6.6 Thermalization of Quantum Systems
187
Therefore, it is useful to consider the limiting case Ny→η as N→∞, where η
is finite. This is easily done if we use Eq. (6.133). If we take the limit N →∞ in
Eq. (6.138), we obtain the following distribution for individual components of the
GOE ensemble of eigenvectors:
F GOE (η) =
1
√
2π
1
√
η
e
−η/2 .
(6.139)
The coefficient is chosen so that
∞
−∞ dη F GOE (η) = 1. The probability density,
F GOE (η), in Eq. (6.139) is called the Porter-Thomas distribution (Porter and
Thomas 1956; Porter 1965) and was first used to describe nuclear reaction widths.
6.6 Thermalization of Quantum Systems
There is a growing body of work which focuses on the mechanisms by which
underlying classical chaos determines the ergodic behavior of quantum systems.
In 1976, Voros (1976a; 1976b) proposed that the density operator for a quantum
system with energy E, whose underlying classical dynamics is chaotic, must have
the form ρ(E)→Cδ(H − E) in the classical limit (H is the classical Hamiltonian).
Following Voros, Berry (1977a; 1977b) made a conjecture on the behavior of the
Wigner function for such a system. If the system was ergodic (chaotic), then the
Wigner function for an N-particle system in an energy eigenstate, when averaged
over a small region of the phase space on the energy surface, would have the form
f (r, p) =
δ(E − H (r, p)
dr
dpδ(E − H (r, p)
,
(6.140)
where r denotes the displacement of the N particles (r 1 , . . . , r N ) and p denotes the
momenta of the N particles. This conjecture of Berry was later used by Srednicki
(1994) to show that a closed quantum system with N hard sphere particles (a chaotic
system) with energy E, is thermalized due to the nature of the eigenstates of such
systems. Below, we give a simplified version of the original Srednicki derivation.
Nearly every statistical mechanics textbook discusses equilibrium properties of
matter, starting with the ideal gas equations of state. An ideal gas equation of state
describes the equilibrium thermal distribution of a “noninteracting” gas of identical
particles (Reichl 2016). For identical classical particles, the thermal distribution is
the Maxwell-Boltzmann distribution; for identical bosons, it is the Bose-Einstein
distribution; and for identical fermions, it is the Fermi-Dirac distribution. However,
the underlying assumption regarding these ideal gases is that they are ergodic.
Ergodicity is the foundation upon which statistical mechanics is built. For an
187
Therefore, it is useful to consider the limiting case Ny→η as N→∞, where η
is finite. This is easily done if we use Eq. (6.133). If we take the limit N →∞ in
Eq. (6.138), we obtain the following distribution for individual components of the
GOE ensemble of eigenvectors:
F GOE (η) =
1
√
2π
1
√
η
e
−η/2 .
(6.139)
The coefficient is chosen so that
∞
−∞ dη F GOE (η) = 1. The probability density,
F GOE (η), in Eq. (6.139) is called the Porter-Thomas distribution (Porter and
Thomas 1956; Porter 1965) and was first used to describe nuclear reaction widths.
6.6 Thermalization of Quantum Systems
There is a growing body of work which focuses on the mechanisms by which
underlying classical chaos determines the ergodic behavior of quantum systems.
In 1976, Voros (1976a; 1976b) proposed that the density operator for a quantum
system with energy E, whose underlying classical dynamics is chaotic, must have
the form ρ(E)→Cδ(H − E) in the classical limit (H is the classical Hamiltonian).
Following Voros, Berry (1977a; 1977b) made a conjecture on the behavior of the
Wigner function for such a system. If the system was ergodic (chaotic), then the
Wigner function for an N-particle system in an energy eigenstate, when averaged
over a small region of the phase space on the energy surface, would have the form
f (r, p) =
δ(E − H (r, p)
dr
dpδ(E − H (r, p)
,
(6.140)
where r denotes the displacement of the N particles (r 1 , . . . , r N ) and p denotes the
momenta of the N particles. This conjecture of Berry was later used by Srednicki
(1994) to show that a closed quantum system with N hard sphere particles (a chaotic
system) with energy E, is thermalized due to the nature of the eigenstates of such
systems. Below, we give a simplified version of the original Srednicki derivation.
Nearly every statistical mechanics textbook discusses equilibrium properties of
matter, starting with the ideal gas equations of state. An ideal gas equation of state
describes the equilibrium thermal distribution of a “noninteracting” gas of identical
particles (Reichl 2016). For identical classical particles, the thermal distribution is
the Maxwell-Boltzmann distribution; for identical bosons, it is the Bose-Einstein
distribution; and for identical fermions, it is the Fermi-Dirac distribution. However,
the underlying assumption regarding these ideal gases is that they are ergodic.
Ergodicity is the foundation upon which statistical mechanics is built. For an