Cellular Automaton-Like Model of Arising Physical-Like Properties
pp a p e r d r a f t Cellular Automaton-Like Model of Arising Physical-Like Properties
M. Pietrow ∗ Institute of Physics, M. Curie-Sk(cid:32)lodowska University, ul. Pl. M. Curie-Sk(cid:32)lodowskiej 1,20-031 Lublin, PolandOctober 26, 2018
Abstract
A simple relation of the order of n abstractobjects generates an n − Introduction:
Cellular automata (CA) are used to de-scribe the behaviour of systems with a wide range of com-plexity from physics to biology [1]. Mainly, the descriptionis functional, but not a structural one (CA rules allow de-scription of some aspects of a system at a high structurallevel without references to the rules from the deeper levelof subsystems).The aim of this presentation is opposite to some extent.One does not require here a compatibility of an introducedmodel with any special real system. Instead, it was as-sumed that there exists a set of some abstract objects anda general quantity, an order, characterising each memberof this set. Based on this, a matrix was constructed whichkeeps relations between these objects. The properties ofthis matrix were examined and some arising similaritiesto physical properties were brought into focus. Further-more, the presented model keeps the compatibility withCA ideas to some extent and tries to adhere to a descrip-tion of a set of physical objects from the structural pointof view.Although most of the ideas here are postulated but notderived from something deeper, it would be promising toconsider the Author’s idea of a simple relation betweensome elementary objects and an introductory model ofhow physical properties arise from it.
Relation matrix ( mrel ): A fundamental feature of asystem of basic objects (thought here as abstract enti-ties, not physical ones; called here elementary objects ) isthe relation between them. One of the simplest relations ∗ e-mail: [email protected] seems to be an ’order’ of these objects. For example, forthree objects there are 3! of their possible arrangements.Now, consider a set of n identical elementary objects. Letus define an n × n matrix mrel i,j ( relation matrix ), whichdescribes the distance (in the meaning of this order) of the i -th relative to the j -th object. For example, mrel , =-1because the 1 st object is one step before the 2 nd one (itproceeds object 2). The mrel for three particles of theorder { , , } is − −
21 0 −
12 1 0 , (1)whereas for the { , , } order we have − −
12 0 11 − . (2) The Eigensystem of the mrel : M rel s have interest-ing properties. Consider a 5 × mrel for an arrangement { , , , , } as an example. Its eigenvalues are { λ = 5 i √ , λ ∗ , , , } , (3)whereas the corresponding eigenvectors are v = { ( − i √ , i √ , (1 + i √ , (4 + i √ , } ,v = v ∗ , (4) v = { , − , , , } , v = { , − , , , } , v = { , − , , , } , where ∗ denotes a complex conjugation.The following is an interesting general rule for mrel (nomatter what its n dimension is). Its n − alwayshave the non-zero values in 3 dimensions only . These vec-tors span the n − physV .The physV seems to be a promising representation of n − n − a r X i v : . [ n li n . C G ] A ug a p e r d r a f t a three-dimensional sub-spaces of a common space. Letus call these vectors with the zero eigenvalues the physicalvectors . Other properties of mrel : Some other interestingproperties of mrel are listed below.1. Permutation of related elementary objects does notchange the eigenvalues of mrel , whereas the eigenvec-tors do change.2. The physical vectors are independent of a in the caseof generalisation of the relation definition in mrel as · · · , − → a − , → a , → a + 1 , → a + 2 , etc.. (5)3. Any mrel ’s sub-matrix of dimension n (cid:48) has n (cid:48) − × mrel s (for the arrangements { , } and { , } )have no physical vectors. Normalised eigenvectors ofthese matrices resemble spin vectors for a spin- par-ticle {− i √ , √ } , { i √ , √ } , (6)whereas these mrel s are proportional to one ofthe Pauli matrices, σ : mrel ( { , } ) = iσ and mrel ( { , } ) = − iσ .For three elementary objects in the relation, there isone physical vector as an eigenvector . In this case, all2 × mrel s gen-erated from permutations of the order { , , } give theeigenvalues from the set {− i, − i, , } and these sub-matrices are a simple combination of the Pauli matri-ces. For dim ( mrel ) > × mrel s seem to be promis-ing operators for spin description.The time evolution of such a system is postulated be-low.5. M rel s are antihermitian (antisymmetric). Some setsof mrel s form a linearly independent set (for example,a subset of three mrel s generated by permutations ofelementary objects). According to the general theory[2], they form a Lie algebra of generators related tosome unitary matrices. This suggests a possibility ofdescription of quantum-like evolution [3] by these ma-trices. When rearrangement of the elementary objects takes place thecomponents of this vector 1 √ { , − , } (7)interchange.
6. Another scheme of a time evolution (called a secondkind ) of a system described by mrel could be sug-gested by the case of 2-dim mrel s which have beenlinked with a spin. Each of the Pauli matrices can bederived from one of them by some elementary opera-tions known from linear algebra (two lines switching,a line multiplication by a number). Thus, the evolutionof mrel in a general sense could be identified with el-ementary operations. In general, swapping lines is notequivalent to permutations of the elementary objects.In the simplest case, one may consider a mrel at eachstep where some two lines could be randomly swapped.However, a more complicated algorithm could be usedas a current mrel generator. A new mrel could beconsidered as a product of up-to-now mrel s that couldchange additionally at some steps by swaps of lines.On the other hand, continuing the idea of relations,for a system of three elementary objects as an exam-ple, their states A , B , C are influenced by each statefrom all these objects in the set. Thus, it could bewritten A = m , A + m , B + m , C,B = m , A + m , B + m , C, (8) C = m , A + m , B + m , C. The matrix m ij here could be identified with mrel .More generally, for a set of consecutive steps t , eq. (8)gives ABC = ( mrel ) t × ABC . (9)The equation above is, in fact, a requirement to find avector [ A, B, C ] T which is unchanged by a projectionby the mrel t operator. Vectors which are the solutionof (9) have interesting properties.As an example, consider the mrel for three elemen-tary objects. Calculate B and C as a function of time(because the rank of any mrel is 2, B and C are A –dependent here). These functions are shown in fig. 1.Additionally, the physical vectors do not change withsteps, whereas the rest of the eigenvectors set oscillatewithin some set of values.The non-zero eigenvalues of mrel t rise logarithmicallywith steps when the system evolves without swaps inbetween inside the matrix–fig. 2. However, when theswaps of lines take place, the non-zero eigenvalue risesmuch faster that logarithmically.One may consider the evolution complicated one stepmore. If one makes some swaps of lines and then solveseq. (9), the result for B and C will approximate asymp-totically some value – fig. 3.On the other hand, if one makes a swap of matrix lines rows or columns, optionally a p e r d r a f t Figure 1: Solution of eq. (9) for 3 × mrel for some t .Figure 2: Logarithm of an absolute of a non-zero eigen-value for 3 × B ( t ), C ( t ) for a 3 × mrel in whichsome swaps of columns and rows precede the evolution.between some steps of evolution and solves eq. (9) af-ter each step then one observes switches to some othervalue for some time (fig. 4). The interesting feature ofthis evolution is that the spectrum of values is discrete(they form a multiplet). Generally, the changes of val-ues do not coincide with the moment of the swap ofthe matrix lines.A discrete spectrum of B ( t ) and C ( t ) is also obtainedwhen one calculates it for any sub-matrix of a larger mrel under evolution.The evolution of a second kind erases the anti-symmetricity of a mrel and thus it is a considerablydifferent scheme. However, the antisymmetricity re-turns after some swaps.7. If the evolution consists in swapping lines, the numberof n − mrel t , the number ofthe physical vectors remains constant.It is interesting to consider physical vectors relating to mrel s representing all permutations of n elementaryobjects. These vectors form sets with non-zero valuesat different three of n positions: ζ : { [ x, y, z, , ... ] } , ζ : { [ x, y, , z, , ... ] } , ζ : { [ x, y, , , z, , ... ] } , etc..Each ζ i points the same network of points located at aplane x + y + z = 0 (blue points in fig. 5; any length ofvectors are possible). The number of points increaseswith n (all points generated by smaller set of n ele-mentary objects are generated by a larger one, too).Furthermore, any swaps of mrel ’s lines produce physi-cal vectors which are a subset of the network given bypermutations of elementary objects (e.g.: red points infig. 5). Moreover, a multiplication of mrel mentionedin the eq. (9) does not give an additional points butthose generated by permutations. Permutations and3 a p e r d r a f t Figure 4: Values of B ( t ), C ( t ) for a four-dimensional mrel where eleven random swaps of columns and rows weremade during the evolution.Figure 5: A part of the network of points (blue) generatedby any ζ i set (see description in the text) for permuta-tions of n =7 elementary objects. This network could beincreased by additional points obtained for larger n . Thered points are a set of coordinates for physical vectorsfor a mrel generated by 1000 random swaps of lines ofan initial mrel for randomly given permutation of n =15elementary objects. Figure 6: The blue points show coordinates of normalisedphysical vectors during the evolution consisting in mrel lines swapping and powering it randomly. The red pointsshow only states which can be occupied when the n = 3problem is considered. These points are vertices of anequilateral triangle. mrel powering (no matter what is done first) give thepoints from the regular structure whose an initial partwas depicted in fig. 5.In fact, any length of the eigenvectors of mrel s arepossible. If one limits to normalized vectors only theset of possible points form a part of a circle centredat (0 , ,
0) with radius 1 and normal vector pointing inthe direction of [1 , ,
1] (blue points in fig. 6).Let us follow the position of the points described by ζ at each step of the evolution consisting on randomswapping lines or powering the matrix. If n > n = 3 (there is only one physical vector) onlyjumps between the points given in red in fig. 6 are pos-sible.To generalise, the physical vectors point a net of placesin a three-dimensional sub-space for each of n − ζ i has its own (’in-ternal’) net of possible states. Although each physicalvector is represented in its own subspace, from thismodel, the coordinates of each possible point obey theequation x + y + z (cid:48) = 0, where x - and y -coordinates canbe regarded as common ones whereas the z (cid:48) -coordinateis set individually for each vector.The further Author’s work will be devoted to check if4 a p e r d r a f t Figure 7: Absolute values of one of non-zero eigenvaluesfor a ten-dimensional mrel during 500 steps of evolutionconsisting in random swaps of columns and rows (but notgoverned by the (9) rule). The time-line resembles one-dimensional random walk.the jumps through the network (for the one particlecase, in particular) could describe a space-time motionof elementary objects in some way.The evolution described in point 6 above resembles rulesobeyed by the CA [1] in general. Its algorithm is an ap-plication of a simple rule (9) at each step (however, whenswaps of matrix lines take place, randomness of choice asa generalisation of CA rules is added). The equivalenceof cells in CA would be matrix elements (or lines) here.Each matrix element changes by application of a rule thatrequires other elements (but not neighbouring ones only).Additionally, in both cases, the mrel evolution and theCA, some values can form a complex pattern of changesin ’time’. Such behaviour is maintained by non-zero eigen-values of mrel s (fig. 7).
Conclusions:
This paper presents a collection of state-ments and hypotheses concerning a relation between ba-sic physical properties, e.g. a number of dimensions ofspace containing physical objects or an evolution process,and relation matrix properties for which some character-istics have been investigated. From a point of view ofthe model presented above, the mrel resembles operatorsin quantum mechanics. Possibly, a permutation groupwould help to find a link. An interesting consequencewould be that the spin-like vector may originate from two-dimensional mrel eigensystems which differ in dimension-ality only from three dimensional physical vectors origi-nating from larger mrel s.The statements do not form a consistent view of linkedconcepts but the Author’s hope is that the interestingproperties of mrel s do reveal a structure resembling CA with quantum-like properties and could be developed fora useful description of physical many-body systems.
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