Entropic dynamical models with unstable Jacobi fields

ENTROPIC DYNAMICAL MODELS WITH UNSTABLE JACOBI FIELDS C. LI1 , L. PENG2 , H. SUN1∗ 1

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China E-mail: [email protected] 2 Department of Applied Mechanics and Aerospace Engineering, Waseda University, Tokyo 169-8555, Japan E-mail: [email protected] Received April 2, 2015

The instability of an entropic dynamical model is considered via Jacobi vector field and the Lyapunov exponent. From the viewpoint of information geometry, geometric structure of the statistical manifold underlying this model is investigated, and we conclude that it is a manifold with constant negative scalar curvature. By use of the Jacobi vector field associated with the geodesics, we study the asymptotic behavior of the geodesic spread on the statistical manifold and reach that it is described by an exponentially divergent Jacobi vector field with respect to time. A positive Lyapunov exponent is also obtained, that explains the local instability of the system as well. Furthermore, submanifolds are studied similarly. Key words: information geometry, entropic dynamics, Jacobi field, instability. PACS: 02.50.Tt, 02.50.Cw, 02.40.-k, 05.45.-a.

1. INTRODUCTION

During the last decades, much attention has been paid on the study of probability models and statistical inference via the methodology of differential geometry. The theory of information geometry is hence developed to describe global properties of sets of probability density functions, i.e. statistical manifolds [1–5]. In light of the information geometry theory, dual geometric structure of a statistical manifold {p(x; θ)} (whose definition is given in subsection (1.1)) can be investigated by considering the Fisher information matrix as its Riemannian metric and dual connections can be introduced. Entropic dynamics (ED) is a theoretical framework constructed on statistical manifolds to explore the possibility of laws of physics - either classical or quantum - might reflect laws of inference rather than laws of nature [6]. It is a combination of inductive inference (maximum relative entropy methods [7–10]) and information geometry. Rather than on an ordinary linear space, dynamics ED is defined on a ∗

Email: [email protected]. (Corresponding author)

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space of probability distributions M , and standard coordinates of the system are replaced by statistical macrovariables [11]. Most intriguing questions pursued in ED include from the possibility of deriving dynamics to purely entropic arguments [12]. In [13], authors explored the possibility of using well established principles of inference to derive Newtonian dynamics from relevant prior information related with an appropriate statistical manifold. Assume there is an irreducible uncertainty in the location of particles so that the state of a particle is defined by a probability distribution. The corresponding configuration space is hence a statistical manifold. The trajectory follows from a principle of the method of maximum entropy. There is no need for additional “physical” postulates. ED reproduces Newton’s mechanics for any number of particles interacting among themselves and with external fields. Both masses of the particles and their interactions are explained as a consequence of the underlying statistical manifold [13]. Moreover, ED is used to explore the possibility of constructing a unified characterization of classical and quantum chaos [15]. Information geometric techniques and inductive inference methods hold great promise for solving computational problems of interest in classical and quantum physics with regard to complexity characterization of dynamical systems in terms of their probability description on curved statistical manifolds [14]. Recently, researchers have devoted to study the chaotic dynamics on curved statistical manifolds based on ED models and obtained fruitful results (see for example [14, 15]). Special focus is devoted to the description of sectional curvature, the Jacobi field intensity and the information geometrodynamical entropy. These quantities serve as powerful information geometric complexity measures for information-constrained dynamics associated with arbitrary chaotic and regular systems defined on statistical manifolds [14]. It is also known that classical complex systems exhibit local exponential instability and are characterized by positive Lyapunov exponents [15–17]. In particular, stability of Jacobi vector fields on manifolds associated with ED models is attracting more and more attention [18–23] as it behaves as a proper indicator of chaoticity [15]. In the present paper, we make reliable macroscopic predictions under the condition that only partial knowledge on the micro-structure of an ED model is available. Complexity of such predictions is quantified in terms of several quantities such as the Jacobi field intensity, on which we will mainly be focused. Here, we consider an ED model whose microstates span on a 3-dimensional space labeled by the variables (x1 , x2 , x3 ) ∈ R+ × R+ × R. Information geometric structures of the relative statistical manifold itself together with its submanifolds are investigated. Instability behaviors of their geodesic spreads are described by considering (the general or approximate) solutions of the Jacobi-Levi-Civita equations. RJP 60(Nos. 9-10), 1249–1262 (2015) (c) 2015 - v.1.3a*2015.11.20

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1.1. THEORY OF INFORMATION GEOMETRY

Definition 1.1 The set M = {p(x; θ)|x ∈ Ω ⊂ Rm , θ ∈ Rn } is called an n-dimensional statistical manifold, whose elements are probability density functions p(x; θ), where θ = (θ1 , θ2 , . . . , θn ) denotes the vector of parameters and can also be considered as a canonical coordinate system of M . Under different situations, other coordinate systems may be introduced. For example, for exponential families given by  p(x; θ) = exp θi hi (x) − φ(θ) + f (x) , (1) with hi , φ and f smooth functions, one may choose the expectation coordinates ηi = ∂i φ(θ) rather than the canonical ones θ. We write ∂i = ∂θ∂ i and the Einstein summation convention is applied here and henceforth. Usually φ(θ) is called the potential function. Definition 1.2 The Fisher information metric (or matrix) (gij ) defined on M is given by components as gij = Eθ [∂i log p(x; θ) ∂j log p(x; θ)],

(2)

where log is the natural logarithm function and Eθ denotes the expectation under coordinate system θ. Its inverse is denoted as (g ij ) = (gij )−1 . Therefore, (M, (gij )) is a Riemannian manifold with a Riemannian metric (gij ). Definition 1.3 The Riemannian connection ∇ with respect to the Riemannian metric (gij ) satisfies Zg(X, Y ) = g(∇Z X, Y ) + g(X, ∇Z Y ), and ∇X Y − ∇Y X = 0, where X,Y,Z are over M and the coefficients of ∇ satisfy 1 Γijk = (∂i gjk + ∂j gki − ∂k gij ), (3) 2 where Γkij = Γijs g sk . (4) Definition 1.4 The components of the Riemannian curvature tensor and Ricci curvature tensor are respectively given by Rijkl = (∂j Γsik − ∂i Γsjk )gsl + (Γjtl Γtik − Γitl Γtjk )

(5)

and Rik = Rijkl g jl . RJP 60(Nos. 9-10), 1249–1262 (2015) (c) 2015 - v.1.3a*2015.11.20

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Definition 1.5 The scalar curvature is defined as the trace of the Ricci curvature tensor. In components, it has expression R = Rik g ik .

(7)

1.2. JACOBI VECTOR FIELDS OF ED MODELS

ED models are fairly useful in inferring “macroscopic predictions” in the absence of detailed knowledge of the microscopic nature of an arbitrary complex system. More explicitly, by “macroscopic prediction” one can get the knowledge of the statistical parameters of the probability distribution function that best reflects what is known about the system. The probability distribution reflects the system in general, not the microstate. This means that one has to select the macrospace of the system manually [14, 26]. To construct an ED model, one needs to identify the associated parameter variables which describe the system and thus the corresponding space of microstates. It is a type of information-constrained dynamics built on curved statistical manifolds M , elements of which are probability distributions p(x; θ) that are in a one-to-one relation with a suitable set of macroscopic statistical variables θ. Such variables θ provide a convenient parametrization of points on M . The set {θ} is called the coordinate parameter of the model. Information geometry provides an appealing tool to illustrate the Riemannian (even dual) geometric structure of the parametric statistical manifolds underlying the ED by considering the Fisher information metric as a Riemannian metric. Hence, the information geometry theory introduced above is suitable for ED models as well. ˙ is Definition 1.6 A curve ξ(t) on M is said to be geodesic if its tangent vector ξ(t) displaced parallel along the curve ξ(t) itself, that is, the geodesic equation ˙ =0 ∇ ˙ ξ(t) (8) ξ(t)

holds. Locally, if the curve is coordinated as ξ(t) = (θ1 (t), θ2 (t), . . . , θn (t)), it can be represented as i j d2 θk k dθ dθ + Γ = 0. (9) ij d t2 dt dt As explained above, the instability (or stability) of the geodesics is completely determined by the curvature. Studying the stability of the dynamics means determining the evolution of perturbations of geodesics. Definition 1.7 The Jacobi vector field J = (δθ1 , δθ2 , . . . , δθn ), that is, the evolution perturbation vector, satisfies the Jacobi-Levi-Civita equations[19, 24, 25] (Some scholars call them the geodesic derivation equations) as follows ∇2 (δθi ) d θk d θl m i + R δθ = 0. kml d t2 dt dt RJP 60(Nos. 9-10), 1249–1262 (2015) (c) 2015 - v.1.3a*2015.11.20

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Here we denote  ∂θi (t; α) δα, (11) ∂α where θi are the solutions of (9), α is a vector of integration constant and the covari2 i) ant derivatives ∇ d(δθ [4, 19, 24] satisfy t2 J i = δθi = δα θi :=



2 k k j ∇2 (δθi ) d2 (δθi ) i d θ d(δθ ) i d θ = + 2Γ δθj + Γ jk jk d t2 d t2 dt dt d t2 (12) s k ∂Γijk d θl d θk j j dθ dθ i l + δθ + Γjk Γls δθ . dt dt ∂θl d t d t Usually for the sake of simplicity, we will introduce some extra constraints on α. The following equality holds between two kinds of the Riemannian curvature tensor representations h ghj . (13) Rijkl = Rikl

2. THE STATISTICAL MANIFOLD M AND THE JACOBI VECTOR FIELD

In this section, we consider an ED model over a 3-dimensional space, with x = (x1 , x2 , x3 ) labeling the space of microstates of the model. Assume all the information relevant to the dynamical evolution of the model can be obtained from the parameters of the probability distribution. For this reason, no other information is required. Each macrostate may be thought as a point of a 4-dimensional statistical manifold with coordinates given by the numerical values of the expectation values < xi > (i=1,2,3) and 4x3 . In other words, the associated coordinate  the variance 1 can be denoted by θ = (θ =< x1 >, θ2 =< x2 >, θ3 =< x3 >, θ4 = 4x3 ) . The joint distribution of two Gamma distributions and one normal distribution is studied, and all the density functions consist of the statistical manifold M = {p(x; θ)} , whose elements are given by


p(x; θ) = p1 x1 ; θ1 p2 x2 ; θ2 p3 x3 ; θ3 , θ4 , where n α o 1  α1 α1 α1 −1 1 x exp − x , 1 1 Γ(α1 ) θ1 θ1 n α o
1  α2 α2 α2 −1 2 p2 x2 ; θ2 = x exp − x 2 2 , Γ(α2 ) θ2 θ2  
1 (x3 − θ3 )2 3 4 p3 x3 ; θ , θ = p exp − . 2(θ4 )2 2π(θ4 )2 RJP 60(Nos. 9-10), 1249–1262 (2015) (c) 2015 - v.1.3a*2015.11.20
p 1 x1 ; θ 1 =

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x ∈ R+ × R+ × R, α1 , α2 > 0, and the Gamma function is Γ(αi ) = RObviously ∞ αi −1 −x e d x. 0 x Thus, the available information can be written in the form of the following information constraint equations Z ∞
x1 p 1 x1 ; θ 1 d x1 , µ1 =< x1 >= Z0 ∞
x2 p 2 x2 ; θ 2 d x2 , µ2 =< x2 >= (15) 0 Z +∞
x3 p3 x3 ; θ3 , θ4 d x3 , µ3 =< x3 >= −∞

and Z

 12

+∞ 2

σ = 4x3 =

3

(x3 − < x3 >) p3 x3 ; θ , θ

4




d x3

.

(16)

−∞

The probability distributions pi in (15) are constrained by the conditions of normalization Z ∞
p1 x1 ; θ1 d x1 = 1, Z0 ∞
p2 x2 ; θ2 d x2 = 1, 0 Z +∞
p3 x3 ; θ3 , θ4 d x3 = 1. −∞

Remark 2.1 Note that we have assumed that information about correlations between the microvariables are not to be considered. The assumption leads to the simplified product rule (14). While the emergence of a correlational structure in the form of constraints among the variables labeling the macrostates of a system would lead to a highly constrained dynamics and to a reduction in the complexity of realising macroscopic predictions [12]. The process of assigning a probability distribution to each state provides M with a metric structure. By a direct calculation, we get Proposition 2.1 The Fisher metric (gij ) on M is  α1  0 0 0 µ1 2 α2  0 0 0  µ2 2 , (gij ) =  (17)  0 0 σ12 0  0 0 0 σ22 RJP 60(Nos. 9-10), 1249–1262 (2015) (c) 2015 - v.1.3a*2015.11.20

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and its inverse is 

µ1 2 α1

  (g ij ) =  0  0 0

0 µ2 2 α2

0 0



0

0

0 σ2 0

 0 .  0 

(18)

σ2 2

From (4), we get Proposition 2.2 The non-vanishing connection coefficients (of the second kind) are Γ111 = −

1 , µ1

Γ222 = −

1 , µ2

Γ433 =

1 , 2σ

(19) 1 1 4 = = − , Γ44 = − . σ σ Therefore, using (5) and (6), the following proposition is obtained Proposition 2.3 The independent non-vanishing components of Riemannian curvature tensor and Ricci curvature tensor are 1 1 1 R3434 = − 4 , R33 = − 2 , R44 = − 2 , (20) σ 2σ σ which amount to a constant negative scalar curvature Γ334

Γ343

R = −1.

(21)

By using the connection coefficients (19), the next proposition is gotten Proposition 2.4 The geodesic equations on manifold M can be represented as   d2 µ1 1 d µ1 2 = , d t2 µ1 d t   d2 µ2 1 d µ2 2 = , d t2 µ2 d t (22)   d2 µ3 2 d µ3 d σ = , d t2 σ dt dt     d2 σ 1 dσ 2 1 d µ3 2 = − , d t2 σ dt 2σ d t and the solutions of above equations can be obtained as follows µ1 (t) = b1 exp{−a1 t}, µ2 (t) = b2 exp{−a2 t}, µ3 (t) = σ(t) =

1 b3 2 2a3 exp{−2a3 t} + b3 exp{−a3 t} 2

b3 exp{−2a3 t} + 8a 2 3

b3 2 8a3 2

+ c1 ,

,

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where ai , bj (i = 1, 2, 3, j = 1, 2, 3) are integration constants. Under the premise of not affecting the properties of the system and for the sake of simplicity, we assume the integration constants satisfy that ai = α > 0 and bj = β > 0 (i, j = 1, 2, 3) and c1 = 0. In order to obtain the behavior of parameters of the Jacobi vector field on M , we consider the parameter family of neighboring geodesics n o FM (α, β) = θl (t; α, β)

l=1,2,3,4

,

(24)

where θ1 (t; α, β) = θ2 (t; α, β) = β exp{−αt}, θ3 (t; α, β) =

β2 2

β 2α exp{−2αt} + 4α β exp{−αt} . θ4 (t; α, β) = β2 exp{−2αt} + 8α 2

,

(25)

Proposition 2.5 The Jacobi-Levi-Civita equations describing the geodesic spread are 1 1 d2 (δθ1 ) 1 d θ d(δθ ) + 2Γ + ∂1 Γ111 11 d t2 dt dt



d θ1 dt

2

δθ1 = 0,

 2 2 2 2 d2 (δθ2 ) dθ 2 d θ d(δθ ) 2 + 2Γ22 + ∂2 Γ22 δθ2 = 0, 2 dt dt dt dt   4 3 3 4 d2 (δθ3 ) 3 d θ d(δθ ) 3 d θ d(δθ ) + 2 Γ + Γ + 34 43 d t2 dt dt dt dt  4 2  4 2 dθ dθ 3 3 3 4 ∂4 Γ34 δθ + Γ43 Γ44 δθ3 dt dt  4 2 1 dθ d θ3 d θ4 4 1 δθ − δθ3 , = R3434 R3434 g33 dt dt g33 dt   3 3 4 4 d2 δθ4 4 d θ d(δθ ) 4 d θ d(δθ ) + 2 Γ33 + Γ44 + d t2 dt dt dt dt  4 2 dθ d θ4 d θ3 3 4 δθ4 + Γ433 Γ334 δθ ∂4 Γ44 dt dt dt  3 2 1 d θ3 d θ4 3 1 dθ = R3434 δθ − R3434 δθ4 , g44 dt dt g44 dt RJP 60(Nos. 9-10), 1249–1262 (2015) (c) 2015 - v.1.3a*2015.11.20

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which can be simplified into d2 (δθ1 ) d(δθ1 ) + 2α + α2 δθ1 = 0, d t2 dt d2 (δθ2 ) d(δθ2 ) + 2α + α2 δθ2 = 0, d t2 dt d(δθ3 ) 16α2 d2 (δθ3 ) d(δθ4 ) + 2α + exp{−αt} + d t2 dt β dt   8α3 2 α − exp{−αt} δθ4 = 0, β

(27)

d(δθ4 ) 8α2 d(δθ4 ) d2 (δθ4 ) + 2α − exp{−αt} + d t2 dt β dt   32α4 8α3 2 α − 2 exp{−αt} δθ4 − exp{−αt}δθ3 = 0. β β This differential system is still very complicated and it is difficult to solve explicitly. Since what we need is the asymptotic behavior of the Jacobi vector field when t goes to infinity, we assume that [18]  d(δθ4 ) 16α2 exp{−αt} = 0, lim t→∞ β dt  3  8α d(δθ3 ) lim exp{−αt} = 0, t→∞ β dt   3 8α exp{−αt}δθ4 = 0. lim t→∞ β 

(28)

Combining (27) with (28), we get d2 (δθ1 ) d(δθ1 ) + 2α + α2 δθ1 = 0, d t2 dt d2 (δθ2 ) d(δθ2 ) + 2α + α2 δθ2 = 0, d t2 dt d2 (δθ3 ) d(δθ3 ) + 2α + α2 δθ4 = 0, d t2 dt d2 (δθ4 ) d(δθ4 ) + 2α + α2 δθ4 = 0. d t2 dt RJP 60(Nos. 9-10), 1249–1262 (2015) (c) 2015 - v.1.3a*2015.11.20

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That can be solved explicitly and the general solutions are δµ1 = (d1 + d2 t) exp{−αt}, δµ2 = (d3 + d4 t) exp{−αt}, δµ3 = (d5 + d6 t) exp{−αt} −

1 d7 exp{−2αt} + d8 , 2α

(30)

δσ = (d5 + d6 t) exp{−αt}, where di (i = 1, 2, . . . , 8) are integration constants. Lemma 2.1 Using J 2 = J i Ji = J i J j gij , the Jacobi vector field JM is described by α2 1 2 α1 (δµ1 )2 + 2 (δµ2 )2 + 2 (δµ3 )2 + 2 (δσ)2 2 µ1 µ2 σ σ α1 α2 2 = 2 (d1 + d2 t) exp{−2αt} + 2 (d3 + d4 t)2 exp{−2αt} µ1 µ2  2 1 d7 + 2 (d5 + d6 t) exp{−αt} − exp{−2αt} + d8 σ 2α 2 + 2 (d5 + d6 t)2 exp{−2αt}. σ

2 JM =

(31)

2 and get (The We keep the leading term in the asymptotic expansion of JM geodesics (25) are applied.) 2 JM ≈

d8 2 β 2 exp{2αt}, 64α4

(32)

thus, we put forward the conclusion that Theorem 2.1 The geodesic spread on M is described by means of an exponentially divergent Jacobi vector field intensity JM . It convincingly indicates the local instability of the ED model. Definition 2.1 In our approach, the quantity λJ M defined as [15]    1 kJM (t)k λJ M = lim ln t→∞ t kJM (0)k

(33)

would play the role of the conventional Lyapunov exponent with JM given approximately by (32). From the definition of λJ M , we can get its concrete expression by direct calculation. Therefore, we derive the result that Theorem 2.2 The quantity λJ M = α further indicates the local instability of the ED model characterized by its positive quality. RJP 60(Nos. 9-10), 1249–1262 (2015) (c) 2015 - v.1.3a*2015.11.20

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3. SUBMANIFOLDS

In this section, we investigate a hypersurface of M , a three dimensional submanifold N = {p1 (x; θ)} with µ3 = 0. Obviously it is still a statistical manifold. All other hypersurfaces and submanifolds with dimensions less than three can be similarly investigated. The probability density function for N is   α1    α2 1 α1 α2 1 α1 α1 −1 pN (x; θ) = x1 exp − x1 · Γ(α1 ) µ1 µ1 Γ(α2 ) µ2     α2 1 x3 2 α2 −1 x2 exp − x2 · √ exp − 2 . µ2 2σ 2πσ 2 Remark 3.1 Note that all the information relevant to the dynamical evolution of N can be obtained from the parameters of the probability distributions again. The detailed discussions are similar to the ones about M . One can refer to the according contents in the above section for more detailed information. Similarly to section 2, we can get Proposition 3.1 The Fisher information metric on N is denoted by   α1 α2 2 (gij ) = diag , , µ1 2 µ2 2 σ 2 and the non-zero components of the connection coefficients are Γ111 = −

1 , µ1

Γ222 = −

1 , µ2

1 Γ333 = − . σ

(34)

A direct calculation suggests that the Riemannian curvature tensor vanishes, that implies the vanishes of the Ricci curvature tensor and the scalar curvature. Therefore, N is a 3-dimensioned submanifold with constant scalar curvature zero. Proposition 3.2 The geodesic equations on N are given by   d2 µ1 1 d µ1 2 = , d t2 µ1 d t   d2 µ2 1 d µ2 2 (35) = , d t2 µ2 d t   d2 σ 1 dσ 2 = , d t2 σ dt whose general solutions are µ1 (t) = r1 exp{−q1 t}, µ2 (t) = r2 exp{−q2 t}, σ(t) = r3 exp{−q3 t}, RJP 60(Nos. 9-10), 1249–1262 (2015) (c) 2015 - v.1.3a*2015.11.20

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where ri and qi (i = 1, 2, 3) are integration constants. For simplicity, we assume qi = A and ri = B > 0 with i = 1, 2, 3. Consider the parameter family of neighboring geodesics n o , (37) FN (A, B) = θl (t; A, B) l=1,2,3

where θi (t; A, B) = B exp{−At}, i = 1, 2, 3. (38) Recall that all components of the Riemannian curvature tensor are zero, then Proposition 3.3 The Jacobi-Levi-Civita equations describing the geodesic spread are  1 2 1 1 d2 (δθ1 ) dθ 1 d θ d(δθ ) 1 + 2Γ11 + ∂1 Γ11 δθ1 = 0, 2 dt dt dt dt  2 2 2 2 dθ d2 (δθ2 ) 2 d θ d(δθ ) 2 (39) + 2Γ22 + ∂2 Γ22 δθ2 = 0, 2 dt dt dt dt  3 2 3 3 d2 (δθ3 ) dθ 3 d θ d(δθ ) 3 + 2Γ33 + ∂3 Γ33 δθ3 = 0, d t2 dt dt dt which can be simplified into the following linear differential system with constant coefficients d2 (δθi ) d(δθi ) + 2A + A2 δθi = 0, i = 1, 2, 3. (40) d t2 dt The general solutions of the equations above are given by δµ1 = (u1 + v1 t) exp{−At}, δµ2 = (u2 + v2 t) exp{−At},

(41)

δσ = (u3 + v3 t) exp{−At}, where ui and vi (i = 1, 2, 3) are integration constants. Lemma 3.1 The Jacobi vector field JN is α2 2 α1 (δµ1 )2 + 2 (δµ2 )2 + 2 (δσ)2 µ1 2 µ2 σ α1 α2 = 2 (u1 + v1 t)2 exp{−2At} + 2 (u2 + v2 t)2 exp{−2At} µ1 µ2 2 + 2 (u3 + v3 t)2 exp{−2At} σ α1 α2 2 = 2 (u1 + v1 t)2 + 2 (u2 + v2 t)2 + 2 (u3 + v3 t)2 B B B 2 :=w1 t + w2 t + w3 ,

2 JN =

2 → ∞. where w1 , w2 , w3 are constants. It is obviously that when t → ∞, JN

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Theorem 3.1 The geodesic spread on N is described by means of a first-order linearly divergent Jacobi vector JN . It explains the local instability of the ED model. Similar to (33), we have Definition 3.1 The quantity λJ N defined as    1 kJN (t)k λJ N = lim (43) ln t→∞ t kJN (0)k would play the role of the Lyapunov exponent with JN given in (42). Consequently, the following theorem can be obtained by analysis Theorem 3.2 The quantity λJ N = 0 demonstrates the local instability of the ED model as well. 4. CONCLUSION

We study an ED model, that is underlain with a four-dimensional statistical manifold M . The geometric structure is introduced from the viewpoint of information geometry by considering the Fisher information metric as a Riemannian metric. The scalar curvature R = −1 is obtained and its negativity provides a strong criterion of local instability. Actually, we only consider the Riemannian structure rather than the dual structure, since the former is sufficient when the instability is taken into account. By discussing the geodesic spread equations with respect to a family of neighboring geodesics, we conclude that the geodesic spread on M is described by means of an exponentially divergent Jacobi vector field, that implies as time goes on, a slight change of some parameter may cause remarkable divergence of the length between two neighboring geodesics. The exponential divergence of the Jacobi vector field intensity JM is a classical feature of chaos. At the same time, the associated positive Lyapunov exponent is obtained and it is a powerful character of local instability. A hypersurface N , which can be viewed as an underlying manifold for another ED model, is investigated similarly and we conclude that its geodesic spread associates with a first-order linearly divergent Jacobi vector field. The associated Lyapunov exponent is also obtained in this case. From the results obtained, the complexity of the system is investigated. It can be generalized to the study of any other submanifolds and they are not necessary to be hypersurfaces. As a remark, we would like to address that the joint probability distribution considered in this paper consists of independent macrostate variables, while the presence of the correlations between macroscopic statistical variables could dramatically change our results [12].

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Acknowledgements. The authors would like to thank Professors Fengxiang Mei and Demeter Krupka for their valuable advices. Special thanks to the referee for the valuable comments. This subject is supported by the National Natural Science Foundations of China (No. 61179031).

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RJP 60(Nos. 9-10), 1249–1262 (2015) (c) 2015 - v.1.3a*2015.11.20