Proof of the ergodic theorem and the H-theorem in quantum mechanics
By J. Neumann. 2010
254 citations recorded by the source corpus.
AbstractIt is shown how to resolve the apparent contradiction between the macroscopic approach of phase space and the validity of the uncertainty relations. The main notions of statistical mechanics are re-interpreted in a quantum-mechanical way, the ergodic theorem and the H-theorem are formulated and proven (without “assumptions of disorder”), followed by a discussion of the physical meaning of the mathematical conditions characterizing their domain of validity.
Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/dd9e74da719af679fc6cd7ce8fd3c7fbdbc97fb1
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In Atomic and Molecular Physics, and Optics
- D. Stanford
- Tobias J. Osborne
- Patrick M. Hayden
- M. B. Hastings
- Nima Lashkari
- S. Shenker
- L. Susskind
- P. Hänggi
In Statistical and Nonlinear Physics
Citations in the corpus
- Colloquium : Statistical mechanics and thermodynamics at strong coupling: Quantum and classical (2019, background): “…systems this is guaranteed by ergodicity (Khinchin, 1949), for quantum systems the problem of thermalization as such is known since long (von Neumann, 1929) but is still under active scrutiny (Deutsch, 1991; Goold et al., 2016; Polkovnikov et al., 2011; Rigol and Srednicki, 2012; Srednicki,…”
- An infalling observer in AdS/CFT (2012, background): “As far as we know, this was first attempted by von Neumann [39] and our “coarse” operators are precisely the “macroscopic” operators of that paper.”
- Towards the fast scrambling conjecture (2011, background): “If the deviation is small at time t, the continuity of the von Neumann entropy implies that H(Ψx(t)) ≤ δ log dimHx for some universal δ – 20 –”
- Normal typicality and von Neumann’s quantum ergodic theorem (2009, unknown): “(von Neumann 1929) There is a (big) constant C1 > 1 such that whenever the two natural numbers D and dν satisfy C1 logD dν D C1 , (46) and U is a Haar-distributed random unitary D ×D matrix, then E D max α6=β=1 ∣ ∣ ∣ dν ∑ γ=1 Uγα(Uγβ) ∗ ∣ ∣ ∣ 2 ≤ logD D , (47) E D max α=1 ( dν ∑ γ=1 |Uγα|2 − dν…”