Normal typicality and von Neumann’s quantum ergodic theorem

@article{Goldstein2010NormalTA,
  title={Normal typicality and von Neumann’s quantum ergodic theorem},
  author={Sheldon Goldstein and Joel L Lebowitz and Christian Mastrodonato and Roderich Tumulka and Nino Zangh{\'i}},
  journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
  year={2010},
  volume={466},
  pages={3203 - 3224}
}
  • S. Goldstein, J. Lebowitz, +2 authors N. Zanghí
  • Published 2010
  • Mathematics, Physics
  • Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
We discuss the content and significance of John von Neumann’s quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e. the statement that, for typical large systems, every initial wave function ψ0 from an energy shell is ‘normal’: it evolves in such a way that |ψt⟩⟨ψt| is, for most t, macroscopically equivalent to the micro-canonical density matrix. The QET… Expand
View on Royal Society
arxiv.org
96 Citations
Highly Influential Citations
3
Background Citations
19
Methods Citations
2

96 Citations

Citation Type

Citation Type

Long-time behavior of macroscopic quantum systems
AbstractThe renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann’s 1929 article on the quantum ergodic theorem. We have found this… Expand
The approach to equilibrium in a macroscopic quantum system for a typical nonequilibrium subspace
We study the problem of the approach to equilibrium in a macroscopic quantum system in an abstract setting. We prove that, for a typical choice of "nonequilibrium subspace", any initial state (from… Expand
Maxwell’s Demon in Quantum Mechanics
TLDR
It is demonstrated by a general state-space argument that Maxwell’s Demon is compatible with classical mechanics, and that the most recent solutions, based on Landauer's thesis, are not general. Expand
Information-theoretic equilibrium and observable thermalization
TLDR
A new notion of thermal equilibrium is proposed, focused on observables rather than on the full state of the quantum system, and it is shown that there is always a class of observables which exhibits thermal equilibrium properties and a recipe to explicitly construct them. Expand
Copenhagen quantum mechanics
In our quantum mechanics courses, measurement is usually taught in passing, as an ad-hoc procedure involving the ugly collapse of the wave function. No wonder we search for more satisfying… Expand
Pure state quantum statistical mechanics
The capabilities of a new approach towards the foundations of Statistical Mechanics are explored. The approach is genuine quantum in the sense that statistical behavior is a consequence of objective… Expand
The Copenhagen Interpretation Born Again
An approach to quantum mechanics is developed which makes the Heisenberg cut between the deterministic microscopic quantum world and the partly deterministic, partly stochastic macroscopic world… Expand
Measurement and ergodicity in quantum mechanics
The experimental realization of successive non-demolition measurements on single microscopic systems brings up the question of ergodicity in quantum mechanics (QM). We investigate whether time… Expand
Pure states statistical mechanics: On its foundations and applications to quantum gravity
The project concerns the study of the interplay among quantum mechanics, statistical mechanics and thermodynamics, in isolated quantum systems. The goal of this research is to improve our… Expand
Quantum ergodicity for a class of non-generic systems
We examine quantum normal typicality and ergodicity properties for quantum systems whose dynamics are generated by Hamiltonians which have residual degeneracy in their spectrum and resonance in their… Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 76 REFERENCES
Long-time behavior of macroscopic quantum systems
AbstractThe renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann’s 1929 article on the quantum ergodic theorem. We have found this… Expand
Quantum macrostates, equivalence of ensembles, and an H-theorem
Before the thermodynamic limit, macroscopic averages need not commute for a quantum system. As a consequence, aspects of macroscopic fluctuations or of constrained equilibrium require a careful… Expand
Quantum statistical ergodic and H-theorems for incompletely specified systems
  • P. Landsberg
  • Mathematics
  • Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
  • 1961
In incompletely specified situations the passage from quantum mechanics to statistical mechanics requires an averaging process. Typically, one has to average a scalar product of a fixed unit vector α… Expand
On the approach to thermal equilibrium of macroscopic quantum systems
In joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zanghi [2, 3, 4], we considered an isolated, macroscopic quantum system. Let H be a micro‐canonical “energy shell,” i.e., a subspace of the… Expand
From Quantum Dynamics to the Canonical Distribution: General Picture and a Rigorous Example
Derivation of the canonical (or Boltzmann) distribution based only on quantum dynamics is discussed. Consider a closed system which consists of a mutually interacting subsystem and a heat bath, and… Expand
Ergodicity Conditions in Quantum Mechanics
In this paper ergodicity conditions for a quantummechanical system are investigated in the line of thought of recent papers of Bocchieri and Loinger and of authors themselves. More precisely, the… Expand
On the quantum-statistical ergodic and H-theorems
  • I. Farquhar, P. Landsberg
  • Mathematics
  • Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
  • 1957
Metrical indecomposability is a condition for the validity of the classical ergodic theorem. Its counterpart in the quantum-statistical theorem is believed to be non-degeneracy of the energy… Expand
Entanglement and the foundations of statistical mechanics
Statistical mechanics is one of the most successful areas of physics. Yet, almost 150 years since its inception, its foundations and basic postulates are still the subject of debate. Here we suggest… Expand
The ergodic behaviour of quantum many-body systems
Synopsis By a pertubation technique adapted to the actual properties of gases and solids (and possibly also of liquids) we have established in previous papers that under suitable conditions a quantum… Expand
Approach to thermal equilibrium of macroscopic quantum systems.
TLDR
It is shown that for "typical" Hamiltonians with given eigenvalues, all initial state vectors psi(0) evolve in such a way that psi(t) is in thermal equilibrium for most times t. Expand
...
1
2
3
4
5
...