MAIS q‐EXPONENCIAIS E q‐GAUSSIANAS. E PARA NÃO DIZER

MAIS q‐EXPONENCIAIS E q‐GAUSSIANAS. E PARA NÃO DIZER

MAIS q EXPONENCIAIS E q GAUSSIANAS.
E PARA NÃO DIZER QUE NÃO FALEI DE
FLORES, A TEORIA DE GRANDES DESVIOS.
additive
nonadditive (if q ≠ 1)
TYPICAL SIMPLE SYSTEMS:
Short-range space-time correlations
e.g., W ( N ) ∝ µ ( µ > 1)
N
Markovian processes (short memory), Additive noise
Strong chaos (positive maximal Lyapunov exponent), Ergodic, Riemannian geometry
Short-range many-body interactions, weakly quantum-entangled subsystems
Linear/homogeneous Fokker-Planck equations, Gausssians
! Boltzmann-Gibbs entropy (additive)
! Exponential dependences (Boltzmann-Gibbs weight, ...)
TYPICAL COMPLEX SYSTEMS:
Long-range space-time correlations
ρ
e.g., W ( N ) ∝ N ( ρ > 0)
Non-Markovian processes (long memory), Additive and multiplicative noises
Weak chaos (zero maximal Lyapunov exponent), Nonergodic, Multifractal geometry
Long-range many-body interactions, strongly quantum-entangled sybsystems
Nonlinear/inhomogeneous Fokker-Planck equations, q-Gaussians
! Entropy Sq (nonadditive)
! q-exponential dependences (asymptotic power-laws)
An entropy is additive if, for any two probabilistically independent
systems A and B,
S ( A + B) = S ( A) + S ( B)
Therefore, since
Sq ( A + B) = Sq ( A) + Sq ( B) + (1 − q) Sq ( A) S q ( B) ,
S BG and SqRenyi (∀q) are additive, and Sq (∀q ≠ 1) is nonadditive .
Consider a system Σ ≡ A1 + A2 + ... + AN made of N (not necessarily independent)
identical elements or subsystems A1 and A2 , ..., A N .
An entropy is extensive if
S(N )
0 < lim
< ∞ , i.e., S ( N ) ∝ N ( N → ∞)
N →∞
N
EXTENSIVITY OF THE ENTROPY (N → ∞)
If W (N )  µ N ( µ > 1)
⇒ S BG (N ) = k B lnW (N ) ∝ N
OK!
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
OK!
(ν > 1; 0 < γ < 1)
δ
⇒ Sδ (N ) = k B ⎡⎣ lnW (N ) ⎤⎦ ∝ N γ
⇒ Sδ =1 γ (N ) ∝ N
IMPORTANT:
µ >> ν
N
δ
OK!
Nγ
>> N ρ if N >> 1
Tackled by
Jacob D. Bekenstein
Stephen W. Hawking
Gary W. Gibbons
Gerard ‘t Hooft
Leonard Susskind
Michael J. Duff
Juan M. Maldacena
Thanu Padmanabhan
Robert M. Wald
and many others
SINCE THE PIONEERING BEKENSTEIN-HAWKING RESULTS,
PHYSICALLY MEANINGFUL EVIDENCE HAS ACCUMULATED
(e.g., HOLOGRAPHIC PRINCIPLE) WHICH MANDATES THAT
lnWblack hole ∝ AREA
THIS IS PERFECTLY ADMISSIBLE AND MOST PROBABLY CORRECT.
HOWEVER,
IS THIS QUANTITY THE THERMODYNAMICAL ENTROPY???
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 )
C. T. (2011)
→ nonadditive if (q,δ ) ≠ (1,1)
C. T. and L.J.L. Cirto (2012), 1202.2154 [cond-mat.stat-mech]
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
pq ( x) ∝
−( x /σ )2
eq
1
≡
⎡⎣1 + (q -1)
1
(x / σ )2 ⎤ q-1
⎦
(q < 3)
A. Plastino and M.C. Rocca, Physica A and Milan J Math (2012)
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)
CONSERVATIVE MC MILLAN MAP:
xn +1 = yn
yn
yn +1 = − xn + 2 µ
+
ε
y
n
2
1 + yn
µ ≠ 0 ⇔ nonlinear dynamics
G. Ruiz, T. Bountis and C. T., Int J Bifurcat Chaos 22, 1250208 (2012)
G. Ruiz, T. Bountis and C. T., Int J Bifurcat Chaos 22, 1250208 (2012)
G. Ruiz, T. Bountis and C. T., Int J Bifurcat Chaos 22, 1250208 (2012)
A
V (r) ∼ − α
r
(r → ∞)
( A > 0, α ≥ 0)
α / d >1
non-integrable if 0 ≤ α / d ≤ 1
integrable if
5
(short-ranged )
(long-ranged )
EXTENSIVE
SYSTEMS
α
4
dipole-dipole
3
α -XY model
NONEXTENSIVE
SYSTEMS
2
Newtonian gravitation
1
0
0
1
2
3
4
d
5
α =2
q =1
α = 0.9
q = 1.58
L.J.L. Cirto, V.R.V. Assis and C. T. (2012),1206.6133 [cond-mat.stat-mech]
α =2
q =1
α = 0.9
q = 1.53
L.J.L. Cirto, V.R.V. Assis and C. T. (2012),1206.6133 [cond-mat.stat-mech]
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
time t
A. Pluchino, A. Rapisarda and C. T., Phys Rev E 87, 022910 (2013)
N = 100; ε = 0.8; σ max = 0.002; τ = 32
Chaotic Regime:
Δdt
stdev
Edge of chaos:
q=1
q=1.54
β=0.5
β=1.47
Δdt
stdev
A. Pluchino, A. Rapisarda and C. T., Phys Rev E 87, 022910 (2013)
q=1.115
β=0.6
Edge of chaos
U. Tirnakli and C. T. (2013)
U. Tirnakli and C. T. (2013)
Kwapien and Drozdz, Phys Rep 515, 115 (2012)
0.6
q-1
0.55
0.5
q − 1 ∝1 / τ
0.1
0.45
0.4
slope = 0.1
0.35
0.3
-3
10
10
-2
-1
10
-1
1/τ (minute )
10
0
PHYSICS
MATHEMATICS
(Large deviation
(Statistical
theory)
mechanics)
q =1
pN ∝ e
(quasi-independent)
=e
q ≠1
(strongly correlated)
−β HN
⎡ HN
− ⎢β
⎣ N
⎤
⎥N
⎦
PN (x)  e
− N r(x)
(N → ∞)
− βq H N
pN ∝ eq
⎡
H ⎤
− ⎢( β q N * ) N* ⎥ N
NN ⎦
⎣
q
=e
− N rq ( x )
PN (x)  eq
(N → ∞)
G. Ruiz and C. T., Phys Lett A 377, 491 (2013)
ln P( N ; n N < x)
G. Ruiz and C. T., Phys Lett A 376, 2451 (2012)
Q −1
q = 1+
γ (3 − Q)
G. Ruiz and C. T., Phys Lett A 376, 2451 (2012)
G. Ruiz and C. T., Phys Lett A 377, 491 (2013)
G. Ruiz and C. T., Phys Lett A 377, 491 (2013)
CONJECTURE:
For all strongly correlated systems whose CLT attractors are Q-Gaussians
(Q > 1), a set ⎡⎣ q > 1, B(x) > 0, rq(lower bound ) (x) > 0, rq(upper bound ) (x) > 0 ⎤⎦
exists such that generically
B(x) ⎡⎣(q − 1) r
(lower bound )
q
(x) ⎤⎦
1
q−1
− rq( lower
q
e
bound )
(x) N
⎤
B(x) ⎡
1
1
2
= 1 ⎢1−
+ o(1 / N ) ⎥
2 (lower bound )
(q − 1) rq
(x) N
⎥⎦
q−1 ⎢
⎣
N
≤ P(N; n N < x)
≤ B(x) ⎡⎣(q − 1) r
(upper bound )
q
(x) ⎤⎦
1
q−1
− rq( upper
q
e
bound )
(x) N
⎤
B(x) ⎡
1
1
2
= 1 ⎢1−
+ o(1 / N ) ⎥
2 (upper bound )
(q − 1) rq
(x) N
⎥⎦
q−1 ⎢
⎣
N
G. Ruiz and C. T., Phys Lett A 377, 491 (2013)
q=1.15
T=0.145
Wong and Wilk, Acta Phys Polon B 43, 2047 (2012)
2
Newton
p
2
e
E = Em= mecc2 ++p2/2m
e
e
2m
e
Newton
Ee = 7 TeV
E =Einstein
me2 c 4 + p 2(1905)
c2
E = (m2e c4 + p2c4)1/2
9
ctro
m < 3 eV/c2
e
ele
8
7
6
5
4
511 KeV/c2
me
n
o
Einstein (1905)
trin
|
|
E = 7 TeV
neu
12
10
1011
1010
9
10
EE 1 108
2 2 − 1107
mmecec
6
10
105
104
103
102
101
100
10 1
10 2
10 3
10 4
3
2
1
0
p [GeV/c]
1
2
3
4
5
6
7
8
9
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
L.J.L. Cirto and C. T. (2012)
See also:
R.N. Costa Filho, M.P. Almeida, G.A. Farias and J.S. Andrade, Phys Rev A 84,
050102(R) (2011)
F.D. Nobre, M.A. Rego-Monteiro and C. T., EPL 97, 41001 (2012)
S.H. Mazharimousavi, Phys Rev A 85, 034102 (2012)
A.R. Plastino and C. T., J Math Phys. 54, 041505 (2013)
R.N. Costa Filho, G. Alencar, B.S. Skagerstam and J.S. Andrade, EPL 101, 10009 (2013)
B.G. Costa and E.P. Borges, preprint (2013)