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Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids

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Please note that terms and conditions apply. J. Stat. Mech. (2014) P12027 - - q q ıguez encia e that, for the one- -Gaussian, focusing q et al ıa Aeroespacial, ıguez limiting probability distributions ) are given by Dirichlet distributions [email protected] Theory and Experiment Theory ). The Dirichlet distributions generalize 2 -Gaussians and Beta distributions via a x -dimensional scale-invariant probabilistic →∞ q β 2,3 and d 023302 − ısicas and Instituto Nacional de Ciˆ N + 1)-dimensional hyperpyramids (Rodr´ defining a normalizable 53 =e d 2 β x β − 1 atica Aplicada a la Ingenier´ and q = 1), the corresponding limiting distributions are , with e d and C Tsallis 2 x ecnica de Madrid, Pza. Cardenal Cisneros s/n, 28040 Madrid, J. Math. Phys. 1 β . It was formerly proved by Rodr´ − q e We show that the rigorous results in statistical mechanics ... ∝ stacks.iop.org/JSTAT/2014/P12027 ıguez [email protected] = 1, 2, d Departamento de Matem´ Centro Brasileiro de Pesquisas F´ Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA for dimensional case ( Gaussians ( of a recentlymodels introduced based family onand of Tsallis Leibniz-like 2012 ( linear transformation. Inregion addition, of we parameters discuss the probabilistically admissible doi:10.1088/1742-5468/2014/12/P12027 Keywords: E-mail: Received 4 November 2014 Accepted for publication 30Published November 22 2014 December 2014 Online at Abstract. A Rodr´ 1 2 Tecnologia de Sistemas Complexos,Janeiro, Rua Brazil Xavier Sigaud3 150, 22290-180 Rio de the so-called Betaconnection distributions to between higher one-dimensional dimensions. Consistently, we make a particularly on theGaussians, the possibility latter ofto corresponding, negative having in temperatures. an both appropriate bell-shaped physical interpretation, and U-shaped Universidad Polit´ Spain

ournal of Statistical Mechanics: of Statistical ournal

2014 IOP Publishing Ltd and SISSA Medialab srl 1742-5468/14/P12027+15$33.00 c Leibniz-like pyramids distributions and a scale-invariant probabilistic model based on Connection between Dirichlet J ⃝ J. Stat. Mech. (2014) P12027 2 8 9 2 4 5 3 7 ], 12 15 14 14 ))) 12 x – ( p 10 ln( ], trapped x d ], motion of 19 – ! 13 k 16 ] the nonadditive − 5 – = 3 BG -Gaussian distributions ], long-range-interacting S q 7 ) has raised much attention ≡ 5 1 S -Gaussians as attractors of the q ], plasma physics [ 15 ], cold atoms in optical lattices [ , 9 1) and scale-invariance (a specific way to 14 ], granular matter [ ≫ 6 -Gaussians (to be introduced in detail later on, N q ], for 2 , usion [ , similarly to the manner through which Gaussians 1 ff N ], solar wind [ R 8 ∝ ∈ ) -Gaussianity (having q q N , ( ] -Gaussians as attractors has been up to now unsuccessful. q S ) q x -statistics [ ( ], a number of probabilistic models have been introduced with q p 1 [ x − 21 q d ! − ]. Nevertheless, the search for a statistical model which provides both 1 and other micro-organisms [ ̸= 1 and -Gaussians have been found in analytical, numerical, experimental and k 26 q q – = ...... 22 plane q 3 S ) emerge when optimizing under appropriate constraints [ β ] =1 =2 ! 2 – 1 d d d q ], among others. 20 -Gaussians and beta distributions 5.1. 5.2. 5.3. The relationship between Limiting probability distributions of a family of scale-invariant models Introduction q The Dirichlet distributions Conclusions Acknowledgments References entropy with Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids q entropy [ in section many-body classical Hamiltonians [ doi:10.1088/1742-5468/2014/12/P12027 Within the framework of 1. Introduction 5. 4. 2. 3. 1. Contents emerge from the Boltzmann–Gibbs (BG) additive entropy ( over damped motion of interacting vortices in type-II superconductors [ observational studies of anomalous di in the standard BGnaturally statistical extend mechanics. Gaussian Physicallysystems. speaking, In distributions fact, for nonergodic and other strongly correlated 6. Hydra viridissima ions [ S introduce correlations in the system toin be recent described years. in section Sincethis [ purpose [ probability distributions inentropic functional the satisfies thermodynamic limit), extensivity (whose associated J. Stat. Mech. (2014) P12027 3 0 (2) (1) > . β 3 function and present = 1, ternary 3 for d d (0, 1) (4) q< ∈ exponential 2 , as expected for the - y π β q 0 (the boundary of the √ √ . We will also comment 0 and 0 otherwise) with 2 q> ; < = q ! 1 − 0, -Gaussian distribution with 0 0 and the bounded interval β 1 z -Gaussian in section q . The one-dimensional case, > q + 1)-dimensional Leibniz-like 5 β < )) > , if 2, q y d β -Gaussians and symmetric Beta z β C q − 1 ) for any value of q> = 4 → (1 q + 3 y ] ( z [1, 3), ) defining a q q 1 − 0 and ∈ 1 and 1 into the interval (0, 1). The probability β − q 4 > > x + ] = q z q< β β 1) (3) ] y x ; D 0 case, thus having a compact support). The expression , and we have made use of the d d 2 − ; +1)-valued variables (binary variables for = 1, 2 and 3 in section 27 x x for ) > d β )) y d ; 1 2 ( 2 d , y } − q 1) ) ) x β (2 ( q 1 2 e ) − 1) β , q ) q β x − N q − , q q ( − − q q ; q q 3 (1 β − − e 2 1 2( β β − C q − ( ( ( √ q √ D B B 1 (1 ! (the symbol [ (1 0 (the negative G β )= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ β q x = 1 ≡ − " < ; 1 + = " -Gaussians are defined as follows: / ) ] 1 β β q , 1 y β x ( − q β , ( = = 2 and so on), based on a family of ( ) q q q C q < G C f x d , | ) stands for the Beta function with lim − -Gaussians only in the one-dimensional (corresponding to binary variables) x y R | q , 2 for ] we introduced a family of x ∈ ( (so a Gaussian distribution may be interpreted as a 22 x/ β B x { [1 + (1 q> -Gaussians are normalizable probability distributions whenever -Gaussians and beta distributions = q In such a context, little attention has been paid to multivariate models. In a recent In this section, we will be mainly concerned with the bounded support case. We shall q ≡ =e Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids q = 1). x 1 x q 2. One-dimensional variables for for the normalization constant is given by interval is reached only in the e D q doi:10.1088/1742-5468/2014/12/P12027 on the relationship between parameters which will allow usdistributions, to up to establish now a unnoticed, connection will between be presented in section hyperpyramids. We showed thatlimit the were corresponding distributions in the thermodynamic paper [ corresponding to the sum of which transforms the bounded support distribution function for the new variable is e literature before). The support is and Gaussian distribution. do the linear change of variable [ case. In the presentDirichlet contribution distributions we show (toa that be the detailed family described demonstration of in for attractors are section the so called where where J. Stat. Mech. (2014) P12027 4 5 ). ), 3 β and , q 1 (0, 1), − ∈ β y plane will -Gaussians , semiplane is q 1 β − 1 – 2 − q -Gaussian with α β q y ) y − (1 1 − 1 ]. In the upper half-plane, 0 values yield α y 28 ) shows such a plane and how < 2 α 1 , β (0, 1) (5) 0. As we will show in section 1 1 plane with vertical axis α ∈ ( > B β y q q – − − q ) and the duplication formula of the 2 1 ) can be reexpressed and we finally get ; 2 0 and having two symmetric inflection )= q 4 2 , any allowed point in a 0 is convex [ 1 − = α 1 T , ) 2 < 1 q< y α . We thus conclude that under change ( α β q ; − , in the 1 = and − y 2 1 ( x 1 (1 q given in ( β -Gaussian. Figure f q α q − q 1 − β , 1 q y 1= C ( ective temperature, the − q q erent ff 1. Shadowed regions correspond to forbidden values of ( 2 − − ff 2 1 α − , q q q -Gaussian with 1 q − − 2 1 -Gaussians, e 1= ' q B − 1 -Gaussian with bounded support transforms into a symmetric Beta α q )= y ( q 0) while any horizontal axis f Figure 1. compact (open). See text for the limiting distributions in the diagonal lines. The support of the uniform distribution in the upper (lower) plane the case in higher dimensions. β plays the role of an e q> 0), with – 1 q − > not β 0 points downwards (being concave for 2 α Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids , > 1 -Gaussians look like depending on their location in it. To start with, any points for be in correspondence withq a di β that are tosystem be is determined associated by with the systems values of at negative temperatures. As the state of a a one-dimensional distribution function with parameters doi:10.1088/1742-5468/2014/12/P12027 Making use of the expression for Gamma function, the normalization constant in ( which is a symmetric Beta distribution ( Since 3. The this is α J. Stat. Mech. (2014) P12027 ] ) ) ) q 5 q } } + 3 3 q ) ) 2 D ( (7) (6) +1 q q − d 3 β with , 1 α α − − → q 1 + − , 3 from 2 2 q D (1 (1 α α < / / ... + 1 ) ) d 1 , -Gaussians q< q q − α 2 x q , α ] +1 − − 3 + , d 1 − α 29 0 are shown for 3 ) (2 (3 α d 1 α ··· ) and considering + − x " " < 3 2 1 + α α ) on the plane. In [ Dir( − β 2 " " q + = 0, a double peaked ( x | 1 | where distributions are ∼ α x β 0 ··· x = 1. In the following we | + | < − ⃗ 3 X 1 β − 3 , α ) x q . 1 x/ 3 6 x/ x C { → = α 0, { + q 2 − 2 = ). We shall use this property in = > j α = q → 2 (1 α d q q 1 i D = x α = 1, the Gaussian is shown. On its − ̸= , D 1 d j q = α d α ). In analogy with the path followed in . Thus, we obtain again a Dirac delta 1 ... x ) 5 , , α i 2 ∞ ··· x α ] and lim , 1 = 0. On the left top quadrant, 0 is kept constant. In the opposite 1 − β 0 1 < 1 α − > we obtain /x β 0 are shown for decreasing values of Beta( x , √ β q d q 0; 0) = ) ) i / i 1 − > ∼ 0 are shown, all of them with bounded support C R 2 α α 1). If two of the parameters equal one a plane surface < i ,1 ( − β ∈ > 3 β Γ X +1 is kept constant. Nevertheless, the same limit can be ). < =1 β ) d i 1 → β ( − d 3 is depicted, whose support is the whole real axis. As 3 q )= q − β -Gaussians for a constant value of +1 =1 x q ) α d i ) √ q ( , G ( , q / 2 β * Γ 1 − α limit when ... − , →∞ , simplex )= α ((1 x 6 2 from left to right. All of them have a bounded support =[ , 1, 1] so in the limit a uniform distribution is obtained instead, as O +1 > 1 . They present a maximum (minimum) at ( 2 1( was made. Thus, the support is d ! →∞ 3 x 1, 1]. Thus, the support remains finite while the area is preserved, so − ( α q D α q +1 > q − , 1 ). + − d )= erent paths given by the graph of any function -Gaussians turns flatter up to the limit 1 3 3 =[ q = 0, while the area is preserved. Thus a Dirac delta function is obtained ff α . The same limiting distribution would be obtained following any limit provided q − ( α = , =[ ... 1 q ) q and , β , -dimensional Dirichlet distribution is defined in the form [ q q 2 D 1 2 D D d )= ... α shows bidimensional Dirichlet distributions for the specified values of pa- α q D α − , , ( ; , = 0. Also, lim 2 1 2 d 1 β α α q x α →−∞ ((1 →∞ →−∞ -Gaussian with 1 , → −∞ . , q q q 1 q D 1 and a constant value of 5 O q α = 1 distribution ( ... , d →∞ 1 Figure In the lower plane, convex q< )= q Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids q x ( ( -Gaussians for a constant value of with lim f curve provided distribution in the with lim we recover the uniformβ distribution. Again, the same limit is obtained for any path with doi:10.1088/1742-5468/2014/12/P12027 limit, with support with lim further increases q right to left. Inright, the a vertical axis, corresponding to The so called 4. Dirichlet distributions shown in figure rameters section is obtained, including the uniform distribution for will be mainly interested in the symmetric case increasing values of where no longer normalizable, with lim the choice in the taken following di the upper half plane, taking delta distribution isthe obtained corresponding as limit of can the be beta seen distribution ( by doing the change ( whenever For generalize Beta distribution todimensional higher marginal dimensions. distributions In of Dirichlet addition, distributions it ( may be shown that one- for are Beta distributions in the form lim J. Stat. Mech. (2014) P12027 ) 6 = 8 (9) (8) q , and Ω d =5 x q =2 3 d Ω 3 α α , ∈ ··· , 1 ) . They have a =3 d x -th one defined 3 0 having all its =1 2 x d 2 , α m x ⃗ α < , Σ 0 the height of the T 0 (again, the border ... = β x , =2 1 β⃗ 2 < 1 α < − q x α erent values of parameters e , β . If we even further take β 1 ff q x -Gaussians with bounded Ω 2 2, q ! / 1 2 when | =( Σ | = ) q> π T q 0). For x − ⃗ 1 q> d , =1 =4 (2 − q 3 3 β > ; C α α = 2 case for which the normalization x ⃗ 0 and 1 β = d = = Σ − 0 and the solid hyperellipsoid > 2 2 T α ,2 α α x -dimensional 0( q β )) β⃗ 2. d > = = y C q − 0 and for 1 1 e < 1, α α − d 0 case). It can be shown that distributions ( β , q< > β q x -Gaussians are plotted in figure > C q< q β 1, − β (1 case. )= ! d , for d xy } x when ( ) R , q q 1 2 ) 3 -Gaussian distribution reads 2 d ) − 1 α q able to be transformed into Dirichlet distributions via a = =2 = ... α (1 q 3 , 3 ( (3 β , if 2 α α . Ω Γ Γ 0 the height of the maximum decreases and the radius of the d 1+ x 3 not = = , R α 1 > 2 2 Bidimensional Dirichlet distributions for di x< ⃗ x α α )= ; β q< Σ = α = = and in the Σ is a positive definite matrix, T ; 1 1 2 are , 2 y q x ⃗ d α α , the corresponding symmetric Dirichlet distribution reads β , α Σ + 2 Ω ( -dimensional / α , x ! q m ( 1 d , ) ) takes the simple expression G Figure 2. α f d ≡ d R 8 x 3 , 1+ α ∈ ... = , 2 whereas for β q< 2 2 α x , In turn the Contrary to the one-dimensional case, Some representative bidimensional 1 = q> Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids x 1 ( only if doi:10.1088/1742-5468/2014/12/P12027 moments defined in the bounded support case and moments up to the { α constant in ( support and change of variables. Let us focus on the simplest support increases for increasing values of minimum (maximum) at the origin for the support is where of minimum increases and the radius of the circular support decreases for increasing values are normalizable for of the support is reached only in the J. Stat. Mech. (2014) P12027 , q 7 N ⃗ (10) X + ··· + 2 ⃗ X + can be associated 1 ⃗ ⃗ X X for simplicity) reads = ) has only symmetry of ) for typical values of I 9 8 ⃗ X 5 4 = 1, while the corresponding =0 = Σ ) via a linear transformation. q 9 1 ,q − ,q 1 + y< )] =1 =1 2 y -Gaussians ( + β β q ) into ( + ) of the triangle support. Even in the x 3 1 2 10 , x 0, 1 3 ( β ) q . ) y> q − − 1 0, -dimensional variables (thus (1 (1 β d − < 4 [1 2 / x> ) 1 2 y q = =9 + − π 2 possible to convert ( ,q x ,q (2 1 β + 1)-valued − Plot of some bidimensional =1 not d ( = β 3 about the barycenter ( )= / β y ) has cylindric symmetry while distribution ( . N -Gaussians (or equivalently simplexes in hyperellipsoids) is out of the π , β q x 10 ( 1, it is q -Gaussian, once diagonalized (we shall take q G and Figure 3. − α = q 1 − ], we introduced a one-parameter family of discrete, scale-invariant (in a sense to be 1 Distribution ( 22 Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids bidimensional doi:10.1088/1742-5468/2014/12/P12027 and is defined on the triangle that is the sum of In [ 5. Limiting probability distributions of a family of scale-invariant models bounded support scope of this paper. Whether there exist highly non trivial changes that transform Dirichlet distributions into case rotation of angle 2 being defined on the circle made explicit below) probabilistic models describing a variable J. Stat. Mech. (2014) P12027 ) ) 8 d N , x 11 11 =1 , (16) (11) (13) (14) ... cients d events ... ffi , ( 2 d . (12) = 1, x n ) stand for ) , , i ν 1 11 cients in ( ... x , N + for 2 ffi d i n , e ⃗ n 1 ( = 1, a pyramid for n 1)-variables subset Γ ' d − . ··· d cients in ( ; with ) nonnegative integers for 2 ) may be displayed in a N . Thus, the sample space ffi e ν ). This condition, trivially ⃗ i i 1 15 n − ⃗ − + X ) 2 N = )): ), that for the simplest ν x n d d ( , random variables n ε 12 ⃗ Γ d , ) n cients ( d , + 1) ... ), with N ν cients. The associated probabilities ε ⃗ , d ...... ffi d , + 2 . , coincides with that of the joint ffi 2 n , + n ⃗ ν x ) is scale-invariant when the functional 2 n N , 1 1, , 1, and 0 (15) 1 +( N n cients ( 1 = x ) − n = n , ) x ν , − ( x > ( 1 N d ffi ν , ( )d N ( d N Γ r ). Coe n 1 ( +1 ) ν , r X . N regions, such that the ( d − Γ d , ν The family of probability distributions ( x 13 ... = α ) . N , , R ≡ d ... d d ; p 2 + ε ⃗ . The multinomial coe ... + ) = n i x n ⃗ , + , ... ν N , n 1 N n ⃗ ( ) , , 1 ν n 2 = 2, ... ··· + x ( N ... =1 " x ( i d i r ! , N , n = d n = N 2 d +2) 1 has, in the thermodynamic limit, symmetric Dirichlet , ) 1, + p ) 1 subset 1 1 n n N x ν ν ) − 0 , ( α , ν − , for X 1 − ( + N + N 2 ν ( ··· n r n +1)( > ( ) e p N , ⃗ n P ν N N - + ( N B ( ··· ( ν ! − = r 1 ) reduce to + 1)-sided dice) with a probability function in the form − Γ ≡ ε ⃗ = + +1 ) . d +1 + i 12 d d ] for details) in the same fashion as the multinomial coe ( N 2 n n ν ⃗ e ⃗ ) ) n n , , n ( , , N n ν ν 22 ( ( N N +1 ...... B N )= = - , , d , and satisfy relation ( r r 2 1 2 + ) i + 1) n n − 1 dimensional marginal distribution of a ( , ε , ⃗ 1 N ν = = + + 1 1 ( , d , N n cients ( n n n n n n ⃗ ) 2 ) ) , ) ) ) , , Γ , , , − x (( ν ν ν ν ν ν e ⃗ ffi ) may be displayed in a hyperpyramid (a triangle for , ( N ( N ( N ( ( ( N N N ... Γ p r p r r p 2 can be cast in the form ( − N 12 1 ... = , and e 0, ⃗ ! , d d 2 d n , events splits into = x , > 1 , = 0, 1, ... 1 , 1 ν N = ε ⃗ ... 2 cients ( x d n n ( , d 1 1 ffi We say that a probabilistic system consisting of Our main claim here is the following: We shall develop our proof separately for increasing values of n ) − , Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids = 2 and so on, see [ ν = 1, N ( N ˜ p with doi:10.1088/1742-5468/2014/12/P12027 5.1. In this case coe with joint probability distribution with the throwing of do, and due to the aforementioned scale-invariance satisfy certain relations among them. Coe d probability distribution of the i characterized by parameter form of the while for the degeneracy arising from the exchangeability of variables traduced in the formcase of reduce to restrictions the on so the called coe Leibniz rule fulfilled in thethe case presence of of independent longe-range variables, correlations. involves, In in our the model, absence the of scale-invariance independence, condition is being the vectors of the canonical basis of where with defined as r belonging to each of them take all the same probability given by the coe where distributions as attractors, with triangle in the planeare as is the case of the Pascal coe J. Stat. Mech. (2014) P12027 ) ) ) 9 n ) , c b a ν ( N )= (17 (17 x (17 , with ( Np ) x ν shows a 0 out of ( ) N ln ) one gets 0 (22) 4 P 16 , as seen in > 0, as above 1. Applying − 15 2 1 > − − β " = ν ν ν > . (18) ) in ⋆ 1 2 x ν = u 16 ; . The convergence )+ " ) q x = ν N 1 and − ,0 2 (1 + n α N N ) ) reduce to m q> x , (0, 1) (20) = = 12 ν − x ∈ 1 u -Gaussian is convex, whereas cients one easily gets + α x d q (1 ) ffi n 1 2 u ν ( − d + ; . (19) ) cients ( B n t 1 , with ) u 1 2 ( ) ) Nx , after some manipulations one gets ( d u − ffi ν ν ]. 3 1 x x ν has a maximum at )+ − xu x ) − − Nf x ,2 e ) 23 1 ν e x × 1 f − ν − ν +2 u + ( − 1 , (1 − ) ν n 0 and the ν − − ν u in [ ) = 100 (dotted lines) together with their B m N − -Gaussians with + ν ( t ) ) ) is apparent. − e n ) q < (1 ν 1 e ν x x + B )= 1 − x − 1 ( , N − ⋆ β − N ν n − ν − u -Gaussian distribution with ν − ) ) , ( ) where (5) (1 ( q x u u ) yields 1 c ν ′′ . (21) -Gaussians pointing downwards are obtained. As (1 P B = 5 and increasing values of − − (1 − and f 1 2 q e e ( 19 ν ν 17 + ) (solid lines). The right panel of figure ν 2, so + + x ) ln(1 0 the demonstration on the attractor of distribution )= in the definition of the Beta function in ( ν − − n x m 20 x +1 Nx u − × ( > transforms probability distribution ( for 0 so ) (1 (1 x − − − N y , and q> ν π e t shows normalized probability distributions ( ν ( n ) ∞ ∞ n x ) , N n 2 ) and ( > 1 × q 0 0 ν − P ( N 0 4 / / ln 18 − β 0 = / − N ≡ x = = ) = (1 Np π N =1 1 ≃ = ) (1 u 2 ( x t ( n n β ) ) Nx )+ , , ( B √ f ν ν ) x ( ( N N 1 and ν " ( r r N ) transforms into a = ≃ ' ) ) − erent values of P (0, 1) we get ν ν 20 1 2 ff m . , , , q< n ∈ ν ν ) , n N = →∞ ( ( ν ). lim ( N ν ) ln(1 N - x r B B 16 x for di ], by means of a more complicated nonlinear change of variable that we will − 27 . For 2 1 we get 2 n/N = , taking into account ( > n ) , ) = (1 d ν ν ⋆ The left panel of figure By doing the change N ( ) given only for positive integer values of Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids u ( 16 doi:10.1088/1742-5468/2014/12/P12027 Let us now turn to the bidimensional case where coe 5.2. corresponding Beta distributions ( detail of the distributions versus We shall now generalize( to any where we have defined Np Beta distribution ( which is a symmetric Beta distribution with parameters distribution ( for shown in [ not show here, it is also possible to obtain section claimed. Next, by only doing the inverse change of ( f Finally, as the change Applying now Stirling approximation to the binomial coe now the Laplace method to integral ( to the corresponding Beta distribution J. Stat. Mech. (2014) P12027 ) ) ) ) " b c a y 10 y + (23) (25 (25 (25 x + x = 5 and ), which 1, ) (in solid ) ln( = 50. For ν 14 x y ( " ) ). + N ν for x y ( x ( , 2, 2 and 10 with n P ) x , / (5) ν N ( and )+( P " y =1 ν Np − ν ,0 x m N the first Beta function − ] as the Pascal trinomial for = u 22 ) transforms as − y e , ) ln(1 n/N 22 n y N − = − 1 u = x x − d ) x m erent values of ν − , =1 n t ff +2 1, d t 1 m ) − versus ν − + ( N u ν , with n n ) = (1 r ) ( , d u ⋆ ν ) u +2 ) ( N u u ) for di = − ( y m ( e ) obey the corresponding relation ( 1 + f 1 24 + Np Nf n − − ) yields a bidimensional Dirichlet distribution 22 x ) e ν m ( t , u + ν n 24 ) , − 2 − m ν . (24) ) ( N − − u ) in the numerator of ( r m e ), with , n (1 − ν 1 y n 1 − ) e , cients ( + − ν − N ν + ( N 1 − ν ffi ) ) +2 r x − + u u m − − . m m ln( e e − m n ) ln(1 +1, + . Coe , − N − − n − y ) , n n ν N N ( N , = − (1 (1 t (Left) Distributions - = 1 the uniform distribution is obtained. r ν x ⋆ 1 " ∞ ∞ ν 0 u = 0 0 − + + / / / m m m , , m = n + n = = ) = 100 together with their corresponding Beta distributions ) , = 100, 200 and 500 compared with the Beta distribution , ν ν 1 . For − n ( N ( N ) = (1 1) a minimum (maximum) is observed at the barycenter of the triangle B N Figure 4. line). (Right) Detail of the top part of the distributions p N r N n u 0, ( > − " f ! shows probability distributions ( ν N m ( m 5 , + B = 2 yields 1, ( n cients do. The corresponding family of probability distributions is n We shall now prove that family ( d ≡ ffi Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids < " 1 ν 0 where now doi:10.1088/1742-5468/2014/12/P12027 with B Figure and may be displayedcoe as a pyramid made of triangular layers [ 1, has a maximum at in the thermodynamic limit. With the change for J. Stat. Mech. (2014) P12027 ) ) ) b c a 11 (27 (27 (27 , with ( ) in the = 50. y ν y + x N + ' . (28) . (26) 1 2 ln m 2 1 − , ) − − ν y ) y + = + x + ) yields ( x ⋆ ( c N n u N ( − ) 25 ) y B y ). The Laplace procedure y + + 2, 1, 2 and 5 and ≡ + / x x ( 2 ( x 1 2 ( 2 1 B =1 =5 =1 y x − )+ ν ν y ν − Ny − x y 1 2 − )= ) for (1 + ⋆ N u 24 ) Nx u ( u has a maximum at y ′′ x d d ) f ) ν − u × yu ( t + x ν d m 2 − ( 1 Nf u − − e − ) 1 − ν u u ), and ) ν e + − (1 y y 1 − e m − e ) + × ν + 1 t − ν x + − x 2 n ν − ( ) ) ) ν y u u 2 y ln(1 (1 ) ln( − − / 1 1 + y e e x − − =2 ν x ν + − − =1 ( x ν + 1 x ν n )= ( − π Probability distributions ( (1 (1 t u ν N 2 1 ( ) ∞ ∞ − . Applying the Laplace method to integral ( 0 0 0 y f y y 0 / / / y − − x + ln = ≃ x = = ) one gets − x y 1 2 2 ) yields 22 c + − B Figure 5. B − x 27 (1 ln )= π ⋆ N x 2 u ( ′′ 0 )= f ⋆ ≃ Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids u 1 ( B doi:10.1088/1742-5468/2014/12/P12027 and in integral ( where now function numerator of ( Following the same steps in the second Beta function f J. Stat. Mech. (2014) P12027 12 =2 ) in =2 (31) ν shows ; 2, 4). 24 ν x 7 ( for f m , n = 50. Larger ) , 1 (30) ν ( N N p . Figure n ν y< − =0 for N m + =2 x ) 2 = 50 (dots). = 2 compared with the . (29) cients one gets α 0, ν ≡ 2 1 m/N , N ffi ) (solid surface) for ν n ) )+ , ) yields ν = y 30 ( N for y> = − ˜ y 29 ) for , P x 1 m x − , α 30 n (1 , and N ; (2) N with ) 1 p y 2 ) and ( − n ) , ν ν N ( ) − N 28 ) ) n/N ˜ y P ν x ν ) 1 ν = ), ( − N + ,2 − ) can be expressed as ν x x 26 ( , transforms probability distribution ( m +3 12 (1 , − 2 1 l B m ν ) + + ν (1 ) + 1 = , Ny ν − versus ν m y n ν ( , the marginal distributions of symmetric Dirichlet ( 1 2 cients ( ,3 y 4 m + B ν , + 1 Ny B ffi ( n ) − , n , ) ν Nx , ν B (2) N n ν ) x x ν p , + ν . 2 ν l − = ν N ( ,2 N 1 , l π )= compared with the corresponding Beta distribution ν ν B = 2 y , which is a bidimensional symmetric Dirichlet distribution with ( − , ) ) 3 Bidimensional Dirichlet distribution ( Nx ν ν N B x ≃ ( α +2 ( m 3 (3 , taking into account ( ) Γ Γ ν . = m m ( − , 2 n m = ) , P n + , ν α N ) ( N n n ν − ) are Beta distributions with parameters p ( = 2 ,2 - )= Figure 6. together with normalized distribution ν 30 1 B N ( y shows the limiting probability distribution ( N ( , α B 1 provide a closer aproach of both distributions. In order to get a one dimensional ) 6 x × B ν ( , ) N 3 ν )= ( ν erent values of = ( y N B ff ! l , = 3, after some algebra, coe , P x d m ( d , Figure ) n ) , ν Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids →∞ ( ν N lim ( N N P distribution ( view of the convergenceof we the shall property resort to stated the in marginal section distributions. As a consequence where values of doi:10.1088/1742-5468/2014/12/P12027 r 5.3. For Next, applying the Stirling approximation to the trinomial coe normalized marginal probability distributions parameters Finally, as the change The convergence is observed as expected. corresponding distribution and di J. Stat. Mech. (2014) P12027 ) ) ), + b a 13 14 n (32) = 2, , (34 ν 1 α − l − l m = 100 and 200 1, − − m , N n n . (35) 1, 1 2 ) − − for ν ( N )+ r n z N , 2 1 ( − ] as the Pascal tetranomial (2) N = y − ˜ B ) P l − 22 z x 1, . (34 N + − ≡ 2 1 − y (1 + − m 1 , ) x N n z ( ) , B ) ν + N z ( N y ) r + z − x ( + ) obey the corresponding relation ( y + N 31 y − +1 − l ) 1, z + x 1 − ) as cients may be approximated via the Stirling x + − m ( ν , ν ffi 1 2 . (33) y 3 n l ) , , ) (1 cients ( + ν )+ ( 2 1 N z ν + m l , z ffi 3 r , + n − ) x , − + ν y ( ν 1 + ( N Nz ) 1 2 y − l r z z x − 1, 1 2 +2 . Coe + − . − + l + Nz (1 x N m , m z ( y N Ny 1 1 2 ) m " N + +1, y − z + , n 1 2 )+ ν l ) , n n y ) x ν + ( − ( N ( z + Marginal probability distributions - + r ν x y B ( Nx z − = 1 m + x N − ≡ ) y l l − , 3 2 , ν y 2 + x ) 2 m m − , , ) B n n + n = 4. − N ) ) , 1 , y x ν ν 2 π x ( N ( N 0, (1 ( + − α Figure 7. compared with the corresponding Beta distribution with parameters p r (2 x ! × × = 3 yields ( (1 ), and l ≃ d , ν π π N N 2 2 . m l , , +3 0 0 n cients do. The corresponding family of probability distributions is l m N , ≃ ≃ ffi + Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids n 2 1 - B doi:10.1088/1742-5468/2014/12/P12027 with On the otherformula hand, as the tetranomial coe corresponding approximations of the Beta functions We shall not go through the details of the demonstration and simply show the m B and may be displayed as a hyperpyramid made of pyramids [ coe which for J. Stat. Mech. (2014) P12027 ) 0 ∈ 14 24 2 < α β = 1 2 (for any -Gaussians α q q> . ν random variables Curado, as well as = ) yields N 4 35 ; (36) α 1 -Gaussians have to do q − = ν plane: that of the -Gaussians have cylindric , which is a tridimensional ) 3 )) but in some alternative ) q ) ) 1, ) (symmetric under changes z α ν ν ν dEMF β ), though having ( ) and ( 22 – 4 (4 24 − b = ,3 q Γ Γ ) based not in the triangular 11 y ν 2 34 ( 24 d> = α being the number of sides of the − B ) ) ), ( x = ν ν cients ( a m ,3 1 ν − -Gaussian limiting distributions in the ffi ( ,2 α 34 q -Gaussians as attractors. In this respect transforms probability distribution ( B ν q ) (1 ( -Gaussians exist whenever l ν 1 q B ,2 1 − = ) ν ν , with ( , J P Gazeau an , bidimensional Dirichlet distributions have z ν B 4 1 , ) /m − ν ν Nz , ν ( π ν , = 1, have instead Dirichlet limiting distributions ( y -dimensional symmetric Dirichlet distribution of B 1 B d m d − ν 3 whereas bidimensional = x / π as a ). Ny 3 )= d , z . This is due to the fact that, even if the 1, where limit for cients (as well of coe n , y , taking into account ( ffi N l , = , x m z< ∝ , ( ) n ) ) , →∞ ν + Nx ν ( N ( y N ) do not have bidimensional p P ( N 3 + m BG N cients with square, pentagonal, hexagonal symmetry and so on (thus x − )= S present sets of probabilities, the entropic functional which is extensive is ffi z n 0, , )= N − y , due to the cancellation of intermediate terms in the product of Beta z ), and can be interpreted as describing negative temperature systems. These > , ν , , d N x y z ( 1. Nevertheless, we should realize that, for , ) limit. Work along this line certainly is welcome. Let us finally mention that for , ν x ↔ y ( N ( , ) P x ν m ( N d> Following the above prescriptions it can be obtained the limiting probability We have analytically shown that discrete distributions ( →∞ P Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids ↔ →∞ lim N with doi:10.1088/1742-5468/2014/12/P12027 Finally, as the change in the BG one, i.e. all the finite- partial financial support from CNPq, Faperj and Capes (Brazilian agencies). symmetric Dirichlet distribution with parameters We have visited an up to now unexplored region of the 6. Conclusions Beta functions is too lengthy and tedious. distribution for anyparameter value of functions, though a general expression for the corresponding approximation of the involved polygon) would be the appropriatem method to get symmetry of trinomial coe n a possible generalization of probability distributions ( symmetry of rotationsymmetry. This of is angle the reason 2 why probability distributions ( as attractors in the for (0, 1) under the linear change ( Acknowledgments We acknowledge fruitful remarks by G Ruiz are strongly correlated, theshrunken set or of modified. all admissible microscopic configurations is not heavily dimension distributions are in correspondence with symmetric Beta distributions with half-plane, showing that a family of U-shaped generalized coe having symmetry of rotation of angle 2 with symmetry. As mentioned in section J. Stat. Mech. (2014) P12027 15 453 260601 1542 105 Models and ( 123301 54 3874 AIP Conf. Proc. 7 Phys. Rev. Lett. 2013 2010 15377 549 J. Math. Phys. 102 293 nAPF 813 (New York: Springer) A 2013 oEMF 110601 73 286 P09006 96 307 Cent. Eur. J. Phys. 191 263 393 217 75 215003 341 Physica A 72 351 95 176 sLMCS 80 B Continuous Multivariate Distributions 34 023302 R2197 347 e F D and Curad 375 Physica 53 J. Stat. Mech. 061111 54 Europhys. Lett. 965 222 055003 E Phys. Rev. Lett. 2361 78 2398 356 A Milan J. Math. E A 52 60 Proc. Natl Acad. Sci. USA 615 100 Eur. Phys. J. 9 E E Phys. Rev. Lett Space Sci. Rev. 063001 J. Plasma Phys. u J P and Rodrigue Physica 102 n K P and Tsallis C 2009 012304 Phys. Rev. Physica Tsallis C 2014 P10023 J. Math. Phys. Astrohphys. Space Sci. 479 Phys. Rev. r J A and Sawada Y 2001 19 AIP Conf. Proc. Moreira A A, Nobr 051402 52 Gazea Nat. Phys. Phys. Rev. Phys. Rev. a L F 2013 67 s D J 2014 Phys. Rev. Lett. A s A F 2005 n D J 1996 a G F T, s V R V and o S 1995 o E M F, J. Stat. Mech. ff Phys. Rev. Lett. Phys. Plasmas Assi chwammle V and Tsallis C 2008 vol 1) (New York: Wiley) J. Stat. Phys. Introduction to Nonextensive Statistical Mechanics Rieu J-P, Glazie Phys. Rev. n J D and Burlag t H J 2010 o A R and Plastino A 1995 a L F and Vina e R G 2009 hyaya A, ıguez A, S ıguez A and Tsallis C 2012 s A S and Tribeche M 2014 o L J L, de Applications Tsallis C and Buckma Antoni M and Ru Burlag Yoon P H 2012 Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids [4] Umarov S and[5] Tsallis Umarov C S, 2007 Tsallis[6] C Plastin and Steinberg S 2008 [1] Tsallis C[2] 1988 Tsallis C[3] 2009 Prato D and Tsallis C 1999 [7] Combe G, Richefeu[8] V, Viggiani Cirt G, Hall S A, Tengattini A and Atma [9] Richardso doi:10.1088/1742-5468/2014/12/P12027 References [10] Lutz E[11] 2003 Douglas P, Bergamini S and Renzoni F 2006 [26] Hilhors [27] Hanel R, Thurner S and Tsallis[29] C 2009 Kotz S, Balakrishnan N and Johnson N L 2000 [23] Rodr´ [24] Moyano L G,[25] Tsallis Thistleton C W and J, Gell-Mann Marsh M J 2006 A, Nelso [19] Bain [20] DeVo [21] Tsallis C, Gell-Mann M and Sato Y 2005 [17] Livadiotis G and[18] McComa Yoon P H, Rhee T and Ryu C-M 2005 [13] Andrade J S,[14] da Upad Silv [15] Sire C[16] and Chavanis Liu P-H B 2008 and Goree J 2008 [12] Lutz E and Renzoni F 2013 [28] Bergeron H, Curad [22] Rodr´