q - CBPF

RESULTADOS TEÓRICOS E EXPERIMENTAIS
RECENTES NO QUADRO DA q-ESTATÍSTICA
Constantino Tsallis
Centro Brasileiro de Pesquisas Físicas
e Instituto Nacional de Ciência e Tecnologia de Sistemas Complexos
Rio de Janeiro
e
Santa Fe Institute, New Mexico, USA
Workshop INCT-SC, Maio 2012
ALGUNS RESULTADOS RECENTES
TEORIA DOS GRANDES DESVIOS
A ENTROPIA DO BURACO NEGRO
∞
1
Sq ↔ ς ( s) ≡ ∑ s =
n =1 n
1
∏
−s
p prime 1 − p
1
1
1
1
1
=
·
·
·
·
L
−s
−s
−s
−s
−s
1 − 2 1 − 3 1 − 5 1 − 7 1 − 11
A. Plastino and M.C. Rocca, 1112.1985 and 1112.1986 [math-ph]
M. Jauregui, C. T. and E.M.F. Curado, JSTAT P10016 (2011)
M. Jauregui and C. T., Phys Lett A 375, 2085 (2011)
H.J. Hilhorst, JSTAT P10023 (2010)
See also:
CENTRAL LIMIT THEOREM
N 1/[α (2-q )] - scaled attractor F ( x) when summing N → ∞ q - independent identical random variables
⎛
1+ q ⎞
with symmetric distribution f ( x) with σ Q ≡ ∫ dx x 2 [ f ( x)]Q / ∫ dx [ f ( x)]Q ⎜ Q ≡ 2q − 1, q1 =
⎟
3− q ⎠
⎝
q =1
[independent ]
q ≠ 1 (i.e., Q ≡ 2q − 1 ≠ 1) [globally correlated ]
F ( x) = Gq ( x) ≡ G(3 q 1−1) / (1+ q 1 ) ( x), with same σ Q of f ( x)
σQ < ∞
(α = 2)
F ( x) = Gaussian G ( x ),
with same σ 1 of f ( x)
Classic CLT
F ( x ) = Levy distribution Lα ( x ),
with same | x | → ∞ behavior
σQ → ∞
(0 < α
⎧G ( x )
⎪
< 2) Lα ( x ) ∼ ⎪⎨ if | x |<< xc (1, α )
1+α
⎪ f ( x ) ∼ Cα / | x |
⎪
if | x |>> xc (1, α )
⎩
with limα → 2 xc (1, α ) = ∞
Levy-Gnedenko CLT
⎧⎪G ( x)
Gq ( x) ∼ ⎨
2/( q −1)
⎪⎩ f ( x) ∼ Cq / | x |
with lim q → 1 xc (q, 2) = ∞
if | x |<< xc (q, 2) ⎫⎪
⎬
if | x |>> xc (q, 2) ⎭⎪
S. Umarov, C. T. and S. Steinberg, Milan J Math 76, 307 (2008)
F ( x ) = Lq,α , with same | x |→ ∞ asymptotic behavior
2(1− q )−α (3− q )
⎧
*
⎪G 2(1− q )−α (1+ q ) ( x ) ∼ C q ,α / | x | 2(1− q )
⎪ 2(1− q )−α (3− q ) , α
⎪
(intermediate regime)
⎪
⎪
Lq,α ~ ⎨
⎪
(1+α )/(1+α q−α )
L
G
(
x
)
∼
C
/
|
x
|
⎪ 2α q −α +3
q ,α
, 2
α +1
⎪
⎪
( distant regime)
⎪
⎩
S. Umarov, C. T., M. Gell-Mann and S. Steinberg
J Math Phys 51, 033502 (2010)
q - GENERALIZED CENTRAL LIMIT THEOREM:
S. Umarov, C.T. and S. Steinberg, Milan J Math 76, 307 (2008)
q-Fourier transform:
Fq [ f ](ξ ) ≡
∞
∫
−∞
eqixξ ⊗q
∞
f ( x ) dx =
∫
ixξ [ f ( x )]q−1
eq
f
( x ) dx
−∞
( q ≥ 1)
(nonlinear!)
For q<1 see K.P. Nelson and S. Umarov, Physica A 389, 2157 (2010)
ON THE INVERSE q-FOURIER TRANSFORM:
1/(2 − q )
⎡2 − q
⎤
f ( y) = ⎢
Fq [ f ( x + y )](ξ , y ) d ξ ⎥
∫
⎣ 2π −∞
⎦
+∞
(1 ≤ q < 2)
Particular case q = 1:
1
f ( y) =
2π
+∞
1
∫−∞ F [ f ( x + y )](ξ , y ) dξ = 2π
+∞
∫ F [ f ( x)](ξ ) e
−iξ y
−∞
M. Jauregui and C. T. (2011)
dξ
M. Jauregui and C. T. (2011)
M. Jauregui and C. T. (2011)
STILL ANOTHER PATH TO INVERT THE q-FOURIER TRANSFORM
A. Plastino and M.C. Rocca, 1112.1985 [math-ph] and 1112.1986 [math-ph]
F(k,q) ≡ ⎡⎣ H (q − 1) − H (q − 2) ⎤⎦
= ⎡⎣ H (q − 1) − H (q − 2) ⎤⎦
∫
∞
−∞
∫
∞
−∞
ikx[ f ( x )]
q
dx e
{
q−1
f (x)
1
q−1 1−q
dx 1+ i(1− q)kx[ f (x)]
}
where H (t) ≡ Heaviside step function
It can be proved that the inverse is uniquely given by
1
f (x) =
2π
∞
2
⎡ lim
⎤ e− ikx dk
F(k,q)
δ
(q
−
1−
ε
)
dq
∫−∞ ⎢⎣ ε→+0 ∫1
⎥⎦
f (x)
CONSERVATIVE MC MILLAN MAP:
xn +1 = yn
yn
yn +1 = − xn + 2 µ
+
ε
y
n
2
1 + yn
µ ≠ 0 ⇔ nonlinear dynamics
Ruiz,
Bountisand
and C.
C. T.
G. G.
Ruiz,
T.T.Bountis
T.
Int J Bifurcat Chaos (2012), in press
Int J Bifurcat Chaos (2011), in press
G. Ruiz, T. Bountis and C. T.
Int J Bifurcat Chaos (2012), in press
G. Ruiz, T. Bountis and C. T.
Int J Bifurcat Chaos (2012), in press
M. Leo, R.A. Leo, P. Tempesta and C. T.
Phys Rev E 85, 031149 (2012)
M. Leo, R.A. Leo, P. Tempesta and C. T.
Phys Rev E 85, 031149 (2012)
D + 2d + 2
q' ≡ 2 − q =
D + 2d + 1
with
1
ν = +d
2
EXTREME EVENTS IN FINANCIAL RECORDS:
27 Aug 2002
23 Oct 2002
Risk function
t +Δt
∫
W (t; Δt ) ≡
∫
t
PQ ( r )dr
⎡
β ( q − 1) Δt ⎤
= 1 − ⎢1 +
∞
⎥
1
+
β
(
q
−
1)
t
⎣
⎦
PQ ( r )dr
t
q −2
q −1
= 1−
− ⎡⎣ β (2− q )⎤⎦( t +Δt )
e1 (2−q )
− ⎡⎣ β (2− q )⎤⎦ t )
e1 (2−q )
U. Tirnakli, H.J. Jensen and C. T., EPL 96, 40008 (2012)
U. Tirnakli, H.J. Jensen and C. T., EPL 96, 40008 (2012)
CORRELATIONS IN COUPLED LOGISTIC MAPS AT THE
EDGE OF CHAOS IN THE PRESENCE OF GLOBAL NOISE
We consider a linear chain of N coupled maps
with periodic boundary conditions in a noisy
environment:
ε
x ti +1 = (1− ε ) f (x ti ) + [ f (x ti−1 ) + f (x ti+1 )] + σ t
2
with ε ∈[0,1]
σ t ∈[0,σ max ]
additive
noise
coupling strength
i
i 2
and f (xt ) = 1− µ (xt ) µ ∈[0,2]
i-th logistic map (i = 1...N)
[zero noise: N.B. Ouchi and K. Kaneko, Chaos 10, 359 (2000)]
Intermittency in the normalized
returns time-series
edge of chaos:
N = 100 − ε = 0.8 − σ = 0.002
global parameter
time returns
Δdt
stdev
Δdt = dt +τ − dt
A. Pluchino, A. Rapisarda and C. T. (2012)
time t
N = 100; ε = 0.8; σ max = 0.002; τ = 32
Edge of chaos:
Chaotic Regime:
q=1.54
β=1.47
Δdt
stdev
Δdt
stdev
q=1.54
β=1.47
A. Pluchino, A. Rapisarda and C. T. (2012)
ε = 0.8; σ max = 0.002; τ = 32
N = 100; ε = 0.8; τ = 32
σ max
N = 100; σ max = 0.002;
τ = 32
1/ N
σN = 100;
ε = 0.8; σ max = 0.002
A.Pluchino, A. Rapisarda and C. Tsallis in preparation τ
A. Pluchino, A. Rapisarda and C. T. (2012)
LHC (Large Hadron Collider)
CMS (Compact Muon Solenoid) detector
~ 2500 scientists/engineers from 183 institutions of 38 countries
PHENIX @ RHIC
q ; 1.10
q ; 1.10
ALGUNS RESULTADOS RECENTES
TEORIA DOS GRANDES DESVIOS
A ENTROPIA DO BURACO NEGRO
PHYSICS
MATHEMATICS
(Theory of large
(Statistical
deviations)
mechanics)
q =1
(quasi-independent)
q ≠1
(strongly correlated)
pN ∝ e
−β HN
PN ( x ) : e
− N r( x)
( N → ∞)
− βq H N
q
pN ∝ e
− N rq ( x )
q
PN ( x ) : e
( N → ∞)
???
STANDARD THEORY - AN EXAMPLE
N independent coins → n heads (n = 0,1, 2,..., N )
[hence
N → ∞ ⇒ Gaussian ]
P ( N = 10; n < 2) >> P ( N = 100; n < 20) >> P( N = 1000; n < 200)
more precisely,lim N →∞ P( N ; n N < x ) = 0
P( N ; n N < x ) =
∑
n n N <x
( 0 ≤ x < 1 2)
1
N!
− N r1 ( x )
≈
e
(N → ∞ )
N
2 n !( N − n )!
where the rate function is given by
r1 ( x ) = ln 2 + x ln x + (1 − x ) ln(1 − x )
pi(0)
REMARK: Relative entropy I1 = − ∑ pi ln
pi
i =1
W
W
1
pi(0) =
yields I1 = ln W + ∑ pi ln pi
W
i =1
r1 ( x ) = I1[W = 2; p1 → x; p2 → (1 − x )]
ln P( N ; n N < x)
G. Ruiz and C. T., 1110.6303 [cond-mat.stat-mech]
r1 ( x )
ln 2 + x ln x + (1 − x) ln(1 − x)
G. Ruiz and C. T., 1110.6303 [cond-mat.stat-mech]
GENERALIZING THE STANDARD THEORY- AN EXAMPLE
N strongly correlated coins → n heads (n = 0,1, 2,..., N )
[such that
N → ∞ ⇒ Q - Gaussian ]
P( N ; n N < x ) =
∑
n n N <x
p N ,n ≡
pQ ( y N ,n )
N
∑p
n =0
Q
( y N ,n )
p N ,n
1
2 − Q −1
with pQ ( z ) ∝ ⎡⎣1 + (Q − 1) z ⎤⎦
and
y N ,n
⎛ n 1⎞
≡ ΔN ⎜ − ⎟
⎝ N 2⎠
⎡⎣ Δ N ≡ δ ( N + 1)γ ;
δ > 0; 0 < γ < 1⎤⎦
Probabilistic model { pN ,n } is determined by (1 ≤ Q < 3, 0<γ ≤ 1, δ > 0) :
lim N →∞ { pN ,n } = Q -Gaussian (∀γ , ∀δ )
We numerically verify that, for N >> 1,
P(N | n N < x) ≈ e
− N rq ( x ; Q , γ , δ )
q
with
and
(for Q ≥ 1)
⎡⎣∝ 1 N 1 ( q −1) if Q > 1⎤⎦
Q −1
1
2
q=
+ 1 ≥ 1 hence
=
−1
γ (3 − Q )
γ ( q − 1) Q − 1
1
q( x = 0) = 2 −
q(0 < x < 1 2)
(0 < x < 1 2)
( ∀γ ; ∀δ )
pi(0)
REMARK: Relative entropy I q = − ∑ pi ln q
pi
i =1
W
1
p =
yields I q = W q −1 ⎡⎣ ln q W − S q ⎤⎦
W
q
q
⎡
x
+
(1
−
x
)
− 1⎤
q −1
I q [W = 2; p1 → x; p2 → (1 − x )] = 2 ⎢ln q 2 −
⎥
q
−
1
⎣
⎦
(0)
i
G. Ruiz and C. T., 1110.6303 [cond-mat.stat-mech]
G. Ruiz and C. T., 1110.6303 [cond-mat.stat-mech]
G. Ruiz and C. T., 1110.6303 [cond-mat.stat-mech]
ALGUNS RESULTADOS RECENTES
TEORIA DOS GRANDES DESVIOS
A ENTROPIA DO BURACO NEGRO
Jacob D. Bekenstein
Stephen W. Hawking
Gerard ‘t Hooft
Leonard Susskind
Stephen Lloyd
Juan M. Maldacena
…
When entropy does not seem extensive
John Maddox, Nature 365, 103 (1993)
Everybody who knows about entropy knows that it is an
extensive property, like mass or enthalpy. [...] Of course,
there is more than that to entropy, which is also a measure
of disorder. Everybody also agrees on that. But how is
disorder measured? [...] So why is the entropy of a black
hole proportional to the square of its radius, and not to the
cube of it? To its surface area rather than to its volume?
ENTROPIES
W
S BG
1
= k B ∑ pi ln
pi
i=1
→ additive
W
1
Sq = k B ∑ pi ln q
pi
i=1
⎛ 1⎞
Sδ = k B ∑ pi ⎜ ln ⎟
⎝ pi ⎠
i=1
W
Sq,δ
C. T. (1988)
(S1 = S BG ) → nonadditive if δ ≠ 1
C. T. (2009)
δ
⎛
1⎞
= k B ∑ pi ⎜ ln q ⎟
pi ⎠
⎝
i=1
W
(S1 = S BG ) → nonadditive if q ≠ 1
δ
(Sq,1 = Sq ; S1,δ = Sδ ; S1,1 = S BG )
→ nonadditive if (q,δ ) ≠ (1,1)
C. T. and L.J.L. Cirto (2011)
1202.2154 [cond-mat.stat-mech]
EXTENSIVITY OF THE ENTROPY (N → ∞)
If W (N )  µ N ( µ > 1)
⇒ S BG (N ) = k B lnW (N ) ∝ N
If W (N )  N ρ ( ρ > 0)
⇒ Sq (N ) = k B ln q W (N ) ∝ [W (N )]1−q ∝ N ρ (1−q)
⇒ Sq=1−1 ρ (N ) ∝ N
If W (N )  ν
Nγ
(ν > 1; 0 < γ < 1)
δ
⇒ Sδ (N ) = k B ⎡⎣ lnW (N ) ⎤⎦ ∝ N γ
⇒ Sδ =1 γ (N ) ∝ N
δ
Hawking, string theory, etc, yield
black hole
SBG
(N ) ≡ k B lnW (N ) ∝ L2 ∝ N 2 3
More generally, we have
SBG (N ) = k B lnW (N ) ∝ Ld−1 ∝ N
d−1
d
(N ∝ L3 )
(d > 1)
hence
W (N ) ∝ Φ(N ) ν
d−1
N d
⎛
⎞
ln Φ(N)
= 0⎟
⎜ with lim N→∞
d−1
⎜⎝
⎟⎠
d
N
d
hence the entropy which is extensive is Sδ with δ =
d −1
i.e.,
Sδ (N ) = k B
W (N )
∑
i=1
black hole
δ =3 2
Consequently S
⎛ 1⎞
pi ⎜ ln ⎟
⎝ pi ⎠
(N ) = k B
W (N )
∑
i=1
d
d−1
(d > 1)
3
2
⎛ 1⎞
pi ⎜ ln ⎟ ∝ N ∝ L3 !!!
⎝ pi ⎠
SYSTEMS ENTROPY SBG
W (N )
(ADDITIVE)
 µN
( µ > 1)
 Nρ
( ρ > 0)
ENTROPY Sq
ENTROPY Sδ
(δ ≠ 1)
(q ≠ 1)
(NONADDITIVE) (NONADDITIVE)
EXTENSIVE
NONEXTENSIVE
NONEXTENSIVE
NONEXTENSIVE
EXTENSIVE
NONEXTENSIVE
Nγ
ν
(ν > 1;
0 < γ < 1)
NONEXTENSIVE NONEXTENSIVE
EXTENSIVE