4 Contact Hamiltonian Systems for Probability Distribution Functions …
79
4.6 Geometric Description of Expectation Variables
Several geometrization of expectation variables and thermodynamic variables in
nonequilibrium states for some models and methods have been proposed. Yet, suffice to say that there remains no general consensus on how best to extend a geometry of equilibrium states to a geometry of nonequilibrium states. In this section, a
geometrization of nonequilibrium states is proposed for the observables associated
with the moment dynamical system defined in Proposition 8.
4.6.1 Geometry of Equilibrium States
Equilibrium states are identified with the Legendre submanifolds generated by functions in the context of geometric thermodynamics [11, 12]. Besides, in the context
of information geometry, equilibrium states are identified with dually flat spaces
[1]. Combining these identifications, one can employ Proposition 1 to discuss information geometric aspects of equilibrium states. To apply Proposition 1 to physical
systems, the coordinate sets x and y are chosen such that x
a and y a form a thermodynamic conjugate pair for each a. Here it is assumed that such thermodynamic
variables can be defined even for nonequilibrium states, and that they are consistent
with those variables defined at equilibrium. In addition to this, the physical dimension of should be equal to that of y a dx
a . Moreover, and its Legendre transform
are chosen as .
Choose an appropriate contact manifold and a convex function on it. Then, it
follows from Proposition 1 that the corresponding dually flat space is induced. To
have such a space, a contact manifold is specified first.
This appropriate contact manifold is the pair (C
O
, λ
O
), where
C
O
:= R
n
× R
n
× R,
and
λ
O
= d −
n
a=1
O a θ dθ
a
.
To have a dually flat space, the function
eq in (4.14) is used as a convex function.
This convex function generates the Legendre submanifold A eq ⊂ C
O as in (4.7)
with =
eq , which is explicitly written in coordinates as
A eq =
( θ, O , , ) ∈ C O
O a θ =
∂ ∂ eq
∂θ a , , = eq (θ), j = 1, . . . , |Γ |
.
Combining the discussions so far and Proposition 1, one has the following.
79
4.6 Geometric Description of Expectation Variables
Several geometrization of expectation variables and thermodynamic variables in
nonequilibrium states for some models and methods have been proposed. Yet, suffice to say that there remains no general consensus on how best to extend a geometry of equilibrium states to a geometry of nonequilibrium states. In this section, a
geometrization of nonequilibrium states is proposed for the observables associated
with the moment dynamical system defined in Proposition 8.
4.6.1 Geometry of Equilibrium States
Equilibrium states are identified with the Legendre submanifolds generated by functions in the context of geometric thermodynamics [11, 12]. Besides, in the context
of information geometry, equilibrium states are identified with dually flat spaces
[1]. Combining these identifications, one can employ Proposition 1 to discuss information geometric aspects of equilibrium states. To apply Proposition 1 to physical
systems, the coordinate sets x and y are chosen such that x
a and y a form a thermodynamic conjugate pair for each a. Here it is assumed that such thermodynamic
variables can be defined even for nonequilibrium states, and that they are consistent
with those variables defined at equilibrium. In addition to this, the physical dimension of should be equal to that of y a dx
a . Moreover, and its Legendre transform
are chosen as .
Choose an appropriate contact manifold and a convex function on it. Then, it
follows from Proposition 1 that the corresponding dually flat space is induced. To
have such a space, a contact manifold is specified first.
This appropriate contact manifold is the pair (C
O
, λ
O
), where
C
O
:= R
n
× R
n
× R,
and
λ
O
= d −
n
a=1
O a θ dθ
a
.
To have a dually flat space, the function
eq in (4.14) is used as a convex function.
This convex function generates the Legendre submanifold A eq ⊂ C
O as in (4.7)
with =
eq , which is explicitly written in coordinates as
A eq =
( θ, O , , ) ∈ C O
O a θ =
∂ ∂ eq
∂θ a , , = eq (θ), j = 1, . . . , |Γ |
.
Combining the discussions so far and Proposition 1, one has the following.