Non-Boltzmann Statistics as an Alternative to Holography
marcelo botta2006, International Journal of Modern Physics A
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An intriguing question related to black hole thermodynamics is that the entropy of a region shall scale as the area rather than the volume. In this essay we propose that the microscopical degrees of freedom contained in a given region of space, are statistically related in such a way that obey a nonstandard statistics, in which case an holographic hypothesis would not be needed. This could provide us with some insight about the nature of degrees of freedom of the geometry and/or the way in which gravitation plays a role in the statistic correlation between the degrees of freedom of a system.
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Non-Boltzmann Statistics as an Alternative to Holography
arXiv:gr-qc/0504004v2 21 Oct 2005
Marcelo Botta Cantcheff†,‡1 and Jose A.C. Nogales‡,⋆,2
†
Instituto de Fı́sica Teórica (IFT/UNESP)
Rua Pamplona 145 - Bela Vista 01405-900- Sao Paulo - SP - Brazil.
‡
Centro Brasileiro de Pesquisas Fı́sicas (CBPF),
Departamento de Teoria de Campos e Partı́culas (DCP),
Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brasil.
⋆
Instituto de Fı́sica, Facultad de Ciencias Puras y Naturales,
Calle 27 Campus Universitario, Universidad Mayor de San Andrés,
Casilla 8635, La Paz, Bolivia
Abstract
An intriguing question related to black hole thermodynamics is that the entropy of a region shall scale as the area rather than the volume. In this essay
we propose that the microscopical degrees of freedom contained in a given region of space, are statistically related in such a way that obey a non-standard
statistics, in which case an holographic hypothesis would be not needed. This
could provide us with some insight about the nature of degrees of freedom of
the geometry and/or the way in which gravitation plays a role in the statistic
correlation between the degrees of freedom of a system.
1
2
[email protected]
[email protected], [email protected]
1
We start this note by making the following crucial observation: when gravitational
interaction of a given system is considered, it is impossible to isolate two subsystems
(unless they are infinitely separated) 3 , so, to define statistical independence of two
subsystems is actually problematic in this context. On the other hand, it is known
that for systems whose microscopical components cannot be considered statistically
independent (and in particular, those being highly correlated), it shall be considered
a non-standard statistics (non-additive or non-extensive)[1]. Consequently, we can
expect a deviation of the standard statistics when gravity is considered an important
part of a system.
In the last two decades, generalizations of the Boltzmann-Gibbs thermodynamics
have been extensively explored [2].
In an ideal system in thermodynamic equilibrium, if the intensive variables are kept
fixed and the side of the system is doubled, the extensive variables should then also
double. This is not true for black holes [3, 4, 5, 6].
From the thermodynamic point of view, we mean macroscopic [7], the entropy of
black holes is proportional to the area of the event horizon
S ∝ A.
We are going to analyze the statistical behavior of the degrees of freedom that compose
this type of systems, in order to have consistency with the relation above.
Let us explain more carefully the difference between additivity and extensivity
[8]. Consider a physical quantity W (i) related with the subsystem i. For N of such
subsystems, W is additive if we have:
W (N) =
N
X
W (i) ,
(1)
i=1
where W (N) ≡ W {i}N
1 . Supposing that all subsystems are equal,
W (N) = NW (1) .
(2)
In its turn, extensivity is defined by:
|W (N)|
<∞.
N →∞
N
lim
3
In contrast with ordinary thermodynamic systems.
2
(3)
The physical quantities that do not fulfill this property are called non-extensive. We
note that all quantities which are additive are also extensive, in fact
|W (N)|
= |W̄ | ,
N →∞
N
lim
(4)
where the mean value, |W̄ |, is finite since W (i) < ∞ ∀i. In particular if the subsystems
are equal, we get limN →∞ |WN(N )| = W (1).
1
Blak Hole thermodynamics and the holographic
interpretation
Roughly speaking, the argument which leads to the holographic principle is that if one
assumes an uniform distribution of the gravitational degrees of freedom in a spatial
region of volume V , or in other words, if, as suggested by quantum gravity, the spacetime is thought as filled of quantum geometric atoms like wormholes or more complex
structures of Planck size ℓp more or less uniformly distributed, then the number of
microstates is ∝ M V , where M is the number of states reachable by each microscopical
component. Then, following the Boltzmann (extensive) statistics, one obtains that
entropy S must scale as the volume V which would lead to a contradiction with the
Bekenstein-Hawking law for the black hole entropy, S = A/4 in Plancks units [3, 5].
So, the holographic principle suggested by tHooft [9] essentially claims that all these
degrees of freedom may be mapped to the boundary of that region, and consequently
the corresponding entropy would scale as the area.
As we can notice, an important part of this argument is the extensive BoltzmannGibbs statistics, and our proposal is precisely based on alternatives to this, in order
to match consistency with the area law and to render the holographic viewpoint not
necessary 4 .
2
Non-standard statistics and consistency with the
area law.
Now, we keep the assumption of uniformity in the distribution of the physical degrees
of freedom in a spatial region of volume V , and drop out the holographic point of view
4
In this approach, we are not worried with the aspects concerning the information paradox which,
in principle, could be avoided by the holographic principle.
3
about the possibility of mapping them to the boundary.
According to this, if we suppose N identical degrees of freedom uniformly distributed in V , we have N ∝ V . Following the area law, we must have S(N) ∝ A ∝
N 2/3 . Thus, our main result is:
S(N) ∼ N 2/3 S(1) ,
(5)
which differs from (2). Substituting this into (3) we get
lim
N →∞
|S(N)|
=0,
N
(6)
which shows the extensivity of the entropy.
Our conclusion is that the statistics that must be considered in a system where the
gravitation plays an important role (as a Black Hole) is non-standard, in particular
the entropy is a non-additive but extensive quantity. The law which rules the non
additivity is given by (5).
3
Correlated Systems and Tsallis Statistics.
Finally, we are going to discuss here a possible realization of the framework described
above. As discussed in [1], for certain systems where the hypothesis of probabilistic
independence is not applicable, the number of states is
Γ ∼ N γ (γ > 0 , N ≫ 1).
(7)
On the other hand, the corresponding expression for the Tsallis entropy is
S=k
Γ1−q − 1
,
1−q
(8)
where q < 1 is a dimensionless parameter.
If the degrees of freedom, geometric or not, are correlated by gravitational interaction such that (7) holds, we obtain that
S ∼ N (1−q)γ (N ≫ 1) .
(9)
Thus, by requiring consistency with (5), we obtain:
(1 − q)γ = 2/3.
4
(10)
As a particular example, if we have a sort of Boolean system where each (distinguishable) degree of freedom may assume the states si = 0, 1 , i = 1, ..., N [9], and the
constraint
N
X
si = 1
(11)
i=1
is also assumed, then, the possible states of the system are given by si = δik , for some
k (1 ≤ k ≤ N). Then, we have Γ = N. Substituting this into (10), as a result one has:
γ ∼ 1 ⇒ q = 1/3.
On the other hand, notice that the q-statistics may be continuously extended to
q > 1 values, and expression (8) becomes
S=k
Γq−1 − 1
.
q−1
(12)
In this case, one can verify a remarkable coincidence with a conjecture due to Tsallis
et al [10, 8], about statistics of systems governed by a potential decreasing with the
inverse square of the distance (as is the case of gravity). By using arguments quite
different from here, they argued that q should be 5/3. But, on the other hand, we
obtain precisely this value (q ∼ 5/3) for the simplified model we are considering above
(with γ ∼ 1).
We conclude this work by stressing that also the problems associated with the
black hole thermodynamical instability could be by-passed by considering certain nonstandard statistics [11].
We would like to thank to CNPq for the invaluable financial help. Special thanks
are due to A.L.M.A. Nogueira and J. A. Helayel by reading the manuscript.
5
References
[1] C. Tsallis, ”What should a statistical mechanics satisfy to reflect nature”,
[arXiv:cond-mat/0403012]
[2] C. Tsallis, J. Stat. Phys. 52, 479-487(1988), ”Possible Generalization of
Boltzmann-Gibbs Statistics”. See related references in
http://www.cbpf.br/GrupPesq/StatisticalPhys/TEMUCO.pdf
[3] J.D. Bekenstein, ”Black Holes and the Second Law”, Lett. Nuovo Cimento 4 , 737
(1972).
[4] J.M. Bardeen, B. Carter, S.W. Hawking, ”4 Laws of Black Hole Mechanics”,
Communications in Mathematical Physics 31,161-170 (1973).
[5] S.W. Hawking, ”Particle Creation by Black Holes”, Communications in Mathematical Physics 43 , 199-220 (1975).
[6] T. Jacobson, ”Thermodynamics of Spacetime- The Einstein Equation of State”
Phys. Rev. Lett. 75,1260-1263 (1995).
[7] M. Botta Cantcheff, On the Microscopical Structure of the Classical Spacetime., hep-th/0408151. T. Padmanabhan, Gravity As Elasticity Of Spacetime: A
Paradigm To Understand Horizon Thermodynamics And Cosmological Constant,
gr-qc/0408051.
[8] C. Tsallis ”Is the entropy Sq extensive or nonextensive?”, proceedings of the 31st
Workshop of the International School of Solid State Physics “Complexity, Metastability and Nonextensivity”, held at the Ettore Majorana Foundation and Centre
for Scientific Culture, Erice (Sicily) in 20-26 July 2004, eds. C. Beck, A. Rapisarda
and C. Tsallis (World Scientific, Singapore, 2005) arXiv/cond-mat/0409631
[9] G. ’T Hooft, ”Dimentional Reduction in Quantum Gravity”, [arXiv:grqc/9310026].
[10] ”Entropic Non-Extensivity: a Possible Measure of Complexity” C. Tsallis , Chaos,
Solitons and Fractals , 13 (2002) 371 arXiv/cond-mat/0010150.
[11] M. Botta Cantcheff and J. Nogales, work in progress.
6
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