Abstract
Abstract
We show that for a system governed by a random-matrix Hamiltonian (a member of the time-reversal invariant Gaussian Orthogonal Ensemble (GOE) of random matrices of dimension N), all functions
Tr
(
A
ρ
(
t
)
)
in the ensemble thermalize: For
N
→
∞
every such function tends to the value
Tr
(
A
ρ
eq
(
∞
)
)
+
Tr
(
A
ρ
(
0
)
)
g
2
(
t
)
. Here
ρ
(
t
)
is the time-dependent density matrix of the system, A is a Hermitean operator standing for an observable, and
ρ
eq
(
∞
)
is the equilibrium density matrix at infinite temperature. The oscillatory function g(t) is the Fourier transform of the average GOE level density and falls off as
1
/
|
t
|
3
/
2
for large t. With
g
(
t
)
=
g
(
−
t
)
, thermalization is symmetric in time. Analogous results, including the symmetry in time of thermalization, are derived for the time-reversal non-invariant Gaussian Unitary Ensemble of random matrices. Comparison with the ‘eigenstate thermalization hypothesis’ of (Srednicki 1999 J. Phys. A: Math. Gen.
32 1163) shows overall agreement but raises significant questions.