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IX International Workshop on
Advanced Computing and Analysis Techniques in Physics Research
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High Energy Accelerator Research
Organization (KEK) 1-1 Oho, Tsukuba, Ibaraki 305-0801 Japan |
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December 1-5, 2003
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Section III, A. Konash, 20 minutes talk, December 3, 11:40-12:00
Numerical experiments with ground on AED theory
Context:
System Theory
- General System Theory
- AED theory S^
Simulations
- Fabri-Perot Interferometer
- Percolation in heterogeneous systems
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Introduction
Nowadays new methods for mathematical definition and for numerical imitations of complex physical phenomena are strongly invited. In our investigations we use mathematical system theory AED, which has origin in the developed at 70th by M.D. Mesarovic and Yasushiko Takahara general system theory (see M.D. Mesarovic and Yasushiko Takahara General System Theory: Mathematical Foundations, Academic Press, New York San Francisco London, 1975). Up to now the AED is well established and recognized as actual mathematical tool with applications in programming [prg1], networks [net1, net2, net3], demography [dem1], economics [econ1], biology [bio1, bio2], physics [phys1, phys2, phys3]. In this report we present our recent results in application of the theory for mathematical definition of the Fabri-Perot Interferometer with chaotic (nonlinear) behavior and the Percolation in heterogeneous systems with realization of these definitions by the object-oriented programming technique in the language Delphi.
General System Theory
The main idea of the General System Theory is to abstract from concrete details of a system and to deal with formal representation of elements and acts in defined environment. Main feature of the theory is the system S with following definition:
S
X
Y
where X is Input Object and Y is Output Object of the System S.
Introduction of Condition
Object C and Function of Reaction R leads to exact
correspondence between the pair of input x[i] X with the condition c[k] C and the output y[j]
Y: R:
X C
-> Y
r [ik] : x[i] c[k] -> y[j]; i 1..I, j 1..J, k 1..K
Function of condition (state) 
produces new condition by current
condition and input: : X C -> C
[ik]: x[i] c[k] -> c[k1]; i
1..I; k, k1 1..K
These relations can be presented as tables:
Production for the Output Object (Reaction of the System R)
| Output \ Input |
x[1] |
x[2] |
x[3] |
... |
x[I-1] |
x[I] |
|
y[1] |
c[1,1] |
c[2,1] |
c[3,1] |
... |
c[I-1,1] |
c[I,
1] |
|
y[2] |
c[1,2] |
c[2,2] |
c[3,2] |
... |
c[I-1,
2] |
c[I,
2] |
|
y[3] |
c[1,2] |
c[2,3] |
c[3,3] |
... |
c[I-1,
3] |
c[I,
3] |
|
... |
... |
... |
... |
... |
... |
... |
|
y[J-1] |
c[1,
J-1] |
c[2,
J-1] |
c[3,
J-1] |
... |
c[I-1,
J-1] |
c[I,
J-1] |
|
y[J] |
c[1,
J] |
c[2,
J] |
c[3,
J] |
... |
c[I-1,
J] |
c[I,
J] |
c[i,j] <-->
c[k]; c C, k
1..K, i 1..I, j 1..J
Production for the Condition
Object ()
| State \ Input |
x[1] |
x[2] |
x[3] |
... |
x[I-1] |
x[I] |
|
c[1] |
c1[1,1] |
c1[2,1] |
c1[3,1] |
... |
c1[I-1,
1] |
c1[I,
1] |
|
c[2] |
c1[1,2] |
c1[2,2] |
c1[3,2] |
... |
c1[I-1,
2] |
c1[I,
2] |
|
c[3] |
c1[1,2] |
c1[2,3] |
c1[3,3] |
... |
c1[I-1,3] |
c1[I,
3] |
|
... |
... |
... |
... |
... |
... |
... |
|
c[K-1] |
c1[1,
K-1] |
c1[2,
K-1] |
c1[3,
K-1] |
... |
c1[I-1,
K-1] |
c1[I,
K-1] |
|
c[K] |
c1[1,
J] |
c1[2,
J] |
c1[3,
J] |
... |
c1[I-1,
K] |
c1[I,
K] |
c1[i,k1]
<--> c[k]; c1, c 1..C, k1,k 1..K, i 1..I
Introduction in consideration conception of Time T for the Objects C, X, Y of the Systems S leads to dynamic reactions:
r[t]: x[t]
c[t]
-> y[t]; x X, c C,
y Y; t,t'
T, t'
>= t
[tt']: c[t] x[tt'] ->
c[t']; c C, x
X, y Y; t,t' T, t' >= t
Composition of the complex system (S0) by collection(connection) of simpler subsystems (S1, S2, S3, S4, S5) is shown on the figure below.
As result, the theory tends to solve the next problems:
AED Theory
AED theory has been developed by S. Novikava at 90th (see, for details, S. Novikava, N. Karcanias, S.Gancharova, P. Buka Mathematical construction in AED theory, Proceedings of IFAC/IFORS/IMACS Symposium on Large Scale Systems, LSS98, Paatrac, Grees, 1998) and has two realizations - S^l and A^l. For our investigations of physical systems and construction of mathematical image of the defined processes we use S^l realization. Below the short introduction to the mathematics is presented, where the systems are denoted by S and l is the index of levelfor the systems.
__________________
is described by following symbol
construction:
- aggregated dynamical realization of the units and acts,
- construction, 
-
co-ordinator, 
- index of level,
.
, 
,
and
are connected by 
and
contain the dynamical realizations and constructions of unit (object) 
, its environment 
(other units of its level), acts
(processes) 
of 
in
and acts 
of
with 
:
, 
;
that is level 
is
discovered in the interlevel connections:
.
contains the dynamical systems
- connections of 
with
other units and acts, and the construction of 
connects
the details of 
(their states 
, inputs 
and outputs 
):
.
 |
 |
The network of
connections of 
,
, 
is
described by Table 1. As can be seen, 
has details, which have
increasing uncertainty in the signs of 
;
sets 
and gets
- outputs of level to the
lower 
and
higher 
levels and inputs from levels 
and 
( 
,
); states 
are own inputs and outputs of level 
: 
Thanks to the
connections in 
any detail of 
is restored by its other details with becoming
uncertainty:
Table 1
 |
States |
Inputs |
Outputs |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
  |
 |
 |
  |
 |
The co-ordinator is described by the following way:
,
that is 
has own
aggregated dynamical realization 
and the
construction 
; the availability of 
(the
connection with higher levels) allows to account and to change 
by its own activity.
Let 
, 
, 
,
, 
,
etc., 
.
Then 
;
is the contraction of field 
on the
and
 |
 |
 |
|
... |
The fields 
are
strata of 
and 
is
outlook in the level space. The knowledge uncertainty of 
is
increased with the distance from 
. Every
level 
of uncertainty on every stratum 
has its own coordinating strategy. The strategies of 
(processes 
) connect the changes of constructions
and 
with the
using of 
. The act of key unit 
creation
in 
is uniting process, the act of 
creation when 
is the highest level is the multiplying process
with the initial unit
.
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Scheme (defintion) of opto-electronic processor in S^l symbols
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References
prg1. Novikava S., Miatliuk K.,Ananich G.,Mazanik L.,Gancharova S., Galavenchik I., Suponitsky W. The Theoretical Model and the Application of Aed-processor. The International Symposium on Neural Networks and Neural Computing, NEURONET'90, Prague, pp. 259-261.(Reprinted by World Scientific Publishing Co., under the title "Theoretical Aspects of Neurocomputing", Vol.1, Singapore, 1991).
net1. S.Novikava, K.Mialtiuk, S.Gancharova, W.Kaliada, A.Ivanov, S.Kritsky, A.Demyanenka, A.Zhybul. Aed Theory and its Realizations by Hierarchical Knowledge Networks. Preprints of the IFAC Conference on Supplementary Ways for Improving International Stability - SWIIS'95, Vienna, Austria, 1995, p.99-106.
net2. S.Novikava, K.Miatliuk, S.Gancharova, A.Ivanow, A.Zhybul, A.Danichaw, P.Buka, V.Siageichick. Aed Theory and Hierarchical Knowledge Networks. Proceedings of the Annual Conference of ICIMS-NOE (E.P.9251) on ASI'96 in Life Cycle Approaches to Production Systems: Management, Control, Supervision. Toulouse, France, 1996, p.377-386.
net3. S.Novikava, S.Gancharova, A.Burawkin, S.Daronin, P.Buka, A.Ivanow, A.Danichaw, A.Demianenka, M.Maroz, A.Michalevich, A.Zagaradnuk. Mathematical Defining of Sway Networks of State and States Unions. Preprints of Eleventh International Conference on Mathematical and Computer Modeling and Scientific Computing. Washington, DC, USA. 1997. p.150.
dem1. W.Nowik, V.Barkalin, W.Zianevich, S.Gancharova, P.Buka, K.Miatliuk, A.Astrowskaya, A.Bahutski, Demographic Units in Hierarchical Coordinates: Construction, Activity and Goals, Preprints of the IFAC Conference on Supplementary Ways for Increasing of International Stability - SWIIS'95, SWIIS'98, Sinaia, Romania, 1998.
econ1. P.Groumpos, S. Novikava, S. Gancharova, A.Zhybul, V. Siarheichyk, K.Miatliuk,. Design&Creating of New Engineering Units in Reconstruction Regions. Proceedings of the Annual Conference of ICIMS-NOE (E.P.9251) on ASI'97 in Life Cycle Approaches
bio1. Novikava S.I.,Miatliuk K.N., Gancharova S.A.,Ananich G.A., Novik V.A., Koleda V.V., Kavalyov V.A., Boika T.N. Aed Technology for Ecological, Social and Engineering Systems Coordination. Proceedings of Eighth International Symposium on Modular Information Computer Systems and Networks, ICS-NET'91, Dubna, pp. 145-152.
bio2. V.Kaliada, V.Rabeka, Iu.Kaminski, E.Kaminskaia. Oncological Defense Maintenance in Dynamic Multilevel Environment. IXth International Conference on Mathematical and Computer Modelling, ICMCM'93, Berkeley, California, USA, p. 128.
phys1. Bagnich S. and Konash A. Computation Study of the Percolation Phenomena in Inhomogeneous Two- and Three- Dimensional systems. J. Phys. A: Math. Gen.36 (2003) 1-13,
phys2 Bagnich S. and Konash A Percolation processes in heterogeneous two-dimensional space Phys. Solid State 42 (2002) 1775.
phys3. Pavel Buka, The Investigation of Photoluminescence in Low-Dimensional Materials with Aed-Processor. The Eleventh International Conference on Superlattices, Microstructures and Microdevices. Book of Abstracts. Hurghada (Red Sea), Egipt, July 27- August 1, 1998
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Simulations
The final result in system theory is realization of the mathematical definition by means of computers or by engineering tools. In our investigations we construct program simulations in object-oriented language Delphi. The language is not so flexible as C++ but easy in programing and allows to prepare actual simulations with common interface under Windows platform. The achievements ware obtained show ability of the simulations to success scientific and engineering demand(requests), have practically convenient features of interactive control and are useful for learning and teaching.
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Simulation of Fabri-Perot Interferometer.
Fabri-Perot Interferometer consists of two mirrors on substrate and nonlinear medium between the mirrors. The physical features of the nonlinear medium depend on intensity of light and temperature between the mirrors, that is why for the same input signal the number of output signals can be obtained. Such behavior allows use the device as logical element for optical computing and input signal amplification. In general with this simulation is demonstrated :
There are two snapshots of the program are presented below.
Snapshots of the simulation
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Percolation in heterogeneous systems
As known, percolation processes and related phenomena play the important role for research of wide range of physical processes. There are rock fracture, fragmentation [1], gelation [2, 3], conduction in random resisting grates [4], strongly inhomogeneous media [5], propagation of forest fires [6, 7] and epidemics [8, 9], electronic properties of doped semiconductors [10], electronic excitation in inhomogeneous systems [11, 12] etc. among them. We report about the program (simulation) with ability for investigation of percolation phenomena in system with any internal heterogeneous structure(proved for random structure), automatic process of decision and ability of interactive control by user with high speed processing and advanced graphical visualization of occurred processes. As previous simulation the program allows to observe in real time the growth of the clusters and research all features for the percolation process.
The new scientific results were obtained by the simulation and were reported at ACAT02 ( for details, see slides http://acat02.sinp.msu.ru/presentations/konash/Slides_ACAT2002.doc) and recently published [ Bagnich S. and Konash A. Computation Study of the Percolation Phenomena in inhomogeneous Two- and Three- Dimensional systems. J. Phys. A: Math. Gen.36 (2003) 1-13, Bagnich S. and Konash A Percolation processes in heterogeneous two-dimensional space Phys. Solid State 42 (2002) 1775].
There are three examples of the investigated objects for matrices with linear size L = 200 in 2D case below. The color of the clusters depends on the cluster's size (the color scale is presented on the snapshot of the program). It is possible to download the program (see end of the report).
|
Homogeneous 2D system L = 200 C_obs = 0.0 |
inhomoeneous. L = 200 l_obs = 1 C_obs = .38 |
inhomoeneous. L = 200 l_obs = 20 C_obs = .52 |
|
matrix
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matrix
|
matrix
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c = .45
|
c = .45
|
c = .45
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percolation threshold c = .598
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percolation threshold c = .933
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percolation threshold c = .71
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picture of the percolation cluster
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picture of the percolation cluster
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picture of the percolation cluster
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References:
1. Broadbent S.R. and Hammersley J.M 1986 Fragmentation Form and Flow in Fractured Media ed R. Engelman and Z. Jarger (Bristol: IPS)
2 Broadbent S.R. and Hammersley J.M 1984 Proc. Int. Topical Conf. Kinetics of Aggregation and Gelation (Athenth, GA) ed F. Family and D.P Landau (Amsterdam: North-Holland)
3 De gennes P. G. 1979 Scalling concepts in Polymer Physics (Ithaca, NY: Cornell University Press)
4. De Gennes P. G. 1983 Percolation Structures Processes (Ann. Inst. Phys. Soc. vol. 5) ed G. Deutsher, R. Zallen and J. Adler (Bristol: Hilger)
5. Shklovsiy B I and Efros A L. 1975 Sov. Phys. Usp. 18 845
6. Stauffer D. 1985 Introduction to Percolation Theory (London: Taylor & Francis)
7. Mackay G and Jan N 1984 J. Phys. A: Math. Gen. 17 L757
8. Grassberger P. 1983 Math. BioSci. 63 157
9. Bunde A., Herrmann H, Margolina A. and Stanley H. E.1985 Phys. Rev. Letters 55 653
10. Sklovskiy B. I. and Efros A. L 1984 Electronic Properties of Doped Semiconductors (NewYork: Springer)
11. Hoshen J and Kopelman R. 1976 J. Chem. Phys.65 2817
12. Kopelman R. 1983 Spectroscopy and Excitation Dynamics in Condensed Molecular Systems ed V.M. Agranovich and R.M. Hochstrasser (Amsterdam: North-Holland)
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Conclusion.
As known, new programming techniques and new mathematical theories promise results with higher precision, advanced simulations and novel applications for practical tasks. We use AED theory for mathematical definition and digital(computer) simulating and have applied one for investigation of Fabri-Perot Interferometer with chaotic behavior and the percolation phenomena in heterogeneous two- and three- dimensional systems. The theory is well established by wide range of applications and lets us obtain new results for the tasks. In general, constructed simulations with ground on the theory demonstrate the ability of the automatic control by the program and the interactive control by user at each step of imitation. This allows to use the theory for research the behavior of investigated system in whole range of the possible states(conditions) both for the system and for the environment. Also the simulations are convenient for goals of learning processes (teaching and engineering). and show fast calculations with ability of real time investigation and presentation of the current data by numbers and colors.
As result, we intend to conclude about preliminary success in application of the S^l realization of the AED theory for important tasks for physical science, learning and engineering. Nevertheless, much wider number of tasks should be investigated with the theory to find out frames of applicability of the methods for tasks in physical researches, and to justify advantages and disadvantages in compare with other techniques. That is the reason of our tend for international and interdisciplinary collaboration.
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This work was done by international team:
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Konas A*,+ |
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Bagnich S.,* |
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Buka P.**, + |
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Kaupuzs J., ++ |
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Mahnke R. +++ |
From the Institutes:
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+++ Universitaet Rostock |
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Download (the simulation of percolation processes for two-dimensional heterogeneous lattices) :
We have designed program for aims of learning and teaching. The program reflects condition of percolation by colors in real time and generates the number of snapshot with step 0.01 in concentration of active(colored) sites. To test the program under windows platform download_simulation. Up to now the interface of the program is only in Cyrillic characters. To start program with default parameters push left buttons one by one and to refresh parameters press right button at the bottom.
Please, share yours remark about the the simulation by e-mail.
- concentration of active sites at the moment of percolation (min..max)
- number of statistics for matrices and for clusters
- also program ask you (is the matrix Ok for investigation)
—
—
(where 
—
active memory ( optically bistable thin-film interferometer ( Fabry -Perot
interferometer) );
—
construction of active memory;
— converter (function generator) of
input signal; 
— its internal construction 
—
—
its construction
—
source of energy of signals (laser), 
—
construction of energy source;
— 
—
construction of receiver
,
, 
,
— 
