Recent theoretical progress on an information geometrodynamical approach to chaos
aa r X i v : . [ m a t h - ph ] O c t Recent theoretical progress on an information geometrodynamical approach to chaos
Carlo Cafaro ∗ Department of Physics, University at Albany-SUNY,1400 Washington Avenue, Albany, NY 12222, USA
In this paper, we report our latest research on a novel theoretical information-geometric frame-work suitable to characterize chaotic dynamical behavior of arbitrary complex systems on curvedstatistical manifolds. Specifically, an information-geometric analogue of the Zurek-Paz quantumchaos criterion of linear entropy growth and an information-geometric characterization of chaotic(integrable) energy level statistics of a quantum antiferromagnetic Ising spin chain in a tilted (trans-verse) external magnetic field are presented.
PACS numbers: 02.50.Tt-Inference methods; 02.40.Ky- information geometry; 05.45.-a- chaos.
I. INTRODUCTION
Research on complexity [1, 2] has created a new set of ideas on how very simple systems may give rise to verycomplex behaviors. Moreover, in many cases, the ”laws of complexity” have been found to hold universally, caringnot at all for the details of the system’s constituents. Chaotic behavior is a particular case of complex behavior andit will be the object of the present work.In this paper we make use of the so-called Entropic Dynamics (ED) [3]. ED is a theoretical framework that arisesfrom the combination of inductive inference (Maximum Entropy Methods (ME), [4, 5]) and Information Geometry(IG) [6]. The most intriguing question being pursued in ED stems from the possibility of deriving dynamics frompurely entropic arguments. This is clearly valuable in circumstances where microscopic dynamics may be too farremoved from the phenomena of interest, such as in complex biological or ecological systems, or where it may justbe unknown or perhaps even nonexistent, as in economics. It has already been shown that entropic arguments doaccount for a substantial part of the formalism of quantum mechanics, a theory that is presumably fundamental [7].Laws of physics may just be consistent, objective ways to manipulate information. Following this line of thought,we extend the applicability of ED to temporally-complex (chaotic) dynamical systems on curved statistical manifoldsand identify relevant measures of chaoticity of such an information geometrodynamical approach to chaos (IGAC).The layout of the paper is as follows. In the section II, we give an introduction to the main features of our IGAC.In section III, we apply our theoretical construct to three complex systems. We emphasize that we have omittedtechnical details some of which can be found in our previous articles [8, 9, 10, 11, 12, 13, 14, 15]. Finally, in sectionIV we present our final remarks and future research directions.
II. GENERAL FORMALISM OF THE IGAC
The IGAC is an application of ED to complex systems of arbitrary nature. ED is a form of information-constraineddynamics built on curved statistical manifolds M S where elements of the manifold are probability distributions { P ( X | Θ) } that are in a one-to-one relation with a suitable set of macroscopic statistical variables { Θ } that providea convenient parametrization of points on M S . The set { Θ } is called the parameter space D Θ of the system.In what follows, we schematically outline the main points underlying the construction of an arbitrary form of en-tropic dynamics. First, the microstates of the system under investigation must be defined. For the sake of simplicity,we assume our system is characterized by an l -dimensional microspace with microstates { x i } where i = 1,..., l . Themain goal of an ED model is that of inferring ”macroscopic predictions” in the absence of detailed knowledge ofthe microscopic nature of arbitrary complex systems. Once the microstates have been defined, we have to select therelevant information about such microstates. In other words, we have to select the macrospace of the system. For thesake of the argument, we assume that our microstates are Gaussian-distributed. They are defined by 2 l -informationconstraints, for example their expectation values µ i and variances σ i . In addition to information constraints, eachGaussian distribution p k ( x k | µ k , σ k ) of each microstate x k must satisfy the usual normalization conditions. Once the ∗ Electronic address: [email protected] microstates have been defined and the relevant information constraints selected, we are left with a set of probabilitydistributions p ( X | Θ) = l Y k =1 p k ( x k | µ k , σ k ) encoding the relevant available information about the system where X is the l -dimensional microscopic vector with components ( x ,..., x l ) and Θ is the 2 l -dimensional macroscopic vectorwith coordinates ( µ ,..., µ l ; σ ,..., σ l ). The set { Θ } define the 2 l -dimensional space of macrostates of the system, thestatistical manifold M S . A measure of distinguishability among macrostates is obtained by assigning a probabilitydistribution P ( X | Θ) ∋ M S to each macrostate Θ . Assignment of a probability distribution to each state endows M S with a metric structure. Specifically, the Fisher-Rao information metric g µν (Θ) [6] defines a measure of distin-guishability among macrostates on M S , with M S being defined as the set of probabilities { p ( X | Θ) } described abovewhere X ∈ R N , Θ ∈ D Θ = [ I µ × I σ ] N . The parameter space D Θ (homeomorphic to M S ) is the direct product ofthe parameter subspaces I µ and I σ , where I µ = ( −∞ , + ∞ ) µ and I σ = (0, + ∞ ) σ in the conventional Gaussiancase. Once M S and D Θ are defined, the ED formalism provides the tools to explore dynamics driven on M S byentropic arguments. Specifically, given a known initial macrostate Θ (initial) (probability distribution), and that thesystem evolves to a final known macrostate Θ (final) , the possible trajectories of the system are examined in the EDapproach using ME methods.The geodesic equations for the macrovariables of the Gaussian ED model are given by nonlinear second order coupledordinary differential equations. They describe a reversible dynamics whose solution is the trajectory between an initialΘ (initial) and a final macrostate Θ (final) . Given the Fisher-Rao information metric, we can apply standard methodsof Riemannian differential geometry to study the information-geometric structure of the manifold M S underlyingthe entropic dynamics. Connection coefficients Γ ρµν , Ricci tensor R µν , Riemannian curvature tensor R µνρσ , sectionalcurvatures K M S , scalar curvature R M S , Weyl anisotropy tensor W µνρσ , Killing fields ξ µ and Jacobi fields J µ can becalculated in the usual way.To characterize the chaotic behavior of complex entropic dynamical systems, we are mainly concerned with the signsof the scalar and sectional curvatures of M S , the asymptotic behavior of Jacobi fields J µ on M S , the existence ofKilling vectors ξ µ and the asymptotic behavior of the information-geometrodynamical entropy (IGE) S M S (see (1)).It is crucial to observe that true chaos is identified by the occurrence of two features: 1) strong dependence on initialconditions and exponential divergence of the Jacobi vector field intensity, i.e., stretching of dynamical trajectories; 2)compactness of the configuration space manifold, i.e., folding of dynamical trajectories. The negativity of the Ricciscalar R M S implies the existence of expanding directions in the configuration space manifold M s . Indeed, since R M S is the sum of all sectional curvatures of planes spanned by pairs of orthonormal basis elements (cid:8) e ρ = ∂ Θ ρ (cid:9) , thenegativity of the Ricci scalar is only a sufficient (not necessary) condition for local instability of geodesic flow. For thisreason, the negativity of the scalar provides a strong criterion of local instability. A powerful mathematical tool weuse to investigate the stability or instability of a geodesic flow is the Jacobi-Levi-Civita equation (JLC-equation) forgeodesic spread. Finally, the asymptotic regime of diffusive evolution describing the possible exponential increase ofaverage volume elements on M s provides another useful indicator of dynamical chaoticity. The exponential instabilitycharacteristic of chaos forces the system to rapidly explore large areas (volumes) of the statistical manifold. It isinteresting to note that this asymptotic behavior appears also in the conventional description of quantum chaos wherethe von Neumann entropy increases linearly at a rate determined by the Lyapunov exponents. The linear increaseof entropy as a quantum chaos criterion was introduced by Zurek and Paz [16, 17]. In our information-geometricapproach a relevant quantity that can be useful to study the degree of instability characterizing ED models is theinformation geometrodynamical entropy (IGE) defined as [9], S M s ( τ ) def = lim τ →∞ log V M s with V M s ( τ ) = 1 τ τ Z dτ ′ Z M s ( τ ′ ) √ gd l Θ (1)and g = | det ( g µν ) | . IGE is the asymptotic limit of the natural logarithm of the statistical weight defined on M s andrepresents a measure of temporal complexity of chaotic dynamical systems whose dynamics is underlined by a curvedstatistical manifold. III. APPLICATIONS OF THE IGAC
As a first example, we apply our IGAC to study the dynamics of a system with l degrees of freedom, eachone described by two pieces of relevant information, its mean expected value and its variance (Gaussian statisti-cal macrostates). The line element ds = g µν (Θ) d Θ µ d Θ ν on M s with µ , ν = 1,..., 2 l is defined by [10, 12], ds = l X k =1 (cid:18) σ k dµ k + 2 σ k dσ k (cid:19) . (2)This leads to consider an ED model on a non-maximally symmetric 2 l -dimensional statistical manifold M s . It is shownthat M s possesses a constant negative Ricci curvature that is proportional to the number of degrees of freedom ofthe system, R M s = − l . It is shown that the system explores statistical volume elements on M s at an exponentialrate. The information geometrodynamical entropy S M s increases linearly in time (statistical evolution parameter)and is moreover, proportional to the number of degrees of freedom of the system, S M s τ →∞ ∼ lλτ . The parameter λ characterizes the family of probability distributions on M s . The asymptotic linear information-geometrodynamicalentropy growth may be considered the information-geometric analogue of the von Neumann entropy growth introducedby Zurek-Paz, a quantum feature of chaos. The geodesics on M s are hyperbolic trajectories. Using the JLC-equation,we show that the Jacobi vector field intensity J M s diverges exponentially and is proportional to the number of degreesof freedom of the system, J M s τ →∞ ∼ l exp ( λτ ). The exponential divergence of the Jacobi vector field intensity J M s is a classical feature of chaos. Therefore, we conclude that R M s = − l , J M s τ →∞ ∼ l exp ( λτ ), S M s τ →∞ ∼ lλτ . Thus, R M s , S M s and J M s behave as proper indicators of chaoticity.In our second example, we employ ED and ” Newtonian Entropic Dynamics ” (NED) [11]. In our special application,we consider a manifold with a line element ds = g µν (Θ) d Θ µ d Θ ν (with µ , ν = 1,..., l ) given by [14, 15], ds = [1 − Φ (Θ)] δ µν d Θ µ d Θ ν , Φ (Θ) = l X k =1 u k ( θ k ) (3)where u k ( θ k ) = − ω k θ k and θ k = θ k ( s ). The geodesic equations for the macrovariables θ k ( s ) are strongly nonlinear and their integration is not trivial. However, upon a suitable change of the affine parameter s used in the geodesicequations, we may simplify the differential equations for the macroscopic variables parametrizing points on the mani-fold M s with metric tensor g µν . Recalling that the notion of chaos is observer-dependent and upon changing the affineparameter from s to τ in such a way that ds = 2 (1 − Φ) dτ , we obtain new geodesic equations describing a set ofmacroscopic inverted harmonic oscillators (IHOs). In order to ensure the compactification of the parameter space ofthe system, we choose a Gaussian distributed frequency spectrum for the IHOs. Thus, with this choice, the foldingmechanism required for true chaos is restored in a statistical (averaging over ω and τ ) sense. Upon integrating thesedifferential equations, we obtain the expression for the asymptotic behavior of the IGE S M s , namely S M s ( τ ) τ →∞ ∼ Λ τ with Λ = l P i =1 ω i . This result may be considered the information-geometric analogue of the Zurek-Paz model used toinvestigate the implications of decoherence for quantum chaos. They considered a chaotic system, a single unstableharmonic oscillator characterized by a potential V ( x ) = − Ω x (Ω is the Lyapunov exponent), coupled to an externalenvironment. In the reversible classical limit [18], the von Neumann entropy of such a system increases linearly at arate determined by the Lyapunov exponent, S (chaotic)quantum ( τ ) τ →∞ ∼ Ω τ .In our final example, we use our IGAC to study the entropic dynamics on curved statistical manifolds induced byclassical probability distributions of common use in the study of regular and chaotic quantum energy level statistics. Itis known [19, 20] that integrable and chaotic quantum antiferromagnetic Ising chains are characterized by asymptoticlogarithmic and linear growths of their operator space entanglement entropies, respectively. In this last example,we consider the information-geometrodynamics of a Poisson distribution coupled to an Exponential bath (spin chainin a transverse magnetic field, regular case) and that of a Wigner-Dyson distribution coupled to a Gaussian bath(spin chain in a tilted magnetic field, chaotic case). Remarkably, we show that in the former case the IGE exhibitsasymptotic logarithmic growth while in the latter case the IGE exhibits asymptotic linear growth. In the regular case,the line element ds on the statistical manifold M s is given by [13, 15], ds = 1 µ A dµ A + 1 µ B dµ B (4)where the macrovariable µ A is the average spacing of the energy levels and µ B is the average intensity of the magneticenergy arising from the interaction of the transverse magnetic field with the spin particle magnetic moment. Insuch a case, we show that the asymptotic behavior of S (integrable) M s is sub-linear in τ (logarithmic IGE growth), S (integrable) M s ( τ ) τ →∞ ∼ log τ . Finally, in the chaotic case, the line element ds on the statistical manifold M s isgiven by [13, 15], ds = 4 µ ′ A dµ ′ A + 1 σ ′ B dµ ′ B + 2 σ ′ B dσ ′ B (5)where the (nonvanishing) macrovariable µ ′ A is the average spacing of the energy levels, µ ′ B and σ ′ B are the averageintensity and variance, respectively of the magnetic energy arising from the interaction of the tilted magnetic fieldwith the spin particle magnetic moment. In this case, we show that asymptotic behavior of S (chaotic) M s is linear in τ (linear IGE growth), S (chaotic) M s ( τ ) τ →∞ ∼ τ . The equations for S (integrable) M s and S (chaotic) M s may be considered as theinformation-geometric analogues of the entanglement entropies defined in standard quantum information theory inthe regular and chaotic cases, respectively. IV. CONCLUSION
In this paper we presented a novel theoretical information-geometric framework suitable to characterize chaotic dy-namical behavior of arbitrary complex systems on curved statistical manifolds. Specifically, an information-geometricanalogue of the Zurek-Paz quantum chaos criterion of linear entropy growth and an information-geometric charac-terization of chaotic (integrable) energy level statistics of a quantum antiferromagnetic Ising spin chain in a tilted(transverse) external magnetic field were presented.The descriptions of a classical chaotic system of arbitrary interacting degrees of freedom, deviations from Gaussianityand chaoticity arising from fluctuations of positively curved statistical manifolds are currently under investigation.Furthermore, we are investigating the possibility to extend the IGAC to quantum Hilbert spaces constructed fromclassical curved statistical manifolds and we are considering the information-geometric macroscopic versions of theHenon-Heiles and Fermi-Pasta-Ulam β -models to study chaotic geodesic flows on statistical manifolds. Finally, theinformation geometry of a chaotic spring pendulum and a periodically perturbed spherical pendulum are currentlyunder investigation. Soft chaos regimes arising in chemical physics are being considered as well. Our objective is tostudy transitions from order to chaos in a floppy molecule using inference methods and information geometry.At this stage of its development, IGAC remains an ambitious unifying information-geometric theoretical constructfor the study of chaotic dynamics with several limitations and unsolved problems [12, 13]. However, based on ourrecent findings, we believe it could provide an interesting, innovative and potentially powerful way to study andunderstand the very important and challenging problems of classical and quantum chaos. Therefore, we believe ourresearch program deserves further investigation and developments. Acknowledgments
The author is grateful to S. A. Ali, Ariel Caticha and Adom Giffin for very useful comments. Thanks are extendedto all MaxEnt 2008 participants in Sao Paulo- Brazil, especially to Rafael M. Gutierrez for his sincere interest andvery important comments on our works. [1] M. Gell-Mann, ”What is complexity?”, Complexity , 1 (1995).[2] D. P. Feldman and J. P. Crutchfield, ”Measures of complexity: why?, Phys. Lett. A238 , 244 (1998).[3] A. Caticha, ”Entropic dynamics”, in
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