Astrophys Space Sci (2006) 305:241–245
DOI 10.1007/s10509-006-9198-5
O R I G I NA L A RT I C L E
Why q-Expectation Values Must be Used in Nonextensive
Statistical Mechanics
Sumiyoshi Abe
Received: 2 May 2006 / Accepted: 2 June 2006
C Springer Science + Business Media B.V. 2006
Abstract There is a controversy in the area of nonextensive
statistical mechanics regarding the form of the expectation
value of a physical quantity. Two definitions have been discussed in the literature: one is the ordinary definition and the
other is the normalized q-expectation value associated with
the escort distribution. Here, it is proved that the normalized
q-expectation value is the correct one to be employed. The
Shore-Johnson theorem is used to show that the formalism
with the normalized q-expectation value is theoretically consistent with minimum cross entropy principle, whereas the
ordinary expectation value has to be excluded.
Keywords q-expectation values . nonextensive statistical
mechanics
1. Introduction
Nonextensive statistical mechanics (Abe and Okamoto,
2001; Kaniadakis et al., 2002; Gell-Mann and Tsallis, 2004;
Kaniadakis and Lissia, 2004) pioneered by Tsallis (1988)
has been attracting continuous interest over the years. It is
expected to offer a unified framework for describing complex systems in their nonequilibrium stationary states, systems with (multi)fractal and self-similar structures, longrange interacting systems, anomalous diffusion phenomena, and so on. The worked examples include cosmic rays
(Tsallis et al., 2003), astrophysics and self-gravitating systems (Lavagno et al., 1998; Taruya and Sakagami, 2003a,b),
dynamical systems at the edge of chaos (Latora et al., 2000;
S. Abe
Institute of Physics, University of Tsukuba, Ibaraki 305-8571,
Japan
e-mail: [email protected]
de Moura et al., 2000; Baldovin and Robledo, 2002; Johal
and Tirnakli, 2004), lattice Boltzmann models (Boghosian
et al., 2003), magnetism of colossal magnetoresistance manganites (Reis et al., 2002), high-energy processes (Walton
and Rafelski, 2000; Bediaga et al., 2000; Beck, 2000;
Alberico et al., 2000; Navarra et al., 2003), quantum groups
(Abe, 1997, 1998, 2003b), cellular aggregates (Upadhyaya
et al., 2001), Lévy flights (Prato and Tsallis, 1999; Abe and
Rajagopal, 2000a), semiclassical dynamics in optical lattices (Lutz, 2004), kinetics of charged particles (Rossani and
Scarfone, 2000), Internet traffic (Abe and Sukuki, 2003a),
earthquakes (Abe and Suzuki, 2003a, 2005; SotolongoCosta and Posadas, 2004), econophysical problems
(Borland, 2002a,b; Kozuki and Fuchikami, 2003), and complex networks (Tadic and Thurner, 2004; Abe and Sukuki,
2004a; Wilk and Wlodarczyk, 2004).
In spite of these successes, the theoretical foundations of
nonextensive statistical mechanics are yet to be established
in some respects. Among others, the problem concerning the
definition of expectation value of a physical quantity seems
to be poorly understood. There are two different definitions
in the literature. One is the frequently employed definition,
which is the normalized q-expectation value (Tsallis et al.,
1998):
U (nor) = H q =
Pi εi ,
(1)
( pi )q
Pi = ,
q
j(pj)
(2)
i
where Pi is called the escort distribution (Beck and Schlögl,
1993; Abe, 2003c) associated with the basic distribution pi
and H is a physical random variable under consideration
(e.g., the system energy) with its ith value εi .q is taken to be
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Astrophys Space Sci (2006) 305:241–245
positive. The other definition is the ordinary one:
U (ord) = H =
−β
pi εi ,
where α and β are the Lagrange multipliers. After eliminating
α, the resulting maximum entropy distribution p̃i(ord) is found
to be
1/(q−1)
p̃i(ord) = 1 + (1 − q) S̃q(ord)
1/(q−1)
q −1 × 1−
,
β εi − Ũ (ord)
q
+
(6)
where
β = i
β
p̃i(ord)
q ,
(7)
and [a]+ ≡ max{0, a}. In Equation (6), S̃q(ord) and Ũ (ord) are
calculated in terms of the distribution in Eq. (6) in the selfreferential manner.
On the other hand, if the normalized q-expectation value
is employed, the functional to be maximized reads
(nor) [ p; α, β] = Sq [ p] − α
−β
1
p̃i(nor) =
First let us recall how nonextensive statistical mechanics depends on the definition of expectation value. In nonextensive statistical mechanics, the Tsallis entropy indexed by q
(Tsallis, 1988)
pi − 1
i
( pi )q εi
i
q
j(pj)
−U
(nor)
.
(8)
Z̃ q(nor)
1/(1−q)
1 − (1 − q)β ∗ εi − Ũ (nor) +
,
1/(1−q)
Z̃ q(nor) = 1 + (1 − q) S̃q(nor)
1/(1−q)
1 − (1 − q)β ∗ εi − Ũ (nor) +
=
,
(9)
(10)
i
kB
q
sq [ p] =
( pi ) − 1
1−q i
(4)
is maximized under appropriate constraints on the normalization condition of the probability distribution and the expectation value of a physical quantity under consideration.
Here and hereafter, the Boltzmann constant, k B , is set equal
to unity for simplicity.
If the ordinary expectation value is employed, then the
functional to be maximized is
i
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(5)
The corresponding maximum entropy distribution is given
by
2. Ordinary Expectation Value and Normalized
q-Expectation Value
[ p; α, β] = Sq [ p] − α
pi εi − U (ord) ,
(3)
which is preferred by some researchers. This may be due
to the fact that the nonextensive statistical mechanical formalisms with these two definitions of expectation value lead
to the stationary distribution of a similar kind (see Sec. 2).
Therefore, it is crucial to identify which the correct definition
is.
It has been shown (Abe and Rajagopal, 2000) that, for
a class of power-law distributions (i.e., the q-exponential
distribution with q > 1), only the normalized q-expectation
value is consistent with the method of steepest descents for
deriving canonical ensemble from microcanonical ensemble.
The situation, however, remains unclear for the other class of
distributions with compact supports (i.e., the q-exponential
distribution with 0 < q < 1).
In this paper, we show that the definition to be employed
in nonextensive statistical mechanics is the normalized qexpectation value. For this purpose, we study the properties of two generalized relative entropies regarding the two
different definitions of expectation value. Then, we use the
Shore-Johnson theorem for minimum cross entropy (relative entropy) principle to prove that the formalism with the
normalized q-expectation value is theoretically consistent,
whereas the ordinary expectation value cannot be employed.
i
i
(ord)
pi − 1
where
β∗ = i
β
p̃i(nor)
q .
(11)
Similarly to the case of the ordinary expectation value, S̃q(nor)
and Ũ (nor) are the values of Sq and U (nor) calculated in terms of
the maximum entropy distribution p̃i(nor) in the self-referential
manner.
Clearly, the Tsallis entropy in Equation (4) and the normalized q-expectation value
tend to the Boltzmann-GibbsShannon entropy S[ p] = − i pi ln pi and the ordinary expectation value in the limit q → 1. Accordingly, both of
Astrophys Space Sci (2006) 305:241–245
243
the distributions in Equations (6) and (9) converge to the
Boltzmann-Gibbs distribution p̃i ∼ exp(−βεi ) in such a limiting case.
It is mentioned that they resemble in their forms (with
opposite signs of the exponents). This point may be a reason
why some researchers think that there is no essential reason
to prefer the normalized q-expectation value and the ordinary
expectation value may be used.
A point to be noticed is that in both cases the following
thermodynamic relations hold:
∂ S̃q(ord)
∂ Ũ (ord)
∂ S̃q(nor)
∂ Ũ (nor)
= β,
(12)
= β,
(13)
which guarantee the existence of the thermodynamic
Legendre-transform structure in both cases. However, it is
still an open problem in nonextensive statistical mechanics what the physical temperature is (Abe, 2001; Abe et al.,
2001).
3. Generalized relative entropies associated with
ordinary expectation value and normalized
q-expectation value
Relative entropy has an important physical meaning as free
energy difference. It is known in mathematical information
theory that there exist two different kinds of relative entropies: one is of the Bregman type (Bregman, 1967) and
the other the Csiszár type (Csiszár, 1972).
The Bregman-type relative entropy (Naudts, 2004) and
the Csiszár-type relative entropy (Abe, 1998; Tsallis, 1998b;
Borland et al., 1998, 1999) associated with the Tsallis entropy
are respectively given by
Iq [ p
r ] =
1 pi [( pi )q−1 − (ri )q−1 ]
q −1 i
−
( pi − ri )(ri )q−1 ,
i
K q [ p
r ] =
(14)
i
pi ln
pi
.
ri
pi
d s[s q−1 − (ri )q−1 ],
(17)
ri
we immediately see its nonnegativity. For K q [ p
r ], it is convenient to rewrite it as
K q [ p
r ] =
1 pi [1 − (ri / pi )1−q ].
1−q i
(18)
Then, using the inequality, (1 − x 1−q )/(1 − q) ≥ 1 − x for
x > 0 and q > 0 with the equality for x = 1, we also see
nonnegativity of K q [ p
r ]. In what follows, we discuss the
physical meanings of Iq [ p
r ] and K q [ p
r ].
Let us take the maximum entropy distributions as the reference distributions in Iq [ p
r ] and K q [ p
r ]. Taking into
account the exponents of the maximum entropy distributions
in Equations (6) and (9) together with the dependencies of
Iq [ p
r ] and K q [ p
r ] on ri , we can expect that Iq [ p
r ] and
K q [ p
r ] may be associated with the formalisms with the ordinary expectation value and the normalized q-expectation
value, respectively. This is indeed the case, as we shall see
below.
Substituting ri = p̃i(ord) into Equation (14), we have
Iq [ p
p̃ (ord) ] = β Fq(ord) − F̃q(ord) ,
(19)
where
Fq(ord) = U (ord) −
1
Sq ,
β
F̃q(ord) = Ũ (ord) −
1 (ord)
S̃ .
β q
(20)
On the other hand, putting ri = p̃i(nor) into Equation (15), we
have
K q [ p
p̃ (nor) ] = β̂
q
p̃i(nor)
Fq(nor) − F̃q(nor) ,
(21)
where
(15)
β̂ = β ∗
( pi )q ,
(22)
i
where ri is a reference distribution (i.e., prior). In the limit
q → 1, both Iq [ p
r ] and K q [ p
r ] tend to the KullbackLeibler relative entropy
H [ p
r ] =
q q −1 i
Iq [ p
r ] =
i
1
1−
( pi )q (ri )1−q ,
1−q
i
Like H [ p
r ], Iq [ p
r ] and K q [ p
r ] are nonnegative and
vanish if and only if pi = ri (∀i). This can be seen as follows. Noticing that Iq [ p
r ] admits the integral representation
(Naudts, 2004)
(16)
Fq(nor) = U (nor) −
1
Sq ,
β̂
F̃a(nor) = Ũ (nor) −
1 (nor).
S̃q
β̂
(23)
Equations (19) and (21) imply that Iq [ p
r ] and K q [ p
r ],
in fact, give the “free energy differences” and, therefore,
are identified with the generalized relative entropies associated with the ordinary expectation value and the normalized
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Astrophys Space Sci (2006) 305:241–245
q-expectation value, respectively. We also mention that the
quantum mechanical counterpart of K q [ p
r ] has recently
been employed to prove the second law of thermodynamics
(Abe and Rajagopal, 2003).
Now, convexity is one of the most important properties to
be satisfied by any types of relative entropies. As can be seen,
Iq [ p
r ] is convex in pi , but not in ri . In marked contrast to this
flaw, K q [ p
r ] is, like the Kullback-Leibler relative entropy
in Equation (16), jointly convex (Abe, 2003a, 2004).
Kq
a
λa p(a) λa r(a) ≤
λa K q [ p(a) r(a) ],
a
a
(24)
where λa > 0 and a λa = 1.
Finally, we mention that, like the Kullback-Leibler relative entropy, K q [ p
r ] is “composable” (Tsallis, 2001),
but Iq [ p
r ] is not. For factorized joint distributions of a
bipartite system (A, B), pi j (A, B) = p(1)i (A) p(2) j (B) and
ri j (A, B) = r(1)i (A)r(2) j (B), K q [ p(1) P(2) r(1)r(2) ] satisfies
the following relation:
K q [ p(1) p(2) r(1)r(2) ] = K q [ p(1) r(1) ] + K q [ p(2) r(2) ]
+(q − 1)K q [ p(1) r(1) ]K q [ p(2) r(2) ],
(25)
whereas such a closed relation does not exist for
Iq [ p(1) p(2) r(1)r(2) ].
Axiom III (System independence): It should not matter
whether one accounts for independent information about
independent systems separately in terms of their marginal
distributions or in terms of the joint distribution.
Axiom IV (Subset independence): It should not matter
whether one treats independent subsets of the states of the
systems in terms of their separate conditional distributions
or in terms of the joint distribution.
Axiom V (Expansibility): In the absence of new information, the prior (i.e., the reference distribution) should not
be changed.
These axioms are natural in the sense that all of them are
fulfilled by the ordinary Kullback-Leibler relative entropy in
Equation (16), which gives rise to the free energy difference
in Boltzmann-Gibbs statistical mechanics.
For the Tsallis entropy in Equation (4), the axioms and
uniqueness theorem are known in the literature (dos Santos,
1997; Abe, 2000). In contrast to this fact, the above set of
axioms is quite general and not very restrictive. Accordingly,
they do not uniquely specify the explicit functional form of
the relative entropy.
Now, the Shore-Johnson theorem (Shore and Johnson
1980, 1981, 1983) states that the relative entropy J [ p||r ]
with the prior ri and the posterior pi satisfying the axioms
I–V must have the following form:
J [ p||r ] =
pi h( pi /ri ),
(26)
i
4. Shore-Johnson theorem selects normalized
q-expectation value
In this section, we shall see by using the Shore-Johnson
theorem that K q [ p
r ] associated with the normalized qexpectation value leads to a consistent framework for minimum cross entropy (i.e., relative entropy) principle, whereas
Iq [ p
r ] corresponding to the ordinary expectation value does
not.
About a quarter a century ago, Shore and Johnson (1980,
1981, 1983) have presented a set of axioms for minimum
cross entropy (i.e., relative entropy) principle. They have
made an attempt to answer to the question: why the correct
rule of inference is to minimize relative entropy, in conformity with a vindication of Jaynes’ claim that every other rule
will lead to contradiction (Uffink, 1995).
The five key axioms are listed as follows.
Axiom I (Uniqueness): If the same problem is solved twice,
then the same answer is expected to result both times.
Axiom II (Invariance): The same answer is expected when
the same problem is solved in two different coordinate
systems, in which the posteriors in the two systems should
be related by the coordinate transformation.
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where h(x) is some function.
It is crucial to recognize that the function h(x) surely exists
for K q [ p||r ]:
h(x) =
1
(1 − x q−1 ) .
1−q
(27)
On the other hand, Iq [ p||r ] cannot be recast to the form
in Equation (26), since Iq [ p||r ] may violate Axiom III.
Therefore, we conclude that the Shore-Johnson theorem
supports the normalized q-expectation value and excludes
the possibility of using the ordinary expectation value from
nonextensive statistical mechanics.
5. Concluding remarks
We have studied the meanings of the two different definitions of expectation value in nonextensive statistical mechanics, i.e., the ordinary expectation value and the normalized q-expectation value. To identify which the correct definition is, we have discussed the corresponding two kinds
of generalized relative entropies, which have the physical
meanings as the “free energy differences”. It was shown that
Astrophys Space Sci (2006) 305:241–245
the Shore-Johnson theorem for minimum cross entropy (i.e.,
relative entropy) principle supports the generalized relative
entropy associated with the formalism with the normalized
q-expectation value and excludes the possibility of using the
ordinary expectation value from nonextensive statistical mechanics. Thus, we conclude that what to be employed in
nonextensive statistical mechanics is not the ordinary expectation value but the normalized q-expectation value.
Acknowledgements The author thanks Hans J. Haubold for inviting
him to the Twelfth United Nations/European Space Agency Workshop
on Basic Space Science (24–28 May 2004, Beijing, P. R. China). This
work was supported in part by the Grant-in-Aid for Scientific Research
of Japan Society for the Promotion of Science.
References
Abe, S., Okamoto, Y. (eds): Nonextensive Statistical Mechanics and Its
Applications. Springer-Verlag, Heidelberg (2001)
Abe, S.: Phys. Lett. A 312, 336 (2003a)
Abe, S.: Phys. Lett. A 324, 507 (2004) (E). See also Abe, S.: Physica
A 344, 359 (2004)
Abe, S., Rajagopal, A.K.: J. Phys. A 33, 8723 (2000a)
Abe, S., Rajagopal, A.K.: J. Phys. A 33, 8733 (2000)
Abe, S., Rajagopal, A.K.: Phys. Rev. Lett. 91, 120601 (2003)
Abe, S., Sukuki, N.: Europhys. Lett. 65, 581 (2004)
Abe, S., Suzuki, N.: J. Geophys. Res. 108(B2), 2113 (2003a)
Abe, S., Suzuki, N.: Phys. Rev. E 67, 016106 (2003a)
Abe, S., Suzuki, N.: Physica A 350, 588 (2005)
Abe, S.: in Proceedings of the 5th International Wigner Symposium,
Vienna, 1997, edited by Kasperkovitz, P., Grau, D. (eds.) (World
Scientific, Singapore, 1998), p. 66. See also Rajagopal, A.K., Abe,
S.: Phys. Rev. Lett. 83, 1711 (1999)
Abe, S.: Phys. Lett. A 224, 326 (1997)
Abe, S.: Phys. Lett. A 244, 229 (1998)
Abe, S.: Phys. Lett. A 271, 74 (2000)
Abe, S.: Phys. Rev. A 68, 032302 (2003)
Abe, S.: Phys. Rev. E 63, 061105 (2001). See also Abe, S., Martı́nez,
S., Pennini, F., Plastino, A.: Phys. Lett. A 281, 126 (2001)
Abe, S.: Phys. Rev. E 68, 031101 (2003c)
Abe, S.: Physica A 300, 417 (2001)
Alberico, W.M., Lavagno, A., Quarati, P.: Eur. Phys. J. C 12, 499 (2000)
Baldovin, F., Robledo, A.: Europhys. Lett. 60, 518 (2002a)
Baldovin, F., Robledo, A.: Phys. Rev. E 66, 045104 (2002b)
Beck, C., Schlögl, F.: Thermodynamics of Chaotic Systems: An Introduction, Cambridge University Press, Cambridge (1993)
Beck, C.: Physica A 286, 164 (2000)
Bediaga, I., Curado, E.M.F., de Miranda, J.M.: Physica A 286, 156
(2000)
245
Boghosian, B.M., Love, P.J., Coveney, P.V., Karlin, I.V., Succi, S.,
Yepez, J.: Phys. Rev. E 68, 025103 (2003)
Borland, L., Plastino, A.R., Tsallis, C.: J. Math. Phys. 39, 6490 (1998)
Borland, L., Plastino, A.R., Tsallis, C.: J. Math. Phys. (E) 40, 2196
(1999)
Borland, L.: Phys. Rev. Lett. 89, 098701 (2002a)
Borland, L.: Quantitative Finance 2, 415 (2002b)
Bregman, L.M.: USSR Comp. Math. Math. Phys. 7, 200 (1967)
Csiszár, I.: Per. Math. Hung. 2, 191 (1972)
de Moura, F.A.B.F., Tirnakli, U., Lyra, M.L.: Phys. Rev. E 62, 6361
(2000)
dos Santos, R.J.V.: J. Math. Phys. 38, 4104 (1997)
Gell-Mann, M., Tsallis, C. (eds.): Nonextensive Entropy–
Interdisciplinary Applications, Oxford University Press, New
York (2004)
Johal, R.S., Tirnakli, U.: Physica A 331, 487 (2004)
Kaniadakis, G., Lissia, M. (eds.): Special Issue of Physica A 340 (2004)
Kaniadakis, G., Lissia, M., Rapisarda, A. (eds.): Special Issue of Physica
A 305 (2002)
Kozuki, N., Fuchikami, N.: Physica A 329, 222 (2003)
Latora, V., Baranger, M., Rapisarda, A., Tsallis, C.: Phys. Lett. A 273,
97 (2000)
Lavagno, A., Kaniadakis, G., Rego-Monteiro, M., Quarati, P., Tsallis,
C.: Astrophysical Letters and Communications 35, 449 (1998)
Lutz, E.: Phys. Rev. A 67, 051402 (2003). See also Abe, S.: Phys. Rev.
E 69, 016102 (2004)
Naudts, J.: Rev. Math. Phys. 6, 809 (2004)
Navarra, F.S., Utyuzh, O.V., Wilk, G., Wlodarczyk, Z.: Phys. Rev. D
67, 114002 (2003)
Prato, D., Tsallis, C.: Phys. Rev. E 60, 2398 (1999)
Reis, M.S., Freitas, J.C.C., Orlando, M.T.D., Lenzi, E.K., Oliveira, I.S.:
Europhys. Lett. 58, 42 (2002)
Rossani, A., Scarfone, A.M.: Physica A 282, 212 (2000)
Shore, J.E., Johnson, R.W.: IEEE Transactions on Information Theory
IT-26, 26 (1980)
Shore, J.E., Johnson, R.W.: IEEE Transactions on Information Theory
IT-27, 472 (1981)
Shore, J.E., Johnson, R.W.: IEEE Transactions on Information Theory
IT-29, 942 (1983)
Sotolongo-Costa, O., Posadas, A.: Phys. Rev. Lett. 92, 048501 (2004)
Tadic, B., Thurner, S.: Physica A 332, 566 (2004)
Taruya, A., Sakagami, M.: Phys. Rev. Lett. 90, 181101 (2003a)
Taruya, A., Sakagami, M.: Physica A 322, 285 (2003b)
Tsallis, C., Anjos, J.C., Borges, E.P.: Phys. Lett. A 310, 372 (2003)
Tsallis, C., Mendes, R.S., Plastino, A.R.: Physica A 261, 534 (1998)
Tsallis, C.: in Ref. Kaniadakis et al. (2002). See also Abe, S.: Phys. Rev.
E 63, 061105 (2001)
Tsallis, C.: J. Stat. Phys. 52, 479 (1988a)
Tsallis, C.: Phys. Rev. E 58, 1442 (1998b)
Uffink, J.: Studies in History and Philosophy of Modern Physics 26B,
223 (1995)
Upadhyaya, A., Rieu, J.P., Glazier, J.A., Sawada, Y.: Physica A 293,
549 (2001)
Walton, D.B., Rafelski, J.: Phys. Rev. Lett. 84, 31 (2000)
Wilk, G., Wlodarczyk, Z.: Acta Physica Polonica B 35, 871 (2004)
Springer