CONEXÕES ENTRE SISTEMAS DINÂMICOS NÃO LINEARES E

CONEXÕES ENTRE SISTEMAS DINÂMICOS NÃO LINEARES E

CONEXÕES ENTRE
SISTEMAS DINÂMICOS NÃO LINEARES E ENTROPIA
QUANDO O MAXIMO EXPOENTE DE LYAPUNOV É ZERO
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Enrico FERMI
Thermodynamics (Dover, 1936)
The entropy of a system composed of several parts is very
often equal to the sum of the entropies of all the parts. This
is true if the energy of the system is the sum of the energies
of all the parts and if the work performed by the system
during a transformation is equal to the sum of the amounts
of work performed by all the parts. Notice that these
conditions are not quite obvious and that in some cases
they may not be fulfilled. Thus, for example, in the case of a
system composed of two homogeneous substances, it will
be possible to express the energy as the sum of the
energies of the two substances only if we can neglect the
surface energy of the two substances where they are in
contact. The surface energy can generally be neglected
only if the two substances are not very finely subdivided;
otherwise, it can play a considerable role.
$*(%&%(#)
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additive
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nonadditive (if q ≠ 1)
DEFINITIONS : q − logarithm :
q − exponential :
x1−q − 1
ln q x ≡
1− q
( x > 0; ln1 x = ln x )
1
1− q
x
eq ≡ [1 + (1 − q) x ]
( e1x = e x )
Hence, the entropies can be rewritten :
equal probabilities
generic probabilities
W
BG entropy
( q = 1)
k ln W
k
∑ pi ln
i=1
W
entropy Sq
(q ∈ R)
k ln q W
1
pi
1
k ∑ pi ln q
pi
i=1
TYPICAL SIMPLE SYSTEMS:
Short-range space-time correlations
e.g., W ( N ) ∝ µ N ( µ > 1)
Markovian processes (short memory), Additive noise
Strong chaos (positive maximal Lyapunov exponent), Ergodic, Euclidean geometry
Short-range many-body interactions, weakly quantum-entangled subsystems
Linear and homogeneous Fokker-Planck equations, Gausssians
! Boltzmann-Gibbs entropy (additive)
! Exponential dependences (Boltzmann-Gibbs weight, ...)
TYPICAL COMPLEX SYSTEMS:
e.g., W ( N ) ∝ N ρ ( ρ > 0)
Long-range space-time correlations
Non-Markovian processes (long memory), Additive and multiplicative noises
Weak chaos (zero maximal Lyapunov exponent), Nonergodic, Multifractal geometry
Long-range many-body interactions, strongly quantum-entangled sybsystems
Nonlinear and/or inhomogeneous Fokker-Planck equations, q-Gaussians
! Entropy Sq (nonadditive)
! q-exponential dependences (asymptotic power-laws)
-  Additive versus Extensive
-  Nonlinear dynamical systems
-  Central Limit Theorem
-  Predictions, verifications, applications
*,*/ %&5=A>B5 &5A71<>=%G6>A4
?175
An entropy is additive if, for any two probabilistically independent
systems A and B,
S ( A + B) = S ( A) + S ( B)
Therefore, since
Sq ( A + B) = Sq ( A) + Sq ( B) + (1 − q) Sq ( A) S q ( B) ,
S BG and SqRenyi (∀q) are additive, and Sq (∀q ≠ 1) is nonadditive .
.*$),*/
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
TYPICAL SIMPLE SYSTEMS (equal probabilities):
W ( N ) ∝ µ N ( N → ∞ ; µ > 1)
⇒ S BG ( N ) = k B ln W ( N ) ∝ N ( EXTENSIVE !)
TYPICAL COMPLEX SYSTEMS (equal probabilities):
W ( N ) ∝ N ρ ( N → ∞ ; ρ > 0)
<< µ N
1− q
⇒ Sq ( N ) = k B ln q W ( N ) = k B
∝N
ρ (1− q )
[W ( N )]
−1
1− q

1
if q = 1 − ρ  ∝ N ( EXTENSIVE !)


HYBRID PASCAL - LEIBNITZ TRIANGLE
1×
(N=0)
1×
(N=1)
1×
(N=2)
(N=3)
1×
(N=4)
(N=5)
1×
1
6
1
5
5×
1
2
1×
1
3
1
1×
4
1
1
2×
1
6
1×
1
3×
12
4×
1
30
1
20
10 ×
1
2
1
3
1
3×
12
6×
1
60
1
30
10 ×
1
1×
4
4×
1
20
1
60
5×
1
5
1
30
N
∑
  rN ,n
n=0  n 
N
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1×
1×
1
6
= 1 (∀ N )
q = 1 SYSTEMS
!
i.e., such that S1 ( N ) ∝ N ( N → ∞)
A=BC#138 1=D1AH
,95==1
Leibnitz triangle
1 

=
p
 N ,0

1
+
N


N independent coins
 pN ,0 = p N 




 with p = 1/ 2 


;;C8A555G1<?;5BBCA93C;HB1C9B6HC85"592=9CIAD;5
Stretched exponential
α
 pN .0 = p N





 with p = α = 1/ 2 


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4
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*#5;;#1==1=4/)1C>&A>3$1C;314)3+)
q ≠ 1 SYSTEMS
i.e., such that Sq ( N ) ∝ N ( N → ∞)
1
q = 1−
d
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Continental Airlines
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| γ |= 1
→ Ising ferromagnet
0 < |γ | <1
γ =0
→ anisotropic XY ferromagnet
→ isotropic XY ferromagnet
λ ≡ transverse magnetic field
L ≡ length of a block within a N → ∞ chain
1ADB>1=4*&8HB(5E
1ADB>1=4*&8HB(5E
Using a Quantum Field Theory result
in P. Calabrese and J . Cardy , JSTAT P 06002 (2004)
we obtain, at the critical transverse magnetic field ,
9 + c2 − 3
qent =
c
with c ≡ central charge in conformal field theory
Hence
Ising and anisotropic XY ferromagnets ⇒ c =
1
⇒ qent = 37 − 6 ≈ 0.0828
2
and
Isotropic XY ferromagnet
⇒ c = 1 ⇒ qent = 10 − 3 ≈ 0.1623
1ADB>1=4*&8HB(5E
(d = 1; T = 0)
BG
(pure magnet with critical transverse field)
9 + c2 − 3
q=
c
1.67
1.67
q = 1−
= 1−
c
ln (2 S + 1)
(random magnet with no field)
A Saguia and MS Sarandy, Phys Lett A 374, 3384 (2010)
Summarizing, for a wide class of quantum systems or subsystems with N elements,
we know that
S BG ( N ) ∝ ln L ∝ ln N
≠N
for d = 1 quantum chains
∝ L
∝ N
≠N
for d = 2 bosonic systems
∝ L2
∝ N 2/3
≠N
for d = 3 black hole
∝ Ld −1
∝ N ( d −1)/ d ≠ N
for d -dimensional bosonic systems
(d > 1; area law)
Ld −1 − 1
∝
≡ ln 2−d L ≠ Ld ∝ N
d −1
(d ≥ 1)
(NONEXTENSIVE!)
For the same class of quantum systems, we expect
S q ( N ) ∝ Ld ∝ N
ent
(d ≥ 1; qent ≠ 1)
(EXTENSIVE!)
(analytically and/or computationally shown for d = 1, 2)
1ADB>1=4*&8HB(5E
SYSTEMS
Short-range
interactions,
weakly entangled
blocks, etc
ENTROPY SBG
(additive)
ENTROPY Sq (q<1)
(nonadditive)
EXTENSIVE
NONEXTENSIVE
Long-range
interactions (QSS), NONEXTENSIVE
strongly entangled
blocks, etc
quarks-gluons, plasma, curved space ...?
EXTENSIVE
-  Additive versus Extensive
-  Nonlinear dynamical systems
-  Central Limit Theorem
-  Predictions, verifications, applications
xt +1 = 1 − axt2
edge of chaos
(0 ≤ a ≤ 2)
fully developed chaos
LOGISTIC MAP:
xt +1 = 1 − a xt
2
(0 ≤ a ≤ 2; −1 ≤ xt ≤ 1; t = 0,1, 2,...)
(strong chaos, i.e., positive Lyapunov exponent)
(a=2)
V. Latora, M. Baranger, A. Rapisarda and C. T., Phys. Lett. A 273, 97 (2000)
We verify
K1 = λ1 ( Pesin − like identity )
where
K1 ≡ limt →∞
S1 (t )
t
and
∆ x(t )
λ
ξ (t ) ≡ lim ∆ x (0)→0
=e
∆ x(0)
1
t
(weak chaos, i.e., zero Lyapunov exponent)
C. T. , A.R. Plastino and W.-M. Zheng, Chaos, Solitons & Fractals 8, 885 (1997)
M.L. Lyra and C. T. , Phys Rev Lett 80, 53 (1998)
V. Latora, M. Baranger, A. Rapisarda and C. T. , Phys Lett A 273, 97 (2000)
E.P. Borges, C. T. , G.F.J. Ananos and P.M.C. Oliveira, Phys Rev Lett 89, 254103 (2002)
F. Baldovin and A. Robledo, Phys Rev E 66, R045104 (2002) and 69, R045202 (2004)
G.F.J. Ananos and C. T. , Phys Rev Lett 93, 020601 (2004)
E. Mayoral and A. Robledo, Phys Rev E 72, 026209 (2005), and references therein
It can be proved that
K q = λq (q − generalized Pesin − like identity )
where
 S q (t ) 
K q ≡ limt →∞ sup 

 t 
and
λq t

∆ x(t ) 
ξ (t ) ≡ sup lim ∆ x (0)→0
 = eq
∆ x(0) 

with
1
1
1
ln α F
=
−
=
1 − q α min α max
ln 2
and
λq =
1
1− q

1
1
1
ln α F ( z ) 
z
=
−
= ( z − 1)
 xt +1 = 1 − a | xt | ⇒

z
l
n
2
α
α
1
q
(
z
)
(
z
)
(
)
−
min
max


EDGE OF CHAOS OF THE LOGISTIC MAP:
(Using result in http://pi.lacim.uqam.ca/piDATA/feigenbaum.txt)
q=
0.244487701341282066198770423404680405234446935490057673670365098632774
9672766558665755156226857540706288349640382728306063600193730331818964
5513410812778097921943860270831944900524658135215031745349520749404481
6546094908744833405672362246648808333307214231898714587299268154849677
4607864821834569063370205946820461899021675321457546117438305008496860
4088469694917043674789915060166464910602178348278899938183825225545823
3803811311803180544823675794499039707439546614634081555316878853503011
3821491411266246328940130370152354936571471269917921021622688833029675
4057806307068223688104320157903521237407354446029700060552504231420280
8919357881123973197797484423515245604092644670957957030465861412956647
9666687743683240492022757393004750895311855179558720483992696896827555
8524450244365268256094237801280330948779544035425248590433797618027118
3000457358555073894113675878440062913563042167454169409213569860320785
9088199859359007319336801069967496707904456092418632112054130547393985
795544410347612222592136846219346009360!
(1018 meaningful digits)
Kqentropy production = λqsensitivity − γ qescape
xt +1 = 1 − a xtz
( z ≥ 1; 0 ≤ a ≤ 2; − 1 ≤ xt ≤ 1)
1
!
1
t
qescape −1
M.A. Fuentes, Y. Sato and C. T., Phys Lett A (2011), in press
SENSITIVITY TO INITIAL CONDITIONS, ENTROPY
AND ESCAPE RATE AT THE ONSET OF CHAOS
M.A. Fuentes, Y. Sato and C. T., Phys Lett A (2011), in press
(λqsensitivity − γ qescape )t
Kqentropy production t
LOGISTIC MAP: EDGE OF CHAOS
odd 2n
q=1.63
beta=6.2
even 2n
q=1.70
beta=6.2
U. Tirnakli, C. Beck and C. T.
Phys Rev E 75, 040106(R) (2007)
U. Tirnakli, C. T. and C. Beck
Phys Rev E 79, 056209 (2009)
EDGE OF CHAOS OF THE LOGISTIC MAP:
qsensitivity = qentropy = 0.244487701341282066198...



q - triplet qrelaxation
= 2.249784109...


= 1.65 ± 0.05
qstationary state
In order to have
( qsensitivity , qrelaxation ,qstationary state ) ≠ (1,1,1)
it seems that we need
maximal Lyapunov exponent = 0
What else do we need?
What relations may exist between these three q-indices?
CONSERVATIVE MC MILLAN MAP:
xn +1 = yn
yn
+
ε
yn +1 = − xn + 2 µ
y
n
2
1 + yn
µ ≠ 0 ⇔ nonlinear dynamics
G. Ruiz, T. Bountis and C. T.
Int J Bifurcat Chaos (2011), in press
( µ , ε ) = (1.6,1.2)
( N = 29 )
18
(N = 2 )
( N = 213 )
(218 ≤ N ≤ 219 )
(λmax ≈ 0.05)
( N = 216 )
(219 ≤ N ≤ 220 )
G. Ruiz, T. Bountis and C. T.
Int J Bifurcat Chaos (2011), in press
− β ( z σ )2
q
p∝e
with ( q, β ) = (1.6, 4.5)
G. Ruiz, T. Bountis and C. T.
Int J Bifurcat Chaos (2011), in press
KURAMOTO MODEL: (N nonlinearly coupled oscillators)
(N=20000; K=2.53)
(N=20000; K=0.6)
G. Miritello, A. Pluchino and A. Rapisarda, Physica A 388, 4818 (2009)
M. Leo, R.A. Leo and P. Tempesta, J Stat Mech P04021 (2010)
q=1.463
M. Leo, R.A. Leo and P. Tempesta, J Stat Mech P04021 (2010)
-  Additive versus Extensive
-  Nonlinear dynamical systems
-  Central Limit Theorem
-  Predictions, verifications, applications
+))$) pq ( x) ∝
−( x /σ )2
eq
1
≡
1 + (q -1)
1
2 q -1
(x / σ ) 
(q < 3)

&A1C>1=4*&8HB(5E
P&(%+*
"$9E1=5="5#581DC51=4'-1=7(5?#1C8&8HB
&>A75B&8HB931
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x ⊗q y ≡  x1− q + y1− q − 1
1
1− q
Properties :
i ) x ⊗1 y = x y
ii ) ln q ( x ⊗q y ) = ln q x + ln q y
5GC5=B9E9CH>6)@
[ whereas ln q ( x y ) = ln q x + ln q y + (1 − q)(ln q x)(ln q y )]
=>=1449C9E9CH>6)@
$("0$*(""#**%(#
)+<1A>E*1=4))C59=25A7#9;1= #1C8
q-Fourier transform:
∞
Fq [ f ](ξ ) ≡
∫
−∞
ixξ
eq ⊗q
∞
f ( x ) dx =
∫
ixξ [ f ( x )]q−1
eq
f
( x ) dx
−∞
( q ≥ 1)
(nonlinear!)
>A@
B55!&$5;B>=1=4)+<1A>EPhysica A 389, 2157 (2010)
$("0$*(""#**%(#
)+<1A>E*1=4))C59=25A7#9;1= #1C8
q-independence:
Two random variables X [ with density f X ( x ) ] and Y [ with density fY ( y )]
having zero q − mean values are said q - independent if
Fq [X+Y](ξ ) =Fq [X](ξ ) ⊗ 1+ q Fq [Y](ξ ) ,
3− q
i.e., if
∫
∞
−∞
dz
eqizξ
∞
∞
ixξ
iyξ



⊗ q f X +Y ( z ) = ∫ dx eq ⊗ q f X ( x ) ⊗(1+ q )/(3− q ) ∫ dy eq ⊗ q fY ( y )  ,
 −∞

 −∞

with
∞
∞
∞
∞
−∞
−∞
−∞
−∞
f X +Y ( z ) = ∫ dx ∫ dy h( x, y ) δ ( x + y − z ) = ∫ dx h( x, z − x ) = ∫ dy h( z − y, y )
where h( x, y ) is the joint density.
if q = 1 , i.e., h( x, y ) = f X ( x ) fY ( y )
independence
q - independence means 
 global correlation if q ≠ 1 , i.e., h( x, y ) ≠ f X ( x ) fY ( y )
CENTRAL LIMIT THEOREM
N 1/[α (2-q )] - scaled attractor ! ( 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 ]
! ( x) = Gq ( x) ≡ G(3 q 1−1) / (1+ q 1 ) ( x), with same σ Q of f ( x)
σQ < ∞
! ( x) = Gaussian G ( x ),
with same σ 1 of f ( x)
(α = 2)
Classic 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)
! ( 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
! ( 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 2α q −α +3 ( x ) ∼ C q ,α / | x |
, 2
α +1


( distant regime)


S. Umarov, C. T., M. Gell-Mann and S. Steinberg
J Math Phys 51, 033502 (2010)
Hilhorst function:



1/( q −1)
C
|
x
|
 q

f A ( x) = 

 0


[H.J. Hilhorst, JSTAT P 10023 (2010)]
( q − 2)/( q −1)
 | x |
{
− A
1/( q − 2)
1 + ( q − 1)  | x |( q −2)/( q −1) − A
2( q −1)/( q − 2)
}
1/( q −1)
if 0 ≤ A <| x |( q −2)/( q −1)
if 0 ≤| x |( q −2)/( q −1) ≤ A
with
∫
∞
−∞
dx f A ( x ) = 1
Particular case: A = 0
1
f0 ( x) =
Cq [1 + ( q − 1) x 2 ]1/( q−1)
q-GENERALIZED INVERSE 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 d ξ
−∞
M. Jauregui and C. T., Phys Lett A 375, 2085 (2011)
q - Gaussian
Hilhorst function
M. Jauregui and C. T.
Phys Lett A 375, 2085 (2011)
WHAT IS THE PHYSICAL MEANING OF q-INDEPENDENCE?
IS IT CONSISTENT WITH (STRICT OR ASYMPTOTIC) SCALE
INVARIANCE?
IF YES, IS IT SUFFICIENT? NECESSARY?
CANDIDATE MODELS FOR q-INDEPENDENCE:
1) N compact-support continuous variables with correlation introduced
(strictly scale-invariant)
through a N-variate covariance matrix
W. Thistleton, J.A. Marsh, K. Nelson and C. T., Cent. Eur. J. Phys. 7, 387 (2009)
(see H.J. Hilhorst and G. Schehr, J Stat Mech (2007) P06003)
2) N binary variables with correlation introduced through the q-product
(strictly scale-invariant)
L.G. Moyano, C. T. and M. Gell-Mann, Europhys Lett 73 (2006) 813
(see H.J. Hilhorst and G. Schehr, J Stat Mech (2007) P06003)
3) N binary variables with correlation introduced through a family of
triangles generalizing the Leibnitz one
(strictly scale-invariant)
A. Rodriguez, V. Schwammle and C. T., J Stat Mech (2008) P09006
R. Hanel, S. Thurner and C. T., Eur Phys J B 72, 263 (2009)
4) N-binary-discretized q-Gaussians
(asymptotically scale-invariant)
A. Rodriguez, V. Schwammle and C. T., J Stat Mech (2008) P09006
q-independence ⇒ scale-invariance ?
i.e.,
∫ dx
N
hN ( x1 , x2 ,..., xN ) = hN −1 ( x1 , x2 ,..., xN −1 ) ?
-  Additive versus Extensive
-  Nonlinear dynamical systems
-  Central Limit Theorem
-  Predictions, verifications, applications
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9*8549BCA92DC9>=>61C><93E5;>39C95B9B11DBB91=
44 ER
99q = 1 +
U0
where
ER ≡ recoil energy
U 0 ≡ potential depth
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45,67,8/09+*3:,;7,<$%9*'&(& *(-,=7,>$(?/(&:,6@53,>$1,A$)) BC:,DDECED,FGEECH
( R 2 ! 0.9985)
( R 2 ! 0.995)
q ! 1"
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!
A
V (r ) ∼ − α
r
(r → ∞)
( A > 0, α ≥ 0)
integrable if
α / d >1
non-integrable if 0 ≤ α / d ≤ 1
(short-ranged )
(long-ranged )
5
EXTENSIVE
SYSTEMS
α
4
dipole-dipole
3
α -XY model
NONEXTENSIVE
SYSTEMS
2
Newtonian gravitation
1
0
0
1
2
3
4
d
5
#
9=5AC91;./<>45;
d = 1 α − XY model
L.J.L. Cirto, V.R.V. Assis and C. T. (2011)
∞
1
Sq ↔ ς ( s) ≡ ∑ s =
n =1 n
1
∏
−s
p prime 1 − p
1
1
1
1
1
=
·
·
·
·
!
−s
−s
−s
−s
−s
1 − 2 1 − 3 1 − 5 1 − 7 1 − 11
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JB395=C9BCB5=79=55AB6A><
9=BC9CDC9>=B>63>D=CA95B
q=1.15
T=0.145
PHENIX @ RHIC
q ≈ 1.10
q-PLANE WAVES:
1) New representation of Dirac delta:
2−q ∞
− ikx
δ ( x) =
dk
e
(1 ≤ q < 2)
q
∫
2π −∞
i.e.,
∫
∞
−∞
dx δ ( x − x0 ) f ( x ) = f ( x0 )
Question: For what class of functions f ( x ) is this so?
M. Jauregui and C. T., J Math Phys 51, 063304 (2010)
2) New representation of π :
Archimedes
(c. 287 BC – c. 212 BC)
M. Jauregui and C. T., J Math Phys 51, 063304 (2010)
standard
Dirac delta
λ
f ( x) ! A | x |
(| x |→ ∞; λ ∈ R)
1
λ
! qmax − 1
A. Chevreuil, A. Plastino and C. Vignat, J Math Phys 51, 093502 (2010)
M. Mamode, J Math Phys 51, 123509 (2010)
A. Plastino and M.C. Rocca, 1012.1223 [math-ph]
M. Jauregui and C. T., Phys Lett A 375, 2085 (2011)
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