Table Of Content

One-parameter Fisher-Rényi complexity: Notion and hydrogenic applications I.V. Toranzoa,b, P. Sańchez-Morenoa,c, L ukasz Rudnickid,e and J.S. Dehesaa,b a Instituto Carlos I de F´ısica Teórica y Computacional, Universidad de Granada, 18071-Granada, Spain b Departamento de F´ısica Atómica, Molecular y Nuclear, Universidad de Granada, 18071-Granada, Spain 7 1 cDepartamento de Matemática Aplicada, Universidad de Granada, 18071-Granada, Spain 0 d 2 Institute for Theoretical Physics, University of Cologne, n Zu¨lpicher Straße 77, D-50937, Cologne, Germany a J eCenter for Theoretical Physics, Polish Academy of Sciences, 6 Aleja Lotników 32/46, PL-02-668 Warsaw, Poland 1 ] h In this work the one-parameter Fisher-Rényi measure of complexity for general d- p dimensional probability distributions is introduced and its main analytic properties are - t n discussed. Then, this quantity is determined for the hydrogenic systems in terms of the a u quantum numbers of the quantum states and the nuclear charge. q [ Keywords: Information theory; Fisher information; Shannon entropy; Rényi entropy; Fisher-Rényi 1 complexity; hydrogenicsystems v 9 9 1 I. INTRODUCTION 4 0 . 1 We all have an intuitive sense of what complexity means. In the last two decades an increasing 0 7 number of efforts have been published [1–12] to refine our intuitions about complexity into 1 : precise, scientific concepts, pointing out a large amount of open problems. Nevertheless there v i X is not a consensus for the term complexity nor whether there is a simple core to complexity. r a Contrary to the Boltzmann-Shannon entropy which is ever increasing according to the second law of thermodynamics, the complexity seems to behave very differently. Various precise, widely applicable, numerical and analytical proposals (see e.g., [13–30] and the monograph [8]) have been donebutthey areyet very far toappropriately formalize theintuitive notion of complexity [11,29]. The latter suggests that complexity should be minimal at either end of the scale. However, a complexity quantifier to take into account the completely ordered and completely disordered limits (i.e., perfect order and maximal randomness, respectively) and to describe/explain the maximum between them is not known up until now. 2 Recently, keeping in mind the fundamental principles of the density functional theory, some statistical measuresofcomplexityhavebeenproposedtoquantifythedegreeofstructureorpattern of finite many-particle systems in terms of their single-particle density, such as the Cra´mer-Rao [23, 26], Fisher-Shannon [18, 21, 24] and LMC (Lo´pez-ruiz, Mancini and Calvet) [12, 17] complexities and some modifications of them [13, 22, 25, 27–29]. They are composed by a two-factor productof entropic measures of Shannon [31], Fisher [6, 32] and Rényi [33] types. Most interesting for quantum systems are those which involve the Fisher information (namely, the Cra´mer-Rao and the Fisher-Shannon complexities, and their modifications [25, 27, 34]), mainly because this is by far the best entropy-like quantity to take into account the inherent fluctuations of the quan- tumwavefunctionsbyquantifyingthegradientcontentofthesingle-particledensityofthesystems. The objetive of this article is to extend and generalize these Fisher-information-based measures of complexity by introducing a new complexity quantifier, the one-parameter Fisher-Rényi complexity, to discuss its properties and to apply it to the main prototype of Coulombian systems, the hydrogenic system. This notion is composed by two factors: a λ-dependent Fisher information (which quantifies various aspects of the quantum fluctuations of the physical wave functions beyond the density gradient, since it reduces to the standard Fisher information for λ = 1) and the Rényi entropy of order λ (which measures various facets of the spreading or spatial extension of the density beyond the celebrated Shannon entropy which corresponds to the limiting case λ 1). → The article is structured as follows. In Section I we introduce the notion of one-parameter Fisher-Rényi measure of complexity. In Section II we discuss the main analytical properties of this complexity, showing that it is bounded from below, invariant under scaling transformations and monotone. In addition the near-continuity and the invariance under replications are also discussed. In Section III,we apply thenew complexity measure to the hydrogenic systems. Finally some concluding remarks are given. II. ONE-PARAMETER FISHER-REŃYI COMPLEXITY MEASURE (λ) In this section the notion of one-parameter Fisher-Rényi complexity C [ρ] of a d-dimensional FR probability density is introduced and its main analytic properties are discussed. This quantity is composed by two entropy-like factors of local (the one-parameter Fisher information of Johnson 3 and Vignat [35], F˜ [ρ]) and global (the λ-order Rényi entropy power [36], N [ρ]) characters. λ λ A. The notion (λ) Theone-parameterFisher-RényicomplexitymeasureC [ρ]oftheprobabilitydensityρ(x), x = FR (x ,x ,...,x ) Rd, is defined by 1 2 d ∈ d 1 d C(λ)[ρ] = D−1F˜ [ρ]N [ρ], λ >max − , , (1) FR λ λ λ d d+2 (cid:26) (cid:27) where D is the normalization factor given as λ 2πdλ−1 Γ(λ−λ1) d2 (d+2)λ−d 2+d(dλ(−λ−1)1) , λ > 1 D =  λ−1 (cid:18)Γ(d2+λ−λ1)(cid:19) 2λ (2) λ 2πdλ−1 Γ(1−1λ−d2) d2 (cid:16)(d+2)λ−d(cid:17)2+d(dλ(−λ−1)1) , max d−1, d < λ < 1. 1−λ Γ( 1 ) 2λ d d+2 (cid:18) 1−λ (cid:19)  (cid:16) (cid:17) n o  This purely numerical factor is necessary to let the minimal value of the complexity be equal to unity, as explained below in paragraph 2.2.1. The F˜ [ρ] denotes the (scarcely known) λ-weighted λ Fisher information [35] defined by −1 F˜ [ρ] = ρλ(x)dx ρλ−2(x) ρ(x)2ρ(x)dx, (3) λ Rd Rd| ∇ | (cid:18)Z (cid:19) Z |∇ρ|2 (which, for λ = 1, reduces to the standard Fisher information F[ρ] = dx), being dx the Rd ρ d-dimensional volume element. Finally, the symbol N [ρ] denotes the λR-Rényi entropy power (see λ e.g., [36]) given as µ 1 ρλ(x)dx d1−λ if λ = 1,0 < λ < , Rd 6 ∞ N [ρ] = (4) λ  e(cid:0)R2dS[ρ] (cid:1) if λ = 1,  where µ = 2+d(λ 1) and S[ρ] := ρ(x)lnρ(x)dx is the Shannon entropy [31]. − − Rd R (λ) The complexity measure C [ρ] has a number of conceptual advantages with respect to the FR Fisher-information-based measures of complexity previously defined; namely, the Cra´mer-Rao and Fisher-Shannon complexity and their modifications. Indeed, it quantifies the combined balance of different (λ-dependent) aspects of both the fluctuations and the spreading or spatial extension of the single-particle density ρ in such a way that there is no dependence on any specific point of the system’s region. The Cra´mer-Rao complexity [23, 26] (which is the product of the standard Fisher information F[ρ] mentioned above and the variance V[ρ] = r2 r 2) measures a single h i−h i aspect of the fluctuations (namely, the density gradient) together with the concentration of the 4 probabilitydensityaroundthecentroid r . TheFisher-Shannoncomplexity [18,21,24],definedby h i CFS[ρ] = F[ρ] e2dS[ρ], quantifies the density gradient jointly with a single aspect of the spreading × givenbytheShannonentropyS[ρ]mentionedabove. Amodificationofthepreviousmeasurebyuse of the Rényi entropy R [ρ] = 1 ln ρλ(x)dx instead of the Shannon entropy, the Fisher-Rényi λ 1−d Rd product of complexity-type, has beeRn recently introduced [25, 27, 34]; it measures the gradient together with various aspects of the spreading of the density. B. The properties Let us now discuss some properties of this notion: bounding from below, invariance under scaling transformations, monotonicity, behavior under replications and near continuity. 1. Lower bound. The Fisher-Rényi complexity measure C(λ)[ρ] fulfills the inequality FR (λ) C [ρ] 1 (5) FR ≥ (for λ > max d−1, d , with λ = 1), and the minimal complexity occurs, as implicitly d d+2 6 proved by Savnaré and Tooscani [36], if and only if the density has the following generalized Gaussian form 1 (C x 2)λ−1, λ > 1 (x)= λ−| | + (6) λ B  (Cλ+|x|2)λ−11, λ < 1 where (x) = max x,0 and C is the normalization constant given by + λ { } − 2(λ−1) C = A d(λ−1)+2, (7) λ λ with Γ( λ ) πd/2 λ−1 , λ >1 Γ(d+ λ ) A = 2 λ−1 λ  Γ( 1 −d)  πd/2 1−λ 2 , d < λ < 1  Γ( 1 ) d+2 1−λ  (λ) Thus, the complexity measure C (ρ) has a universal lower bound of minimal complexity, FR that is achieved for the family of densities (x). λ B 2. Invariance under scaling and translation transformations. The complexity measure (λ) C (ρ) are scaling and translation invariant in the sense that FR (λ) (λ) C [ρ ] = C [ρ], λ, (8) FR a,b FR ∀ 5 where ρ (x) = adρ(a(x b)), with a R and b Rd. To prove this property we follow the a,b − ∈ ∈ lines of Savaré and Toscani [36]. First we calculate the generalized Fisher information of the transformed density, obtaining −1 F˜ [ρ ] = adλρλ(a(x b))dx λ a,b Rd − (cid:18)Z (cid:19) a2d(λ−2)ρ2(λ−2)(a(x b))ad+1[ ρ](a(x b))2adρ(a(x b))dx × Rd − | ∇ − | − Z −1 = ad(λ−1)+2 ρλ(y)dy ρ2λ−4(y) ρ(y)2ρ(y)dy Rd Rd |∇ | (cid:18)Z (cid:19) Z ad(λ−1)+2F˜ [ρ], λ λ ≡ ∀ Note that in writing the first equality we have used that ρ (x)2 = ad+1[ ρ](a(x b))2. a,b |∇ | | ∇ − | Then, we determine the value of the λ-entropy power of the density ρ (x) which turns out a,b to be equal to 2+d(λ−1) d(1−λ) N [ρ ] = adλρλ(a(x b))dx λ a,b Rd − (cid:18)Z (cid:19) 2+d(λ−1) d(1−λ) = ad(λ−1) ρλ(y)dy Rd (cid:18) Z (cid:19) a−d(λ−1)−2N [ρ], λ λ ≡ ∀ In particular, we have 2 N [ρ ] = exp adρλ(a(x b))ln[adρλ(a(x b))]dx 1 a,b −d Rd − − (cid:20) Z (cid:21) 2 = exp ρ(y)ln[adρ(y)]dy −d Rd (cid:20) Z (cid:21) 2 = exp (dlna+S[ρ]) −d (cid:20) (cid:21) a−2N [ρ], 1 ≡ Finally, from Eq. (1) and the values of F˜ [ρ ] and N [ρ ] just found, we readily obtain λ a,b λ a,b the wanted invariance (8). 3. Monotonicity. The existence of a non-trivial operation with interesting properties under which a complexity measure is monotonic [11] is a valuable property of the measure in question from the axiomatic point of view. To show the monotonic behavior of the Fisher- (λ) Rényi complexity C (ρ) we make use of the so-called rearrangements, which represent a FR 6 useful tool in the theory of functional analysis and, among other applications, have been used to prove relevant inequalities such as Young’s inequality with sharp constant. Two of the main properties of rearrangements is that they preserve the Lp norms, which implies that the rearrangements of a probability density give rise to another probability density, and that they make everything spherically symmetric. The second feature makes the rearrangement operation relevant for quantification of statistical complexity [11], since a spherically symmetric variant of a probability density can in an atomic context be viewed as less complex. Then, we introduce the definition of this operation as well as its effects over the entropic quantities that make up our complexity measure. Let f be a real-valued function, f : Rn [0, ) and A = x : f(x) t . The symmetric t → ∞ { ≥ } decreasing rearrangement of f is defined as ∞ f∗(x) = χ{x∈A∗}dt, (9) t Z0 with χ{x∈A∗t} = 1 if x ∈A∗t and 0 otherwise. At represents the super-level set of the function f and A∗ (which denotes the symmetric rearrangement of a set A Rn) is the Euclidean ⊂ ball centered at 0 such as Vol(A∗) =Vol(A). The central idea of this transformation is to build up f∗ from the rearranged super-level sets in the same manner that f is built from its super-level sets. As a by-product from its construction, f∗ turns out to be a spherically symmetric decreasing function (i.e. f∗(x) = f∗(x ) and moreover f∗(b) < f∗(a) b > a, where a,b A∗) which means that for any | | ∀ ∈ t function f :Rn [0, ) and all t 0 → ∞ ≥ x :f(x)> t ∗ = x :f∗(x) > t , (10) { } { } or in other words, that for any measurable subset B [0, ), the volume of the sets ⊂ ∞ x :f(x) B and x : f∗(x) B are the same. { ∈ } { ∈ } It is known [37] that under this transformation and for any p [0,1) (1, ] the Rényi and ∈ ∪ ∞ Shannon entropies remain unchanged, i.e. R [ρ] = R [ρ∗], S[ρ]= S[ρ∗] (11) p p if both S[ρ] and S[ρ∗] are well defined, where lim R [ρ] = S[ρ]. The invariance of the p→1 p Rényi entropy follows from the preservation of the Lp norms via rearrangements and the proofoftheinvarianceoftheShannonentropyisdonein[37]. Moreover, WangandMadiman 7 [37] consider the Fisher information, finding that the standard Fisher information decreases monotonically under rearrangements, i.e. F[ρ] F[ρ∗]. (12) ≥ LetusnowconsiderthebiparametricFisher-like information, I [f], ofaprobability density β,q function f(x) which is defined [38] by f(x) β I [f]= fβ(q−1)+1(x) |∇ | f(x)dx (13) β,q Rd f(x) Z (cid:16) (cid:17) with q 0, β > 1. Then one notes that the one-parameter Fisher information, F˜ [ρ], given λ ≥ by (3) can be expressed in terms of the previous quantity with β =2 and q λ as ≡ ρλ−2(x) ρ(x)2ρ(x)dx I [ρ] F˜ [ρ] = Rd| ∇ | = 2,λ . (14) λ R Rdρλ(x)dx Nλ[ρ]µd(1−λ) R On the other hand, considering the transformation ρ = u(x)k with k = β , the bipara- β(q−1)+1 metric Fisher information becomes I = u(x)βdx (15) β,q Rd|∇ | Z also known as the β-Dirichlet energy of u(x). If k = 2, note that the function u(x) corresponds to a quantum-mechanical wave function. By using the symmetric decreasing rearrangement to the density function ρ, the well-known Po´lya-Szego¨ inequality states that I [ρ] = uβ I [ρ∗] = u∗ β, (16) β,q β,q Rd|∇ | ≥ Rd|∇ | Z Z which implies that the minimizer of the left side is necessarily radially symmetric and decreasing, so the extremal function belongs to the subset of radially symmetric probability densities, and is represented by the generalized Gaussian given in (6). Now by taking into account(14)andtheinvarianceoftheRényientropy(andthereforetheRényientropypower, N [ρ]) upon rearrangements one obtains the monotonic behavior of F˜ [ρ] as λ λ I [ρ] I [ρ∗] F˜ [ρ] = 2,λ F˜ [ρ∗] = 2,λ , (17) λ Nλ[ρ]µd(1−λ) ≥ λ Nλ[ρ∗]µd(1−λ) Finally, this observation together with (1)allows us to obtain the monotonic behavior of this (λ) complexity measure C (ρ) proved by rearrangements, i.e. FR C(λ)(ρ) C(λ)(ρ∗), (18) FR ≥ FR 8 where the inequality is saturated for the generalized Gaussian, ρ(x) = (x), which also λ B meansthatthesymmetricrearrangementofageneralized Gaussiangivesanothergeneralized Gaussian,i.e. rearrangementspreservethissubsetofradiallysymmetricprobabilitydensities Bλ∗(x) = Bλ′(x). 4. Behavior under replications. Let us now study the behavior of the Fisher-Rényi com- (λ) plexity C (ρ) under n replications. We have found that for one-dimensional densities FR ρ(x), x R with bounded support, this complexity measure behaves as follows: ∈ C [ρ˜]= n2C [ρ], (19) FR FR where the density ρ˜representing n replications of ρ is given by n ρ˜(x) = ρm(x); ρm(x) = n−12ρ n21(x bm) , − mX=1 (cid:16) (cid:17) where the points b are chosen such that the supports Λ of each density ρ are disjoints. m m m Then, the integrals n (ρ˜(x))λ−2ρ˜′(x)2ρ˜(x)dx = (ρ (x))λ−2ρ′ (x)2ρ (x)dx | | | m m | m ZΛ m=1ZΛm X n = n−λ+1 (ρ(y))λ−2ρ′(y)2ρ(y)dy = n−λ+2 (ρ(y))λ−2ρ′(y)2ρ(y)dy, | | | | m=1 ZΛ ZΛ X and n n (ρ˜(x))λdx = (ρ˜m(x))λdx = n−λ+21 (ρ(y))λdy = n−λ−21 (ρ(y))λdy, ZΛ m=1ZΛm m=1 ZΛ ZΛ X X 1 where the change of variable y = n2(x bm) has been performed. − Thus, the two entropy factors (the generalized Fisher information and the Rényi entropy (λ) power) of the Fisher-Rényi measure C (ρ) gets modified as FR F˜λ[ρ˜] = n3−2λF˜λ[ρ], Nλ[ρ˜] = nλ+21Nλ[ρ], (20) so that from these two values and (1) we finally have the wanted behavior (19) of the Fisher- Rényicomplexityundernreplications. Althoughthishasbeenprovedintheonedimensional case, similar arguments hold for general dimensional densities. 5. Near-continuity behavior. Let us now illustrate that the Fisher-Rényi complexity is not near continuous by means of a one-dimensional counter-example. Recall first that a 9 functional G is near continuous if for any ǫ > 0 exist δ > 0 such that, if two densities ρ and ρ˜ are δ-neighboring (i.e., the Lebesgue measure of the points satisfying ρ(x) ρ˜(x) δ is | − | ≥ zero), then G[ρ] G[ρ˜] < ǫ. Now, let us consider the δ-neighboring densities | − | 2 sin2(x), π x 0, ρ(x) = − ≤ ≤ π  0, elsewhere,  and  sin2(x), π x 0, − ≤ ≤ 2 ρ˜(x) = δsin2 x , 0 < x δ5π, π(1+δ6)  δ5 ≤   0, (cid:0) (cid:1) elsewhere.     Due to the increasing oscillatory behaviour of ρ˜ for x (0,δ5π) as δ tends to zero, the ∈ generalized Fisher information F˜ grows rapidly as δ decreases, while the Rényi entropy power tends to a constant value. Then, the more similar ρ and ρãre, the more different are (λ) their values of C . Therefore, the Fisher-Rényi complexity measure is not near continuous. FR III. THE HYDROGENIC APPLICATION (λ) In this section we determine the one-parameter Fisher-Rényi complexity measure C , given FR by (1), for the probability density of hydrogenic atoms consisting of an electron bound by the Coulomb potential, V(r)= Z, where Z denotes the nuclear charge, r ~r = 3 x2 and the −r ≡ | | i=1 i position vector ~r = (x1,x2,x3) is given in spherical polar coordinates as (r,θ,φq) P(r,Ω), Ω S2. ≡ ∈ Atomic units are used. The hydrogenic states are well known to be characterized by the three quantum numbers n,l,m , with n= 0,1,2,..., l = 0,1,...,n 1 and m = l, l+1,...,l. They { } − − − have the energies E = Z2 , and the corresponding quantum probability densities are given by n −2n2 ρ (~r) = ρ (r˜) Θ (θ,φ) (21) n,l,m n,l l,m where r˜ = 2Zr, and the symbols ρ (r˜) and Θ (θ,φ) are the radial and angular parts of the n n,l l,m density, which are given by 4Z3ω (r˜) ρ (r˜)= 2l+1 [L(2l+1)(r˜)]2 (22) n,l n4 r˜ n−l−1 b and Θ (θ,φ) = Y (θ,φ)2, (23) l,m l,m | | 10 respectively. Inaddition, Lα(x) denotes theorthonormalLaguerrepolynomials[39]with respectto n the weight function ω = xαe−x on the interval [0, ), and Y (θ,φ) are the well-known spherical α b ∞ l,m harmonics which can be expressed in terms of the Gegenbauer polynomials, Cm(x) via n 1 Y (θ,φ) = (l+ 12)(l−|m|)![Γ(|m|+ 12)]2 2 eimφ(sinθ)|m|C|m|+21(cosθ), (24) l,m 21−2|m|π2(l+ m )! l−|m| ! | | (λ) where 0 θ π and 0 φ 2π. Let us now compute the complexity measure C [ρ ] of the ≤ ≤ ≤ ≤ FR n,l,m hydrogenic probability density which, according to (1), is given by 2 1 −1 C(λ)[ρ ] = D−1F˜ [ρ ]N [ρ ] D−1I I (cid:16)3(1−λ) (cid:17), (25) FR n,l,m λ λ n,l,m λ n,l,m ≡ λ 1 2 whereD is thenormalization constant given by (2)andthe symbolsI and I denote theintegrals λ 1 2 I = [ρ (~r)]λ−2 ρ (~r)2ρ (~r)d3~r = [ρ (~r)]2λ−3 ρ (~r)2d3~r, (26) 1 n,l,m n,l,m n,l,m n,l,m n,l,m | ∇ | |∇ | Z Z ∞ I = [ρ (~r)]λ d3~r = [ρ (r˜)]λ r2dr [Θ (θ,φ)]λ dΩ, (27) 2 n,l,m n,l l,m Z Z0 ZΩ which can be solved by following the lines indicated in Appendix A. In the following, for simplicity and illustration purposes, we focus our attention on the compu- tation of the complexity measure for two large, relevant classes of hydrogenic states: the (ns) and the circular (l = m = n 1) states. − 1. Generalized Fisher-Rényi complexity of hydrogenic (ns) states. Inthis case, Θ (θ,φ)= Y (θ,φ)2 = 1 sothat thethree angular integrals can betrivially 0,0 | 0,0 | 4π determined, and the radial integrals simplify as 24λ−3Z6λ−4 I(rad) = (2λ 1)−1 (n,0,λ) (28) 1a n10λ−6 − G 24λ−3Z6λ−4 1 I(rad) = (2λ 1)−1Φ 0,0,2(2λ 1), n 1 , 1 ; ,1 (29) 1b n10λ−6 − 0 − { − } { } 2λ 1 (cid:18) (cid:26) − (cid:27)(cid:19) 22λ−3Z3(λ−1) 1 I(rad)(λ) = λ−3Φ 2,0,2λ, n 1 , 1 ; ,1 , (30) 2 n5λ−3 0 { − } { } λ (cid:18) (cid:26) (cid:27)(cid:19) with 1 (n,0,λ) = (2λ 1)−2 Φ 2,0,2(2λ 1), n 1,...,n 1 , 1,...,1 ; ,1 0 G − − { − − } { } 2λ 1 " (cid:18) (cid:26) − (cid:27)(cid:19) 1 +4Φ 2,0,2(2λ 1), n 1,...,n 1,n 2,n 2 , 1,...,1,2,2 ; ,1 0 − { − − − − } { } 2λ 1 ! (cid:26) − (cid:27) 1 +4Φ 2,0,2(2λ 1), n 1,...,n 1,n 2 , 1,...,1,2 ; ,1 . 0 − { − − − } { } 2λ 1 !# (cid:26) − (cid:27) (31)