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Riemannian geometry of fluctuation theory: An introduction
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2016 J Phys.: Conf Ser 720 012005
(http://iopscience.iop.org/1742-6596/720/1/012005)
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Trang 2Riemannian geometry of fluctuation theory:
An introduction
Luisberis Velazquez
Departamento de F´ısica, Universidad Cat´ olica del Norte, Av Angamos 0610, Antofagasta, Chile
E-mail: lvelazquez@ucn.cl
Abstract. Fluctuation geometry was recently proposed as a counterpart approach of
Riemannian geometry of inference theory (information geometry ), which describes the
geometric features of the statistical manifoldM of random events that are described
by a family of continuous distributions dpξ(x|θ) This theory states a connection
among geometry notions and statistical properties: separation distance as a measure
of relative probabilities, curvature as a measure about the existence of irreducible statistical correlations, among others In statistical mechanics, fluctuation geometry arises as the mathematical apparatus of a Riemannian extension of Einstein fluctuation theory, which is also closely related to Ruppeiner geometry of thermodynamics Moreover, the curvature tensor allows to express some asymptotic formulae that account for the system fluctuating behavior beyond the gaussian approximation, while curvature scalar appears as a second-order correction of Legendre transformation between thermodynamic potentials.
1 Introduction
Riemannian geometries defined on statistical manifolds establish a direct correspondence among statistical properties of a parametric family of continuous distributions:
and geometrical notions of certain statistical manifolds M and P associated to them.
The advantage of these formalisms is that they enable a direct application of powerful tools of Riemannian geometry for statistical analysis There exist two possible Riemannian geometries in the framework of continuous distribution (1) The first one
is Riemannian geometry of inference theory, which is widely known as information
geometry [1] Distance notion of this geometry:
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Published under licence by IOP Publishing Ltd 1
Trang 3Figure 1. Continuous distributions dp ξ (x |θ) and dpˇ(ˇx |θ) are diffeomorphic
distributions, that is, a same abstract distribution dp(ϵ |E) expressed into two different
coordinate representations of the abstract statistical manifoldM.
establishes a statistical separation between two close distributions of parametric family
(1), which characterizes distinguishing probability of these distribution during an
statistical inference of control parameters (θ, θ + dθ) The second one is Riemannian geometry of fluctuation theory, or more briefly, fluctuation geometry [2, 3, 4] Its distance
notion:
ds2= g ij (x |θ)dx i dx j (3)
establishes a statistical separation between two close values (x, x + dx) of a random quantity ξ for a given member of parametric family (1).
A great advantage of differential geometry is the possibility to perform a
coordinate-free treatment An important concept here is the notion of diffeomorphic distributions
[4] There are those distributions whose random quantities ξ and ζ are related by a diffeomorphism ϕ : ξ → ˇξ, that is, a bijective and differentiable map that leaves invariant
their respective probability distributions (see scheme in Fig.1):
ϕ : dp ξ (x |θ) = dpˇ(ˇx |θ) ⇒ ρˇ(ˇx |θ) = ρ ξ (x |θ)
∂ ˇ ∂x x −1 (4)
All these distributions are regarded as different representations of a same abstract
distribution defined on the manifolds M y P.
A simple example of transformation among random quantities is the one associated
with Box-Muller transformation [5]:
ζ1=√
−2 ln (ξ1) cos(2πξ2) and ζ2 =√
−2 ln (ξ1) sin(2πξ2) (5)
which is employed to generate Gaussian random numbers ζ1 and ζ2 from uniform
random numbers ξ1 and ξ2 Continuous distribucions whose associated manifoldsM are
Trang 4diffeomorphic to the real one-dimensional spaceR are always diffeomorphic distributions because of the only possible Riemannian geometry for these manifolds is the Euclidean
one In particular, Gaussian distribution:
dp ξ (x |µ, σ) = √1
2πσexp
[
−(x − µ)2/2σ2]
dx, −∞ < x < +∞. (6)
Cauchy distribution:
dpˇ(ˇx |ν, γ) = 1
π
γdˇ x
γ2+ (ˇx − ν)2, −∞ < ˇx < +∞. (7)
Bimodal Gaussian distribution:
dp˜(˜x |µ, σ) = 1
2√
2πσ
{ exp[
−(˜x − µ)2/2σ2]
+ exp[
−(˜x + µ)2/2σ2]}
d˜ x, −∞ < ˜x < +∞.
(8) are fully equivalent from this geometric perspective, namely, all they can be regarded as different representations of a same abstract distribution Of course, not all distributions
can be regarded as diffeomorphic distributions For random quantities ξ whose abstract
statistical manifoldM has a dimension n ≥ 2 are possible the notions of curvature and
statistical correlations In particular, distributions family [4]:
dp ξ (x, y |θ) = Z (θ)1 exp
[
−1
2(x
2+ y2)
]
θdxdy
2π√
x2+ y2+ θ2 (9) with normalization constant:
Z (θ) = √ πe1θ2√ θ
2erfc
(
θ
√
2
)
(10)
can be associated with curved geometry of surface of revolution represented in Fig.2 This family cannot be map to the product of two Gaussian distributions:
dp ζ (x, y |σ) = 1
2πσexp
[
−(x2+ y2)
/2σ]
because of this last has Euclidean geometry of two-dimensional real space R2 Geometrical non-equivalence means that distributions (9) cannot be decomposed into the product of two independent distributions
2 Fundamental equations and results of fluctuation geometry
For the sake of simplicity in notations, let us hereinafter omit the subindex of random
quantity ξ in all mathematical expressions. Riemannian structure of the statistical manifold M allows us to introduce the invariant volume element dµ(x|θ):
dµ(x |θ) =√|g ij (x |θ)/2π|dx, (12)
3
Trang 5R3
0 -θ
dz dr
dt z
t θ
Figure 2 The geometry of the statistical manifoldM associated with the distributions
family (9) is fully equivalent to curved geometry defined on the revolution surface represented here
which replaces the ordinary volume element dx (Lebesgue measure) that is employed
in equation (1) The notation |T ij | represents the determinant of a given tensor T ij of
second-rank, while the factor 2π has been introduced for convenience Additionally, one can define the probabilistic weight [3]:
ω(x |θ) = ρ(x|θ)√|2πg ij (x |θ)|, (13)
which is a scalar function that arises as a local invariant measure of the probability Although the mathematical form of the probabilistic weight ω(x |θ) depends on the
coordinates representations of the statistical manifolds M and P; the values of this
function are the same in all coordinate representations Using the above notions, the family of continuous distributions (1) can be rewritten as follows:
which is a form that explicitly exhibits the invariance of this family of distributions
The notion of probability weight ω(x |θ) can be employed to redefine the notion of
information entropy for continuous distributions [6]:
S d [ω |g, M] = −
∫
M ω(x|θ) log ω(x|θ)dµ(x|θ). (15)
as a global invariant measure that depends on the metric tensor g ij (x |θ) of the manifold
M The quantity I(x|θ):
Trang 6represents a local invariant measurement of the information content, where differential
entropy (15) exhibits the same value for all diffeomorphic distributions Introducing the
information potential S(x|θ) as the negative of the information content (16):
S(x|θ) = log ω(x|θ) ≡ −I(x|θ), (17) the metric tensor can be rewritten as follows [3]:
g ij (x |θ) = −D i D j S(x|θ) = − ∂2S(x|θ)
∂x i ∂x j + Γk ij (x |θ) ∂S(x|θ)
where D i is the covariant derivative associated with the Levi-Civita affine connections
Γk ij (x |θ) [10]:
Γk ij (x |θ) = g km (x |θ)1
2
[
∂g im (x |θ)
∂x j +∂g jm (x |θ)
∂x i − ∂g ij (x |θ)
∂x m
]
Covariant set of differential equations (18) can be rewritten into the alternative form:
g ij (x |θ) = − ∂2log ρ(x |θ)
∂x i ∂x j + Γk ij (x |θ) ∂ log ρ(x |θ)
∂x k +∂Γ
k
jk (x |θ)
∂x i − Γ k
ij (x |θ)Γ l
kl (x |θ) (20)
in terms of probability density According to expression (18), the metric tensor g ij (x |θ)
defines a positive definite distance notion (3), while the information potentialS(x|θ) is
locally concave everywhere This last behavior guarantees the uniqueness of the point
¯
x where the information potential reaches a global maximum, that is, the uniqueness of
the point of global maximum ¯x of the probabilistic weight ω(x |θ).
The main consequence derived from equation (18) is the possibility to rewrite the
distributions family (14) into the following Riemannian gaussian representation [2, 3]:
dp(x |θ) = Z(θ)1 exp
[
−1
2ℓ
2
θ (x, ¯ x)
]
where ℓ θ (x, ¯ x) denotes the separation distance between the arbitrary point x and the
point ¯x with maximum information potential S(x|θ) (the arc-length ∆s of the geodesics
that connects these points) Moreover, the negative of the logarithm of gaussian partition functionZ(θ) defines the so-called gaussian potential:
which appears as the first integral of the problem (18):
P(θ) = S(x|θ) +1
2ψ
2(x |θ). (23)
Here, ψ2(x |θ) = ψ i (x |θ) ψ i (x |θ) = g ij (x |θ)ψ i (x |θ) ψ j (x |θ) is the square norm of
covariant vector field ψ i (x |θ) defined by the gradient of the information potential S (x|θ):
ψ i (x |θ) = −D i S (x|θ) ≡ −∂S (x|θ) /∂x i (24)
5
Trang 7The factor 2π of definition (12) guarantees that the gaussian partition function Z(θ)
drops the unity when the Riemannian structure of the manifold M is the same of
Euclidean real spaceRn
Riemannian gaussian representation (46) rephrases the distributions family (1) in term of geometric notions of the manifold M According to this result, the distance
ℓ θ (x, ¯ x) is a measure of the occurrence probability of a deviation from the state ¯ x with
maximum information potential This result can be obtained combining equations (14) and (23) with the following the identity:
ψ2(x |θ) ≡ ℓ2
This last relation is a consequence of the geodesic character of the curves x g (s) ∈ M
derived from the following set of ordinary differential equations [3]:
dx i g (s)
ds = υ
Here, υ i (x |θ) = g ij (x |θ)υ j (x |θ) is the contravariant form of the unitary vector field
υ i (x |θ) associated with the vector field (24):
υ i (x |θ) = ψ i (x |θ) /ψ (x|θ) , (27)
while the parameter s is the arc-length of the curve x g (s) It is easy to check that this
unitary vector field obeys the geodesic differential equation:
υ j (x |θ)D j υ i (x |θ) = υ j (x |θ) [g ij (x |θ) − υ i (x |θ)υ j (x |θ)] ≡ 0. (28) Identity (25) follows from the directional derivatives:
d S (x g (s) |θ)
ds ≡ ψ(x g (s) |θ) and d2S (x g (s) |θ)
which can be obtained from equation (26)
Let us now talk about the notion of curvature of fluctuation geometry The affine connections Γk ij = Γk ij (x |θ) are employed to introduce of the curvature tensor
R l
ijk = R l
ijk (x |θ) of the manifold M:
R ijk l = ∂
∂X iΓl jk − ∂
∂X jΓl ik+ Γl imΓm jk − Γ l
Generally, the affine connections Γk ij (x |θ) and the metric tensor g ij (x |θ) are independent
entities of Riemannian geometry However, the knowledge of the metric tensor allows to introduce natural affine connections: the Levi-Civita connections (19) These affine connections are also referred to in the literature as the metric connections or the Christoffel symbols The same ones follow from the consideration of a torsion-free
covariant differentiation D i that obeys the condition of Levi-Civita parallelism [10]:
Trang 8Figure 3. Curvature characterizes the deviation of local geometric properties of a manifold from the properties of the Euclidean geometry
Using the Levi-Civita connections, the curvature tensor can be expressed in terms of the
metric tensor g ij (x |θ) and its first and second partial derivatives Additionally, one can
introduce the Ricci curvature tensor R ij (x |θ):
R ij (x |θ) = R k
as well as the curvature scalar R(x |θ):
R(x |θ) = g ij (x |θ)R k
kij (x |θ) = g ij (x |θ)g kl (x |θ)R kijl (x |θ). (33)
According to Riemannian geometry [10], the curvature scalar R(x |θ) is the only invariant
derived from the first and second partial derivatives of the metric tensor g ij (x |θ).
The curvature tensor characterizes the deviation of local geometric properties of a manifold M from the properties of the Euclidean geometry (see scheme of Fig.3) For
example, the volume of a small sphere about a point x has smaller (larger) volume (area) than a sphere of the same radius defined on the n-dimensional real space Rn when the
scalar curvature R(x |θ) is positive (negative) at that point Quantitatively, this behavior
is described by the following approximation formulae:
Vol[
S(n −1) (x |ℓ) ⊂ M]
Vol[
S(n −1) (x |ℓ) ⊂ R n ] = 1 − R(x |θ)
6(n + 2) ℓ
Area[
S(n −1) (x |ℓ) ⊂ M]
Area[
S(n −1) (x |ℓ) ⊂ R n ] = 1 − R(x |θ)
6n ℓ
where the notation S(m) (x |ℓ) represents a m-dimensional sphere with small radius ℓ
centered at the point x Accordingly, the local effects associated with the curvature
of the manifold M appears as second-order (and higher) corrections of the Euclidean
formulae The corresponding asymptotic formula for distribution (46) using spherical
coordinates (ℓ, q) for radius ℓ sufficiently small:
dp(ℓ, q |θ) = Z(θ)1
[
1− 1
24ℓ
2F(q|θ) + O(ℓ4)
]
dp G (ℓ, q |θ). (36)
7
Trang 9Figure 4 Schematic representation about the Cartesian product of manifolds.
Here, dp G (ℓ, q |θ) denotes the spherical coordinate representation of a gaussian
distribution associated with the local Euclidean properties of the manifold M at the
point ¯x with maximum information potential:
dp G (ℓ, q |θ) = exp
(
−1
2ℓ
2
)
ℓ n −1 dℓ
√
2π
√
κ αβ 2π (q)
where κ αβ (q) = ¯ g ij ξ α i (q)ξ β j (q) The (n − 1) vector fields ξ α (q) = {
ξ i α (q)}
are obtained
from the unitary vector field e(q) ={
e i (q)} associated with the spherical coordinates at the point ¯x as follows:
ξ α i (q) = ∂e
i (q)
F(q|θ) is a function on the spherical coordinates q defined as follows:
F(q|θ) = ¯ R ijkl κ αβ (q)S α ij (q)S kl β (q), (39)
which is referred to as the spherical function. Moreover, ¯R ijkl = R ijkl(¯x |θ) is the
curvature tensor evaluated at the point ¯x, while the quantities S α ij (q) are defined as:
S α ij (q) = e i (q)ξ α j (q) − e j (q)ξ i α (q). (40) Curvature of statistical manifold M is directly related to the notion of irreducible
statistical correlations Specifically, it is said that a continuous distribution dp(x |θ)
exhibits a reducible statistical dependence if it possesses a diffeomorphic distribution
dp(ˇ x |θ) that admits to be decomposed into independent distribution functions dp (i)(ˇx i |θ)
for each coordinate as follows:
dp(ˇ x |θ) =
n
∏
i=1
Otherwise, the distribution function dp(x |θ) exhibits an irreducible statistical dependence The existence (or nonexistence) of a reducible statistical dependence for
a given distributions family (1) is fully equivalent to the existence (or nonexistence) of
a Cartesian decomposition of its associated statistical manifold M into two (or more)
independent statistical manifolds
{
A (i)
θ
} :
M = A(1)⊗ A(2) ⊗ A (l) (42)
Trang 10A given manifold A is said to be an irreducible manifold when the same one does not
admit the Cartesian decomposition (42) Moreover, a given Cartesian decomposition (42)
is said to be an irreducible Cartesian decomposition if each independent manifold A (k)is
an irreducible manifold In general, the question about the Cartesian decomposition of
a Riemannian manifold into independent manifolds with arbitrary dimensions is better
phrased and understood in the language of holonomy groups The relation of holonomy
of a connection with the curvature tensor is the main content of Ambrose-Singer theorem, while de Rham theorem states the conditions for a global Cartesian decomposition [10].
The flat character of the statistical manifold M implies the existence of a reducible
statistical dependence for the family of distributions (1), while its curved character implies the existence of an irreducible statistical dependence
3 Relevance in statistical mechanics
Redefining information potential (17) in units of Boltzmann constant k, the probability
distribution (14) can be rewritten as follow:
dp(x|θ) = exp [S(x|θ)/k] dµ(x|θ). (43)
Formally, this expression represents a sort of covariant extension of Einstein postulate
of classical fluctuation theory [11], where the information potential S(x|θ) is identified
with the thermodynamic entropy of closed system (up to the precision of an additive constant) Hereinafter, the coordinates x = (x1, x2, , x n) are the relevant macroscopic
observables of the closed system, e.g.: the internal energy U , the volume V , the total
angular momentum M, the magnetization M, etc Moreover, θ represents the set of
control parameters of the given situation of thermodynamic equilibrium The metric
tensor g ij (x |θ) of fluctuation geometry:
g ij (x |θ) = −D i D j S(x|θ) = −∂ i ∂ i S(x|θ) + Γ k
ij (x |θ)∂ k S(x|θ) (44) establishes a constraint between the entropy S(x|θ) and the metric tensor g ij (x |θ) of
the statistical manifold M of macroscopic observables x Expression (44) provides a
generalization for the thermodynamic metric tensor of Ruppeiner geometry [7, 8]:
g ij(¯x) = − ∂2S(¯ x |θ)
while the Riemannian Gaussian representation:
dp(x |θ) = Z(θ)1 exp
[
− 1
2k ℓ
2
θ (x, ¯ x)
]
is an exact improvement of Gaussian approximation of classical fluctuation theory [11]:
dp(x|θ) ≃ exp[−g ij(¯x)(x − ¯x) i (x − ¯x) j /2k] √
|g ij(¯x|θ)/2πk|d n x. (47) According to the asymptotic formula (36), curvature characterizes deviation exact distribution beyond Gaussian approximation for thermodynamical fluctuations
9
... Z(θ)drops the unity when the Riemannian structure of the manifold M is the same of< /i>
Euclidean real spaceRn
Riemannian gaussian representation (46) rephrases... (k)is
an irreducible manifold In general, the question about the Cartesian decomposition of
a Riemannian manifold into independent manifolds with arbitrary dimensions... always diffeomorphic distributions because of the only possible Riemannian geometry for these manifolds is the Euclidean
one In particular, Gaussian distribution:
dp ξ