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, ..


Contenido

1 Generalized information entropies depending on the proba-bility

2 Superstatistics and Gravitation

3 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.




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)




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)




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)




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.




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)




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)




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




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)




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)




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)




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).




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)




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.




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)





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.





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)




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.



generalized entropy(ies) depending only on the probability ... · octavio obregon´ departamento de...

Documents