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Evidences for nonextensivity conjugation in hadronic scattering systems
29 March 2001
Physics Letters B 503 (2001) 263–270 www.elsevier.nl/locate/npe
Evidences for nonextensivity conjugation in hadronic scattering systems D.B. Ion, M.L.D. Ion National Institute for Physics and Nuclear Engineering, NIPNE-HH, Bucharest, P.O. Box MG-6, Romania Received 28 June 2000; received in revised form 5 October 2000; accepted 18 January 2001 Editor: R. Gatto
Abstract In this Letter the degrees of nonextensivity of the hadronic quantum scattering systems is investigated by using Tsallis-like scattering entropies SJ (p) and Sθ (q), p, q ∈ R. Then, evidences for strong nonextensive statistical effects, with nonextensivity indices 1/2 p 0.6, as well as, for the (1/2p + 1/2q = 1)-nonextensivity conjugation, are obtained by fitting the pion– nucleus experimental entropies with the PMD-SQS-optimal state predictions. The interpretation of these strong nonextensive statistical effects as indirect manifestations of the presence of the quarks and gluons in the hadronic quantum system is suggested. 2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Ca; 25.80.Dj; 12.40.Ee
1. Introduction The whole theory of equilibrium statistical mechanics is based on a variational formalism for entropy (see, e.g., Refs. [1,2]). The Boltzmann–Gibbs statistical mechanics and standard thermodynamics do not seem to be universal. They have domains of applicability quite poorly known nowadays. A generalization of these statistical formalisms to nonextensive quantum systems is now available (see also Ref. [3]). A partially successful global description of the hadronic interactions (especially hadroproduction process) has been provided through a thermodynamical equilibrium approach of Hagedorn [4]. A nonextensive generalization of this approach was presented in Ref. [5]. Moreover, other applications of the nonextensive statisti-
E-mail address: [email protected] (D.B. Ion).
cal approach in cosmic rays and elementary particles are presented in Refs. [6–13]. Recently, we investigated [6–9] the nonextensive quantum statistics of the hadronic scattering states by introducing the Tsallislike scattering entropies SJ (p) and Sθ (q), p, q ∈ R, corresponding to the systems of the angular momentum states [J-states] and the quantum angular scattering states [θ -states], respectively. Then, we proved, not only [θ, J ]-entropic uncertainty relations, but also the optimal entropic upper bounds for each Tsallislike scattering entropy Sθ (q), SJ (p) in terms of optimal entropies Sθo1 (q), SJo1 (p), derived from the principle of minimum distance in the space of quantum states (PMD-SQS) [14]. In general, the nonextensivity is expected to emerge in the quantum hadronic systems (e.g., in pion–nucleus quantum scattering states) if long-range interactions, memory effects or fractal space–time structures, are present. Hence, by the investigation of the nonextensivity of quantum hadronic
0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 1 5 5 - 1
264
D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
systems we can obtain indirect informations about the presence of the quarks and gluons having a strongcoupling long-range regime of quantum chromodynamics. The pion–nucleon and pion–nucleus quantum scattering systems are the most suitable for such investigations since in these cases the experimental entropies Sθ (q), SJ (p) can be obtained with high accuracy from the available phase shifts analyses [15]. In this Letter the nonextensivity of the quantum scattering states, as well as the (θ, J )-nonextensivity conjugation, are investigated in the pion–nucleus scattering. Strong experimental evidences for the (θ, J )nonextensivity conjugation is experimentally obtained with high accuracy (CL 99%) by fitting the pion– nucleus entropies with the “optimal state” predictions derived from the principle of minimum distance in the space of states (PMD-SQS). So, we introduce concept of nonextensivity conjugation in quantum physics in terms of Tsallis-like entropies SJ (p) and Sθ (q). We present the first experimental determination of the (p, q)-nonextensivity indices for the quantum systems of (J and θ )-systems of states from the pion–nucleus scatterings. Hence, using 49-sets of pion–nucleus and 88-sets pion–nucleon experimental phase shifts [15], we obtain a strong experimental evidence (CL 99%) for the (1/p + 1/q = 2)-nonextensivity conjugation for (J, θ )-nonextensive quantum statistics with 0.50 p 0.60.
2. Nonextensive statistical description of hadron–nucleus scatterings We start the nonextensive statistical description of the hadron–nucleus scatterings by introduction of the [Sθ (q), SJ (p)]-scattering entropies for the case of spinless particles. As is known, the scattering amplitude f (x) of spinless particles is developed in partial amplitudes as follows f (x) =
L
(2l + 1)fl Pl (x),
l=0
x ∈ [−1, +1],
fl ∈ C,
(1)
where L + 1 is the number of partial amplitudes fl , Pl (x), l = 0, 1, . . . , L, are Legendre polynomials. Hence, the [SL (p), Sθ (q)]-scattering entropies are
defined as follows p (2l + 1)pl /(p − 1), SL (p) = 1 −
p ∈ R,
L SL (1) = − (2l + 1)pl ln pl
+1 q /(q − 1), Sθ (q) = 1 − dx P (x)
(2)
j =0
q ∈ R,
−1
+1 Sθ (1) = − dx P (x) ln P (x),
(3)
−1
where the probability distributions P (x) and pl , are defined by pl = 4π|fl |2 /σel ,
L
(2l + 1)pl = 1,
j =0
2π dσ (x), P (x) = σel dΩ
+1 dx P (x) = 1.
(4)
−1
The indices p = 1 or q = 1 can be interpreted as a measure of nonextensivity and will be called the nonextensivity indices. Therefore, the scattering entropies {SL (p), Sθ (q)} allow us to investigate the (extensive or nonextensive)-statistical behavior of the [J or θ ]-quantum scattering states considered as canonical ensembles. So, corresponding to [SL (p), Sθ (q)]scattering entropies we introduce the J -nonextensive quantum statistics and θ -nonextensive quantum statistics which are describing the statistical behavior of canonical ensembles composed of angular-momentum quantum states and θ -angular quantum states, respectively.
3. (1/2p + 1/2q = 1)-nonextensivity conjugation as a new principle in quantum scattering Now, a natural but fundamental question can be addressed, namely, what kind correlation (if it exists) is expected to be observed between the nonextensivity indices p and q corresponding to the Tsallis-like J nonextensive quantum statistics described by SJ (p) and θ -nonextensive quantum statistics described by
D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
Sθ (q)? The answer at this question is given by the following result. If the Fourier transform (1) is a bounded map T : L2p → L2q , then the (p, J )-statistics and (q, θ )statistics are correlated via the relation 1 1 (5) + = 1, or q = p/(2p − 1), 2p 2q while the norm estimate M of this map is given by M≡
p−1 1−q
Tf 2q 2 2p = 2 2q .
f 2p
(6)
Proof. The proof of this remarkable result is based on the definition of the Fourier transformation (1) as a bounded map from the space Lp to the space Lq and also on the Lp -estimate-theorems with p = 2p and q = 2q. So, we show that the results given by Eq. (5) are direct consequences of the Riesz–Thorin interpolation theorem [13]. Indeed, let T be the Fourier transform defined by the scalar scattering amplitude divided by the respective L2 -norms and let us redefine the scattering amplitudes in such a way that the square modulus correspond to the respective probabilities P (x) and pJ , respectively. Then, we obtain the “probability functions” 2 f (x) , P (x) = ϕ(x) ϕ(x) ≡ (7)
f (x) 2 and fl ϕl ≡
(8) , pl = |ϕl |2 . (2l + 1)|fl |2 Hence, the Fourier transform (1) take the form 1 (2l + 1)ϕl Pl (x) ϕ(x) = √ (9) 2 and, conversely 1 ϕl = √ 2
+1 ϕ(x)Pl (x) dx.
(10)
−1
Now, by using Eqs. (9) and (10) and taking into account that |Pl (x)| 1, it is easy to prove the inequalities 1/2 1/2 sup P (x) (11) 2−1/2 (2l + 1)pl , sup[pl ]1/2 2
+1 −1/2 −1
P (x)
1/2
and (2l + 1)pl = P (x) dx = 1. +1
(12)
(13)
−1
This means that in both cases we have T : L1 → L∞ with norm M1 = 2−1/2 ,
(14)
T : L2 → L2 with norm M2 = 1.
(15)
Then, using the Riesz–Thorin interpolation theorem for the complex-valued functions (see Ref. [16]) we obtain T : Lp → Lq with norm M ≡
T ϕ q
ϕ p
(16)
with (1/p ) = (1 − t)/1 + t/2, (1/q ) = (1 − t)/∞ + t/2.
(17)
Hence, eliminating the parameter t and using the relations p = 2p and q = 2q, we get not only the condition 1 1 + = 1, 2q 2p
t=
1 1 1 − +1= , 2q 2p q
(18) p−1
but also the norm-estimate (6) M M11−t M2t = 2 2p . By the norm estimate (6) we proved the following inequalities 1 1 + (1 − q)Sθ (q) 2q p−1 1 2 2p 1 + (1 − p)SL (p) 2p , 1 1 + (1 − q)SL (q) 2q p−1 1 2 2p 1 + (1 − p)Sθ (p) 2p ,
(19)
(20)
valid for any 1/2 < p 1 and 1/2p + 1/2q = 1. Proof. Indeed, by the definitions of the norms in the L2p and L2q spaces, in our case we have
ϕl 2t =
dx,
265
(2l + 1)|ϕl |2t = (2l + 1)plt
1 = 1 + (1 − t)SL (t) 2t ,
t ≡ p or q,
(21)
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D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
and
tively. These distributions are given by:
ϕ(x) = 2t
+1 +1 t ϕ(x) 2t dx = P (x) dx
−1
plme = plo1
1 = 1 + (1 − t)Sθ (t) 2p ,
t ≡ p or q, (22)
since pl ≡ |ϕl |2 , and P (x) ≡ |ϕ(x)|2 [see Eqs. (7), (8)]. This means that by the result (6) we proved the inequalities (19), (20) since by definitions of the Lp norms from Eq. (6) and the definitions (2)–(4) we have M=
1 1 = , 2K(1, 1) (Lo + 1)2 pJme = 0, for l Lo + 1 =
−1
P
me
[K(x, 1)]2 (x) = P (x) = , K(1, 1) o1
K(x, 1) =
[1 + (1 − q)Sθ (q)] 2q 1
[1 + (1 − p)SL (p)] 2p
2
1−p 2p
1/2
(23)
ϕl L2q
◦ 1 ◦ P Lo +1 (x)+ P Lo (x) , 2 K(1, 1) = (Lo + 1)2 /2 =
(29)
◦
ϕ(x) L2p [1 + (1 − q)SL (q)] 2q [1 + (1 − p)Sθ (p)]
1 2p
2
1−p 2p
(24)
for the inverse Fourier transformation (10). From this result, in the limit p → 1, we obtained in Ref. [10] the (θ, l)-entropic uncertainty relations: ln 2 Sθ + SL
o 1 (2l + 1)Pl (x) 2
L
ϕl L2p
1
=
(28)
where the reproducing kernel K(x, y) is given by
ϕ(x) L2q
for direct Fourier transformation (9), and = M
1 l Lo , 2 (27)
and
1
=
for
or Var[θ ] · Var[l] 2,
(25)
where the statistical variances Var[X] are defined as Var[X] = exp{SX }, X ≡ θ, L.
4. PMD-SQS-optimal distributions as equilibrium distributions of quantum scattering states Now, we can use the solution to the maximumentropy problem
max SJ (p), Sθ (q) dσ when σel = fixed and (26) (1) = fixed dΩ as criterion for the determination of the equilibrium distributions plme and P me (x), of the J -quantum scattering states and θ -quantum scattering states, respec-
d Pl (x), while the optimal angular where Pl (x) ≡ dx momentum Lo is given by: 1/2 4π dσ (1) −1 . Lo = integer (30) σel dΩ
Proof. Indeed, solving the problem (26) via Lagrange multipliers we obtain that singular solution λ0 = 0 exists and is just given by the [SJo1 (p), Sθo1 (q)]-optimal entropies obtained from the PMD-SQS-optimal state f o1 (x) = f (1)
K(x, 1) K(1, 1)
(31)
[ principle of minimum distance in the space of quantum states], where f (x), x ∈ [−1, 1], is the scattering amplitude (1). So, the solution of the problem (6) is given by [10] SLmax (p) = SLo1 (p) = 1 − [2K(1, 1)]1−p (p − 1)−1 , Sθmax (q) = Sθo1 (p)
(32)
+1 [K(x, 1)]2 q = 1 − dx (q − 1)−1 . K(1, 1) −1 (33)
D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
267
Table 1 o1 (p)]-scattering entropies calculated from pion–nucleus The values of the χL2 (p) obtained from Eqs. (34), (35) by using values of [SL (p), SL phase shift analyses [15]
p
p q = (2p−1)
0.538
7.0
π 0 He
π 0C
π 0O
π 0 Ca
χJ2 (p)/12
χJ2 (p)/13
χJ2 (p)/6
χJ2 (p)/14
0.152
0.143min
0.182
0.048
0.545
6.0
0.150
0.147
0.179
0.047
0.550
5.5
0.148
0.150
0.177
0.046
0.556
5.0
0.146
0.154
0.175
0.045
0.563
4.5
0.144
0.159
0.173
0.045
0.571
4.0
0.142
0.168
0.170
0.044min
0.583
3.5
0.139
0.180
0.168
0.045 0.048
0.600
3.0
0.135
0.201
0.167min
0.625
2.5
0.132
0.238
0.169
0.055
0.667
2.0
0.130min
0.312
0.179
0.075
0.700
1.75
0.130
0.381
0.193
0.096
0.750
1.50
0.135
0.498
0.221
0.135
0.833
1.25
0.150
0.727
0.284
0.213
1.00
1.00
0.201
1.201
0.456
0.406
1.33
0.80
0.377
2.543
0.958
0.900
1.50
0.75
0.507
3.387
1.298
1.204
2.00
0.67
1.370
42.00
3.00
0.60
5.599
23.51
6.055
8.862
11.67
8.045
5. Determination of nonextensivity indices from pion–nucleus experimental data
(c) the χJ2 (p) or/and χθ2 (q)-test functions defined by
5.1. Numerical results for nonextensivity (p, q)-indices
χX2 (k) =
The nonextensivity (p, q)-indices can be obtained by fit the experimental data on SJ (p) and Sθ (q) with the PMD-SQS-optimal state predictions (32), (33). Then, the man objective is to get the value of p and q for which we obtain the best fit. For the investigation of this important problem we used the experimental pion–nucleus [15] experimental phase-shifts for to obtain: (a) the experimental values of the scattering entropies SL (p) [Eq. (2)], Sθ (q) [Eq. (3)] as well as of the optimal angular momenta Lo [Eq. (30)], (b) the optimal entropies SLo1 (q) [Eq. (32)], Sθo1 (q) [Eq. (33)],
nexp [SX (k)]i − [S o1 (k)]i 2 X
i=1
X ≡ J, θ ;
o1 ] ['SX i
,
k ≡ p, q,
(34)
and o1 o1 'SX (k) = SX (k) L
o
o1 − SX (k) L +1
o −1
(35)
o1 where [SX (k)]Jo ±1 are the values of the PMD-SQSoptimal entropies calculated with the optimal angular momenta Lo ± 1, respectively. The results obtained in this way are presented in Fig. 1 and Tables 1–3.
268
D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
Table 2 The values of the χθ2 (q) obtained from Eqs. (34), (35) by using values of [Sθ (q), Sθo1 (q)]-scattering entropies calculated from pion–nucleus phase shift analyses [15] π 0 He
π 0C
π 0O
π 0 Ca
p
p q = (2p−1)
χθ2 (q)/12
χθ2 (q)/13
χθ2 (q)/6
χθ2 (q)/14
0.538
7.0
< 10−3 min
0.001min
< 10−3 min
< 10−3 min
0.545
6.0
< 10−3
0.002
< 10−3
< 10−3
0.550
5.5
< 10−3
0.003
< 10−3
0.001
5.0
< 10−3
0.005
0.001
0.002
4.5
< 10−3
0.007
0.002
0.004
0.556 0.563 0.571
4.0
0.001
0.012
0.004
0.006
0.583
3.5
0.002
0.022
0.007
0.010
0.600
3.0
0.005
0.042
0.016
0.020
0.625
2.5
0.011
0.092
0.037
0.045
0.667
2.0
0.028
0.257
0.101
0.123
0.700
1.75
0.047
0.494
0.184
0.231
0.750
1.50
0.084
1.085
0.370
0.485
0.833
1.25
0.163
2.796
0.843
1.174
1.00
1.00
0.348
2.225
3.284
1.33
0.80
0.679
23.71
8.613
5.335
8.145
1.50
0.75
0.807
30.91
6.707
2.00
0.67
1.082
48.11
3.00
0.60
1.371
69.14
5.2. Strong experimental evidence of (1/p + 1/q = 2)-nonextensivity correlation Now, we can give an “experimental” answer to the fundamental question: what kind correlation (if it exists) is expected to be observed between the nonextensivity indices p and q corresponding to the (p, J )-nonextensive statistics described by SJ (p) and (q, θ )-nonextensive statistics described by Sθ (q)? [We remember that the “mathematical” answer is given by Eq. (5)]. Indeed, from Fig. 1 as well as from the Table 3 we see that the experimental data on the scattering entropies SL (p) and Sθ (q) are simultaneously in excellent agreement (CL > 99%) with the [SLo1 (p), Sθo1 (q)]-optimal state predictions if the nonextensivities p and q of the (J and θ )-statistics are correlated via Riesz–Thorin relation: 1/p + 1/q =
9.905 13.62
10.29 15.25 20.90
2 (or q = p/(2p − 1)). So, the best fit is obtained (see Table 3) for the conjugate pairs p and q = p/(2p − 1) with the values of p in the range 0.5 p 0.6. In fact a significant departures from “optimality” is observed only for pairs with p > 2. These results can be compared with those obtained in Ref. [13] for pion– nucleon scattering case. Then, we can see that the nonextensive statistical behaviors of the corresponding bosonic (π 0 A)-quantum scattering states are very similar (see again Fig. 1) with those of the fermionic (π 0 P )-quantum scattering states.
6. Conclusions In summary, in this paper by using 49-sets of pion– nucleus phase shifts analyses [15], we obtained the
D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
269
Table 3 o1 (p)]-scattering The values of the χ 2 = {χJ2 [p] + χθ2 [p/(2p − 1)]}/[2nD + 1] obtained from Eqs. (34), (35) by using values of [SL (p), SL entropies calculated from pion–nucleus phase shift analyses [15]
p
p q = (2p−1)
0.538
7.0
π 0 He
π 0C
π 0O
π 0 Ca
χ 2 (p)/25
χ 2 (p)/27
χ 2 (p)/13
χ 2 (p)/29
0.073
0.069min
0.084
0.024
0.545
6.0
0.072
0.072
0.083
0.023
0.550
5.5
0.071
0.074
0.082
0.023min
0.556
5.0
0.070
0.076
0.081
0.023
0.563
4.5
0.069
0.080
0.081
0.023 0.024
0.571
4.0
0.068
0.087
0.080min
0.583
3.5
0.068
0.098
0.081
0.027
0.600
3.0
0.067min
0.117
0.085
0.033
0.625
2.5
0.069
0.159
0.095
0.048
0.667
2.0
0.076
0.274
0.129
0.096
0.700
1.75
0.085
0.421
0.174
0.158
0.750
1.50
0.105
0.762
0.273
0.299
0.833
1.25
0.150
1.696
0.520
0.669
1.00
1.00
0.263
1.237
1.781
1.33
0.80
0.507
12.63
4.725
2.904
4.367
1.50
0.75
0.631
16.51
3.694
5.550
2.00
0.67
1.177
43.38
3.00
0.60
3.346
44.61
following important results. Also, we used the nonextensive statistics for the description of quantum scattering systems from the pion–nucleus scattering. We proved the principle of nonextensivity conjugation in nonextensive quantum scattering statistics. This new concept in quantum physics, is experimentally tested (see Table 3) with high accuracy (CL 99%) by using 49 sets of pion–nucleus phase shifts [15]. The best fits of the experimental scattering entropies SL (p) with PMD-SQS-optimal state predictions SLo1 (p) are obtained (see the upper part of Fig. 1 and Table 1) for values of the nonextensivity index p in the interval 1/2 p 0.6 while, the best fits of the experimental scattering entropies Sθ (q) with the PMD-SQS-optimal state predictions Sθo1 (q) are obtained (see lower part of Fig. 1 and Table 2) for values of the nonexten-
7.366 11.67
11.64 13.97
sivity index q in the interval 3 q < ∞. Moreover, we obtained strong experimental evidences that the nonextensive statistical behaviors of the corresponding bosonic (π 0 A)-quantum scattering states are very similar with those of the fermionic (π 0 P )-quantum scattering states (see again Fig. 1). The experimental evidence obtained here for the nonextensive statistical (p = 1 and q = 2p/(1 − p)) behavior of the (L and θ )-quantum scatterings states can be interpreted as an indirect manifestation of the presence of the quarks and gluons in the pion–nucleus scattering system having a strong-coupling long-range regime of quantum chromodynamics. We remark that our results on the nonextensive statistical behavior of the quantum scattering in pion–nucleus scatterings are consistent with recent results of Ref. [13] and also with
270
D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
those from Ref. [17] about the nonextensivity index (around p = 1/2) obtained from radial density profiles in pure-electron plasmas in Penning traps. Finally, we note that further investigations are needed since this strong nonextensive statistical behavior of the quantum scattering, discovered here with high accuracy (CL > 99%), can be signatures for the presence of long-range interactions, memory effects or fractal space–time structures, in the hadronic interactions.
References
Fig. 1. The experimental results on [SJ (p), Sθ (p/ (2p − 1))]-Tsallis-like entropies for (π 0 A)-bosonic scattering states and (π 0 p)-fermionic quantum states are compared with the PMD-SQS-optimal state predictions [Sθo1 (p), SJo1 (p/ (2p − 1))]. No substantial differences between the statistical behavior of (π 0 A)-bosonic states and (π 0 p)-fermionic states are observed. The full curve represent optimal state PMD-SQS-entropies.
[1] M. Ohya, D. Petz, Quantum Entropy and Its Use, SpringerVerlag, Berlin, 1993, p. 279, and references therein. [2] E.T. Jaynes, Phys. Rev. 106 (1957) 620; E.T. Jaynes, Phys. Rev. 108 (1957) 171; S. Kullback, Information Theory and Statistics, Willey, New York, 1959, p. 37; J.E. Shore, R.W. Johnson, IEEE Trans. Inf. Theory IT-26 (1980) 26. [3] C. Tsallis, J. Stat. Phys. 52 (1988) 479; C. Tsallis, Phys. Lett. A 195 (1994) 329; M. Lyra, C. Tsallis, Phys. Rev. Lett. 80 (1998) 53; C. Tsallis, Brazilian. J. Phys. 29 (1999) 1, and references therein. [4] R. Hagedorn, Suppl. Nuovo Cimento 3 (1965) 147. [5] I. Bediaga, E.M.F. Curado, J. Miranda, A nonextensive thermodynamical equilibrium in e+ e− → hadrons, hepph/99005255. [6] D.B. Ion, M.L.D. Ion, Phys. Rev. Lett. 81 (1998) 5714. [7] D.B. Ion, M.L.D. Ion, Phys. Rev. E 60 (1999) 5261. [8] M.L.D. Ion, D.B. Ion, Phys. Rev. Lett. 83 (1999) 463. [9] D.B. Ion, M.L.D. Ion, Phys. Lett. B 466 (1999) 27. [10] G. Wilk, Z. Wlodarcsyk, Nucl. Phys. A 75 (1999) 191. [11] D.B. Walton, J. Rafelski, Phys. Rev. Lett. 84 (2000) 31. [12] M.L.D. Ion, D.B. Ion, Phys. Lett. B 474 (2000) 395. [13] M.L.D. Ion, D.B. Ion, Phys. Lett. B 482 (2000) 57. [14] D.B. Ion, Phys. Lett. B 376 (1996) 282. [15] J. Frolich, H.G. Schlaile, L. Streit, H.F.K. Zingl, Z. Phys. A 302 (1981) 89; O. Dumbrais, J. Frolich, U. Klein, H.G. Schlaile, Phys. Rev. C 29 (1984) 581; J. Frolich, H. Pilkuhn, H.G. Schlaile, Nucl. Phys. A 415 (1984) 399; B. Brinkmoller, H.G. Schlaile, Phys. Rev. C 48 (1993) 1973; H.G. Schlaile, Phys. Rev. C 55 (1997) 2584; G. Höhler, F. Kaiser, R. Koch, E. Pietarinen, Physics Data, Handbook of Pion–Nucleon Scattering, Vol. 12-1, 1979. [16] J. Bergh, J. Lofstrom, Interpolation Spaces, Springer-Verlag, Berlin, 1976, pp. 2–5. [17] B.M. Boghosian, Phys. Rev. E 53 (1996) 4754; C. Anteneodo, C. Tsallis, J. Mol. Liq. 71 (1997) 255; B.M. Boghosian, Brazilian J. Phys. 29 (1999) 91.
Physics Letters B 503 (2001) 263–270 www.elsevier.nl/locate/npe
Evidences for nonextensivity conjugation in hadronic scattering systems D.B. Ion, M.L.D. Ion National Institute for Physics and Nuclear Engineering, NIPNE-HH, Bucharest, P.O. Box MG-6, Romania Received 28 June 2000; received in revised form 5 October 2000; accepted 18 January 2001 Editor: R. Gatto
Abstract In this Letter the degrees of nonextensivity of the hadronic quantum scattering systems is investigated by using Tsallis-like scattering entropies SJ (p) and Sθ (q), p, q ∈ R. Then, evidences for strong nonextensive statistical effects, with nonextensivity indices 1/2 p 0.6, as well as, for the (1/2p + 1/2q = 1)-nonextensivity conjugation, are obtained by fitting the pion– nucleus experimental entropies with the PMD-SQS-optimal state predictions. The interpretation of these strong nonextensive statistical effects as indirect manifestations of the presence of the quarks and gluons in the hadronic quantum system is suggested. 2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Ca; 25.80.Dj; 12.40.Ee
1. Introduction The whole theory of equilibrium statistical mechanics is based on a variational formalism for entropy (see, e.g., Refs. [1,2]). The Boltzmann–Gibbs statistical mechanics and standard thermodynamics do not seem to be universal. They have domains of applicability quite poorly known nowadays. A generalization of these statistical formalisms to nonextensive quantum systems is now available (see also Ref. [3]). A partially successful global description of the hadronic interactions (especially hadroproduction process) has been provided through a thermodynamical equilibrium approach of Hagedorn [4]. A nonextensive generalization of this approach was presented in Ref. [5]. Moreover, other applications of the nonextensive statisti-
E-mail address: [email protected] (D.B. Ion).
cal approach in cosmic rays and elementary particles are presented in Refs. [6–13]. Recently, we investigated [6–9] the nonextensive quantum statistics of the hadronic scattering states by introducing the Tsallislike scattering entropies SJ (p) and Sθ (q), p, q ∈ R, corresponding to the systems of the angular momentum states [J-states] and the quantum angular scattering states [θ -states], respectively. Then, we proved, not only [θ, J ]-entropic uncertainty relations, but also the optimal entropic upper bounds for each Tsallislike scattering entropy Sθ (q), SJ (p) in terms of optimal entropies Sθo1 (q), SJo1 (p), derived from the principle of minimum distance in the space of quantum states (PMD-SQS) [14]. In general, the nonextensivity is expected to emerge in the quantum hadronic systems (e.g., in pion–nucleus quantum scattering states) if long-range interactions, memory effects or fractal space–time structures, are present. Hence, by the investigation of the nonextensivity of quantum hadronic
0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 1 5 5 - 1
264
D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
systems we can obtain indirect informations about the presence of the quarks and gluons having a strongcoupling long-range regime of quantum chromodynamics. The pion–nucleon and pion–nucleus quantum scattering systems are the most suitable for such investigations since in these cases the experimental entropies Sθ (q), SJ (p) can be obtained with high accuracy from the available phase shifts analyses [15]. In this Letter the nonextensivity of the quantum scattering states, as well as the (θ, J )-nonextensivity conjugation, are investigated in the pion–nucleus scattering. Strong experimental evidences for the (θ, J )nonextensivity conjugation is experimentally obtained with high accuracy (CL 99%) by fitting the pion– nucleus entropies with the “optimal state” predictions derived from the principle of minimum distance in the space of states (PMD-SQS). So, we introduce concept of nonextensivity conjugation in quantum physics in terms of Tsallis-like entropies SJ (p) and Sθ (q). We present the first experimental determination of the (p, q)-nonextensivity indices for the quantum systems of (J and θ )-systems of states from the pion–nucleus scatterings. Hence, using 49-sets of pion–nucleus and 88-sets pion–nucleon experimental phase shifts [15], we obtain a strong experimental evidence (CL 99%) for the (1/p + 1/q = 2)-nonextensivity conjugation for (J, θ )-nonextensive quantum statistics with 0.50 p 0.60.
2. Nonextensive statistical description of hadron–nucleus scatterings We start the nonextensive statistical description of the hadron–nucleus scatterings by introduction of the [Sθ (q), SJ (p)]-scattering entropies for the case of spinless particles. As is known, the scattering amplitude f (x) of spinless particles is developed in partial amplitudes as follows f (x) =
L
(2l + 1)fl Pl (x),
l=0
x ∈ [−1, +1],
fl ∈ C,
(1)
where L + 1 is the number of partial amplitudes fl , Pl (x), l = 0, 1, . . . , L, are Legendre polynomials. Hence, the [SL (p), Sθ (q)]-scattering entropies are
defined as follows p (2l + 1)pl /(p − 1), SL (p) = 1 −
p ∈ R,
L SL (1) = − (2l + 1)pl ln pl
+1 q /(q − 1), Sθ (q) = 1 − dx P (x)
(2)
j =0
q ∈ R,
−1
+1 Sθ (1) = − dx P (x) ln P (x),
(3)
−1
where the probability distributions P (x) and pl , are defined by pl = 4π|fl |2 /σel ,
L
(2l + 1)pl = 1,
j =0
2π dσ (x), P (x) = σel dΩ
+1 dx P (x) = 1.
(4)
−1
The indices p = 1 or q = 1 can be interpreted as a measure of nonextensivity and will be called the nonextensivity indices. Therefore, the scattering entropies {SL (p), Sθ (q)} allow us to investigate the (extensive or nonextensive)-statistical behavior of the [J or θ ]-quantum scattering states considered as canonical ensembles. So, corresponding to [SL (p), Sθ (q)]scattering entropies we introduce the J -nonextensive quantum statistics and θ -nonextensive quantum statistics which are describing the statistical behavior of canonical ensembles composed of angular-momentum quantum states and θ -angular quantum states, respectively.
3. (1/2p + 1/2q = 1)-nonextensivity conjugation as a new principle in quantum scattering Now, a natural but fundamental question can be addressed, namely, what kind correlation (if it exists) is expected to be observed between the nonextensivity indices p and q corresponding to the Tsallis-like J nonextensive quantum statistics described by SJ (p) and θ -nonextensive quantum statistics described by
D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
Sθ (q)? The answer at this question is given by the following result. If the Fourier transform (1) is a bounded map T : L2p → L2q , then the (p, J )-statistics and (q, θ )statistics are correlated via the relation 1 1 (5) + = 1, or q = p/(2p − 1), 2p 2q while the norm estimate M of this map is given by M≡
p−1 1−q
Tf 2q 2 2p = 2 2q .
f 2p
(6)
Proof. The proof of this remarkable result is based on the definition of the Fourier transformation (1) as a bounded map from the space Lp to the space Lq and also on the Lp -estimate-theorems with p = 2p and q = 2q. So, we show that the results given by Eq. (5) are direct consequences of the Riesz–Thorin interpolation theorem [13]. Indeed, let T be the Fourier transform defined by the scalar scattering amplitude divided by the respective L2 -norms and let us redefine the scattering amplitudes in such a way that the square modulus correspond to the respective probabilities P (x) and pJ , respectively. Then, we obtain the “probability functions” 2 f (x) , P (x) = ϕ(x) ϕ(x) ≡ (7)
f (x) 2 and fl ϕl ≡
(8) , pl = |ϕl |2 . (2l + 1)|fl |2 Hence, the Fourier transform (1) take the form 1 (2l + 1)ϕl Pl (x) ϕ(x) = √ (9) 2 and, conversely 1 ϕl = √ 2
+1 ϕ(x)Pl (x) dx.
(10)
−1
Now, by using Eqs. (9) and (10) and taking into account that |Pl (x)| 1, it is easy to prove the inequalities 1/2 1/2 sup P (x) (11) 2−1/2 (2l + 1)pl , sup[pl ]1/2 2
+1 −1/2 −1
P (x)
1/2
and (2l + 1)pl = P (x) dx = 1. +1
(12)
(13)
−1
This means that in both cases we have T : L1 → L∞ with norm M1 = 2−1/2 ,
(14)
T : L2 → L2 with norm M2 = 1.
(15)
Then, using the Riesz–Thorin interpolation theorem for the complex-valued functions (see Ref. [16]) we obtain T : Lp → Lq with norm M ≡
T ϕ q
ϕ p
(16)
with (1/p ) = (1 − t)/1 + t/2, (1/q ) = (1 − t)/∞ + t/2.
(17)
Hence, eliminating the parameter t and using the relations p = 2p and q = 2q, we get not only the condition 1 1 + = 1, 2q 2p
t=
1 1 1 − +1= , 2q 2p q
(18) p−1
but also the norm-estimate (6) M M11−t M2t = 2 2p . By the norm estimate (6) we proved the following inequalities 1 1 + (1 − q)Sθ (q) 2q p−1 1 2 2p 1 + (1 − p)SL (p) 2p , 1 1 + (1 − q)SL (q) 2q p−1 1 2 2p 1 + (1 − p)Sθ (p) 2p ,
(19)
(20)
valid for any 1/2 < p 1 and 1/2p + 1/2q = 1. Proof. Indeed, by the definitions of the norms in the L2p and L2q spaces, in our case we have
ϕl 2t =
dx,
265
(2l + 1)|ϕl |2t = (2l + 1)plt
1 = 1 + (1 − t)SL (t) 2t ,
t ≡ p or q,
(21)
266
D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
and
tively. These distributions are given by:
ϕ(x) = 2t
+1 +1 t ϕ(x) 2t dx = P (x) dx
−1
plme = plo1
1 = 1 + (1 − t)Sθ (t) 2p ,
t ≡ p or q, (22)
since pl ≡ |ϕl |2 , and P (x) ≡ |ϕ(x)|2 [see Eqs. (7), (8)]. This means that by the result (6) we proved the inequalities (19), (20) since by definitions of the Lp norms from Eq. (6) and the definitions (2)–(4) we have M=
1 1 = , 2K(1, 1) (Lo + 1)2 pJme = 0, for l Lo + 1 =
−1
P
me
[K(x, 1)]2 (x) = P (x) = , K(1, 1) o1
K(x, 1) =
[1 + (1 − q)Sθ (q)] 2q 1
[1 + (1 − p)SL (p)] 2p
2
1−p 2p
1/2
(23)
ϕl L2q
◦ 1 ◦ P Lo +1 (x)+ P Lo (x) , 2 K(1, 1) = (Lo + 1)2 /2 =
(29)
◦
ϕ(x) L2p [1 + (1 − q)SL (q)] 2q [1 + (1 − p)Sθ (p)]
1 2p
2
1−p 2p
(24)
for the inverse Fourier transformation (10). From this result, in the limit p → 1, we obtained in Ref. [10] the (θ, l)-entropic uncertainty relations: ln 2 Sθ + SL
o 1 (2l + 1)Pl (x) 2
L
ϕl L2p
1
=
(28)
where the reproducing kernel K(x, y) is given by
ϕ(x) L2q
for direct Fourier transformation (9), and = M
1 l Lo , 2 (27)
and
1
=
for
or Var[θ ] · Var[l] 2,
(25)
where the statistical variances Var[X] are defined as Var[X] = exp{SX }, X ≡ θ, L.
4. PMD-SQS-optimal distributions as equilibrium distributions of quantum scattering states Now, we can use the solution to the maximumentropy problem
max SJ (p), Sθ (q) dσ when σel = fixed and (26) (1) = fixed dΩ as criterion for the determination of the equilibrium distributions plme and P me (x), of the J -quantum scattering states and θ -quantum scattering states, respec-
d Pl (x), while the optimal angular where Pl (x) ≡ dx momentum Lo is given by: 1/2 4π dσ (1) −1 . Lo = integer (30) σel dΩ
Proof. Indeed, solving the problem (26) via Lagrange multipliers we obtain that singular solution λ0 = 0 exists and is just given by the [SJo1 (p), Sθo1 (q)]-optimal entropies obtained from the PMD-SQS-optimal state f o1 (x) = f (1)
K(x, 1) K(1, 1)
(31)
[ principle of minimum distance in the space of quantum states], where f (x), x ∈ [−1, 1], is the scattering amplitude (1). So, the solution of the problem (6) is given by [10] SLmax (p) = SLo1 (p) = 1 − [2K(1, 1)]1−p (p − 1)−1 , Sθmax (q) = Sθo1 (p)
(32)
+1 [K(x, 1)]2 q = 1 − dx (q − 1)−1 . K(1, 1) −1 (33)
D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
267
Table 1 o1 (p)]-scattering entropies calculated from pion–nucleus The values of the χL2 (p) obtained from Eqs. (34), (35) by using values of [SL (p), SL phase shift analyses [15]
p
p q = (2p−1)
0.538
7.0
π 0 He
π 0C
π 0O
π 0 Ca
χJ2 (p)/12
χJ2 (p)/13
χJ2 (p)/6
χJ2 (p)/14
0.152
0.143min
0.182
0.048
0.545
6.0
0.150
0.147
0.179
0.047
0.550
5.5
0.148
0.150
0.177
0.046
0.556
5.0
0.146
0.154
0.175
0.045
0.563
4.5
0.144
0.159
0.173
0.045
0.571
4.0
0.142
0.168
0.170
0.044min
0.583
3.5
0.139
0.180
0.168
0.045 0.048
0.600
3.0
0.135
0.201
0.167min
0.625
2.5
0.132
0.238
0.169
0.055
0.667
2.0
0.130min
0.312
0.179
0.075
0.700
1.75
0.130
0.381
0.193
0.096
0.750
1.50
0.135
0.498
0.221
0.135
0.833
1.25
0.150
0.727
0.284
0.213
1.00
1.00
0.201
1.201
0.456
0.406
1.33
0.80
0.377
2.543
0.958
0.900
1.50
0.75
0.507
3.387
1.298
1.204
2.00
0.67
1.370
42.00
3.00
0.60
5.599
23.51
6.055
8.862
11.67
8.045
5. Determination of nonextensivity indices from pion–nucleus experimental data
(c) the χJ2 (p) or/and χθ2 (q)-test functions defined by
5.1. Numerical results for nonextensivity (p, q)-indices
χX2 (k) =
The nonextensivity (p, q)-indices can be obtained by fit the experimental data on SJ (p) and Sθ (q) with the PMD-SQS-optimal state predictions (32), (33). Then, the man objective is to get the value of p and q for which we obtain the best fit. For the investigation of this important problem we used the experimental pion–nucleus [15] experimental phase-shifts for to obtain: (a) the experimental values of the scattering entropies SL (p) [Eq. (2)], Sθ (q) [Eq. (3)] as well as of the optimal angular momenta Lo [Eq. (30)], (b) the optimal entropies SLo1 (q) [Eq. (32)], Sθo1 (q) [Eq. (33)],
nexp [SX (k)]i − [S o1 (k)]i 2 X
i=1
X ≡ J, θ ;
o1 ] ['SX i
,
k ≡ p, q,
(34)
and o1 o1 'SX (k) = SX (k) L
o
o1 − SX (k) L +1
o −1
(35)
o1 where [SX (k)]Jo ±1 are the values of the PMD-SQSoptimal entropies calculated with the optimal angular momenta Lo ± 1, respectively. The results obtained in this way are presented in Fig. 1 and Tables 1–3.
268
D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
Table 2 The values of the χθ2 (q) obtained from Eqs. (34), (35) by using values of [Sθ (q), Sθo1 (q)]-scattering entropies calculated from pion–nucleus phase shift analyses [15] π 0 He
π 0C
π 0O
π 0 Ca
p
p q = (2p−1)
χθ2 (q)/12
χθ2 (q)/13
χθ2 (q)/6
χθ2 (q)/14
0.538
7.0
< 10−3 min
0.001min
< 10−3 min
< 10−3 min
0.545
6.0
< 10−3
0.002
< 10−3
< 10−3
0.550
5.5
< 10−3
0.003
< 10−3
0.001
5.0
< 10−3
0.005
0.001
0.002
4.5
< 10−3
0.007
0.002
0.004
0.556 0.563 0.571
4.0
0.001
0.012
0.004
0.006
0.583
3.5
0.002
0.022
0.007
0.010
0.600
3.0
0.005
0.042
0.016
0.020
0.625
2.5
0.011
0.092
0.037
0.045
0.667
2.0
0.028
0.257
0.101
0.123
0.700
1.75
0.047
0.494
0.184
0.231
0.750
1.50
0.084
1.085
0.370
0.485
0.833
1.25
0.163
2.796
0.843
1.174
1.00
1.00
0.348
2.225
3.284
1.33
0.80
0.679
23.71
8.613
5.335
8.145
1.50
0.75
0.807
30.91
6.707
2.00
0.67
1.082
48.11
3.00
0.60
1.371
69.14
5.2. Strong experimental evidence of (1/p + 1/q = 2)-nonextensivity correlation Now, we can give an “experimental” answer to the fundamental question: what kind correlation (if it exists) is expected to be observed between the nonextensivity indices p and q corresponding to the (p, J )-nonextensive statistics described by SJ (p) and (q, θ )-nonextensive statistics described by Sθ (q)? [We remember that the “mathematical” answer is given by Eq. (5)]. Indeed, from Fig. 1 as well as from the Table 3 we see that the experimental data on the scattering entropies SL (p) and Sθ (q) are simultaneously in excellent agreement (CL > 99%) with the [SLo1 (p), Sθo1 (q)]-optimal state predictions if the nonextensivities p and q of the (J and θ )-statistics are correlated via Riesz–Thorin relation: 1/p + 1/q =
9.905 13.62
10.29 15.25 20.90
2 (or q = p/(2p − 1)). So, the best fit is obtained (see Table 3) for the conjugate pairs p and q = p/(2p − 1) with the values of p in the range 0.5 p 0.6. In fact a significant departures from “optimality” is observed only for pairs with p > 2. These results can be compared with those obtained in Ref. [13] for pion– nucleon scattering case. Then, we can see that the nonextensive statistical behaviors of the corresponding bosonic (π 0 A)-quantum scattering states are very similar (see again Fig. 1) with those of the fermionic (π 0 P )-quantum scattering states.
6. Conclusions In summary, in this paper by using 49-sets of pion– nucleus phase shifts analyses [15], we obtained the
D.B. Ion, M.L.D. Ion / Physics Letters B 503 (2001) 263–270
269
Table 3 o1 (p)]-scattering The values of the χ 2 = {χJ2 [p] + χθ2 [p/(2p − 1)]}/[2nD + 1] obtained from Eqs. (34), (35) by using values of [SL (p), SL entropies calculated from pion–nucleus phase shift analyses [15]
p
p q = (2p−1)
0.538
7.0
π 0 He
π 0C
π 0O
π 0 Ca
χ 2 (p)/25
χ 2 (p)/27
χ 2 (p)/13
χ 2 (p)/29
0.073
0.069min
0.084
0.024
0.545
6.0
0.072
0.072
0.083
0.023
0.550
5.5
0.071
0.074
0.082
0.023min
0.556
5.0
0.070
0.076
0.081
0.023
0.563
4.5
0.069
0.080
0.081
0.023 0.024
0.571
4.0
0.068
0.087
0.080min
0.583
3.5
0.068
0.098
0.081
0.027
0.600
3.0
0.067min
0.117
0.085
0.033
0.625
2.5
0.069
0.159
0.095
0.048
0.667
2.0
0.076
0.274
0.129
0.096
0.700
1.75
0.085
0.421
0.174
0.158
0.750
1.50
0.105
0.762
0.273
0.299
0.833
1.25
0.150
1.696
0.520
0.669
1.00
1.00
0.263
1.237
1.781
1.33
0.80
0.507
12.63
4.725
2.904
4.367
1.50
0.75
0.631
16.51
3.694
5.550
2.00
0.67
1.177
43.38
3.00
0.60
3.346
44.61
following important results. Also, we used the nonextensive statistics for the description of quantum scattering systems from the pion–nucleus scattering. We proved the principle of nonextensivity conjugation in nonextensive quantum scattering statistics. This new concept in quantum physics, is experimentally tested (see Table 3) with high accuracy (CL 99%) by using 49 sets of pion–nucleus phase shifts [15]. The best fits of the experimental scattering entropies SL (p) with PMD-SQS-optimal state predictions SLo1 (p) are obtained (see the upper part of Fig. 1 and Table 1) for values of the nonextensivity index p in the interval 1/2 p 0.6 while, the best fits of the experimental scattering entropies Sθ (q) with the PMD-SQS-optimal state predictions Sθo1 (q) are obtained (see lower part of Fig. 1 and Table 2) for values of the nonexten-
7.366 11.67
11.64 13.97
sivity index q in the interval 3 q < ∞. Moreover, we obtained strong experimental evidences that the nonextensive statistical behaviors of the corresponding bosonic (π 0 A)-quantum scattering states are very similar with those of the fermionic (π 0 P )-quantum scattering states (see again Fig. 1). The experimental evidence obtained here for the nonextensive statistical (p = 1 and q = 2p/(1 − p)) behavior of the (L and θ )-quantum scatterings states can be interpreted as an indirect manifestation of the presence of the quarks and gluons in the pion–nucleus scattering system having a strong-coupling long-range regime of quantum chromodynamics. We remark that our results on the nonextensive statistical behavior of the quantum scattering in pion–nucleus scatterings are consistent with recent results of Ref. [13] and also with
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those from Ref. [17] about the nonextensivity index (around p = 1/2) obtained from radial density profiles in pure-electron plasmas in Penning traps. Finally, we note that further investigations are needed since this strong nonextensive statistical behavior of the quantum scattering, discovered here with high accuracy (CL > 99%), can be signatures for the presence of long-range interactions, memory effects or fractal space–time structures, in the hadronic interactions.
References
Fig. 1. The experimental results on [SJ (p), Sθ (p/ (2p − 1))]-Tsallis-like entropies for (π 0 A)-bosonic scattering states and (π 0 p)-fermionic quantum states are compared with the PMD-SQS-optimal state predictions [Sθo1 (p), SJo1 (p/ (2p − 1))]. No substantial differences between the statistical behavior of (π 0 A)-bosonic states and (π 0 p)-fermionic states are observed. The full curve represent optimal state PMD-SQS-entropies.
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