q-statistics on symmetric generalized binomial distributions

Transcription

1 q-statistics on symmetric generalized binomial distributions G. Ruiz López a, C. Tsallis b a Universidad Politénica de Madrid (Spain). b Centro Brasileiro de Pesquisas Fisicas and ational Institute of Science and Technology for Complex Systems (Brazil); Santa Fe Institute (USA).

2 Physical scenario of a possible Large deviation Theory (LDT) generalization a) Standard many-body Hamiltonian system in thermal equilibrium (T ) BG weight: LDT probability: LDT probability: P ( ) e r e β H e β r [ H ] P ( ) e, r, β = T (short-range + ergodic = extensive energy) r b) d-dimensional classical system: 2-body interactions Long ranged ( 0 α d ): ( BG relative entropy per particle) P ( ) e, r B rate function V() r q z z z z z e e ( ) q q + qz e = e ; ln z = ln z = ln z q q βh βh = β H ( β = β, H = H, = ln α d ) [ ] ( ) ( ) r q q q q -rate function r α ( β H intensive variable) [G. Ruiz & C. Tsallis, Phys. Lett.A 376 (202) ] [G. Ruiz & C. Tsallis, Phys. Lett. A 377 (203) ]

3 Sequence of Probability of win : η umber of win outcomes: Binomial distribution and LDT umber of outcomes containing wins : Wea Law of large numbers: Rate at which limit is ained: independent trials (two outcomes) [ 0,] ; Probability of loss : ( η) + = 0, =,..., = C,! = =!( )! ( ) Probability of the ratio of wins per trial p( / ) = p η = 2 P < x p = x lim ln P ; < x = [ xln x+ ( x) ln( x) + ln 2 ] r( x) lim P ; > ε = 0 ε > 0 lim ; < = 0 2 P x x 2 r ( x): relative entropy ( ) ( ) ; 0 /2 2 : < x : < x Probability of obtaining wins : p ( )! = η ( η) ( )!! ; < r x P x e ( ) Large Deviation Principle (r : rate function)

4 Sequence of Generalized Binomial distribution nonnegative strictly increasing real numbers, χ { } = x p ( )! = η ( η) Properties: ( p ) preserves the symmetry win-loss: invariant under ( -, η -η ) ( ) p represent probabilities of wins in a sequence of correlated trials: ( ) a) p =, η [0,] ( ) = 0 generating functions of p ( ) b) p 0,, η [0,] ηα ( η) α qn n α ( ) F( η ;) t t ; F( η ;) t F( η ;) t = ( t/ α), α > 0 p = n= 0 xn! α Fulfills the Leibniz triangle rule S BG extensive (q entropy =) Ordinary binomial limit ( α ) α / Expectation value = η ; Variance ( σ ) ( ηα, ) = η( η) + α Spetial case: q-exponential generating function Properties: ( )!! p ( η x ) = ( η ) ( η ) ( )! q q x! x! where: 2 0 ( )! x! xx x, x! = η q polynomial of degree [ H. Bergeron, E. M. F. Curado, and J. P. Gazeau, J. Math. Phys. 54, (203) 2330]

5 A) Finite sequence of nonnegative strictly increasing real numbers umber of win outcomes: Ratio of wins per trial + = 0, =,..., = 0 / { } 0 χ = x = Probability of ratio of wins per trial / : p ( ) ( / ) = ( ηα, ) p α ( ) ( / ) = p p Binomial distribution limit ormalized distribution of umerical results strongly suggest that, α, : p η = α = η = α ( ) ( / ; / 2, ) p ( / 2, ) umerical values of q : : β p p ( / ) = pmaxe q max β x where e q [ β ( qx ) ] 2 ( q) ormalized distribution evolves as q-gaussians, The value of q increases with, α. 2 q ( α) lim q( α, ) = α 2 q α = 2

6 Ratio of wins per trial B) Infinite sequence of nonnegative strictly increassing real numbers χ { } = x /2 ( ) Centered and scaled probabilities, p colapses when = superdiffusion process ( > / 2) (q<)-gaussian ractors in probability space, G p ( ) q( / ηα, ) = lim ( ηα, ) Probability of deviation from a ratio of wins per trial P < x = p p dx ( ) ( ; ηα, ) ( ηα, ) ( / ) : < x 0 x lim p ( ηα, ) = cte ηα, ( ) = /2 The Wea Law of large numbers is OT satified: P ( ) ; > ε ηα, = 2 p ( ηα, ) lim P ; > ε ηα, 0 ε> 0, ηα,. 2 2 : < x ε 2

7 P( ; < x ηα, ) distributions ( < ) q-exponential decay: P( ; < x ηα, ) e r q q=q(α,) dependence: Slopes dependence? r= rx (,, ηα, )... q( ηα,, ) = + (2 < α < < q( α ) 2) η ηα = < /2 2 2 α = = 2 + = η = /2 q( α) q ( α) q( α) q ( α) η= /2 η= /2

8 P( ; < x ηα, ) distributions ( < ) r= rx ( = 0.5,, = 0.5 η = 0.5, α = 0) : ( ) : q-exponential decay: P < x = e max ( x,, ) ( ) ( ; ηα, ) p ( ηα, ) = 0 r ( max ηα, ) q= + ηα

9 Conclusions ( We analyzed a set of generalized binomial distributions, p ) ( ηα ; ), that can be interpreted as the probability of having wins (probability η) and - losses (probability -η) in a sequence of correlated trials. They preserve the symmetry win-loss, and fulfill the Leibniz triangle rule ( q entropy =). umerical results strongly suggest that increasing and for fixed values of α= α 0, the probability distribution function of the ratio of wins per trial evolves as a q-gaussian, and q(α 0, ) increases with. umerical results strongly suggest that the ractors ( ) of the probability distribution function of the ratio of wins per trial are q-gaussians (q<). The value of q ( α) lim q( α, ) is analytically obtained 2 as a function of α as q ( α) =, recovering the binomial distribution limit when α α 2 The generalized binomial distributions represent a superdiffusion process. The generalized binomial distributions do not satisfy the Wea Law of large numbers. Large-deviation-lie properties are satisfied by P( ; < x ηα, ) distributions ( < ). In particular: Probability deviation q-exponentially decays. The value of q is obtained as a function of the parameters of the model (η, α): q( ηα, ) = + ηα on-trivial dependence of the rate of q-exponential decay, r= rx (,, ηα, ), is analyzed. The q-gaussian ractor q ( α), and the q-exponential parameter of η=/2 models q( ηα, )( α ), satisfy 2 ( ) + = ( ) =2+ q α = q α η = /2 2 q( α) q( α) q ( α) q ( α) 2 η= /2 η= /2 Large-deviation-lie properties of P( ; < x ηα, ) distributions ( > ) are being studied, and so is the behavior of deviation probabilities form the expectation value ( > and <).