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Riemannian Geometry of Fluctuation Theory: an Introduction Home Search Collections Journals About Contact us My IOPscience Riemannian geometry of fluctuation theory: An introduction This content has been downloaded from IOPscience. Please scroll down to see the full text. 2016 J. Phys.: Conf. Ser. 720 012005 (http://iopscience.iop.org/1742-6596/720/1/012005) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 80.82.77.83 This content was downloaded on 05/03/2017 at 12:31 Please note that terms and conditions apply. You may also be interested in: Curvature of fluctuation geometry and its implications on Riemannian fluctuation theory L Velazquez Effective Electromagnetic Parameters and Absorbing Properties for Honeycomb Sandwich Structures with a Consideration of the Disturbing Term Hu Ji-Wei, He Si-Yuan, Rao Zhen-Min et al. On the compressibility of a classical one-component plasma Y Rosenfeld Bremsstrahlung radiation from a non-relativistic pair plasma Liang Guo, Jian Zheng, Bin Zhao et al. Thermodynamic curvature: pure fluids to black holes George Ruppeiner A MD-based method to calculate free energy for crystalline structures Y Long, J Chen, Y G Liu et al. Extended irreversible thermodynamics and runaway electrons in plasmas D Jou, M Ferrer and J E Llebot Macroscopic fluctuation theory of local collisional dynamics Raphaël Lefevere Fluctuation conductance and the Berezinskii-Kosterlitz-Thouless transition in two dimensional epitaxial NbTiN ultra-thin films Makise K, Terai H, Yamashita T et al. XIX Chilean Physics Symposium 2014 IOP Publishing Journal of Physics: Conference Series 720 (2016) 012005 doi:10.1088/1742-6596/720/1/012005 Riemannian geometry of fluctuation theory: An introduction Luisberis Velazquez Departamento de F´ısica,Universidad Cat´olicadel Norte, Av. Angamos 0610, Antofagasta, Chile E-mail: [email protected] 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 manifold M of random events that are described by a family of continuous distributions dpξ(xjθ). 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: dpξ(xjθ) = ρξ(xjθ)dx (1) 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: 2 α β ds = gαβ(θ)dθ dθ (2) Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 XIX Chilean Physics Symposium 2014 IOP Publishing Journal of Physics: Conference Series 720 (2016) 012005 doi:10.1088/1742-6596/720/1/012005 j j Figure 1. Continuous distributions dpξ(x θ) and dpξˇ(ˇx θ) are diffeomorphic distributions, that is, a same abstract distribution dp(ϵjE) expressed into two different coordinate representations of the abstract statistical manifold M. 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: 2 i j ds = gij(xjθ)dx dx (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 ! ξˇ, that is, a bijective and differentiable map that leaves invariant their respective probability distributions (see scheme in Fig.1): −1 @xˇ ϕ : dpξ(xjθ) = dpˇ(ˇxjθ) ) ρˇ(ˇxjθ) = ρξ(xjθ) : (4) ξ ξ @x 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]: p p 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 manifolds M are 2 XIX Chilean Physics Symposium 2014 IOP Publishing Journal of Physics: Conference Series 720 (2016) 012005 doi:10.1088/1742-6596/720/1/012005 diffeomorphic to the real one-dimensional space R are always diffeomorphic distributions because of the only possible Riemannian geometry for these manifolds is the Euclidean one. In particular, Gaussian distribution: [ ] 1 2 2 dpξ(xjµ, σ) = p exp −(x − µ) =2σ dx; −∞ < x < +1: (6) 2πσ Cauchy distribution: 1 γdxˇ dp (ˇxjν; γ) = ; −∞ < xˇ < +1: (7) ξˇ π γ2 + (xˇ − ν)2 Bimodal Gaussian distribution: { [ ] [ ]} 1 2 2 2 2 dp~(~xjµ, σ) = p exp −(~x − µ) =2σ + exp −(~x + µ) =2σ dx;~ −∞ < x~ < +1: ξ 2 2πσ (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 manifold M has a dimension n ≥ 2 are possible the notions of curvature and statistical correlations. In particular, distributions family [4]: [ ] 1 1 2 2 θdxdy dpξ(x; yjθ) = exp − (x + y ) p (9) Z (θ) 2 2π x2 + y2 + θ2 with normalization constant: ( ) p 1 θ2 θ θ Z (θ) = πe 2 p erfc p (10) 2 2 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: 1 [ ( ) ] dp (x; yjσ) = exp − x2 + y2 =2σ dxdy (11) ζ 2πσ 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(xjθ): q dµ(xjθ) = jgij(xjθ)=2πjdx; (12) 3 XIX Chilean Physics Symposium 2014 IOP Publishing Journal of Physics: Conference Series 720 (2016) 012005 doi:10.1088/1742-6596/720/1/012005 3 RRR MMM z dz dr -θ 0 dt θ t Figure 2. The geometry of the statistical manifold M 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 jTijj represents the determinant of a given tensor Tij of second-rank, while the factor 2π has been introduced for convenience. Additionally, one can define the probabilistic weight [3]: q !(xjθ) = ρ(xjθ) j2πgij(xjθ)j; (13) which is a scalar function that arises as a local invariant measure of the probability. Although the mathematical form of the probabilistic weight !(xjθ) 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: dp(xjθ) = !(xjθ)dµ(xjθ); (14) which is a form that explicitly exhibits the invariance of this family of distributions. The notion of probability weight !(xjθ) can be employed to redefine the notion of information entropy for continuous distributions [6]: Z Sd [!jg; M] = − !(xjθ) log !(xjθ)dµ(xjθ): (15) M as a global invariant measure that depends on the metric tensor gij(xjθ) of the manifold M. The quantity I(xjθ): I(xjθ) = − log !(xjθ) (16) 4 XIX Chilean Physics Symposium 2014 IOP Publishing Journal of Physics: Conference Series 720 (2016) 012005 doi:10.1088/1742-6596/720/1/012005 represents 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(xjθ) as the negative of the information content (16): S(xjθ) = log !(xjθ) ≡ −I(xjθ); (17) the metric tensor can be rewritten as follows [3]: @2S(xjθ) @S(xjθ) g (xjθ) = −D D S(xjθ) = − + Γk (xjθ) ; (18) ij i j @xi@xj ij @xk where Di is the covariant derivative associated with the Levi-Civita affine connections k j Γij(x θ) [10]: [ ] 1 @g (xjθ) @g (xjθ) @g (xjθ) Γk (xjθ) = gkm(xjθ) im + jm − ij : (19) ij 2 @xj @xi @xm Covariant set of differential equations (18) can be rewritten into the alternative form: @2 log ρ(xjθ) @ log ρ(xjθ) @Γk (xjθ) g (xjθ) = − + Γk (xjθ) + jk − Γk (xjθ)Γl (xjθ) (20) ij @xi@xj ij @xk @xi ij kl in terms of probability density.
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