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IN PRINCIPLE, this theory should play the same role for such systems
as the Boltzmann-Gibbs theory for open systems placed in a heat bath. All
the results are presented as functions of total energy of the system E (instead
of temperature T in canonical approach). Moreover, in finite closed systems
there are new very interesting phenomena like huge enhancement of effects
produced by weak interactions.
As is known, quantum statistical laws are derived from systems of
infinite numbers of particles, or for systems in a heat bath, and their applicability
to isolated finite systems of a few particles
n is at least questionable. However, if one considers such system, the
density of multi-particle eigenstates increases exponentially with energy, and
rapidly
becomes exponentially large [a exp(n)]. The spacing between the energy
levels is exponentially small, and the residual interaction between particles (even
if not too strong) becomes greater than the energy interval. As a
consequence, large numbers of simple many-particle basis states (shell-model basis
states, Slater determinates) are strongly mixed by the residual interaction into
very complicated superpositions chaotic eigenstates.
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The components of chaotic eigenstates can be treated as
random variables. This means the onset of dynamical Quantum Chaos.
In this situation one can develop statistical methods to describe the system.
Of course, the statistical distributions for this system may be quite different
from the standard canonical distributions; e.g., application of the usual
Fermi-Dirac or Bose-Einstein formulae may give incorrect results. Only in the
limit of a large number of particles this theory should yield standard
statistical mechanical and thermodynamic relations.
Our theory includes the following aspects:
- The structure of chaotic eigenstates
- Criteria of equilibrium ("Quantum chaos")
- Partition function for closed systems (at a given total energy E)
- Occupation numbers n2(E) (for orbital |s>) and expectation values
of operators
- Matrix elements and transition amplitudes between chaotic states
- Theory of enhancement of weak interactions in chaotic states.
Victor Flambaum &
F.M. Izrailev |
