108
F. Barbaresco
*
( )
,
( )
( )
( )
with
and
S Q
Q
S Q
Q
Q
β
β
β
β
β
=
−Φ
∂Φ
∂
=
∈
=
∈
∂
∂
g
g
(5.39)
In the Souriau Lie groups thermodynamics model, β is a “geometric” (Planck)
temperature, element of Lie algebra gof the group, and Q is a “geometric” heat,
element of the dual space of the Lie algebra
*
g of the group. Souriau has proposed a
Riemannian metric that we have identified as a generalization of the Fisher metric:
I (β) =
g β
with g β ([β, Z 1 ], [β, Z 2 ]) = ˜
β (Z 1 , [β, Z 2 ])
(25)
with ˜
β (Z 1 , Z 2 ) = ˜
(Z 1 , Z 2 ) +
Q, ad Z 1 (Z 2 )
where ad Z 1 (Z 2 ) = [Z 1 , Z 2 ]
(5.40)
Souriau has proved that all co-adjoint orbit of a Lie Group given by
{
}
*
*
*
,
subset of ,
F
g
Ad F g G
F
Ο =
∈
∈
g
g carries a natural homogeneous symplectic
structure by a closed G-invariant 2-form. If we define K = Ad
∗
g =
Ad g −1
∗ and
K ∗ (X ) = −(ad X )
∗ with:
( )
*
* ( )
X
K X
ad
= −
with:
1
*
* ,
,
,
,
,
g
g
Ad F Y
F Ad Y
g G Y
F
−
=
∀ ∈
∈
∈
g
g
(5.41)
where if X ∈g,
1
( )
g
Ad X
gXg
−
=
∈g, the G-invariant 2-form is given by the following
expression:
(
)
( )
[ ]
,
,
, ,
, ,
X
Y
F
ad F ad F
B X Y
F X Y X Y
σ Ω
=
=
∈ g
(5.42)
Souriau Foundamental Theorem is that «Every symplectic manifold on which a
Lie group acts transitively by a Hamiltonian action is a covering space of a coadjoint
orbit». We can observe that for Souriau model, Fisher metric is an extension of this
2-form in non-equivariant case:
g β ([β, Z 1 ], [β, Z 2 ]) = ˜
(Z 1 , [β, Z 2 ]) + Q, [Z 1 , [β, Z 2 ]]
(5.43)
The Souriau additional term ˜
(Z 1 , [β, Z 2 ]) is generated by non-equivariance
through Symplectic cocycle. The tensor ˜
used to define this extended Fisher metric
is defined by the moment map J (x), application from M(homogeneous symplectic
manifold) to the dual space of the Lie algebra
*
g , given by: