Generalized entropy(ies) depending only on the probability ... · Octavio Obregon´ Departamento de...
Transcript of Generalized entropy(ies) depending only on the probability ... · Octavio Obregon´ Departamento de...
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
Generalized entropy(ies) depending only onthe probability; Gravitation, AdS/CFT, ..
Octavio Obregon
Departamento de Fısica, Universidad de Guanajuato.
December, 2014
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
Contenido
1 Generalized information entropies depending on the proba-bility
2 Superstatistics and Gravitation
3 AdS-CFT, Ryu and Takayanagi proposal
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
Contenido
1 Generalized information entropies depending on the proba-bility
2 Superstatistics and Gravitation
3 AdS-CFT, Ryu and Takayanagi proposal
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
Contenido
1 Generalized information entropies depending on the proba-bility
2 Superstatistics and Gravitation
3 AdS-CFT, Ryu and Takayanagi proposal
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
Generalized information entropies depending on theprobability
Boltzman-Gibbs (BG) statistics works perfectly well for classicalsystems with short range forces and relatively simple dynamicsin equilibrium.SUPERSTATISTICS: Beck and Cohen considered nonequilib-rium systems with a long term stationary state that possessa spatio temporally fluctuating intensive quantity (temperature,chemical potential, energy dissipation). More general statisticswere formulated.
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
The macroscopic system is made of many smaller cells that aretemporarily in local equilibrium, � is constant. Each cell is largeenough to obey statistical mechanics.But has a different � assigned to it, according to a general dis-tribution f(�), from it one can get on effective Boltzmann factor
B(E) =
ˆ 1
0d�f(�)e
��E
, (1)
where E is the energy of a microstate associated with each ofthe considered cells. The ordinary Boltzmann factor is recoveredfor
f(�) = �(� � �0). (2)
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
One can, however, consider other distributions. Assume a � (or�
2), distribution depending on a parameter p
l
, to be identified
with the probability associated with the macroscopic configura-tion of the system.
f
pl(�) =1
�0pl
�
⇣1pl
⌘✓
�
�0
1
p
l
◆ 1�plpl
e
��/�0pl, (3)
Integrating over �
B
pl(E) = (1 + p
l
�0E)
� 1pl. (4)
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
By defining S = k
P⌦l=1 s(pl) where p
l
at this moment is an arbi-trary parameter, it was shown that it is possible to express s(x)
and a generic internal energy by
s(x) =
ˆx
0dy
↵+ E(y)
1� E(y)/E
⇤ , (5)
and
u(x) = (1 + ↵/E
⇤)
ˆx
0
dy
1� E(y)/E
⇤ , (6)
where E(y) is to be identified with the inverse function Bpl (E)´10 dE
0Bpl (E
0)
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
One selects f(�) ! B(E) ! E(y) ! s(x) and u(x) are thencalculated.
For the distribution �(�2), we have shown,
S = k
⌦X
l=1
(1� p
pll
). (7)
Its expansion gives
�S
k
=
⌦X
l=1
"p
l
ln p
l
+
(p
l
ln p
l
)
2
2!
+
(p
l
ln p
l
)
3
3!
+ · · ·#, (8)
where the first term corresponds to the usual Shannon entropy.
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
The same infinite expansion is obtained by making an extensionof the Replica trick by including higher order derivative terms
�S
k
=
X
k�1
1
k!
lim
n!k
@
k
@n
k
Tr⇢
n
. (9)
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
The corresponding functional including restrictions is given by
� =
S
k
� �
⌦X
l=1
p
l
� �
⌦X
l=1
p
pl+1
l
E
l
, (10)
where the first restriction corresponds toP⌦
l=1 pl = 1 and thesecond one concerns the average value of the energy and � and� are Lagrange parameters. By maximizing �, p
l
is obtained as
1 + ln p
l
+ �E
l
(1 + p
l
+ p
l
ln p
l
) = p
�pll
, (11)
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
1 2 3 4 5bEl
0.2
0.4
0.6
0.8
1.0
pl
Figure: Comparison of the two probabilities. Blue dotted linecorresponds to pl = f(�El), Eq. (10), and red dashed line to thestandard one pl = e
��El
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
Assume now the equiparable condition p
l
=
1⌦ , remember
�S
k
=
⌦X
k=1
p
l
ln p
l
�! S
B
k
= ln⌦. (12)
In our case
S = k⌦
1� 1
⌦
1⌦
�, (13)
in terms of SB
, the Boltzmann entropy
S
k
=
S
B
k
� 1
2!
e
�SB/k
✓S
B
k
◆2
+
1
3!
e
� 2SBk
✓S
B
k
◆3
· · · . (14)
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
1 2 3 4 5 6 7W
0.5
1.0
1.5
Entropy
Figure: Entropies as function of ⌦ (small). Blue dashed and reddotted lines correspond to S
k , Eq.(12) and, S�
k , respectively (pl = 1/⌦
equipartition)
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
200 400 600 800 1000W
3
4
5
6
7Entropy
Figure: Entropies as function of ⌦ (large). Blue Dashed and reddotted lines correspond to S
k , Eq. (12) and S�
k , respectively (pl = 1/⌦
equipartition)
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
The known generalized entropies depend on one or several pa-rameters. The one proposed here depends only on p
l
. Otherentropies depending only on p
l
can be proposed. Take, as anexample, the Kaniadakis entropy
S
= �k
⌦X
l
p
1+
l
� p
1�
l
2
. (15)
This entropy reduces to the Shannon entropy for = 0. Inspiredin this, we propose
S = �k
⌦X
l
p
pll
� p
�pll
2
, (16)
which expansion gives also a first term the Shannon entropy.There is more...PRE 88, 062146 (2013).
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
Superstatistics and Gravitation
According with Ted Jacobson’s (and also E. Verlinde) proposal,we can reobtain gravitation from the entropy for a modified en-tropy
S =
A
4l
2p
+ s, (17)
one gets a modified Newton’s law
F = �GMm
R
2
1 + 4l
2p
@s
@A
�
A=4⇡R2
. (18)
Coming back to Eq.(14) and identifying S
B
=
A
4l
2p
we get
F = �GMm
R
2+
GMm⇡
l
2p
1� ⇡R
2
2l
2p
�e
�⇡R2
l2p. (19)
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
Superstatistics and Gravitation
Generalized gravitation?, basically (Jacobson), Clausius relation
�Q
T
=
2⇡
~
ˆT
ab
k
a
k
b
(��)d�d
2A , (20)
and�S
B
= ⌘
ˆR
ab
k
a
k
b
(��)d�d
2A , (21)
one gets Einstein’s Equations.In our case, even approximated,
�S = �S
B
(1� S
B
). (22)
A generalization with integrals?Entropy 2010, 12, 2067.
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
AdS-CFT, Ryu and Takayanagi proposal
Now, von-Neumann entropy
S
A
= �trA⇢Alog⇢A, ⇢A = trB| ih |. (23)
For one-dimensional (1D) quantum many-body systems at criti-cality (i.e. 2D CFT) it is known
S
A
=
c
3
· log✓
L
⇡↵
sin
✓⇡ l
L
◆◆, (24)
where l and L are the length of the subsystem A and the totalsystem A[B respectively ↵ is a UV cutoff (lattice spacing), c isthe central charge of the CFT.For an infinite system
S
A
=
c
3
log
l
↵
(25)
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
AdS-CFT, Ryu and Takayanagi proposal
Ryu and Takayanagi proposed
S
A
=
Area of�A
4G
(d+2)N
, (26)
where �
A
is the d dimensional static minimal surface in AdSd+2
whose boundary is given by @A, and G
(d+2)N
is the d+2 dimen-sional Newton constant. Intuitively, this suggests that the min-imal surface �
A
plays the role of a holographic screen for anobserver who is only accesible to the subsystem A. They showexplicitly the relation (25) in the lowest dimensional case d = 1,where �
A
is given by a geodesic line in AdS3.
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
AdS-CFT, Ryu and Takayanagi proposal
For general d this also seems to work. A more general treat-ment and a proof of the conjecture were given by Maldacenaand Lewkowycz.Which is the generalization of this conjecture if
S = T
r
[I � ⇢
⇢
], (27)
by means of
Tr⇢
n
A= b
16 (
1n�n)
(Cardy-Calabrese), (28)
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..
Generalized information entropies depending on the probability Superstatistics and Gravitation AdS-CFT, Ryu and Takayanagi proposal
with c = 1 and b results to be b = e
3S0 , now utilizing the modifiedReplica trick one gets
S = S0�e
� 34S0 S0
2!
5
2
2
6S0 +
1
2
3
�+e
� 43S0 S0
3!
5
3
3
6S
20 +
5
3
4S0 +
1
3
2
�.
(29)
Collaborators: N. Cabo, R. Castaneda, J.L Lopez, A. Martınez,J.Torres, S. Zacarıas.
Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.
Generalized entropy(ies) depending only on the probability; Gravitation, AdS/CFT, ..