Appendix F
Gaussian Ensembles
Below we derive some key properties of the Gaussian ensembles. There are three
Gaussian ensembles that have been studied in great detail. They include the
Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE),
and the Gaussian Symplectic Ensemble (GSE). Each of these ensembles extremize
information about the properties of a quantum system for a particular set of
symmetries.
GOE describes systems that invariant under rotation and time reversal, with
dynamics governed by a real symmetric Hamiltonian matrix. GUE describes
systems that are invariant under rotation but not time reversal (for example if a
magnetic field is present), and has dynamics governed by a complex Hermitian
Hamiltonian matrix. GSE describes systems with spin−
1
2 particles that are not
rotationally invariant but are time reversal invariant (strong spin-orbit coupling but
no magnetic field), and have dynamics governed by quaternion-real Hamiltonian
matrices.
In the sections below, we describe some properties of these three Gaussian
ensembles.
F.1 Gaussian Orthogonal Ensemble (GOE)
A number of key properties of the Gaussian Orthogonal Ensemble were discussed
in Chap. 6. However, because the derivation of the cluster functions for GOE are
somewhat tedious, the derivation has been moved to the sections below.
As we showed in Chap. 6, the GOE probability density, in polar form, takes the
form
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3
467
Gaussian Ensembles
Below we derive some key properties of the Gaussian ensembles. There are three
Gaussian ensembles that have been studied in great detail. They include the
Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE),
and the Gaussian Symplectic Ensemble (GSE). Each of these ensembles extremize
information about the properties of a quantum system for a particular set of
symmetries.
GOE describes systems that invariant under rotation and time reversal, with
dynamics governed by a real symmetric Hamiltonian matrix. GUE describes
systems that are invariant under rotation but not time reversal (for example if a
magnetic field is present), and has dynamics governed by a complex Hermitian
Hamiltonian matrix. GSE describes systems with spin−
1
2 particles that are not
rotationally invariant but are time reversal invariant (strong spin-orbit coupling but
no magnetic field), and have dynamics governed by quaternion-real Hamiltonian
matrices.
In the sections below, we describe some properties of these three Gaussian
ensembles.
F.1 Gaussian Orthogonal Ensemble (GOE)
A number of key properties of the Gaussian Orthogonal Ensemble were discussed
in Chap. 6. However, because the derivation of the cluster functions for GOE are
somewhat tedious, the derivation has been moved to the sections below.
As we showed in Chap. 6, the GOE probability density, in polar form, takes the
form
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3
467