$ yields a wave function $\psi_1$ for system 1 that is random with distribution $GAP(\rho_1)$, and (ii) canonical typicality, the fact that when system 2 is a heat bath, then for a typical pure state $\psi$ of the composite, the reduced density matrix $\rho_1$ of system 1 is canonical. The talk is based on joint work with S. Goldstein, J.L. Lebowitz, R. Tumulka and N. Zangh\`ı.

@incollection{statphys23_0638, abstract = {Associated with any density matrix $\rho$ on a Hilbert space, there is a GAP measure, a natural probability measure $GAP(\rho)$ on the unit sphere of the Hilbert space, having covariance $\rho$. A system whose wave function is random with distribution $GAP(\rho)$ is a system which, according to quantum mechanics, is in the state $\rho$. For a system in thermal equilibrium with canonical density matrix $\rho_\beta \propto \exp(-\beta H)$, its wave function should be regarded as random, with distribution $GAP(\rho_\beta)$ on the unit sphere of the system's Hilbert space. The proof of this claim is our main result. Crucial ingredients are: (i) the general claim that for a system 1 in interaction with a much larger system 2 and such that the composite is in a pure state $\psi$ for which the reduced density matrix of system 1 is fixed to be $\rho_1$, then a typical such $\psi$ yields a wave function $\psi_1$ for system 1 that is random with distribution $GAP(\rho_1)$, and (ii) canonical typicality, the fact that when system 2 is a heat bath, then for a typical pure state $\psi$ of the composite, the reduced density matrix $\rho_1$ of system 1 is canonical. The talk is based on joint work with S. Goldstein, J.L. Lebowitz, R. Tumulka and N. Zangh\`\i.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Mastrodonato, C.}, biburl = {https://www.bibsonomy.org/bibtex/21ccfd7417907399e836fb10f990ff57f/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {9c723055af13f75459df7bd21f761387}, intrahash = {1ccfd7417907399e836fb10f990ff57f}, keywords = {canonical ensemble function gap gaussian group haar measure measures mechanics on quantum statphys23 topic-1 typical unitary wave}, month = {9-13 July}, timestamp = {2007-06-20T10:16:25.000+0200}, title = {Typicality of the GAP Measure}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=638}, year = 2007 }