deformed binomial distributions

Evaldo M. F. Curado Centro Brasileiro de Pesquisas Físicas/Rio de Janeiro, Brazil

Collaborators: Jean Pierre Gazeau (Univ. Paris Diderot, Paris, France), Herve Bergeron (Univ. Paris Sud, Orsay, France) and

Ligia M. C. S. Rodrigues (CBPF, Rio de Janeiro, Brazil)

foundations of complexity - rio de janeiro - 2015

outline

• uncorrelated system - binomial distribution

• correlated system - deformed binomial distribution

• mathematical formalism to construct a deformed binomial distribution

• extensivity - analytical studies of Boltzmann-Gibbs, Tsallis and Rényi entropies for some deformed binomial distributions

• extensivity and correlation - discussion

Rudolf Clausius

"I prefer going to the ancient languages for the names of impor- tant scientific quantities, so that they mean the same thing in all living tongues. I propose, accordingly, to call S the entropy of a body, after the Greek word [τρoπη], ‘transformation’. I have designedly coined the word entropy to be similar to energy, for these two quantities are so analogous in their physical significance, that an analogy of denominations seems to me helpful”

R. Clausius, Annalen der Physik und Chimie 7 (1865) 23

1) Die Energie der Welt ist konstant 2) Die Entropie der Welt strebt einem Maximum zu

dS =dQ

T

S = S0 +

ZdQ

T

(59)

(60)

• macroscopic quantity

• μ-space

• filled with points representing the N particles that comprise the gas, where each possible distribution is called a complexion {wi}

W =PPP

logP ⇠ �X

i

wi logwi

wj / exp (�j✏/✏̄) ✏̄ = �✏/N

Boltzmann related log P with the expression of 1872’s paper and with Clausius entropy

• partitioned into many small and disjoint (6-dimensional) rectangular cells, each one having energy 0, ✏, 2✏, 3✏, · · · , p✏

• macrostate of the gas can then be described as the number of particles that occupy each of these rectangular regions of the μ-space w0, w1, w2, · · · , wp

(Permutabilitätsmass)P / N !

w0!w1! · · ·wp!

pX

i=0

wi = NpX

i=0

wii✏ = �✏

no dynamical assumption is made!

• the assumption that the total energy can be expressed in the form E = ∑i ni εi means that the energy of each particle depends only on the cell in which it is located, and not the state of other particles. This can only be maintained, independently of the number N, if there is no interaction at all between the particles. The validity of the argument is thus really restricted to ideal gases

• Boltzmann suggests at the end of the paper that the same argument might be applicable also to dense gases and even to solids

• beginning of statistical mechanics

Claude Shannon - information theory

p.379

microscopic quantity

Edwin T. Jaynes

Rényi entropy

Proc. Fourth Berkeley Symp. on Math. Statist. and Prob., Vol. 1 (Univ. of Calif. Press, 1961), 547-561http://projecteuclid.org/euclid.bsmsp/1200512181

Gibbs: microcanonical ensemble

one macroscopic entropy …

many entropic forms depending on the probability of the microstates

BG, Rényi, Tsallis, Kaniadakis, Abe, Hanel-Thurner, …

independent systems

uncorrelated system - binomial distribution

• n independent trials with two possible outcomes - “win” or “loss”

p(n)k (⌘) =

✓nk

◆⌘k (1� ⌘)n�k =

n!

(n� k)! k!⌘k (1� ⌘)n�k

probability to have k wins in n trials - regardless the order

$(n)k = ⌘k(1� ⌘)n�k ) $(n�1)

k = $(n)k +$(n)

k+1

Leibniz triangle rule

⌘ 2 [0, 1]

p 1� p1

p2 (1� p)2

p3 3p2(1� p) 3p(1� p)2 (1� p)3

2p(1� p)

4p3(1� p) 6p2(1� p)2 4p(1� p)3 (1� p)4p4

1

Pascal and Leibniz rules

limit of large n

✓n

k

◆⌘

k(1� ⌘)

n�k ⇠ 1p2⇡nx(1� x)

exp

n

✓�x log

x

⌘

�� (1� x) log

1� x

1� ⌘

�◆�

⌘

nx

(1� ⌘)

n(1�x) ! exp [n (x log [⌘] + (1� x) log [1� ⌘])]

✓n

k

◆!

k=nx

✓n

nx

◆⇠ 1p

2⇡nx(1� x)

exp [n (�x log[x]� (1� x) log[1� x])]

nX

k=0

✓n

k

◆⌘

k

(1� ⌘)

n�k !k=nx

n

Z 1

0dx

1p2⇡nx(1� x)

exp[nf(x, ⌘)]

! np2⇡n⌘(1� ⌘)

Z 1

�1dx exp

� n

2⌘(1� ⌘)

(x� ⌘)

2

�= 1Laplace's method

) P(n)k =

✓n

k

◆⌘

k(1� ⌘)

n�k !r

n

2⇡⌘(1� ⌘)

exp

� n

2⌘(1� ⌘)

(x� ⌘)

2

�

limit distribution is a Gaussian

binomial case

P(n)k =

✓n

k

◆⌘k(1� ⌘)n�k

nX

k=0

P(n)k = 1

$(n)k =

P(n)k�nk

� = ⌘k(1� ⌘)n�k

SBG(⌘) = �nX

k=0

✓n

k

◆$(n)

k log($(n)k ) = �

nX

k=0

P(n)k log($(n)

k )

⌘ 2 [0, 1]

SBG(⌘) = �(< k > log ⌘+ < n� k > log(1� ⌘))

= �n [⌘ log ⌘ + (1� ⌘) log(1� ⌘)] SBG(1/2) = n log 2

SBG is extensive

nX

k=0

✓n

k

◆k$(n)

k (⌘) = n⌘ =< k >

binomial case - Sq entropy

Sq =1�

Pi p

qi

q � 1! Sq =

1�P

k

�nk

� ⇣$(n)

k

⌘q

q � 1

nX

k=0

✓n

k

◆⇣$(n)

k (⌘)⌘q

=nX

k=0

✓n

k

◆⌘qk(1� ⌘)q(n�k) = (⌘q + (1� ⌘)q)n

= exp [n log (⌘q + (1� ⌘)q)]

0 < q < 1 ! Sq / exp [n log (⌘q + (1� ⌘)q)]

q > 1 ! Sq ! 1

q � 1

Sq is not extensive

systems with independent events - binomial

Sq =1�

Pnk=0

�nk

�pqn,k

q � 1

50 100 150 200

20

40

60

80

100

120

140

n

Sq

q=1

q=1.05

q=0.95(⌘ = 1/2)

BG linear with n!

SBG = �nX

k=0

p(n)k log

p(n)k�nk

�p(n)k =

✓n

k

◆⌘k(1� ⌘)n�k

pn,k ⌘p(n)k�nk

�

N (t) = et

⇠ n log 2

binomial case - Rényi entropy

S(q)R (⌘) =

1

1� qlog

"nX

k=0

✓n

k

◆⇣$(n)

k

⌘q#

S(q)R (⌘) =

n

1� qlog [⌘q + (1� ⌘)q] S(q)

R (1/2) = n log 2

(0 < q < 1)

nX

k=0

✓n

k

◆⇣$(n)

k (⌘)⌘q

=nX

k=0

✓n

k

◆⌘qk(1� ⌘)q(n�k) = (⌘q + (1� ⌘)q)n

= exp [n log (⌘q + (1� ⌘)q)]

SR is extensive as well!

deformed binomial distribution => correlation between events

Laplace - de Finetti modification of the binomial law

P(n)k =

✓n

k

◆$(n)

k

binomial ! $(n)k = ⌘k(1� ⌘)n�k

binary exchangeable stochastic process

e$(n)k :=

Z 1

0dy yk(1� y)n�kg(y) where

Z 1

0dy g(y) = 1

binary correlated systemeP(n)k :=

✓n

k

◆e$(n)k

Leibniz triangle rulee$(n�1)k = e$(n)

k + e$(n)k+1

Laplace (1774)

de Finetti (1937)

limit distribution and extensivity

Hanel, Thurner, Tsallis, EPJB (2009)

⇢(x) = g(x)

Boltzmann-Gibbs entropy is extensive

Rényi

eS(q)R [g] =

1

1� qlog

"nX

k=0

✓n

k

◆⇣e$(n)k

⌘q#

eS(q)R [g] ⇠ n log 2

SR

n200 400 600 800 1000

200

400

600

g(y) =8

⇡

py(1� y)

q = 1/2

two microscopic entropies are extensive for the Laplace-de Finetti case

H Bergeron, EMFC, JP Gazeau, Ligia MCS Rodrigues, Physica A 441 (2016) 23

systems "more correlated”

• several kind of deformations have been studied:

• L. G. Moyano, C. Tsallis, M. Gell-Mann, Europhys. Lett. 73 (2006) 813

• A. Rodriguez, V. Schwammle, C. Tsallis, J. Stat. Mech. (2008) P09006

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

• A. Rodriguez, C. Tsallis, J. Math. Phys 53 (2012) 023302

• G. Ruiz, C. Tsallis, J. Math. Phys. (2014)

• G. Sicuro

correlations -> deformed binomial distribution

correlated events - deformed binomial

• n correlated trials with two possible outcomes - “win” or “loss”

xn! = x1x2 · · ·xn, x0! ⌘ 1

x0 = 0 xn > xn�1x0, x1, x2, x3, · · · , xn, · · ·

deformed probability to have k wins in n trials - induced by correlations

8n 2 N,nX

k=0

p(n)k (⌘) = 1. (xn ! n ) qk ! ⌘

k)

p(n)k (⌘) =xn!

xn�k!xk!qk(⌘)qn�k(1� ⌘)

probabilistic interpretation

qk(⌘) has to be nonnegative for all ⌘ 2 [0, 1]

program

- choose a sequence: x0, x1, x2, … , xn, …

- construct the generalized exponential N(t)

- construct the functions qn(η) using

q0(⌘) = 1

q1(⌘) = ⌘

- construct the function pk(n)(η)

- this procedure does not work in general!

qn(⌘) + qn(1� ⌘) =n�1X

k=0

✓n

k

◆qk(⌘)qn�k(1� ⌘)

N (t) =

1X

n=0

t

n

xn!

!

example of the wrong route

x0 = 0, x1 = 1� ✏, x2 = 2� ✏, · · · , xn = n� ✏, · · ·

✏ = 0.10.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

q2(⌘)

⌘

⌘

q2(⌘)

0.02 0.04 0.06 0.08 0.10

0.001

0.002

0.003

0.004

0.005

0.006

p(n)k (⌘) =xn!

xk!xn�k!qk(⌘)qn�k(1� ⌘)

⌘ ! 1� ⌘ k ! n� k

nX

k=0

p(n)k (⌘) = 1

solving the positiveness problem by means of generating functions

↦ symmetric distributions

N (t) =

1X

n=0

t

n

xn!

!

8n, k 2 N, 8⌘ 2 [0, 1], p(n)k (⌘) � 0

G(⌘; t) :=1X

n=0

qn(⌘)

xn!t

n

G(⌘; t)G(1� ⌘; t) = N (t)

nX

k=0

p(n)k (⌘) = 1 p(n)k (⌘) =xn!

xn�k!xk!qk(⌘)qn�k(1� ⌘)

N (t) =

1X

n=0

t

n

xn!

!

N (t) =

1X

k=0

qk(⌘)tk

xk!

! 1X

m=0

qm(1� ⌘)tm

xm!

!

=1X

k=0

1X

m=0

qk(⌘)

xk!

qm(1� ⌘)

xm!t

k+m

=1X

n=0

nX

k=0

xn!

xk!xn�k!qk(⌘)qn�k(1� ⌘)

!t

n

xn!

8n, k 2 N, 8⌘ 2 [0, 1], p(n)k (⌘) � 0

G(⌘; t) :=1X

n=0

qn(⌘)

xn!t

n G(⌘; t)G(1� ⌘; t) = N (t)

G(⌘; t) = ±p

N (t)e�(⌘,1�⌘;t)

�(x, y; t) = ��(y, x; t)

�(x, y; t) = (x� y)'(t)simplest case:

G(0; t) = 1 , G(1; t) = N (t)

N (t) = e2�(1,0;t) = e2'(t) G(⌘; t) = e(2⌘)'(t) = (N (t))⌘

8⌘ 2 [0, 1], N (t)⌘ =1X

n=0

qn(⌘)t

n

xn!

N (t) =

1X

n=0

t

n

xn!

!

all an > 0, n � 2

N (t) = 1 + t+ a2t2 + · · · )

⌃

p(n)k (⌘) =xn!

xk!xn�k!qk(⌘)qn�k(1� ⌘)

⌘ ! 1� ⌘ k ! n� k

nX

k=0

p(n)k (⌘) = 1GN ,⌘(t) = N (t)⌘ =

1X

n=0

qn(⌘)

xn!t

n

solving the positiveness problem by means of generating functions

JMP 53 (2012) JMP 54 (2013)

⌃0

(F (t) = a1t+ a2t2 + · · · )

> 0 � 0

↦ symmetric distributions

⌃+ = {N 2 ⌃ | 8⌘ 2 [0, 1[, qn(⌘) > 0} =�eF |F 2 ⌃0

main theorem:

N (t) =

1X

n=0

t

n

xn!

!

example 1 - q-exponential

xn! = ↵

n �(↵)n!

�(n+ ↵)=

↵

nn!

(↵)n

qn(⌘) =�(↵)

�(n+ ↵)

�(n+ ↵⌘)

�(↵⌘)=

(↵⌘)n(↵)n

q0(⌘) = 1 q1(⌘) = ⌘

p(n)k (⌘) =xn!

xn�k!xk!qk(⌘)qn�k(1� ⌘)

N (t) =

✓1� t

↵

◆�↵

, ↵ > 0

xn =n↵

n+ ↵� 1, lim

n!1xn = ↵

N (t) =

1X

n=0

t

n

xn!

!

p(n)k (⌘) =

✓n

k

◆�(↵)

�(⌘↵)�((1� ⌘)↵)

�(⌘↵+ k)�((1� ⌘)↵+ n� k)

�(↵+ n)

Pólya-Markov distribution

G. Pólya (1923): urn scheme. From a set of b blue balls and r red balls contained in an urn one extracts one ball and return it to the urn together with c balls of the same color. The probability to select in the urn k blue balls after the n-th trial is given by with p(n)k (⌘)

⌘ =b

b+ r↵ =

b+ r

c

(special cases: c = 0; c = �1)

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

(↵ = 5)

q1(⌘) =⌘

q2(⌘) =1

6⌘(1 + 5⌘)

q3(⌘) =1

42⌘(1 + 5⌘)(2 + 5⌘)

q4(⌘) =1

336⌘(1 + 5⌘)(2 + 5⌘)(3 + 5⌘)

q5(⌘) =1

3024⌘(1 + 5⌘)(2 + 5⌘)(3 + 5⌘)(4 + 5⌘)

qk(⌘)

⌘

asymptotic behavior at large n

nX

k=0

p(n)k

! n

Z 1

0dx p(n)

k=nx

= 1

Wigner law -> q-Gaussian (q=(α-4)/(α-3) and β=2(α-2))

limiting distribution after centering

pnnx

pnn/2

⇠ 2↵�2x

12 (↵�2)(1� x)

↵2 �1 !

x!y+1/2

�1� 4y2

� 12 (↵�2)

x 2 [0, 1]pnk=nx

⇠ 1

n

(x)↵⌘�1(1� x)↵(1�⌘)�1 �[↵]

�[↵⌘]�[↵(1� ⌘)]

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

pnnx

pnn/2

x

Leibniz triangle rule is strictly obeyed

nX

k=0

p(n)k (⌘) =nX

k=0

✓n

k

◆$n

k = 1

$n�1k = $n

k +$nk+1

n=1000, 2000 and Wigner

hkin = n⌘

Boltzmann-Gibbs and Sq entropies

SBG = �nX

k=0

✓n

k

◆p(n)k�nk

�log

p(n)k�nk

� S(n)q =

1�Pn

k=0

�nk

�✓p(n)k

(nk)

◆q

q � 1

20 40 60 80 100

20

40

60

SBG

S1 .05

S0 .95

n

α = 3, η = 1/2

SBG ⇠ n

(↵)� ⌘ (↵⌘)� (1� ⌘) (↵(1� ⌘))� 1

↵

�

SBG ⇠ 0.552961n (↵ = 3; ⌘ = 1/2) SR ⇠ n ln 2

example 2 - Abel-type polynomials

N (t) = e�↵W (�t/↵) , ↵ > 0

W (t)eW (t) = t W-Lambert’s function

xn! = n!↵

n�1

(n+ ↵)n�1

xn =n↵

n+ ↵

✓1� 1

n+ ↵

◆n�2

limn!1

xn = ↵/e

p(n)k (⌘) =

✓n

k

◆⌘(1� ⌘)

(⌘ + k/↵)k�1(1� ⌘ + (n� k)/↵)n�k�1

(1 + n/↵)n�1

qn(⌘) = ⌘

�⌘ + n

↵

�n�1

�1 + n

↵

�n�1q0(⌘) = 1, q1(⌘) = ⌘

asymptotic behavior at large n

P(n)k=nx

⇠ 1

n

3/2

↵ ⌘ (1� ⌘)p2⇡ x

3/2 (1� x)3/2

nX

k=0

! n

Z 1

0dx

✏ =A

nA =

8(↵⌘(1� ⌘))2

⇡

⇠large n

1

n

1/2

↵ ⌘ (1� ⌘)p2⇡

lim✏!0

Z 1�✏

✏

dx

x

3/2 (1� x)3/2

⇠large n

lim✏!0

4↵⌘(1� ⌘)p2⇡

1pn✏

(large n) n

Z 1

0dxP(n)

k=nx

= 1

limiting distribution after centering

P(n)nx

P(n)n/2

⇠ 1

8[x(1� x)]3/2(large n) !

x!y+1/21

(1� 4y2)3/2

q = 5/3 � = �6

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

n = 20000

↵ = 20

⌘ = 1/2

P(n)nx

P(n)n/2

⇠ 1

8[x(1� x)]3/2

P(n)nx

P(n)n/2

x

Leibniz triangle rule is asymptotically obeyed, large n

nX

k=0

p(n)k (⌘) =nX

k=0

✓n

k

◆$n

k = 1

$n�1k ' $n

k +$nk+1(large n)

50 100 150 200

200

400

600

800

1000

n

SBG

2↵p2⇡⌘(1� ⌘)

pn ' 5.26392

pn

↵ = 5

⌘ = 0.3

10000 20000 30000 40000 50000

200

400

600

800

1000

SBG

n

SBG = �nX

k=0

✓n

k

◆p(n)k�nk

�log

p(n)k�nk

� ⇠ 2p2⇡ ↵ ⌘ (1� ⌘)

pn

10 20 30 40 50

0.48

0.50

0.52

0.54

n

slope

7 8 9 10

4.5

5.0

5.5

6.0

6.5

7.0

logSBG

log n

↵ = 5

⌘ = 0.3

SBG = �nX

k=0

✓n

k

◆p(n)k�nk

�log

p(n)k�nk

� ⇠ 2p2⇡ ↵ ⌘ (1� ⌘)

pn

Rényi entropy

SRe;q =

1

1� qlog

"nX

k=0

✓n

k

◆ p(n)k�nk

�!q#

⇠ n log 2

(↵ = 3, ⌘ = 1/2, q = 1/2)

n

SRe;q

5000 10000 15000 20000

2000

4000

6000

8000

10000

12000

14000

• deformation of the binomial distribution (correlations) with a probabilistic interpretation

• binomial -> BG and Rényi extensives

• Laplace-de Finetti -> BG and Rényi extensives

• q-exponential case -> BG and Rényi extensives

• Abel-type case -> BG non-extensive; Rényi extensive

• correlations can change the extensive entropy - numerical and analytical calculations

• for a given system there is more than one entropic form, function of the probabilities of the microscopic states, which are extensive

• how can we choose the correct entropic form to associate with the thermodynamical entropy of a system?

final comments

• S. Twareque Ali, L. Balkova, E. M. F. Curado, J. P. Gazeau, M. A. Rego Monteiro, Ligia M. C. S. Rodrigues and K. Sekimoto, J. Math. Phys. 50 (2009) 043517

• E. M. F. Curado, J. P. Gazeau, Ligia M. C. S. Rodrigues, Phys. Scr. 82 (2010) 038108

• E. M. F. Curado, J. P. Gazeau, Ligia M. C. S. Rodrigues, J. Stat. Phys. 146 (2012) 264.

• H. Bergeron, E. M. F. Curado, J. P. Gazeau, Ligia M. C. S. Rodrigues, J. Math. Phys. 53 (2012) 103304

• H. Bergeron, E. M. F. Curado, J. P. Gazeau, Ligia M. C. S. Rodrigues, J. Math. Phys. 54 (2013) 123301.

• H. Bergeron, E. M. F. Curado, J. P. Gazeau, Ligia M. C. S. Rodrigues, Physica A 441 (2016) 23.

• H. Bergeron, E. M. F. Curado, J. P. Gazeau, Ligia M. C. S. Rodrigues, arXiv:1412.0581v1 (2014); submitted to J. Math. Phys. (2015)

references