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Approach to Equilibrium of Quantum Systems

Approach to Equilibrium of Quantum Systems. Christian Mastrodonato (Università di Genova) In joint work with: S. Goldstein, J.L. Lebowitz, R. Tumulka N. Zanghì. Quantum Macrostates. Set of commuting macroscopic observables. Macroscopic Decomposition of the System Hilbert Space.

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Approach to Equilibrium of Quantum Systems

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  1. Approach toEquilibrium of Quantum Systems Christian Mastrodonato (Università di Genova) In joint work with: S. Goldstein, J.L. Lebowitz, R. Tumulka N. Zanghì

  2. Quantum Macrostates Set of commuting macroscopic observables Macroscopic Decomposition of the System Hilbert Space Quantum Macrostates

  3. Some Notations Time Average Haar Average Macroscopic Projectors Dimensionalities

  4. von Neumann’s Ergodic Theorem Averaging with respect to Macroscopic Observers (1927) (von Neumann) (1958) (Bocchieri and Loinger)

  5. Bocchieri and Loinger Ergodic Relation Averaging with respect to initial states (1959) “If dν<< 1 (ν = 1, . . . ,N), then for the overwhelming majority of the initial states the probabilities <ψ(t)|Pν|ψ(t)> differ insignificantly from dν/D at the overwhelming majority of time instants and large deviations from these values will last very short time intervals”

  6. A New Perspective Averaging with to respect to Hamiltonians

  7. Approach to Equilibrium Equilibrium Macrostate By Chebychev Inequality For a “typical” Hamiltonian A Quantum System is in Equilibrium for most of the time and for the overwhelming majority of Hamiltonians

  8. Generalizable? What does it happen for general macrostates ? By Large Deviation Theory For a “typical” Hamiltonian The probability to find a Quantum System in a macrostate Hν is dν/D for most of the time istants and for the overwhelming majority of Hamiltonians

  9. Work in Progress Analytical Approach Numerical Approach In joint work with R. Mosca and L. Cassettari (DIPTEM, Università di Genova)

  10. Bibliography • P. Bocchieri, A. Loinger: Ergodic Theorem in Quantum Mechanics, Phys. Rev. 111, 668, 1958. • P. Bocchieri, A. Loinger: Ergodic Foundation of Quantum Statistical Mechanics, Phys. Rev. 114, 948, 1959. • S. Goldstein, J.L. Lebowitz, C. Mastrdonato, R. Tumulka, P. Zanghì: Energy Eigenstates and the Approach to Equilibrium of Quantum Systems, in preparation. • J.L. Lebowitz: Microscopic Origins of Irreversible Macroscopic Behavior: An Overview, Physica A 263, 516-527, 1999. • J. von Neumann: Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik, Z. Physik 57, 30, 1929. • J. von Neumann: Mathematical Foundation of Quantum Mechanics, Princeton University Press, 1955. Translation of Mathematische Grundlagen der Quantenmechanick, Springer-Verlag, Berlin, 1932.

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