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Fractality in subatomic systems
Chaos, Solitons and Fractals 12 (2001) 97±100
www.elsevier.nl/locate/chaos
Fractality in subatomic systems J.-C. Leclerc, C. Rioux, R.J. Slobodrian * D epartement de physique, Universit e Laval, Qu ebec, Que., Canada G1K 7P4 Accepted 13 September 1999
Abstract Subatomic systems have unit components in permanent motion. Images in con®guration space calculated for such systems are shown to have fractal characteristics. The fractal dimension is an invariant of the motion. The plausibility of extending fractal concepts to subnucleon structures is discussed. Ó 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Static physical fractals are currently being studied at scales ranging from atomic dimensions to those of the known universe and pervade all of science [1]. Computer simulations have reproduced quite adequately such fractals. Particularly relevant for the present work on subatomic systems are three-dimensional fractals of metallic elements with spherical monomers, obtained using evaporation±condensation techniques [2]. The monomers range in size from tens of nanometers to micrometers and can be put in evidence with electron and atomic force microscopes (EM and AFM, respectively). These aggregates are, of course, static systems and the images appeal directly to intuition and allow quantitative interpretation.The monomers possess substructures noticeable as hemispherical bumps in images obtained with a novel Hitachi EM that does not require previous metallisation of surfaces, lending credence to the observations. Furthermore, X-ray analysis has shown that there is also a sub-microcrystalline structure [2]. Recently, remarkable images of simulated particle and nuclear physics systems have been published [3] and Fig. 1(a) shows a two-dimensional image of the uranium atom nucleus. Such images are representations of `unobservable' systems and one may question their `reality', yet in fact both, semiclassical and quantum mechanical formulations, are based upon the concept of particles (nucleons in the case of uranium) and their positions and/or mechanical momenta [4]. Lagrangians and Hamiltonians form the backbone of the simulations in Ref. [3]. Subatomic systems possess components in permanent motion at variance with static physical fractal aggregates. 2. Fractality of the representations of uranium atoms and nuclei Self-similarity of physical systems in some way is presently the accepted criterion for the characterization of a fractal [5]. The fractal dimension D is a generalization of the discrete dimensionality of ordinary spaces extended to non-dense spaces (ensembles of points), 1 it provides a measure of the occupation of dense
* 1
Corresponding author. Hausdor, Besicovitch and Kolmogorov are credited with laying the foundations of this generalization, see [6].
0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 9 ) 0 0 1 7 4 - 5
98
J.-C. Leclerc et al. / Chaos, Solitons and Fractals 12 (2001) 97±100
space by a fractal. Thus it can be determined by a simple method known as `ball counting' which consists in covering three-dimensional aggregates with spheres or cubes of variable linear parameter L (radius or side) and counting the numbers N(L). The fractal dimension is given by D lim
L!0
ln N L : ln 1=L
1
Self-similar systems yield a linear relation between the numerator and the denominator of (1). It is easy to verify that the slope is a measure of D. Reciprocally, linearity of the relation con®rms the fractality of a non-dense system. Employing the described procedure in Fig. 1(a) which is a two-dimensional projection of the uranium nucleus (fractality is a projective invariant) one obtains D 1:87 0:05 from the linear diagram of lnN l vs ln 1=L, which con®rms the fractality as shown in Fig. 1(b). The nucleons of Ref. [3] exhibit clearly a substructure in the context of quark±gluon models (see for example [7,8]). Fig. 2(a) shows a
Fig. 1. Fractality of the simulation of the uranium nucleus: (a) image obtained from Ref. [3], it is a projection on a plane; (b) diagram of the results from the ball counting method. Fig. 2. Stochastic simulations of the uranium nucleus and its electrons: (a) the nucleus; (b) the electrons, the nucleus is shown augmented at the center by a factor of 1000.
J.-C. Leclerc et al. / Chaos, Solitons and Fractals 12 (2001) 97±100
99
simpler image of uranium with spherical nucleons (monomers) having, within error, a similar fractal dimension as Fig. 1(a), perfectly suitable for the uranium nucleus within nuclear theory as given by the optical model. It is also possible to represent the stochastic nature of the instantaneous positions of the electrons in the same manner. Fig. 2(b) shows the two-dimensional image of such a calculation. This set is also fractal. Clearly D depends on the assumed electron radius. Fractality is independent of electron radii over a large range of values. For Fig. 2(b) D 1:78 0:05. Scaling down the radius by a factor of two results in D 1:15 0:05. It is clear that the fractal dimensions of these systems in con®guration space are time invariant. Non-dense states of matter like gases and vapors, consisting similarly of components in permanent motion, are likely to exhibit fractal characteristics which could be simulated as shown above for nucleons and electrons.The fractal dimension would express the occupation of vacuum (empty space) by the atoms or molecules. This exercise could be repeated for subnucleon structures of quarks and gluons, although the number of virtual particles involved is not clearly de®ned and the adequacy of fractal concepts may be less clear and open to speculation due mainly to the surmised relevance of non-local and non-linear phenomena. However, the continued repetition of non-static fractality into extremely small scales cannot be ruled out. Thus, protons and neutrons may indeed be fractal systems of quarks and gluons. A paper summarizing works done in this direction by Nottale, Ord and others is that of Argyris et al. [9], where the signatures of fractal space in quantum physics and cosmology are reviewed. 3. Discussion The realm of atomic phenomena (at scales of 1 nm and below) is believed to be well described by quantum mechanics. The stochastic nature of these phenomena is re¯ected in the probabilistic interpretation of the wave function, due to Max Born [10]. Current work on fractal aggregation has yielded directly visible evidence of the relevance of stochastic phenomena at hyperatomic scales (up to micrometers and beyond), where a transition of the foundations of mechanics from quantum to classical principles should take place. Precise knowledge of such transition is lacking and the correspondence principle has become questionable. Imaging with electron and atomic force microscopes allows physical intuition to play an increasing role in the study of matter at this transition region. The stochastic nature of the motion of units forming static fractal aggregates is experimentally veri®ed and easily accepted. Semiclassical physics [11] may prove to be useful as a ®rst step for the understanding of structures and phenomena in this transition region. In contradistinction, direct images are not available for subatomic systems. Stochastic calculations in con®guration space may be adequate for simulation of many particle systems where motions are mostly incoherent. The study of subnucleon systems is still evolving and it is not clear that relativistic quantum mechanics holds the key to their understanding. Very short-range forces and con®nement seem to escape conventional principles and wisdom. Thus, instead of departing from dierential (or integral) equations requiring continuity to determine the wave function of a system, subject to a reinterpretation as a stochastic quantity, it may be advantageous to start directly from a stochastic basis for its description [12]. Indeed there are presently far-reaching new approaches to the underlying spacetime structure of phenomena and possible fractal characteristics, videt licet polymerized spacetime [13]. 4. Concluding remarks We express agreement with the views of Refs. [3,4] on the value of visualization of subatomic systems in con®guration space, which allow intuition to regain a relevant role in their understanding. Fractal concepts are applicable and the fractal dimensions can be obtained from the calculated images. They are timeinvariant quantities related to the topology of the systems. Within the realm of quantum mechanics, the description in momentum space should also yield ensembles with fractal characteristics, as position and momentum space descriptions are topologically equivalent in one-to-one correspondence.
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J.-C. Leclerc et al. / Chaos, Solitons and Fractals 12 (2001) 97±100
Acknowledgements The support of the Canadian Space Agency is warmly thanked as well as the invaluable assistance of Stephane Desjardins, scienti®c manager of our project at the agency. Universite Laval and Fond pour la Formation de Chercheurs et l' Aide a la Recherche ± Quebec have also provided support to this work and are heartily thanked. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Bunde A, Havlin S, editors. Fractals in science. Berlin: Springer, 1994. Deladurantaye P, Rioux C, Slobodrian RJ. Chaos, Solitons & Fractals 1997;8:1693±708. Cohen E. Nucl Phys News 1995;5:20±2. Cohen E. Comput Phys Commun 1992;70:441±6. Mandelbrot B. Les objets fractals: forme, hasard et dimension, 3rd edn. Paris: Flammarion, 1989. Edgar Gerald A, editor. Classics on fractals. Reading, MA: Addison-Wesley, 1993. Hughes IS. Elementary particles, third ed. Cambridge: The Cambridge University Press, 1991. Kane G. Modern elementary particle physics. Reading, MA: Addison-Wesley, 1993. Argyris J, Ciubotariu CI, Weingaertner WE. Fractal space signatures in quantum physics and cosmology I. Chaos, Solitons & Fractals 2000;11:1671±719. Born M, Atomic physics. eighth ed., and references therein, revised by: Blin Stoyle RJ, Radclie, JM. Glasgow: Blackie & Sons, 1969. Brack M, Bhaduri Rajat K. Semiclassical physics. Reading, MA: Addison-Wesley, 1997. El Naschie MS. Int J Theoret Phys 1998;37(12):2935±51. El Naschie MS. Chaos Solitons & Fractals 1999;10:567±80.
www.elsevier.nl/locate/chaos
Fractality in subatomic systems J.-C. Leclerc, C. Rioux, R.J. Slobodrian * D epartement de physique, Universit e Laval, Qu ebec, Que., Canada G1K 7P4 Accepted 13 September 1999
Abstract Subatomic systems have unit components in permanent motion. Images in con®guration space calculated for such systems are shown to have fractal characteristics. The fractal dimension is an invariant of the motion. The plausibility of extending fractal concepts to subnucleon structures is discussed. Ó 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Static physical fractals are currently being studied at scales ranging from atomic dimensions to those of the known universe and pervade all of science [1]. Computer simulations have reproduced quite adequately such fractals. Particularly relevant for the present work on subatomic systems are three-dimensional fractals of metallic elements with spherical monomers, obtained using evaporation±condensation techniques [2]. The monomers range in size from tens of nanometers to micrometers and can be put in evidence with electron and atomic force microscopes (EM and AFM, respectively). These aggregates are, of course, static systems and the images appeal directly to intuition and allow quantitative interpretation.The monomers possess substructures noticeable as hemispherical bumps in images obtained with a novel Hitachi EM that does not require previous metallisation of surfaces, lending credence to the observations. Furthermore, X-ray analysis has shown that there is also a sub-microcrystalline structure [2]. Recently, remarkable images of simulated particle and nuclear physics systems have been published [3] and Fig. 1(a) shows a two-dimensional image of the uranium atom nucleus. Such images are representations of `unobservable' systems and one may question their `reality', yet in fact both, semiclassical and quantum mechanical formulations, are based upon the concept of particles (nucleons in the case of uranium) and their positions and/or mechanical momenta [4]. Lagrangians and Hamiltonians form the backbone of the simulations in Ref. [3]. Subatomic systems possess components in permanent motion at variance with static physical fractal aggregates. 2. Fractality of the representations of uranium atoms and nuclei Self-similarity of physical systems in some way is presently the accepted criterion for the characterization of a fractal [5]. The fractal dimension D is a generalization of the discrete dimensionality of ordinary spaces extended to non-dense spaces (ensembles of points), 1 it provides a measure of the occupation of dense
* 1
Corresponding author. Hausdor, Besicovitch and Kolmogorov are credited with laying the foundations of this generalization, see [6].
0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 9 ) 0 0 1 7 4 - 5
98
J.-C. Leclerc et al. / Chaos, Solitons and Fractals 12 (2001) 97±100
space by a fractal. Thus it can be determined by a simple method known as `ball counting' which consists in covering three-dimensional aggregates with spheres or cubes of variable linear parameter L (radius or side) and counting the numbers N(L). The fractal dimension is given by D lim
L!0
ln N L : ln 1=L
1
Self-similar systems yield a linear relation between the numerator and the denominator of (1). It is easy to verify that the slope is a measure of D. Reciprocally, linearity of the relation con®rms the fractality of a non-dense system. Employing the described procedure in Fig. 1(a) which is a two-dimensional projection of the uranium nucleus (fractality is a projective invariant) one obtains D 1:87 0:05 from the linear diagram of lnN l vs ln 1=L, which con®rms the fractality as shown in Fig. 1(b). The nucleons of Ref. [3] exhibit clearly a substructure in the context of quark±gluon models (see for example [7,8]). Fig. 2(a) shows a
Fig. 1. Fractality of the simulation of the uranium nucleus: (a) image obtained from Ref. [3], it is a projection on a plane; (b) diagram of the results from the ball counting method. Fig. 2. Stochastic simulations of the uranium nucleus and its electrons: (a) the nucleus; (b) the electrons, the nucleus is shown augmented at the center by a factor of 1000.
J.-C. Leclerc et al. / Chaos, Solitons and Fractals 12 (2001) 97±100
99
simpler image of uranium with spherical nucleons (monomers) having, within error, a similar fractal dimension as Fig. 1(a), perfectly suitable for the uranium nucleus within nuclear theory as given by the optical model. It is also possible to represent the stochastic nature of the instantaneous positions of the electrons in the same manner. Fig. 2(b) shows the two-dimensional image of such a calculation. This set is also fractal. Clearly D depends on the assumed electron radius. Fractality is independent of electron radii over a large range of values. For Fig. 2(b) D 1:78 0:05. Scaling down the radius by a factor of two results in D 1:15 0:05. It is clear that the fractal dimensions of these systems in con®guration space are time invariant. Non-dense states of matter like gases and vapors, consisting similarly of components in permanent motion, are likely to exhibit fractal characteristics which could be simulated as shown above for nucleons and electrons.The fractal dimension would express the occupation of vacuum (empty space) by the atoms or molecules. This exercise could be repeated for subnucleon structures of quarks and gluons, although the number of virtual particles involved is not clearly de®ned and the adequacy of fractal concepts may be less clear and open to speculation due mainly to the surmised relevance of non-local and non-linear phenomena. However, the continued repetition of non-static fractality into extremely small scales cannot be ruled out. Thus, protons and neutrons may indeed be fractal systems of quarks and gluons. A paper summarizing works done in this direction by Nottale, Ord and others is that of Argyris et al. [9], where the signatures of fractal space in quantum physics and cosmology are reviewed. 3. Discussion The realm of atomic phenomena (at scales of 1 nm and below) is believed to be well described by quantum mechanics. The stochastic nature of these phenomena is re¯ected in the probabilistic interpretation of the wave function, due to Max Born [10]. Current work on fractal aggregation has yielded directly visible evidence of the relevance of stochastic phenomena at hyperatomic scales (up to micrometers and beyond), where a transition of the foundations of mechanics from quantum to classical principles should take place. Precise knowledge of such transition is lacking and the correspondence principle has become questionable. Imaging with electron and atomic force microscopes allows physical intuition to play an increasing role in the study of matter at this transition region. The stochastic nature of the motion of units forming static fractal aggregates is experimentally veri®ed and easily accepted. Semiclassical physics [11] may prove to be useful as a ®rst step for the understanding of structures and phenomena in this transition region. In contradistinction, direct images are not available for subatomic systems. Stochastic calculations in con®guration space may be adequate for simulation of many particle systems where motions are mostly incoherent. The study of subnucleon systems is still evolving and it is not clear that relativistic quantum mechanics holds the key to their understanding. Very short-range forces and con®nement seem to escape conventional principles and wisdom. Thus, instead of departing from dierential (or integral) equations requiring continuity to determine the wave function of a system, subject to a reinterpretation as a stochastic quantity, it may be advantageous to start directly from a stochastic basis for its description [12]. Indeed there are presently far-reaching new approaches to the underlying spacetime structure of phenomena and possible fractal characteristics, videt licet polymerized spacetime [13]. 4. Concluding remarks We express agreement with the views of Refs. [3,4] on the value of visualization of subatomic systems in con®guration space, which allow intuition to regain a relevant role in their understanding. Fractal concepts are applicable and the fractal dimensions can be obtained from the calculated images. They are timeinvariant quantities related to the topology of the systems. Within the realm of quantum mechanics, the description in momentum space should also yield ensembles with fractal characteristics, as position and momentum space descriptions are topologically equivalent in one-to-one correspondence.
100
J.-C. Leclerc et al. / Chaos, Solitons and Fractals 12 (2001) 97±100
Acknowledgements The support of the Canadian Space Agency is warmly thanked as well as the invaluable assistance of Stephane Desjardins, scienti®c manager of our project at the agency. Universite Laval and Fond pour la Formation de Chercheurs et l' Aide a la Recherche ± Quebec have also provided support to this work and are heartily thanked. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Bunde A, Havlin S, editors. Fractals in science. Berlin: Springer, 1994. Deladurantaye P, Rioux C, Slobodrian RJ. Chaos, Solitons & Fractals 1997;8:1693±708. Cohen E. Nucl Phys News 1995;5:20±2. Cohen E. Comput Phys Commun 1992;70:441±6. Mandelbrot B. Les objets fractals: forme, hasard et dimension, 3rd edn. Paris: Flammarion, 1989. Edgar Gerald A, editor. Classics on fractals. Reading, MA: Addison-Wesley, 1993. Hughes IS. Elementary particles, third ed. Cambridge: The Cambridge University Press, 1991. Kane G. Modern elementary particle physics. Reading, MA: Addison-Wesley, 1993. Argyris J, Ciubotariu CI, Weingaertner WE. Fractal space signatures in quantum physics and cosmology I. Chaos, Solitons & Fractals 2000;11:1671±719. Born M, Atomic physics. eighth ed., and references therein, revised by: Blin Stoyle RJ, Radclie, JM. Glasgow: Blackie & Sons, 1969. Brack M, Bhaduri Rajat K. Semiclassical physics. Reading, MA: Addison-Wesley, 1997. El Naschie MS. Int J Theoret Phys 1998;37(12):2935±51. El Naschie MS. Chaos Solitons & Fractals 1999;10:567±80.