an entropy depending only on the probability (or the ... · (shannon) entropy. octavio obregon´...

The Entropy and Superstatistics H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick Attempts to relate this Entropy to Gravitation and Holography

An Entropy depending only on the Probability(or the Density Matrix)

Octavio Obregon

Departamento de Fısica, Universidad de Guanajuato.

December, 2016

Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.

An Entropy depending only on the Probability (or the Density Matrix)


The Entropy and Superstatistics


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 possessesa 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, β is constant. Each cell is largeenough to obey statistical mechanics.

But has a different β assigned to it, according to a general dis-tribution f(β), from it one can get on effective Boltzmann factor

B(E) =

ˆ ∞0

dβf(β)e−βE , (1)

where E is the energy of a microstate associated with each ofthe considered cells. The ordinary Boltzmann factor is recoveredfor

f(β) = δ(β − β0). (2)





One can, however, consider other distributions. Assume a Γ (orχ2), distribution depending on a parameter pl, to be identifiedwith the probability associated with the macroscopic configura-tion of the system.

fpl(β) =1

β0plΓ(

1pl

) ( β

β0

1

pl

) 1−plpl

e−β/β0pl , (3)

Integrating over β

Bpl(E) = (1 + plβ0E)− 1pl . (4)





By defining S = k∑Ω

l=1 s(pl) where pl at this moment is an arbi-trary parameter, it was shown that it is possible to express s(x)by

s(x) =

ˆ x

0dy

α+ E(y)

1− E(y)/E∗, (5)

whereE(y) is to be identified with the inverse function Bpl (E)´∞0 dE′ Bpl (E

′).





One selects f(β)→ B(E)→ E(y), S(x) is then calculated.

For our distribution Γ(χ2),

S = k

Ω∑l=1

(1− ppll ). (6)

Its expansion gives

−Sk

=

Ω∑l=1

[pl ln pl +

(pl ln pl)2

2!+

(pl ln pl)3

3!+ · · ·

], (7)

where the first term corresponds to the usual Boltzmann-Gibbs(Shannon) entropy.





The corresponding functional including restrictions is given by

Φ =S

k− γ

Ω∑l=1

pl − βΩ∑l=1

ppl+1

l El, (8)

where the first restriction corresponds to∑Ω

l=1 pl = 1 and thesecond one concerns the average value of the energy and γand β are Lagrange parameters.

By maximizing Φ, pl is obtained as

1 + ln pl + βEl(1 + pl + pl ln pl) = p−pll . (9)





Assume now the equipartition condition pl = 1Ω , remember

−Sk

=

Ω∑k=1

pl ln pl −→SBk

= ln Ω. (10)

In our case

S = kΩ

[1− 1

Ω1Ω

], (11)

in terms of SB (the Boltzmann entropy), S reads

S

k=SBk− 1

2!e−SB/k

(SBk

)2

+1

3!e−

2SBk

(SBk

)3

· · · . (12)





1 2 3 4 5 6 7W

0.5

1.0

1.5

Entropy

Figure 1: Entropies as function of Ω (small). Blue dashed and reddotted lines correspond to S

k , and SB

k , respectively (pl = 1/Ωequipartition).




H-Theorem

H-Theorem

The usual H-theorem is established for the H function definedas

H =

ˆd3pf ln f. (13)

The new H function can be written as

H =

ˆd3p(ff − 1). (14)

Considering the partial time derivative, we have

∂H

∂t=

ˆd3r[ln f + 1]ef ln f ∂f

∂t. (15)




H-Theorem

Using the mean value theorem for integrals, and realizing thatthe factor ef ln f is always positive, it follows from the conven-tional H-theorem that the variation of the new H-function withtime satisfies

∂H

∂t≤ 0. (16)




Distribution and their Associated Boltzmann Factors


For the fpl(β), Γ(χ2) distribution we have shown that the Boltz-mann factor can be expanded for small plβ0E, to get

Bpl(E) = e−β0E[1 +

1

2plβ

20E

2 − 1

3p2l β

30E

3 + . . .]. (17)

We follow now the same procedure for the log-normal distribu-tion,

fpl(β) =1

√2πβ[ln(pl + 1)]

12

exp

−

[ln β(pl+1)12

β0]2

2 ln(pl + 1)

. (18)





The Boltzmann factor can be obtained at leading order, for smallvariance of the inverse temperature fluctuations,

Bpl(E) = e−β0E[1 +

1

2plβ

20E

2 − 1

6p2l (pl + 3)β3

0E3 + . . .

]. (19)

In general, the F -distribution has two free constant parameters.

We consider, particulary the case in which one of these parame-ters is chosen as v = 4. For this value we define a F -distributionas

fpl(β) =Γ(8pl−1

2pl−1

)Γ(4pl+1

2pl−1

) 1

β20

(2pl − 1

pl + 1

)2 β

(1 + ββ0

2pl−1pl+1 )

(8pl−1

2pl−1

) . (20)





Once more the associated Boltzmann factor can not be evalu-ated in a closed form, but for small variance of the fluctuationswe obtain

Bpl(E) = e−β0E[1 +

1

2plβ

20E

2 +1

3pl

5pl − 1

pl − 2β3

0E3 + . . .

]. (21)

As can be observed the first correction term to all these Boltz-mann factor is the same. This will imply that their associatedentropies also coincide up to the first correction term.




The Generalized Replica Trick


The corresponding generalized entanglement entropy to the en-tropy (6) is given by

S

k= Tr(I − ρρ), (22)

with ρ the density matrix, but this exactly corresponds to a ”nat-ural” generalization of the Replica trick namely

−Sk

=∑k≥1

1

k!limn→k

∂k

∂nkTrρn. (23)





As shown, by example, by C. Pasquale and J. Cardy for severalexamples of 2dCFT’s

TrρnA = cnbc( 1n−n)

6 (24)

where ρA = TrBρ for a system composed of the subsystemsA and B, with cn a constant, b a parameter depending on themodel and c the central charge.





Then the usual Von Neumann entropy (k = 1) SA

− ∂

∂nTrρn|n=1 =

c

3ln b = SA. (25)

Our generalized Replica trick taking the second and third deriva-tion will give

S = SA+e−3SA

4

16SA

(1+

25

8SA

)−e− 4SA

3

6SA

(1

27+

5

181SA+

125

729S2A

)+. . . .

(26)The correction terms are exponentially suppressed.




Attempts to relate this Entropy to Gravitation andHolography

According to Ted Jacobson’s (and also E. Verlinde) proposal, wecan reobtain gravitation from the entropy, for a modified entropy

S =A

4l2p+ s, (27)

one gets a modified Newton’s law

F = −GMm

R2

[1 + 4l2p

∂s

∂A

]A=4πR2

. (28)




Coming back to our entropy and identifying SB =A

4l2pwe get

F = −GMm

R2+GMmπ

l2p

[1− π R2

2l2p

]e−π R

2

l2p . (29)

Generalized gravitation? From Clausius relation (Jacobson)δQ

T=

2π

~

ˆTabk

akb(−λ)dλd2A , (30)

andδSB = η

ˆRabk

akb(−λ)dλd2A , (31)

one gets Einstein’s Equations.In our case, approximately one gets

δS = δSB(1− SB). (32)

A nonlocal gravity?Octavio Obregon Departamento de Fısica, Universidad de Guanajuato.



Now, von-Neumann entropy

SA = −trAρAlogρA, ρA = trB|Ψ〉〈Ψ|. (33)

For a 2dCFT with periodic boundary conditions

SA =c

3· log

(L

παsin

(π l

L

)), (34)

where l and L are the length of the subsystem A and the totalsystem A∪B respectively α is a UV cutoff (lattice spacing), c isthe central charge of the CFT.For an infinite system

SA =c

3log

l

α. (35)




Ryu and Takayanagi proposed

SA =Area ofγA

4G(d+2)N

, (36)

where γA 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 minimal surface γA plays therole of a holographic screen for an observer who is only acce-sible to the subsystem A. They show explicitly the relation (35)in the lowest dimensional case d = 1, where γA is given by ageodesic line in AdS3.




Let us explore the generalization of this conjecture if

S = Tr[I − ρρ]. (37)

As we have shown the entropy we propose gives further termsalso functions of SA which are polynomials exponentially sup-pressed. So on the gravitational side (the AdS3 spacetime) isnot clear how to generalize it, i.e. which modified theory of grav-itation would provide a spacetime solution for which one couldconstruct a minimal surface area corresponding to the 2dCFTgeneralized entropy.

Moreover, recently Xi Dong found that a derivative of holographicRenyi entropy Sn = 1

1−n lnTrρn with respect to n, gives the areaSA of a bulk codimensional-2 cosmic brane.




The classical action I(Bn) on the replicated bulk Bn is related tothe area of the brane as ∂nI(Bn) ≡ ∂n(n−1

n )Sn = An4GN

1n2 , for A2,

A2

4Gn= 2S

′′2 + 2S

′2 − S2. (38)

Our entropy corresponds to

S = S0 +1

2e−S2

[−S′′2 − 2S

′2 + (S

′2 + S2)2

], (39)

where the quantities S2, S′2 and S′′2 can be written vs. I(Bn) andits derivatives:

S2 = 2(I(B2)− I(B1)),

S′2 = 2∂nI(Bn)|n=2 − I(B2) + I(B1), (40)S′′2 = −∂nI(Bn)|n=2 + 2(I(Bn)− I(B1)) + 2∂2

nI(Bn)|n=2.




References

[1] Superstatistics and Gravitation. O. Obregon. Entropy 2010,12(9), 2067-2076.[2] Generalized information entropies depending only on the prob-ability distribution. O. Obregon and A. Gil-Villegas. Phys. Rev.E 88, 062146 (2013).[3] Generalized information and entanglement entropy, gravita-tion and holography. O. Obregon. Int.J.Mod.Phys. A30 (2015)16, 1530039.[4] Generalized entanglement entropy and holography. N. CaboBizet and O. Obregon. arXiv:1507.00779.[5] H-theorem and Thermodynamics for generalized entropiesthat depend only on the probability. O. Obregon, J. Torres-Arenasand A. Gil-Villegas. arXiv:1610.06596.




[6] Modified entropies, their corresponding Newtonian forces,potentials, and temperatures. A. Martınez-Merino, O. Obregonand M.P. Ryan (2016), sent to Phys. Rev. D (2016).[7] Newtonian black holes: Particle production, ”Hawking” tem-perature, entropies and entropy field equations. A. Martınez-Merino, O. Obregon and M.P. Ryan (2016), sent to Class. andQ. Grav.[8] Gravitothermodynamics modified by a generalized entropy.J. L. Lopez, O. Obregon, and J. Torres-Arenas (2016).[9] Nanothermodinamics from a generalized entropy dependingonly on the probability. O. Obregon and C. Ramırez (2016).



an entropy depending only on the probability (or the ... · (shannon) entropy. octavio obregon´...

Documents