The information geometric descriptions of
denormalized thermodynamic manifolds
Shicheng Zhang, Huafei Sun
Abstract. In view of information geometry,
state space of thermodyPthe
n
namic parameters S = {ρ|ρ = Z −1 exp{− i=1 βi Fi , Tr ρ = 1} has been
investigated. Here the α-geometric structures of the denormalization of
S called Se = {e
ρ|e
ρ = f (τ )ρ, f (τ ) > 0, Tr ρ = 1} is investigated. The covariant derivative and the α-curvature tensor is studied. Therefore, the
relation of the α-geometric structures between S and Se are obtained. The
results were obtained in [8] when α = 0 and in [2] when f (τ ) = 1.
M.S.C. 2000: 53C15, 62B10.
Key words: Thermodynamic parameters; information geometry; α-curvature tensor.
1
Introduction
Information geometry ([1]) originated from investigating the geometric structures of
the manifold which consists of probability density functions and it has various applications such as in statistical inference and neural networks.
Recently, some authors ([7, 6, 5, 3, 4, 2]) considered the geometric structure of the
space of thermodynamic parameters, which forms a manifold called S. This manifold
characterizes a given physical system. One of the main results they obtained is to
give the Riemannian Gaussian curvature of the manifold S. In the present paper,
we firstly define the α-connection and obtain the α-Gaussian curvature which will
becomes the Riemannian Gaussian curvature when α = 0. Ingarden ([3]), Janyzek
([4]), Zheng Z. [10] and other authors reaching a Riemannian metric by
Pstatistical
n
method. For a given equilibrium density operator ([7]) ρ = Z −1 exp{−
Pn i=1 βi Fi },
where F1 , F2 , · · · , Fn are linear independent operators, Z = Tr exp{− i=1 βi Fi } is
a partition function and β = (β1 , β2 , · · · , βn ) are classical real parameters (statistical
temperature, press, magnetic field), which describes the environment of a physical
system. The thermodynamic parameters set S = {ρ| Tr ρ = 1} can be regarded as a
differential manifold equipped with a Riemannian metric
gij = Tr[ρ(∂i ln ρ)(∂j ln ρ)]
for the case of commuting operators Fi , where Tr denotes the trace of the matrix,
and ∂i means the partial derivative with respect to the parameter βi .
Applied Sciences, Vol.12, 2010, pp.
158-166.
c Balkan Society of Geometers, Geometry Balkan Press 2010.
°
The information geometric descriptions ...
159
In [1], Amari investigated the denormalization of statistical model. Here we consider the denormalization of thermodynamic parameters model, a more general manifold, namely
Se = {e
ρ|e
ρ = f (τ )ρ, f (τ ) > 0, Tr ρ = 1},
where f (τ )(> 0) is an arbitrary-order differentiable function. Then Se is a manifold
which contains S as a submanifold with dim Se = dim S + 1. Se is called the denormalization of S. By calculating the covariant derivative and the α-curvature tensor, we
e At last, we use a
obtain the relation of the α-geometric structures between S and S.
example to illustrate our conclusions.
2
The information geometric structures of the manifold Se
Parameterizing the elements of Se as ρe by a coordinate system [β, τ ] = (β1 , · · · , βn , τ )
and letting

 2 ρe1−α
2
(α 6= 1)
(α)
e
l := 1 − α

ln ρe
(α = 1),
e (α) are
the components of the α-metric ge(α) and the coefficients of the α-connection ∇
represented respectively by
(α)
(2.1)
geij = Tr(∂ie
l(α) ∂j e
l(−α) ),
(2.2)
e (α) = Tr(∂i ∂j e
Γ
l(α) ∂k e
l(−α) ).
ijk
∂
∂
We denote its natural basis by ∂ei = ∂β
, ∂eτ = ∂τ
and use µ, ν, λ, γ to denote the
i
subscripts of the natural basis, that is, µ, ν, λ, γ ∈ {1, · · · , n, n + 1}, while i, j, k, l ∈
{1, · · · , n}. For the case of commuting operators Fi , from (2.1), we see that
geµν = Tr(e
ρ∂eµ ln ρe∂eν ln ρe).
By a calculation, we get
geij = f (τ )gij ,
geiτ = 0,
geτ τ = f (τ )−1 (∂τ f (τ ))2 .
The Riemannian metric of Se and its inverse are given by
µ
¶
µ
f (τ )G
0
f (τ )−1 G−1
−1
e
e
(2.3) G =
, G =
0
f (τ )−1 (∂τ f (τ ))2
0
¶
0
.
f (τ )(∂τ f (τ ))−2
The square of the arc length of S defined by ds2 = gij dβi dβj . Thus, the square of
the arc length of Se is given by
de
s2 = f (τ )ds2 + f (τ )−1 (∂τ f (τ ))2 dτ 2 .
Note that the volume element of S is given by
p
dV = det(G)dβ1 Λ · · · Λdβn ,
160
Shicheng Zhang, Huafei Sun
we have the relation between the volume elements dV and dVe
q
e
dVe = det(G)dβ
1 Λ · · · Λdβn Λdτ
p
n−1
= (f (τ )) 2 ∂τ f (τ ) det(G)dβ1 Λ · · · Λdβn Λdτ
= dV (f (τ ))
n−1
2
df (τ ).
e (α) for Se satisfy
Proposition 1. The α-connection coefficients of ∇
e (α) = − 1 + α ∂τ f (τ )gij ,
Γ
ij,τ
2
(α)
(α)
e
e
e (α) = 0,
Γ
=Γ
= 0,
Γ
e (α) = f (τ )Γ(α) ,
Γ
ij,k
ij,k
(2.4)
e (α) = Γ
e (α) = 1 − α ∂τ f (τ )gik ,
Γ
iτ,τ
τ i,τ
iτ,k
τ i,k
τ τ,k
2
1+α
e (α)
Γ
f (τ )−2 (∂τ f (τ ))3 + f (τ )−1 ∂τ f (τ )∂τ ∂τ f (τ ),
τ τ,τ = −
2
and
e (α)τ = − 1 + α f (τ )(∂τ f (τ ))−1 gij ,
Γ
ij
2
(α)τ
(α)τ
e
e
e (α)k
Γ
=Γ
= 0,
Γ
= 0,
ττ
e (α)k = f (τ )Γ(α)k ,
Γ
ij
ij
e (α)k = Γ
e (α)k = 1 − α f (τ )−1 ∂τ f (τ )δik ,
(2.5) Γ
iτ
τi
iτ
τi
2
1+α
e (α)τ
Γ
=−
f (τ )−1 ∂τ f (τ ) + (∂τ f (τ ))−1 ∂τ ∂τ f (τ ),
ττ
2
where the α-connection coefficients of ∇(α) for S satisfies
(α)
Γijk (β)
=
1−α
∂i ∂j ∂k ln Z.
2
Proof. By (2.2) and notice that
1−α
∂ei ∂ej e
l(α) = f (τ ) 2 ∂i ∂j l(α) ,
1−α
fτ ∂eie
∂
l(α) =
f (τ )∂τ f (τ )∂i l(α) ,
2
3+α
1−α
1+α
1−α
1+α
fτ ∂
fτ e
∂
l(α) = −
f (τ )− 2 (∂τ f (τ ))2 ρ 2 + f (τ )− 2 ∂τ ∂τ f (τ )ρ 2 ,
2
we can obtain proposition 1.
These equations enable us to verify that the following relations hold for the coe (on S),
e respectively:
variant derivatives ∇ (on S) and ∇
(2.6)
e e Ye = (∇X Y )∼ − 1 + α (∂τ f (τ ))−1 < X,
e Ye > ∂eτ ,
∇
X
2
(2.7)
e e ∂eτ = ∇
ee X
e = 1 − α f (τ )−1 ∂τ f (τ )X,
e
∇
X
∂τ
2
(2.8)
¸
·
e e ∂eτ = − 1 + α f (τ )−1 ∂τ f (τ ) + (∂τ f (τ ))−1 ∂τ ∂τ f (τ ) ∂eτ ,
∇
∂τ
2
The information geometric descriptions ...
161
]
where X and Y are arbitrary vector fields on S, and (· · · )∼ denotes (·
· · ).
From proposition 1, we can obtain the following
e
Theorem 1. S is (-1)-autoparallel in S.
f be its denormalization. For
Theorem 2. Let M be a submanifold of S and M
every α ∈ R, the following conditions (i) and (ii) are equivalent.
(i) M is α-autoparallel in S.
f is α-autoparallel in S.
e
(ii) M
(α)
(α)
e
Proof. Let ∇ and ∇ be the α-connections on Se and S as above. Noting that
fτ | f by vector fields {Xi } on
f can be represented as hi X
fi + h0 ∂
every vector field on M
M
i
0
M and functions {h } and h on M , we see from (2.7) and (2.8) that condition (ii) is
e (α) Ye ∈ T (M
f) for all X, Y ∈ T (M ). On the other hand,
equivalent to stating that ∇
e
X
we have for all X, Y ∈ T (M ),
e (α) Ye ∈ T (M
f) ⇔ (∇(α) Y )∼ ∈ T (M
f) ⇔ ∇(α) Y ∈ T (M ),
∇
X
X
e
X
where the first equivalence follows from (2.6) and the second one is obvious. Therefore,
(i) and (ii) are equivalent.
Remark 1. The result is Theorem 2.10 in [1] when f (τ ) = τ .
Proposition 2. The components of the α-curvature tensor of Se are given by
2
e(α) = f (τ )R(α) − 1 − α f (τ )(gil gjk − gjl gik ),
R
ijkl
ijkl
4
(α)
(α)
e
e
R
=
R
=
0,
ijkτ
iτ kτ
where the components of α-curvature tensor of S satisfy
(α)
Rijkl =
1 − α2
(∂k ∂m ∂i ln Z∂j ∂l ∂n ln Z − ∂k ∂m ∂j ln Z∂i ∂l ∂n ln Z)g mn .
4
Proof. Since
fγ Γ
e(α) = (∂
e (α)ν − ∂f
e (α)ν
e (α) e (α)ω e (α) e (α)ω
R
β Γγλ )gνµ + (Γγωµ Γβλ − Γβωµ Γγλ ),
βγλµ
βγ
combining (2.3), (2.4) and (2.5), we obtain
2
e(α) = f (τ )R(α) + 1 − α f (τ )(gil gjk − gjl gik ),
e(α) = 0,
R
R
ijkl
ijkl
iτ kτ
4
e(α) = 1 + α ∂τ f (τ )(∂i gjk − ∂j gik + Γ(α) − Γ(α) ).
R
jkl
ikj
ijkτ
2
From (2.2), we get
(α)
Γijk (β) = Tr[ρ(∂i ∂j ln ρ)(∂k ln ρ)] +
1−α
1−α
Tr[ρ(∂i ln ρ)(∂j ln ρ)(∂k ln ρ)] =
∂i ∂j ∂k ln Z.
2
2
162
Shicheng Zhang, Huafei Sun
On the other hand, from (2.1), we get
∂k gij
=
=
=
∂k (Tr[ρ∂i ln ρ∂j ln ρ])
Tr[ρ∂k ln ρ∂i ln ρ∂j ln ρ] + Tr[ρ∂k ∂i ln ρ∂j ln ρ] + Tr[∂i ln ρρ∂k ∂j ln ρ]
∂i ∂j ∂k ln Z.
From above we can get
e(α) = 0
R
ijkτ
and
(α)
Rijkl =
1 − α2
(∂k ∂m ∂i ln Z∂j ∂l ∂n ln Z − ∂k ∂m ∂j ln Z∂i ∂l ∂n ln Z)g mn .
4
This finishes the proof of proposition 2.
Remark 2. Clearly manifold S and Se are ±1-flat manifolds.
By a direct calculation, we obtain the α-sectional curvatures and the α-scalar
curvature of Se
(2.9)
e (α) =
K
ijij
e(α)
R
1 − α2
ijij
(α)
= (f (τ ))−1 Kijij −
(f (τ ))−1 ,
geii gejj − geij geji
4
and
2
e(α) = R
e(α) geµν geλγ = f (τ )−1 R(α) + 1 − α f (τ )−1 (n − n2 ).
R
µνλγ
4
When n = 2, from (2.9), we see that the α-Gaussian curvature of Se satisfies
¯
¯
¯ ∂12 ln Z
∂12 ∂2 ln Z ¯¯
∂13 ln Z
−1
2
−1
2 ¯
f
(τ
)
(1
−
α
)
e (α) =
¯ ∂1 ∂2 ln Z ∂12 ∂2 ln Z ∂1 ∂22 ln Z ¯ − f (τ ) (1 − α ) .
K
¯
¯
4 det(G)2
4
¯ ∂22 ln Z
∂23 ln Z ¯
∂1 ∂22 ln Z
Remark 3. The result is the conclusion in [8] when α = 0.
Here, we will use an example to verify our conclusion above.
¤
£ P2
Example 1. Suppose that S = {ρ|ρ = Z −1 exp − i=1 βi Fi }, and the matrices
Fi are given by




0 0 0
1 −1 0
0 0 .
F1 =  1 1 0 , F2 =  0
1 1 0
−1 1 0
Taking f (τ ) = τ > 0, we get
2
£ X
¤
Se = {e
ρ|e
ρ = τ Z −1 exp −
βi Fi }.
i=1
An equilibrium density ρ ∈ S can be represented by

−β2
β2
1
−β1
ρ = exp  β1
Z
β1 + β2 −β1 − β2

0
0  ,
0
The information geometric descriptions ...
163
and the partition functions of ρ is Z = exp[−β1 ] + exp[−β2 ] + 1. Under the coordinate
system (β1 , β2 ), the geometric metrics of S are given by
G = (gij ) = (∂i ∂j ln Z),
det(G) = ∂12 ln Z∂22 ln Z − (∂1 ∂2 ln Z)2 =
and
(α)
R1212
¯
¯ ∂12 ln Z
1 − α ¯¯
∂1 ∂2 ln Z
=−
4 det(G) ¯¯
∂22 ln Z
2
=
(α)
K1212 =
exp 2(β1 + β2 )
,
(exp(β1 ) + exp(β2 ) + exp(β1 + β2 ))3
∂13 ln Z
∂12 ∂2 ln Z
∂1 ∂22 ln Z
¯
∂12 ∂2 ln Z ¯¯
∂1 ∂22 ln Z ¯¯ =
∂23 ln Z ¯
(1 − α2 ) exp 2(β1 + β2 )
,
4(exp(β1 ) + exp(β2 ) + exp(β1 + β2 ))3
(α)
R1212
1 − α2
=
,
det(G)
4
R(α) =
2
1 − α2
(α)
R1212 =
.
det(G)
2
Under the coordinate system (β1 , β2 , τ ), we obtain the geometric metrics of Se
corresponding to those of S:


µ
¶
−τ ∂1 ∂2 ln Z
0
−τ ∂12 ln Z
τG
0
e =  −τ ∂1 ∂2 ln Z
0 =
−τ ∂22 ln Z
,
G
0 τ −1
0
0
τ −1
and
2
e(α) = τ R(α) − 1 − α τ det(G) = 0,
R
1212
1212
4
(α)
(α)
(α)
e
e
e
e(α) = R
e(α) = 0,
R1τ 1τ = R121τ = R212τ = R
1τ 2τ
2τ 2τ
2
e (α) = τ −1 K (α) − 1 − α τ −1 = 0.
K
1212
1212
4
The scalar curvature of Se satisfies
¯
¯
¯ ∂12 ln Z
∂12 ∂2 ln Z ¯¯
∂13 ln Z
−1
2
−1
2 ¯
τ
(1
−
α
)
e(α) =
¯ ∂1 ∂2 ln Z ∂12 ∂2 ln Z ∂1 ∂22 ln Z ¯ − τ (1 − α ) = 0.
R
¯
2 det(G)2 ¯¯
2
∂23 ln Z ¯
∂1 ∂22 ln Z
∂22 ln Z
3
The approximation of the equilibrium density
In this section, we firstly define a Kullback-Leibler divergence, which is different from
the general distance of two points in the manifold.
e the Kullback-Leibler
Definition 1. Let ρe and σ
e be two points in manifold S,
divergence between two states is defined by
(3.1)
D(e
ρ|e
σ ) = Tr[e
ρ(ln ρe − ln σ
e)].
So, we have the relation of the Kullback-Leibler divergences between S and Se
(3.2)
D(e
ρ|e
σ ) = f (τ )D(ρ|σ).
164
Shicheng Zhang, Huafei Sun
Definition 2. Let
½
f=
M
(3.3)
σ
e|e
σ=
f (τ )Zσ−1
exp{−
r
X
¾
θj E j }
j=1
f is a submanifold
be an r-dimensional manifold, where r ≤ n and Ej is related to Fi . M
e
of S.
f is a ±1-flat manifold, where θ = (θ1 , · · · , θr ) is the e–affine
Clearly submanifold M
f
coordinate system of M .
e the projection onto M
f is defined by
Definition 3. For ρe ∈ S,
Y
f∗ =
(3.4)
σ
ρe = arg min D(e
ρ|e
σ ).
e
ρ
e∈S
So we have the following proposition.
f is σ ∗ .
Proposition 3. Suppose that the projection from a point ρe in Se onto M
∗
Then σ can be considered as a optimal approximation of the equilibrium density ρe in
f, and σ ∗ is a solution of the following differential equation
M
(3.5)
−
∂
ln Zσ = Tr[ρEi ].
∂θi
Proof. From (3.1) and (3.2) we have
D(e
ρ|e
σ) =
Tr[e
ρ(ln ρe − ln σ
e)] = f (τ )D(ρ|σ)
n
r
X
X
= f (τ ) Tr[ρ(−
βi Fi − ln Zρ +
θj Ej + ln Zσ )]
i=1
j=1
½ X
¾
n
r
X
f (τ ) −
βi Tr[ρFi ] − ln Zρ +
θj Tr[ρEj ] + ln Zσ .
=
i=1
j=1
Since
∂D(e
ρ|e
σ (θ, τ ))
∂
∂
= −f (τ )
Tr[ρ ln σ] = f (τ )(Tr[ρEi ] +
ln Zσ ) = 0,
∂θi
∂θi
∂θi
we obtain (3.5). This finishes the proof of Proposition 3.
Example 2. Let S be a 3-dimensional manifold and take
½
Se =
ρe = f (τ )Zρ−1 exp −
3
X
¾
βi Fi
½
=
¾
ρe = f (τ )ρ ,
i=1
where β = (β1 , β2 , β3 )
given by

1
F1 =  0
0
is the e-affine coordinate system of S, and the matrixes Fi are
0
0
0


0
0
0 , F2 =  0
0
0


0 0
0
1 0 , F 3 =  0
0 0
0
0
0
0

0
0 .
1
The information geometric descriptions ...
165
f of Se satisfies
Then submanifold M
½
f=
M
σ
e = f (τ )Zσ−1 exp −
2
X
¾
θj Ej
½
=
¾
σ
e = f (τ )σ ,
j=1
where θ = (θ1 , θ2 ) is the e-affine
given by

1
E1 =  0
0
coordinate system of M , and the matrixes Ej are
0
1
0


0 0
0
0  , E2 =  0 0
0 0
0

0
0 .
1
So the equilibrium density ρe and σ
e can be written as


exp{−β1 }
0
0
f (τ ) 
,
0
exp{−β2 }
0
ρe =
Zρ (β)
0
0
exp{−β3 }
and


exp{−θ1 }
0
0
f (τ ) 
,
0
exp{−θ1 }
0
σ
e=
Zσ (θ)
0
0
exp{−θ2 }
respectively.
From (3.5), we have
2 exp{−θ1 }
exp{−β1 } + exp{−β2 }
=
,
2 exp{−θ1 } + exp{−θ2 }
exp{−β1 } + exp{−β2 } + exp{−β3 }
exp{−θ2 }
exp{−β3 }
=
.
2 exp{−θ1 } + exp{−θ2 }
exp{−β1 } + exp{−β2 } + exp{−β3 }
So we obtain the coordinates θi of the optimal approximation σ ∗ of ρe as
θ1∗ = − ln
4
exp{−β1 } + exp{−β2 }
, θ2∗ = β3 .
2
Conclusions
In this paper, we investigate the structures of the state space of the thermodynamic
parameters based upon the information geometric approach to the denormalization
Se of S. By calculating the covariant derivative and the α-curvature tensor, we obe At last, we study
tain the relation of the α-geometric structures between S and S.
the approximation of the equilibrium density and use two examples to illustrate our
conclusions.
Acknowledgement. This subject is supported by the National Natural Science
Foundation of China (No. 10871218).
166
Shicheng Zhang, Huafei Sun
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Authors’ addresses:
Shicheng Zhang
Department of Mathematics, Beijing Institute of Technology,
Beijing, 100081 China.
School of Mathematics, Xuzhou Normal University,
Xuzhou, Jiangsu 221116, China.
E-mail: zhangshicheng@126.com
Huafei Sun
Department of Mathematics, Beijing Institute of Technology, Beijing, 100081 China
E-mail: huaf.sun@gmail.com