The Phase Space Elementary Cell in Classical and Generalized Statistics

0.14 MB·English

by  Piero Quarati

Entropy2013,15,1-x; doi:10.3390/——
OPENACCESS
entropy
ISSN 1099-4300
www.mdpi.com/journal/entropy
Article
The Phase Space Elementary Cell in Classical and
5 Generalized Statistics
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Piero Quarati 1,2,* andMarcello Lissia2

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J 1 DISAT,Politecnico diTorino,C.so DucadegliAbruzzi24, TorinoI-10129,Italy

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2 IstitutoNazionalediFisicaNucleare, Sezione diCagliari, MonserratoI-09042,Italy;
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E-Mail: [email protected]
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e * Authorto whomcorrespondenceshouldbeaddressed; E-Mail: [email protected];
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Tel.: [phone];Fax: [phone].
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t Received: 05 September2013;in revisedform: 25 September2013/Accepted: 27 September2013 /
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. Published: xx
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n Abstract: In the past, the phase-space elementary cell of a non-quantized system was
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set equal to the third power of the Planck constant; in fact, it is not a necessary assumption.
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Wediscusshowthephasespacevolume,thenumberofstatesandtheelementary-cellvolume

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of a system of non-interacting N particles, changes when an interaction is switched on and
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3 thesystembecomesorevolvestoasystemofcorrelatednon-Boltzmannparticlesandderives
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4 theappropriateexpressions. Evenifweassumethatnowadaysthevolumeoftheelementary
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cell is equal to the cube of the Planck constant, h3, at least for quantum systems, we show
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1. that there is a correspondence between different values of h in the past, with important
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and, in principle, measurable cosmological and astrophysical consequences, and systems
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1 with an effective smaller (or even larger) phase-space volume described by non-extensive
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generalized statistics.
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r Keywords: classicalstatisticalmechanics;thermodynamics
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1. Introduction
In amicrocanonicalformulation,theentropy isproportionaltothelogarithmofthenumberofstates,
W, accessibletothesystems.
In quantum mechanics, we can count the number of microstates W corresponding to a microstate of
totalenergy, E. ThisnumberofmicrostatesW is oftencalled themultiplicityofthemicrostate.
Entropy2013,15 2
In classical mechanics, the state of an N-particle system is represented in a 6N dimensional
coordinate-momentum space (Γ space). Boltzmann solved the problem of infinities when counting the
multiplicity of continuous states by dividing the phase-space in elementary cells, whose volume ∆Ω
remains arbitrary inclassicalmechanics.
The advent of quantum mechanics provided the framework to determine the volume of the smallest
elementary cell, which is givenby the cube of thePlanck constant: h3. This lowerbound to thevolume
oftheelementarycell is directlyconnected totheuncertaintyprinciple.
In classical statisticalmechanics, thearbitrary volumeoftheelementary cell does not affect physical
quantities. Infact, aslongas theentropyofthesystemisproportionaltothelogarithmofthenumberof
states: W = Ω/(∆Ω)N
Ω
S = kln (1)
(∆Ω)NN!
where Ωisthevolumeofthephasespaceaccessibleto thesystem,entropydifferences:
∆S = S(A)−S(B) (2)
donotdependonthevolumeoftheelementarycell∆Ω,andtherefore,allphysicalquantitiesthatdepend
on entropydifferentials,suchas theheat capacity:
∂S
C = T (3)
V
∂T
(cid:18) (cid:19)(V,N)
orthepressure:
∂S
P = T (4)
∂V
(cid:18) (cid:19)(E,N)
are similarlyindependentofthechoiceoftheelementary cellvolume.
However,thechemicalpotential:
∂S
µ = −T (5)
∂N
(cid:18) (cid:19)(E,V)
does depend on ∆Ω. Therefore, the tendency of particles to diffuse along the gradient of densities or,
more generally, of chemical potential appears to be affected by the elementary volume of the the phase
space. Furthermore, the elementary cell enters also in the Sackur-Tetrode entropy of a mono-atomic
classical idealgas:
3/2
S 5 V E m
= +ln (6)
kN 2 N N 3πh2
" #
(cid:18) (cid:19)
where E is the internal energy of the gas and m the mass of the particles of the system. In fact, this
entropy incorporatesquantumeffects through,forinstance, thethermal wavelength 3πh2N/(mE).
Up to now, we have considered classical systems composed of non-interacting classical particles
p
or of particles subjected to short-range forces that do not alter the basic one-particle structure of the
excitations and that can be absorbed in a few effective parameters. This kind of systems can be
described by the Maxwell-Boltzmann (MB) statistics, and their excitation distribution is obtained by
themaximizationmethod.
Entropy2013,15 3
The number of different microstates w accessible to a system of N identical particles is obtained by
dividingbyN! thenumberofmicrostatesW ofacorrespondingsystemofdistinguishableparticles:
W
w = (7)
N!
which is greatly reduced with respect to W. According to Gibbs, all microstates in this microcanonical
ensemblehaveequal probabilities.
In classical statistical mechanics, the statement (∆Ω)N = h3N is justified considering the product of
an N independent quantum one-particle system or from the pure dimensionalpoint of view: the Planck
constanthasdimensionsofthetwo-dimensionalphase-spacevolumex×p,andeachparticleoccupiesa
volume∆Ω = h3. Apossibledimensionlessfactor cannotbeexcludedclassically.
In conclusion,inclassicaldiscreteand continuumstatisticalmechanics,thevolumeofan elementary
cell is not determined a priori; it is dimensionless in units of the cube of the Planck constant and could
spanalargesetofvalues. Thechoicethatthecellhasavolumeexactlyequaltoh3 isplausible,butonly
theexperimentcouldpossiblydeterminethisconstantin classicalmechanics.
Thereexistcomplexnaturalandartificialsystemscharacterizedbylong-rangespace-timecorrelations
whose dynamics cover only a multi-fractal subset of the whole phase space and develop non-Gaussian
correlations, at least for practical finite equilibration times. These kinds of systems can be effectively
describedbygeneralizedstatisticalmechanicsthatareacontinuousdeformationoftheusualone[1,2]. A
particularlysimplesuchgeneralizedapproachistheonethatusesthenon-extensiveTsallisentropy[3,4]
in its continuous and quantum versions. We have already studied modifications of the phase-space
elementary cell in the context of non-extensive statistical mechanics [5]; this work develops such an
approach on amoregeneral basis.
Recently,S.Abe[6]usedthischoice∆Ω = h3,butonlyforclassicaldiscretesystems. Hearguesthat
variation of entropy or other measurable quantities in non-extensive systems do not depend on h3 only
indiscretesystems. Onthecontrary,entropyorothermeasurablequantitiesincontinuumnon-extensive
systems depend on h3 against the hypothesis that the systems are classical. We shall elaborate on this
pointin Section3.
In this paper, we show how the phase-space volume can be deformed and how the non-extensive
expression of the number of states changes from the one of an ideal non-interacting particle system.
We obtain first the value of the deformed phase space volume. Then, we deduce the elementary cell
volumeas afunctionoftheentropicdeformationparameter.
Even quantum systems, which appear to have a lower bound to the phase-space volume, due to the
uncertainty principle, could have an effective volume of the elementary cell slightly smaller than h3 in
non-Boltzmann systems with a fractal measure of the volume. In addition, there exists the possibility
that the Planck constant and, therefore, the volume h3 were different in the past. Both possibilities
would have measurable cosmological and astrophysical consequences. Entropic uncertainty relations
are alsoconsidered.
2. The DeformationofPhaseSpace Volume
The very large number of microstates makes both the phase-space and elementary-cell volume
practicallyinsensitivetosmallvariationsofthedistributionfunction. Infact,thelikelihoodofdeviations
Entropy2013,15 4
is exponentially small, with the exponent proportional to the large number of degrees of freedom.
Howeverinteractionscan createstrongcorrelationsthatstronglyreducetheeffectivenumberofdegrees
offreedom relevantat finitetime-scales.
Since the equilibrium (maximum) value, W , is much larger than the values of the number of
0
microstates W associated with distributions even slightly different from equilibrium, one usually
identifies the state of maximum probability with the real state that is experimentally measured
(Sommerfeld [7], p. 218). However, we will show how going from an extensive classical system to a
non-extensiveone,i.e.,fromapureMBdistributiontodistributionswithpower-lawtails,smallchanges
ofthephase-spacevolumecan influencethedescriptionofthesystem.
The effect on Ω of variations from the Maxwellian distribution function n of the i-th state can be
io
found in the paper by Bohm and Schutzer [8]. In the book of Diu et al. [9], it is shown that in classical
physics,theelementarycellisthePlanckconstantbycomparingthesequenceoftheadmissiblestatesto
thesequenceofstatesin quantummechanics.
Long-range interactions, correlations, memory effects, a fractal space, elementary length or discrete
time are some of the physical reasons systems could not be consistentlydescribed by MB distributions.
Such systems can often be better described by generalized statistics, as developed in the last few years.
In fact, one might also view these effective descriptions as non-equilibrium metastable states: if the
lifetimeis longenough,thefinal thermodynamicalequilibriumis notexperimentallyrelevant.
For the sake of simplicity, we discuss only the so-called q-statistics [3,4]. However, similar
conclusions could be reached with other power-law deformations of the MB statistics, such as
the κ-statistics [10,11], which offers a mathematically attractive context, symmetry properties, and
interestingrelationswiththeformalismofrelativity.
Criticisms to the use of non-extensive, generalized statistics can be found, for instance, in works by
Gross [12]. In Shalizi [13], for instance, one can find criticisms of the use of approaches that adopt
deformed, non-extensiveorgeneralized statistics.
In thefollowing,we showhowthe phase-spacevolumevariesfor smalldeviationsfrom thestandard
distributionfunctionusingthespecificformalismofthegeneralized q-statistics.
The phase space volume for a system of N non-interacting Boltzmann particles (or molecules) is
defined by:
N! 1
Ω = (∆Ω )N (8)
0 0
n !n !···N!
10 20
forinteractingnon-Boltzmannparticles, wedefine:
N! 1
Ω = (∆Ω )N (9)
q q
n !n !···N!
1q 2q
where n and n are, respectively, the standard and deformed distributions. The number of accessible
i0 iq
states is:
N! Ω
W = = N! 0 (10)
0 n !n !··· (∆Ω )N
10 20 0
and:
N! Ω
W = = N! q (11)
q n !n !··· (∆Ω )N
1q 2q q
We areinterested insmalldeviationsfromthestandarddistributionsand define: δn = n −n .
i iq i0
Entropy2013,15 5
UsingtheStirlingformula,lnW = ln(N!)− ln(n !)canbeexpandedaroundthevalueW (n )
q i iq 0 i0
that maximizesthephasevolumeΩ at theMBdistribution,whichis expressedas:
0 P
n = A e−ai with a ≡ βǫ (12)
i0 M i i
where
3/2
N β
A = ∆Ω (13)
M 0
V 2πm
(cid:18) (cid:19)
and thenormalizationissuch that n = N. Inthisexpansion:
i i0
P
1 δn2
lnW = N lnN − n lnn − (1+lnn )δn − i +··· (14)
q i0 i0 i0 i
2 n
! i0
i i i
X X X
terms linear in δn vanish, because of the maximum conditions, and retaining the leading contribution,
i
the explicit expressions of W and Ω for a gas that deviates only slightly from the MB distribution,
q q
assumingforthemomentan unchangedelementary cell, are:
1 δn2
W = W exp − i (15)
q 0
2 n
i0 !
i
X
and:
1 δn2
Ω = Ω exp − i (16)
q 0
2 n
i0 !
i
X
Thedeformed distributionis:
n = A A e−βǫ−(1/2)(1−q)(βǫ)2 (17)
iq M q
with n = N where:
i iq
15
A = 1+ (1−q)+o(1−q)2 (18)
P q
8
Forq ≈ 1, wehave:
15 (1−q)
n = n 1+ (1−q)− (βǫ )2 (19)
iq i0 i
8 2
(cid:18) (cid:19)
(δn )2 (1−q)2 15 2
i = n −(βǫ )2
i0 i
n 4 4
i i0 i (cid:20) (cid:21)
X X
(1−q)2 15 2 15
= N − h(βǫ )2i+h(βǫ )4i (20)
i i
4 4 2
" #
(cid:18) (cid:19)
(1−q)2 15 2 1515 945 45
= N − + = N(1−q)2
4 4 2 4 16 4
" #
(cid:18) (cid:19)
where in the preceding equations, we have calculated the average values with the
unperturbed distribution.
Therefore, weobtain:
W = W e−(45/8)N(1−q)2 (21)
q 0
Entropy2013,15 6
From W /W = (Ω /Ω )(∆Ω /∆Ω )N (see, for instance, Equations (10) and (11)), this change
q 0 q 0 0 q
can be interpreted microscopically as a change of an elementary cell leaving the phase-space
volumeunchanged:
∆Ω = ∆Ω e(45/8)(1−q)2 (22)
q 0
Because the correction factor to W depends on (q −1)2, ∆Ω is always larger than ∆Ω . However, if
q q 0
we had assumed that the number of accessible states were independent of q, Ω and (∆Ω )N would
q q
change in the same way, and one could obtain an elementary cell smaller than ∆Ω ; in particular,
0
∆Ω = ∆Ω exp(−(45/8)(1−q)2. Here, we just remark that W and W have different values only to
q 0 0 q
the second order in an expansion in (1−q). In the definition of the standard entropy, we use W , while
0
in the definition of the q-entropy, we should use the q-logarithm of W . However, the leading linear
q
correction comes onlyfromtheq-logarithm,andonecan useW .
0
3. ClassicalVolumeCell, Continuous States andNon-MaxwellianStatistics
When computing the entropy of a classical system with states parameterized by continuous
parameters, coarse graining, i.e. a finite cell size, overcomes the difficulties of defining the
correct measure of probability densities. Abe [6] argues that generalized statistical mechanics with
non-logarithmic entropies can be applied only to physical systems with discrete degrees of freedom.
In fact, the elementary volume necessary to regularize the sum over states in continuous Hamiltonian
systemsdoesnotdisappearinthefinalresultsfornon-additiveentropicfunctions: iftheelementarycellis
assumedtobeh3 fromtheunderlingquantizedformulation,classicalresultswouldcontainthequantum
constanth3. Thisreasoningspurred an interestingdiscussion[14,15]betweenAbeand Andresen.
We remark that a discrete cell can appear also in classical mechanics, for instance, due to effective
coarse graining with a volumenot necessarily equal to h3. In addition, the same constant h can have an
interpretationalsoin aclassicalcontext[16–18].
Weconcludethatbothcontinuousanddiscreteclassicalsystemscancontainaphasespaceelementary
cell that in classical physics can have any value: in particular, its volume can be h3. Therefore,
both continuous and discrete system can be described by generalized statistical mechanics that permit
elementary cellswitha volumedifferent fromh3.
4. The Generalized Number ofStates andthe Generalized Elementary CellVolume
We consider the standard classical phase space Ω , which is maximum when the global
0
thermodynamical equilibrium distribution n is the MB distribution for the i-th state. The elementary
i0
phasespacecell is∆Ω ,and N isthenumberofparticlesofthesystem.
0
Let us definethedimensionlessquantity:
Ω W
MN ≡ 0 = 0 = w (23)
0 (∆Ω )N N! 0
0
which is the measure of the 3D phase-space volume relativeto the volumeof the one-particle cell ∆Ω
0
and wherew isthenumberofmicrostates.
0
Entropy2013,15 7
Analogouslyintheq-deformedformalism,wedefinetheq-volumeΩ andtheq elementarycell∆Ω
q q
and introducethedimensionlessmeasure:
Ω W
M⊗qN = q = q = w (24)
q (∆Ω )N N! q
q
where wehaveused theq-power(see, forinstance,[19]):
M⊗qN = NM(1−q) −N +1 1/(1−q) (25)
q q
TheexpansionofM⊗qN forN ≫ 1 aro(cid:2)und thevalueq = 1(cid:3)(seeAppendixA fordetails)gives:
q
q −1
M⊗qN = MN 1+ N2(lnM )2 (26)
q q 2 q
(cid:20) (cid:21)
2
W q −1 W
= q × 1+ ln 0 (27)
N! 2 N!
" #
(cid:18) (cid:19)
where, inside the logarithm, we have used M = M to the order in 1−q considered. If we divide the
q 0
aboveequationby thecorrespondingequationwiththeindexofzero (q = 1), wefind:
M⊗qN W q −1 W 2 Ω ∆Ω N
q = q × 1+ ln 0 = q 0 (28)
MN W 2 N! Ω ∆Ω
0 0 " (cid:18) (cid:19) # (cid:18) 0(cid:19)(cid:18) q(cid:19)
where thelast equalityderivesfrom definitions(23)and (24)ofMN andM⊗qN. Therefore:
0 q
−1/N 2 −1/N 1/N
W q−1 W Ω
∆Ω = ∆Ω q 1+ ln 0 q (29)
q 0
W 2 N! Ω
(cid:18) 0(cid:19) " (cid:18) (cid:19) # (cid:18) 0(cid:19)
Inserting the deformed expression of W , as derived in Equation (21), and ascribing the whole
q
dependence on q to the micro-physics, i.e. Ω = Ω , as it appears reasonable if the thermodynamical
q 0
limitexists,weobtain,finally:
−1/N
2
q −1 W
∆Ω = ∆Ω e(45/8)(q−1)2 × 1+ ln 0 . (30)
q 0
2 N!
" #
(cid:18) (cid:19)
The expressionoftheq-deformed elementary cell is madeoftwo factors: thefirst is theelementary cell
derived withouttheuse ofq-algebra(see Equation(22)); thesecond represents thecorrection due to the
q-algebra. Note that the last relation shows how ∆Ω can be smaller or larger than ∆Ω , depending on
q 0
thevalueofq.
In AppendixB,we alsoconsiderthecorrespondingentropywithafew numericalexamples.
5. Entropic Uncertainty RelationsandElementary Cell
The above sections show the link between elementary-cell volume and the uncertainty principle. It
is well known that generalizations and extensions of the original uncertainty relations of Heisenberg
have been proposed involving Shannon, Re´nyi and, later, Tsallis entropy. To be more clear: quantum
Entropy2013,15 8
mechanical uncertainty relations for momentumand position were expressed as inequalities in terms of
Shannon, Re´nyiorTsallisentropy.
Uncertainty phase space volume is related to the phase-space elementary cell that therefore can be
expressed in terms of Shannon, Re´nyi and Tsallis entropy, giving us the elementary volume for these
generalized statistics[20–27].
Let us introduce the experimental phase-space elementary cell volume ∆Ω as the volume of phase
E
spacedeterminedbytheresolutionofthemeasuringinstruments. Ontheotherhand,theelementarycell
volume,asdiscussedintheprevioussections,isrelatedtotheHeisenberguncertaintyprincipleanddoes
not dependon theaccuracy ofourmeasuringinstruments.
In theShannon informationformalism,theentropyis:
H = − p lnp (31)
k k
k
X
where p is the dimensionless position distribution for the k state (κ = 1). The quantity, H, is the
k B
positionforShannon. Analogously,H can alsobeexpressed formomentum.
In theRe´nyiframework [23], positionandmomentumentropy havetheexpressions:
1
Hx = ln pα (32)
α α k
k
X
1
Hp = ln pβ (33)
β β k
k
X
whereHxandHprepresenttheuncertaintyinpositionandmomentumforRe´nyistatisticsinthequantum
α β
world. Whentheentropicparameter α orβ goesto one,onerecoverstheShannon entropy.
Shannonconnectedthecontentofthemeasureofinformationwithaprobabilitydistribution,inserting
theset ofprobabilitiesobtainedfromquantummechanicsintotheinformationentropy.
We definethedimensionlessrelativevolumeofthecell: ω
S
3
∆Ω δxδp
ω = E = (34)
S
∆Ω h
0 (cid:18) (cid:19)
This quantity decreases as we enter into the quantum regime. In the Shannon frame, we have identified
h3 withthestandardelementary cell∆Ω .
0
Fromtheentropicuncertaintyrelations,wecangiveanalyticalexpressionsofωintheShannon,Re´nyi
and Tsallisstatistics.
In quantum mechanics and in information theory, the entropic uncertainty is defined as the sum of
temporal and spectral entropies. The uncertainty principle of Heisenberg can be expressed as a lower
boundonthesumoftheseentropies. Thisismuchstrongerthanthestandardstatementoftheuncertainty
Heisenberg principlein termsoftheproduct ofstandarddeviations.
TheShannon entropicuncertaintyrelation can bewritten: as [23]
1
Hx +Hp ≥ 1−ln2− lnω (35)
S
3
By invertingtheaboverelation,weobtain:
ω ≥ e3−ln8e−3(Hx+Hp) (36)
S
Entropy2013,15 9
Thelowerquantummechanicsboundon(Hx+Hp)isfoundforω = 1: (Hx+Hp) ≥ 1−ln2 ≈ 0.307.
Correspondingto ω < 1(> 1),onehas lowerboundsgreater(smaller)than1−ln2.
Let us report the following limiting cases (these limits are unphysical, because they violate the
Heisenberg principle):
• if∆Ω goes tozero, then (Hx +Hp)goes toinfinity(completelackofinformation);
E
• if(Hx +Hp) = 0 (perfect information),thenω = exp(3−ln8).
S
TheRe´nyi entropicrelationis:
1 lnα lnβ 1
Hx +Hp ≥ − + −ln2− lnω (37)
α β 2 1−α 1−β 3 R
(cid:18) (cid:19)
where ω = ∆Ω /∆Ω , ∆Ω is the elementary cell in the Re´nyi frame, and the entropic parameters
R E R R
mustverifyα−1 +β−1 = 2. Therefore, fromEquation(7)of[23], wederive:
1
ωR ≥ e−3(Hαx+Hβp)e−32(1ln−αα+1ln−ββ) (38)
8
If the same measure of uncertainty is used for both variables, position and momentum, we obtain
from Equation(46)of[23]:
1 1
ωRˆ ≥ 8e−3(Hβx+Hβp)β3 (2β −1)(3/2−3β)/(1−β) (39)
with1/2 ≤ β ≤ 1.
The cell ∆Ω can easily be derived in terms of the sum of position and momentum uncertainty and
R
oftheexperimentalphase-space elementarycell volume.
Adetailedanalysisintermsofthephase-spaceelementarycellneedstobeperformed. Atthemoment,
we say that the volumeof theelementary cell depends on theeffectivestatisticsrelevant for thespecific
physicalsystem,anditsvalueinunitsofh3 canbesmallerthanone;insomecases,thiseffectivevolume
can be even much bigger than one. In addition, the Planck constant, and, therefore, the fine-structure
constanth¯c/e2 ≈ 137.036,could havehad valuesdifferentfrom theonepresentlymeasured.
Existinggeneralizationsoftheuncertaintyprincipledonotmakereferencetotheimplicationsforthe
volume of the elementary cell in quantum dynamics. Parallel modifications of the elementary cell can
be studied within classical dynamics, where the scale h3 is not derived from an uncertainty principle.
For instance T. H. Boyer interprets h as the scale of random classical zero-point radiation that is one of
thesolutionsoftheMaxwellequations[16–18].
Westresstheimportantrelationbetweennon-extensivestatisticsandmodificationsoftheelementary
volumeofphasespace.
Inthefollowing,wesummarizeuncertaintyentropicrelationsderivedfromBialynicki-Birula[23]and
Wilk [24,26] using also the condition of the non-negativityof momentum entropy and position entropy
and oftheright-handsidesofinequalities(7)and (46)of[23]and (19)and (20)of[24]
ForShannon, wehave:
1/3
2 ∆Ω 2
e(Hx+Hp) ≥ 0 ≥ (40)
e δxδp e
Entropy2013,15 10
ForRe´nyi, wehave:
from Equation(7)of[23]ifα 6= β withα ≥ 1 and β ≤ 1:
1/3
∆Ω
2e21(1ln−αα+1ln−ββ)e(Hαx+Hβp) ≥ R ≥ 2e21(1ln−αα+1ln−ββ) (41)
δxδp
where 1/α+1/β = 2;
from Equation(46)of[23], ifα = β:
1/3
∆Ω
2β(2β −1)β1−−1β/2eHβx+Hβp ≥ Rˆ ≥ 2β(2β −1)β1−−1β/2 (42)
δxδp
where 1/2 ≤ β ≤ 1.
ForTsallis,wehavetwo relationswithα 6= β (α > β): ifη < 1 (from Equation(19)of[24]):
1 1/3 1 α/(α−1)
β 2(α−1) ∆Ωq β 2(α−1) 1
2β ≥ ≥ 2β (43)
α δxδp α 1+(1−α)(Hx +Hp)
(cid:18) (cid:19) (cid:18) (cid:19) " β β #
whileifη > 1 (from Equation(20)of[24]):
1 1/3 1 α/(α−1)
α 2(1−α) ∆Ωq α 2(1−α) 1
2β ≥ ≥ 2β (44)
β δxδp β 1+(α−1)(Hx +Hp)
(cid:18) (cid:19) (cid:18) (cid:19) " β β #
where:
(α−1)/α
1/(2α)
β δxδp
η(α,β) = (2β)(α−1)/α (45)
α ∆Ω1/3!
(cid:18) (cid:19) q
Thedistributionofmomentumandpositionwithdifferententropicparametersaretakenwiththesame
elementary cell ∆Ω .
q
The meaning of the above relations can be summarized as follows. Once the effective statistical
framework (Shannon, Re´nyi, Tsallis or others), and, therefore, the corresponding volume of the
elementary cell, is determined by the physical characteristics of the system, the only meaningful
measurementsoftwoconjugatequantities,suchaspositionandmomentum,arethosewhoseuncertainty
product satisfiestheaboverelations.
6. Astrophysicaland CosmologicalImplications
Experiments have been arranged in astrophysics to measure the cosmic red shift, the variation of the
fine-structure constant in space and time, and to observe a cosmological rate of change of the Planck
constant: results are under scrutiny and do not appear to give conclusive answers. Cosmic acceleration
should be verified by future ground-based laboratory experiments, where dark matter and dark energy
are lookedfor.
It is believed that phase-space structures with a volume smaller than h3 do not exist or do not
matter; however, Zurek [28] has shown that sub-Planck structures exist in the quantum version of
classically chaotic systems. This situation is allowed in non-extensivestatistical mechanics [29], where
an elementary cell smaller than the Planck constant can be admitted at particular values of the entropic
parameters.