The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
An Entropy depending only on the Probability
(or the Density Matrix)
Octavio Obregón
Departamento de Fı́sica, Universidad de Guanajuato.
December, 2016
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
The Entropy and Superstatistics
The Entropy and Superstatistics
Boltzman-Gibbs (BG) statistics works perfectly well for classical
systems with short range forces and relatively simple dynamics
in equilibrium.
SUPERSTATISTICS: Beck and Cohen considered nonequilibrium systems with a long term stationary state that possesses
a spatio-temporally fluctuating intensive quantity (temperature,
chemical potential, energy dissipation). More general statistics
were formulated.
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
The Entropy and Superstatistics
The macroscopic system is made of many smaller cells that are
temporarily in local equilibrium, β is constant. Each cell is large
enough to obey statistical mechanics.
But has a different β assigned to it, according to a general distribution f (β), from it one can get on effective Boltzmann factor
ˆ ∞
B(E) =
dβf (β)e−βE ,
(1)
0
where E is the energy of a microstate associated with each of
the considered cells. The ordinary Boltzmann factor is recovered
for
f (β) = δ(β − β0 ).
(2)
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
The Entropy and Superstatistics
One can, however, consider other distributions. Assume a Γ (or
χ2 ), distribution depending on a parameter pl , to be identified
with the probability associated with the macroscopic configuration of the system.
fpl (β) =
1
β0 pl Γ
1
pl
β 1
β0 pl
1−pl
pl
e−β/β0 pl ,
(3)
.
(4)
Integrating over β
− p1
Bpl (E) = (1 + pl β0 E)
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
l
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
The Entropy and Superstatistics
P
By defining S = k Ω
l=1 s(pl ) where pl at this moment is an arbitrary parameter, it was shown that it is possible to express s(x)
by
ˆ
s(x) =
x
dy
0
α + E(y)
,
1 − E(y)/E ∗
where E(y) is to be identified with the inverse function
(5)
´∞
0
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Bpl (E)
.
dE 0 Bpl (E 0 )
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
The Entropy and Superstatistics
One selects f (β) → B(E) → E(y), S(x) is then calculated.
For our distribution Γ(χ2 ),
S=k
Ω
X
(1 − ppl l ).
(6)
l=1
Its expansion gives
"
#
Ω
X
(pl ln pl )2 (pl ln pl )3
S
pl ln pl +
+
+ ··· ,
− =
k
2!
3!
(7)
l=1
where the first term corresponds to the usual Boltzmann-Gibbs
(Shannon) entropy.
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
The Entropy and Superstatistics
The corresponding functional including restrictions is given by
Ω
Φ=
Ω
X
X p
S
−γ
pl − β
pl l+1 El ,
k
l=1
(8)
l=1
P
where the first restriction corresponds to Ω
l=1 pl = 1 and the
second one concerns the average value of the energy and γ
and β are Lagrange parameters.
By maximizing Φ, pl is obtained as
l
1 + ln pl + βEl (1 + pl + pl ln pl ) = p−p
l .
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
(9)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
The Entropy and Superstatistics
Assume now the equipartition condition pl =
1
Ω,
remember
Ω
X
S
SB
− =
pl ln pl −→
= ln Ω.
k
k
(10)
k=1
In our case
1
S = kΩ 1 − 1 ,
ΩΩ
in terms of SB (the Boltzmann entropy), S reads
SB
1
S
=
− e−SB /k
k
k
2!
SB
k
2
1 2SB
+ e− k
3!
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
SB
k
(11)
3
··· .
(12)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
The Entropy and Superstatistics
Entropy
1.5
1.0
0.5
W
1
2
3
4
5
6
7
Figure 1: Entropies as function of Ω (small). Blue dashed and red
dotted lines correspond to Sk , and SkB , respectively (pl = 1/Ω
equipartition).
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
H-Theorem
H-Theorem
The usual H-theorem is established for the H function defined
as
ˆ
H = d3 pf ln f.
(13)
The new H function can be written as
ˆ
H = d3 p(f f − 1).
Considering the partial time derivative, we have
ˆ
∂f
∂H
= d3 r[ln f + 1]ef ln f
.
∂t
∂t
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
(14)
(15)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
H-Theorem
Using the mean value theorem for integrals, and realizing that
the factor ef ln f is always positive, it follows from the conventional H-theorem that the variation of the new H-function with
time satisfies
∂H
≤ 0.
(16)
∂t
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
Distribution and their Associated Boltzmann Factors
Distribution and their Associated Boltzmann Factors
For the fpl (β), Γ(χ2 ) distribution we have shown that the Boltzmann factor can be expanded for small pl β0 E, to get
1
1
Bpl (E) = e−β0 E 1 + pl β02 E 2 − p2l β03 E 3 + . . . .
2
3
(17)
We follow now the same procedure for the log-normal distribution,
fpl (β) = √
1
2
1
1
exp
2πβ[ln(pl + 1)] 2
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
−
[ln β(plβ+1)
]2
0
2 ln(pl + 1)
.
(18)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
Distribution and their Associated Boltzmann Factors
The Boltzmann factor can be obtained at leading order, for small
variance of the inverse temperature fluctuations,
1
1
Bpl (E) = e−β0 E 1 + pl β02 E 2 − p2l (pl + 3)β03 E 3 + . . . . (19)
2
6
In general, the F -distribution has two free constant parameters.
We consider, particulary the case in which one of these parameters is chosen as v = 4. For this value we define a F -distribution
as
l −1
Γ 8p
2pl − 1 2
β
2pl −1 1
fpl (β) =
(20)
8p −1 .
4pl +1 β 2
l
p
+
1
Γ 2pl −1 0
l
β 2pl −1 2pl −1
(1 + β0 pl +1 )
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
Distribution and their Associated Boltzmann Factors
Once more the associated Boltzmann factor can not be evaluated in a closed form, but for small variance of the fluctuations
we obtain
1
1 5pl − 1 3 3
Bpl (E) = e−β0 E 1 + pl β02 E 2 + pl
β0 E + . . . . (21)
2
3 pl − 2
As can be observed the first correction term to all these Boltzmann factor is the same. This will imply that their associated
entropies also coincide up to the first correction term.
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
The Generalized Replica Trick
The Generalized Replica Trick
The corresponding generalized entanglement entropy to the entropy (6) is given by
S
= T r(I − ρρ ),
k
(22)
with ρ the density matrix, but this exactly corresponds to a ”natural” generalization of the Replica trick namely
−
X 1
S
∂k
=
lim
T rρn .
k
k! n→k ∂nk
(23)
k≥1
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
The Generalized Replica Trick
As shown, by example, by C. Pasquale and J. Cardy for several
examples of 2dCFT’s
T rρnA = cn b
1 −n)
c( n
6
(24)
where ρA = T rB ρ for a system composed of the subsystems
A and B, with cn a constant, b a parameter depending on the
model and c the central charge.
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
The Generalized Replica Trick
Then the usual Von Neumann entropy (k = 1) SA
−
∂
c
T rρn |n=1 = ln b = SA .
∂n
3
(25)
Our generalized Replica trick taking the second and third derivation will give
−3SA
4
− 4SA
1
5
125 2
25
e 3
SA 1+ SA −
SA
+
SA +
SA +. . . .
S = SA +
16
8
6
27 181
729
(26)
The correction terms are exponentially suppressed.
e
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
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
According to Ted Jacobson’s (and also E. Verlinde) proposal, we
can reobtain gravitation from the entropy, for a modified entropy
S=
A
+ s,
4lp2
(27)
one gets a modified Newton’s law
GM m
2 ∂s
F =−
1 + 4lp
.
R2
∂A A=4πR2
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
(28)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
Coming back to our entropy and identifying SB =
A
we get
4lp2
2
GM m GM mπ
π R2 − πlR2p
F =−
+
1
−
e
.
R2
lp2
2lp2
(29)
Generalized gravitation? From Clausius relation (Jacobson)
ˆ
δQ
2π
=
Tab k a k b (−λ)dλd2 A ,
(30)
T
~
and
ˆ
δSB = η Rab k a k b (−λ)dλd2 A ,
(31)
one gets Einstein’s Equations.
In our case, approximately one gets
δS = δSB (1 − SB ).
(32)
A nonlocal gravity?
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
Now, von-Neumann entropy
SA = −trA ρA logρA ,
ρA = trB |ΨihΨ|.
For a 2dCFT with periodic boundary conditions
c
L
πl
SA = · log
sin
,
3
πα
L
(33)
(34)
where l and L are the length of the subsystem A and the total
system A ∪ B respectively α is a U V cutoff (lattice spacing), c is
the central charge of the CFT.
For an infinite system
SA =
c
l
log .
3
α
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
(35)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
Ryu and Takayanagi proposed
SA =
Area ofγA
(d+2)
,
(36)
4GN
where γA is the d dimensional static minimal surface in AdSd+2
(d+2)
whose boundary is given by ∂A, and GN
is the d+2 dimensional Newton constant.
Intuitively, this suggests that the minimal surface γA plays the
role of a holographic screen for an observer who is only accesible to the subsystem A. They show explicitly the relation (35)
in the lowest dimensional case d = 1, where γA is given by a
geodesic line in AdS3 .
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
Let us explore the generalization of this conjecture if
S = Tr [I − ρρ ].
(37)
As we have shown the entropy we propose gives further terms
also functions of SA which are polynomials exponentially suppressed. So on the gravitational side (the AdS3 spacetime) is
not clear how to generalize it, i.e. which modified theory of gravitation would provide a spacetime solution for which one could
construct a minimal surface area corresponding to the 2dCFT
generalized entropy.
Moreover, recently Xi Dong found that a derivative of holographic
1
ln T rρn with respect to n, gives the area
Rényi entropy Sn = 1−n
SA of a bulk codimensional-2 cosmic brane.
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
bn ) on the replicated bulk B
bn is related to
The classical action I(B
An 1
n−1
b
the area of the brane as ∂n I(Bn ) ≡ ∂n ( n )Sn = 4G
2 , for A2 ,
N n
A2
00
0
= 2S2 + 2S2 − S2 .
4Gn
Our entropy corresponds to
h 00
i
1
0
0
S = S0 + e−S2 −S2 − 2S2 + (S2 + S2 )2 ,
2
(38)
(39)
bn ) and
where the quantities S2 , S20 and S200 can be written vs. I(B
its derivatives:
b2 ) − I(B
b1 )),
S2 = 2(I(B
bn )|n=2 − I(B
b2 ) + I(B
b1 ),
S20 = 2∂n I(B
S200
(40)
2
b
b
b
b
= −∂n I(Bn )|n=2 + 2(I(Bn ) − I(B1 )) + 2∂n I(Bn )|n=2 .
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
References
[1] Superstatistics and Gravitation. O. Obregón. Entropy 2010,
12(9), 2067-2076.
[2] Generalized information entropies depending only on the probability distribution. O. Obregón and A. Gil-Villegas. Phys. Rev.
E 88, 062146 (2013).
[3] Generalized information and entanglement entropy, gravitation and holography. O. Obregón. Int.J.Mod.Phys. A30 (2015)
16, 1530039.
[4] Generalized entanglement entropy and holography. N. Cabo
Bizet and O. Obregón. arXiv:1507.00779.
[5] H-theorem and Thermodynamics for generalized entropies
that depend only on the probability. O. Obregón, J. Torres-Arenas
and A. Gil-Villegas. arXiv:1610.06596.
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.
The Entropy and Superstatistics
H-Theorem Distribution and their Associated Boltzmann Factors The Generalized Replica Trick
[6] Modified entropies, their corresponding Newtonian forces,
potentials, and temperatures. A. Martı́nez-Merino, O. Obregón
and M.P. Ryan (2016), sent to Phys. Rev. D (2016).
[7] Newtonian black holes: Particle production, ”Hawking” temperature, entropies and entropy field equations. A. Martı́nezMerino, O. Obregón and M.P. Ryan (2016), sent to Class. and
Q. Grav.
[8] Gravitothermodynamics modified by a generalized entropy.
J. L. López, O. Obregón, and J. Torres-Arenas (2016).
[9] Nanothermodinamics from a generalized entropy depending
only on the probability. O. Obregón and C. Ramı́rez (2016).
Octavio Obregón
An Entropy depending only on the Probability (or the Density Matrix)
Departamento de Fı́sica, Universidad de Guanajuato.