A scale invariant probabilistic model based on Leibniz-like pyramids Antonio Rodríguez 1,2 1 Dpto. Matemática Aplicada y Estadística. Universidad Politécnica.

A scale invariant probabilistic model

based on Leibniz-like pyramids

Antonio Rodríguez1,2

1Dpto. Matemática Aplicada y Estadística. Universidad Politécnica de Madrid2Grupo Interdisciplinar de Sistemas Complejos

http://www.upm.es/canalUPM/archivo/imagenes/logos/color/escupm01_new.eps

Outline One-dimensional model.

Scale invariant triangles. q-entropy.

Two-dimensional model. Scale invariant tetrahedrons. Conditional and marginal distributions.

Generalization to arbitrary dimension.

Conclusions.

scale invariance

extensivity

q-gaussiani

ty

1 2 1 1 2 1, , , , , ,N N N N Np x x x dx p x x x

A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

joint probability distributionN-1 variables

probability distributionvariables

marginal N-1

joint N

scale invariance

One-dimensional model.

x1

p 1-p

1 0

N=1

N distinguisable 1d-binary independent variables


1 2 NX X X X

1


x1

p2 p (1-p)

p (1-p) (1-p)2

1 0

1

0

x2

N=2




x1

p2 p (1-p)

p (1-p) (1-p)2

1 0

1

0

x2

p

1-p

p 1-p

N=2



p3 p2(1-p)

p2(1-p) p(1-p)2

N=3


p2 p(1-p)

p(1-p) (1-p)2

p2(1-p) p(1-p)2

p(1-p)2 (1-p)3

x3=1

x3=0


p3

p2(1-p) p(1-p)2

p2(1-p)

N=3


p2 p(1-p)

(1-p)2p(1-p)

(1-p)3

p2(1-p) p(1-p)2

p(1-p)2 1 p 1-p

N=0

N=1 N=2


,N nr

p3 p2(1-p) p(1-p)2N=3


p2 p(1-p) (1-p)2

(1-p)3

1 p 1-p

N=0

N=1 N=2

,N nr

+

+

+

, , 1 1,N n N n N nr r r Leibniz rule


p3 p2(1-p) p(1-p)2N=3


p2 p(1-p) (1-p)2

(1-p)3

1 p 1-p

N=0

N=1 N=2

,N nr,N n

N

np

111

2 11

3 31 1

Pascal triangle

CLT

N

2

1

2 (1- )

2 (1 )( )Np p

x Np

Np pp x e


Binomial distribution Gaussian

p3 p2(1-p) p(1-p)2N=3

Scale invariant triangles

p2 p(1-p) (1-p)2

(1-p)3

1 p 1-p

N=0

N=1 N=2

,N nr

0,0r

1,0r 1,1r

2,0r 2,1r 2,2r

3,0r3,1r 3,2r

3,3r


N=3

N=0

N=1 N=2

0,0r

1,0r 1,1r

2,0r 2,1r 2,2r

3,0r3,1r 3,2r

3,3r

1

1

121

12

1

4

1

4

1

3

1

31

6

1

2

1

2


Leibniz triangle

1

(1),

1

1N n

Nr

nN


N=3

N=0

N=1 N=2

1

1

121

12

1

4

1

4

1

3

1

31

6

1

2

1

2

(1),

1

1N nrN


Leibniz triangle

(1),N n

Np

n

N

2xe

1

3 1

41

4


N=3

N=0

N=1 N=2

1

1

121

12

1

4

1

4

1

3

1

31

6

1

2

(2),N nr


Leibniz triangle

1

2


1

21

2

1

1

53

10

3

101

51

5

1

101

10

(1),N nr

N=3

N=0

N=1 N=2

1

1

10

1

51

5

3

10

3

101

5

1

2

(2),N nr

1

10

2

7

3

142

7 5

28

5

283

28

3

28


1

2


11

2

1

2

(3),N nr



(1)2( 1), 1( )

, (1)2( 1), 1

N nN n

rr

r

; 1, 2,

( ),N nr ( )

,N n

Np

n

Scale invariant triangles1

2N

N

n

2 1

4(2 1)i jX X

0


( ),N nr ( )

,N n

Np

n

N

lim

21; ( 1)

1q


12 1

2

nx

N


2

lim lim( ) x

q qq x C e

( ),N nr ( )

,N n

Np

n

N

2

lim lim( ) x

q qq x C e

lim 0q lim 1/ 2q lim 3/ 4q

lim 8 / 9q

lim 1q 500N



( , )

( , );

B n N n

B


R. Hanel, S. Thurner and C. Tsallis. Eur. Phys. J. B 72, 263 (2009)

0

lim

21; ( 1)

1q

12 1

2

nx

N

lim

2 31; ( 0)

2 1q

/ 2

1/ 2( )

n Nx

n N n

(1)2( 1), 1

(1)2( 1), 1

N nr

r

; 1, 2, ( ),N nr

scale invariance

extensivity

q-gaussiani

ty

?

for 1q

?

q-entropy ( )

,01

( )1

qN

N nn

q

Nr

nS N k

q

ent 1q

scale invariance

extensivity

q-gaussiani

ty

?

for 1q

?





Conclusions

Two dimensional model

N=1

N distinguisable independent variables

1

2d-ternary

(x1 , y1)

p q

(1 , 0) (0 , 1)

1-p-q

(0 , 0)

1 1 2 2, , , ,N NX Y X Y X Y X Y

A. Rodríguez and C. Tsallis, J. Math. Phys 53, 023302 (2012)

Two dimensional model

N=2

N distinguisable 2d-ternary independent variables

(x1 , y1)

p2 pq

pq q2

(1 , 0) (0 , 1)

(1 , 0)

(0 , 1)

(x2 , y2)

(0 , 0)

(0 , 0)

p(1-p-q) q(1-p-q) (1-p-q) 2

p(1-p-q)

q(1-p-q)

p q 1-p-q

p

q 1-p-q

A. Rodríguez and C. Tsallis, J. Math. Phys 53, 023302 (2012)

N=2

N=0

N=1

, ,N n mr

1-p-qp q

p2

pq q2

p(1-p-q) (1-p-q) 2N=3

1

q3 (1-p-q) 3

p3

p2q pq2

p2(1-p-q)

p(1-p-q) 2

q2(1-p-q) q (1-p-q) 2

pq (1-p-q) q(1-p-q)

N=2

p2

1N=0

N=1

, ,N n mr

N=3

p

1-p-qq

p(1-p-q)

(1-p-q) 2q(1-p-q)q2

pq

q3

p2q pq2

p2(1-p-q)

q2(1-p-q) q (1-p-q) 2

pq (1-p-q)

p3

p(1-p-q) 2

(1-p-q) 3

++ +

+

+

+

+

+

++ ++

, , , 1, 1 , , 1 1, , 1N n m N n m N n m N n mr r r r

Generalized Leibniz rule

N=2

p2

1N=0

N=1

, ,N n mr

q2

q

q3

N=3

, , ,N n m

N

n mp

Pascal pyramidTrinomial distribution

1

1

1

CLT

N

111 2

2( , )

TX Xp x y e

2d-Gaussian

;,

N N n m

n m n m m

1-p-q1

p(1-p-q)

q(1-p-q) (1-p-q) 21

pq

2

p3

p2(1-p-q)

p(1-p-q) 2

p2q pq2

q (1-p-q) 2 q2(1-p-q)

363

3pq (1-p-q)

1 3

2

p

1

2

1

33

1(1-p-q) 3

1

N=2

p2

p

1N=0

N=1

, ,N n mr

1-p-q

p(1-p-q)

(1-p-q) 2q(1-p-q)q2

pq

q

q3 (1-p-q) 3

p3

p2q pq2

p2(1-p-q)

p(1-p-q) 2

q2(1-p-q) q (1-p-q) 2

pq (1-p-q) N=3

1,0,0r

2,1,0r

1,1,0r

0,0,0r

2,0,1r2,0,0r

1,0,1r

2,0,2r 2,1,1r 2,2,0r

3,0,2r

3,0,0r

3,2,0r3,0,1r

3,1,1r3,0,3r 3,1,2r 3,2,1r

3,1,0r

3,3,0r

N=2

N=0

N=1

, ,N n mr

N=3

1,0,0r

2,1,0r

1,1,0r

0,0,0r

2,0,1r2,0,0r

1,0,1r

2,0,2r 2,1,1r

3,0,2r

3,0,0r

3,2,0r3,0,1r

3,1,1r3,0,3r 3,1,2r 3,2,1r

3,1,0r

3,3,0r

1

(1), ,

2

,( 1)( 2)N n m

Nr

n mN N

1

2,2,0r

1

3 1

3

1

3

1

6

1

6

1

12

1

61

12

1

12

1

10

1

101

10

1

301

30 1

30

1

30

1

30

1

301

60

Leibniz-like pyramid

N=2

N=0

N=1

, ,N n mr

N=3

1

(1), ,

2

,( 1)( 2)N n m

Nr

n mN N

11

3 1

3

1

3

1

6

1

6

1

12

1

61

12

1

12

1

10

1

101

10

1

301

30 1

30

1

30

1

30

1

301

60

1

6

1

61

6

1

10

1

10

1

10

1

10

1

10

1

10

1

101

10

(1) (1), , , ,

2

, ( 1)( 2)N n m N n m

Np r

n m N N

Leibniz-like pyramid

0 m n N

(1), ,N n mp

n

m

30N

N=2

N=0

N=1

N=3

11

3 1

3

1

3

1

6

1

6

1

12

1

61

12

1

12

1

101

10

1

301

30 1

30

1

30

1

30

1

301

60

1

10

Leibniz pyramid

(1), ,N n mr

111

3 1

3

1

3

(2), ,N n mr

(2), ,N n mp

n

m

50N

N=2

N=0

N=1

N=3

11

3 1

3

1

3

1

7

1

7

2

21

1

72

21

2

21

1

141

14

1

281

28 1

30

1

28

1

28

1

28

1

14

1

42

(2), ,N n mr

1

1

3 1

3

1

3

1

(3), ,N n mr

(3), ,N n mp

n

m

50N

( ), ,

( , ) ( 2 , )

( , ) ( , 2 )N n m

B n m B n m N n m

B Br

Scale invariant pyramids(1)

3( 1), 1, 1( ), , (1)

3( 1), 1, 1

N n mN n m

rr

r

; 1, 2, ; 0

( ) ( ), , , ,,N n m N n m

Np r

n m

(3), ,N n mr

2 2 22

9(3 1)i j i jX X X Y

0

1

, 3N

N

n m

( ), ,

( , ) ( 2 , )

( , ) ( , 2 )N n m

B n m B n m N n m

B Br

Scale invariant pyramids

; 0

( ) ( ), , , ,,N n m N n m

Np r

n m

N

2D q-Gaussian?





Conclusions

Conditional distributions

, /3, /3n m N N

( ), ,N n mp

n

m

3

Nn

Conditional distributions

( )

, ,3

NN n

p

n

2

4

5

3

( ) ( ), | ,N n m N m np p

lim

2

1q

300N

Marginal distributions

N=3

0n

1n

2n

3n

0m

1m

2m

3m

0n m

1n m

2n m

3n m

( ) ( ), ,,

0

N n

N n mN nm

p p

( ) ( ), ,,

0

N m

N n mN mn

p p

( ) ( ), ,,

0

n m

N k n m kN n mk

p p

( )3,0,2p ( )

3,2,0p

( )3,0,1p

( )3,1,2p ( )

3,2,1p

( )3,1,0p

( )3,0,0p

( )3,1,1p

( )3,0,3p ( )

3,3,0p

Marginal distributions The three directions yield

identical nonsymmetric scale-invariant distributions.100N

Marginal distributions

2n m 0n m 2n m

3n m 1n m 1n m 3n m

( ) ( ), ,,

0

n m

N k n m kN n mk

p p

( )3,2,0p

( )3,1,0p

( )3,0,0p

( )3,0,1p

( )3,0,2p ( )

3,1,1p

( )3,1,2p ( )

3,3,0p ( )3,2,1p ( )

3,0,3p

Marginal distributions The direction yields a symmetric non scale-invariant distribution

Joint distribution

2 20 10

scale invariance

extensivity

q-gaussiani

ty

?

q-entropy

( ), ,0

1,

( )1

qN

N n mn m

q

Nr

n mS N k

q

ent 1q





Conclusions

Scale invariant hyperpyramids

N distinguisable independent variables3d-cuaternary

N=2N=0 N=1

N=3

1

(1), , ,

6

, ,( 1)( 2)( 3)N n m l

Nr

n m lN N N

Leibniz-like hyperpyramid

1

60 1

601

60

1

60

1

60

1

120

1

60 1

120

1

20

1

201

20

1

601

60 1

60

1

60

1

60

1

601

120

1

120

1

20

1

20 1

201

20

1

10

1

10

1

20

1

10 1

201

20

1

10

1

4 1

41

4

1

41

(1)4( 1), 1, 1, 1( )

, , , (1)4( 1), 1, 1, 1

N n m lN n m l

rr

r

( ) ( ) ( ) ( ) ( ), , , , 1, 1, , , 1, 1 , , 1, 1, , 1,N n m l N n m l N n m l N n m l N n m lr r r r r

(d + 1)-sided dice d-dimensional variable

d-dimensional (d+1)-ary variable taking values

Leibniz hyperpyramid

( ) 1 2d NX X X X ��

:iX

��

1 2, , , ,0.de e e��

1 2

1

(1), , , ,

1 2

!, , ,( 1)( 2) ( )lN n n n

d

Ndr

n n nN N N d





Conclusions

Conclusions and future work

We have generalized to an arbitrary dimension a one-dimensional discrete probabilistic model which, for one dimension, yields q-gaussians in the thermodynamic limit.

Our two-dimensional model, though containing one-dimensional conditional distributions yielding q-gaussians doesn’t seem to yield bidimensional q-gaussians as limiting probability distributions for .

The case of binary variables is special !!! The formulation of a probabilistic model yielding

multidimensional q-gaussians in the thermmodynamic limit is still an open question.

The relationship between scale invariance, q-gaussianity and extensivity is still an open question.

N

Thanks!

A scale invariant probabilistic model based on Leibniz-like pyramids Antonio Rodríguez 1,2 1 Dpto. Matemática Aplicada y Estadística. Universidad Politécnica.

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