Symmetry Analysis of Discrete Dynamical Systems

Transcription

1 Symmetry Analysis of Discrete Dynamical Systems Talk at CADE 2007, February 20-23, Turku, Finland Vladimir Kornyak Laboratory of Information Technologies Joint Institute for Nuclear Research February 23, 2007 V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

2 Outline 1 Introduction 2 Discrete Relations 3 Lattices and Their Symmetries 4 Influence of Lattice Symmetry On Dynamics 5 Symmetric Local Rules 6 Computer Program And Its Functionality 7 Reversibility 8 Lattice Models On Symmetric Lattices 9 Summary V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

3 Motivations Symmetry analysis of finite discrete systems can be complete in contrast to continuous systems where only negligible small part of all thinkable symmetries is considered: point and contact Lie, Bäcklund and Lie Bäcklund, sporadic instances of non-local symmetries etc. Many hints from quantum mechanics and quantum gravity that discreteness is more suitable for physics at small distances than continuity which arises only as a logical limit in considering large collections of discrete structures V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

4 Preliminaries Discrete Relations on Abstract Simplicial Complexes Our recent observation. Any collection of discrete points valued by states from finite set posseses some kind of locality: such collection has naturally a structure of abstract simplicial complex. This construction is generalization of both system of polynomial equations cellular automaton The talk is about discrete dynamical systems on symmetric lattices More specifically, we consider cellular automata with symmetric local rules a special case of discrete relations on abstract simplicial complex V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

5 Discrete Relations Relations (basic definitions) Collection of N points : symbolically δ = {x 1,..., x N } q i states of ith point: Q i = { si 1,..., } sq i i or in standard notation Q i = {0,..., q i 1} Relation on δ is arbitrary subset R δ Q δ Q δ is symbolical notation for Cartesian product Q 1 Q N Extension of relation R δ to larger set of points τ δ: R τ = R δ Q τ\δ Consequence of relation R δ is arbitrary subset S δ R δ Proper consequence of R δ is relation P α on proper subset α δ such that P α Q δ\α R δ (extention of projection is usual consequence) V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

6 Decomposition of Relation Canonical Decomposition R δ can be represented as combination of its proper consequences P δ 1,..., P δm and principal factor PR δ R δ = PR ( m ) δ P δ i Q δ\δ i i=1 Principal Factor PR δ = R ( ) m δ Q δ \ P δ i Q δ\δ i i=1 Canonical decomposition imposes structure of Abstract Simplicial Complex on R δ V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

7 Compatibility of System of Relations Compatibility Condition (= Base Relation) of any set of relations R δ 1,..., R δn is intersection of their extentions to common domain δ = n δ i i=1 n B δ = R δ i Q δ\δ i i=1 V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

8 Decomposition And Compatibility Analysis Algorithms are very simple and efficient main loops include only bitwise operations NOT OR XOR AND operators in C & Example Compatibility analysis of relations defining rules for Conway s Game of Life takes < 1 sec (in fact, zero within timer uncertainty) Maple Gröbner basis package applied to these relations expressed by polynomials over F 2 produces essentially the same result for hours depending on ordering V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

9 When Polynomials Work Any system of discrete relations R δ 1,..., R δn can be represented as a set of polynomial equations if 1 The sets of states at all points in union δ = {x 1,..., x N } of domains δ 1,..., δ n are the same: Q 1 = = Q N = Q 2 Number of elements in Q is a power of a prime: q = p k Then relations correspond to sets of zeros for some polynomials from ring F q [x 1,..., x N ] and Gröbner basis technique can be applied to compatibility analysis of the system of relations V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

10 When System Of Relations Is Cellular Automaton 1 collection of points x 1,..., x N can be represented as union of connected congruent simplices with the same relation (local rule) on each simplex 2 the local rule is functional relation, i.e., value of one of the points in simplex is determined unambiguously by values of other points V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

11 Lattices and Their Symmetries Space of cellular automaton is lattice L = k-valent (k-regular) graph GL Symmetry of lattice L is graph automorphism group Aut (G L ) In applications it is assumed often that lattice is embedded in some continuous space in this case notion of dimension of lattice makes sense Reasons to study trivalent two-dimensional lattices carbon molecules of special interest known as fullerenes, nanotubes and graphenes are 3-valent 2-dimensional structures this is the simplest case except bivalent Wolfram s elementary automata whose geometry is rather primitive V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

12 Examples of Symmetric Trivalent Lattices V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

13 Remark. Cube Is Smallest Graphene Distinction between lattice and its graph The same graph forms 4-gonal (6 tetragons) lattice in sphere S 2 and 6-gonal (4 hexagons) lattice in torus T 2 V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

14 Functions On Lattices And Their Orbits Q = {0,..., q 1} is set of states of lattice vertices Q L is space of Q-valued functions on L Aut (G L ) acts non-transitively on Q L splitting this space into orbits of different sizes Burnside s lemma counts total number of orbits N g cycles N orbits = 1 Aut(G L ) g Aut(G L ) is number of cycles in group element g q Ng cycles, V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

15 Some Numerical Information On Symmetric 3-valent Lattices, Their Groups And Orbits Lattice V (G L ) Aut(G L ) N multistates = q V (G L) N orbits Tetrahedron Cube Dodecahedron Buckyball Graphene 3 4 Torus Graphene 3 4 Klein bottle V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

16 Influence of Lattice Symmetry On Dynamics Second Law of Thermodynamics For Any Causal Dynamical System trajectories can be directed only from larger orbits to smaller orbits or to orbits of the same size periodic trajectories must lie within the orbits of equal size V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

17 Symmetric Local Rules k-valent neighborhood is a k-star graph x3 x4 x1 x2 this is trivalent neighborhood Local rule x k+1 = f (x 1,..., x k, x k+1 ) k-symmetry is symmetry over k outer points x 1,..., x k Number of k-symmetric rules N q S k = q (k+q 1 q 1 )q V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

18 k-symmetric Rules And Generalized Game Of Life Generalized k-valent Life rule is a binary rule described by two arbitrary subsets B, S {0, 1,..., k} containing conditions for the x k x k transitions of the forms 0 1 and 1 1 Proposition For any k the set of k-symmetric binary rules coincides with the set of k-valent Life rules Hence there are 3 ways to express binary local rule x k+1 = f (x 1,..., x k, x k+1 ) by bit string (Example: Rule 86 = ) by Birth/Survival lists (B123/S0) by polynomial over F 2 (x 4 = x 4 + x 1 + x 2 + x 3 + x 1 x 2 + x 1 x 3 + x 2 x 3 + x 1 x 2 x 3 ) V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

19 Equivalence With Respect To Permutations Of States Renaming of q states of automaton leads essentially to the same rule Burnside s lemma counts number of non-equivalent rules in k-valent binary case this number is N k = 2 2k k For trivalent rules N 3 = 136 V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

20 Computer Program And Its Functionality Program is written in C language Input Graph of lattice G L = {N 1,..., N n } Set of local rules R = {r 1,..., r m } Control parameters Program constructs automorphism group Aut (G L ) phase portraits of automata modulo Aut(G L ) for all rules from R Manipulating control parameters we can select automata with specified properties, e.g., reversibility, Hamiltonian conservation etc. search automata producing specified structures, e.g., limit cycles, isolated cycles, Gardens of Eden etc. V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

21 Small Lattices: Phase Portraits For Rule 86 Tetrahedron Attractor (Sink) Isolated 2-cycles Hexahedron (Cube) Sink Attractors Limit 2-cycles Isolated cycles Limit 4-cycles 24 V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

22 Large Lattices: Concise Outputs Instead Of Phase Portraits Dodecahedron Graphene 3 4 Torus Klein Bottle Period N isolated N limit N isolated N limit V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

23 Quantum Gravity Difficulties: Irreversibility of Gravity: information loss (=dissipation) at horizon of black hole Reversibility and Unitarity of standard Quantum Mechanics G. t Hooft s approach to reconcile Gravity with Quantum Mechanics 1 Discrete degrees of freedom at Planck distance scales 2 States of these degrees of freedom form primordial basis ( t Hooft s term) of (nonunitary) Hilbert space 3 Equivalence classes of states: two states are equivalent if they evolve into the same state after some lapse of time 4 Equivalence classes form basis of unitary Hilbert space and now evolution is described by time-reversible Schrödinger equation In our terminology this corresponds to transition to limit cycles: after short time of evolution limit cycles become physically indistinguishable from reversible isolated cycles V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

24 Search For Reversibility Reversibility is rare property tending to disappear with growth of complexity of lattice Two rules trivially reversible on all lattices 85 or x 4 = x or x 4 = x 4 Tetrahedron, 6 additional reversible rules 43 or x 4 =x 4(x 2 x 3 +x 1 x 3 +x 1 x 2 +x 3 +x 2 +x 1 )+x 1 x 2 x 3 +x 2 x 3 +x 1 x 3 +x 1 x 2 +x 3 +x 2 +x or x 4 =x 3 + x 2 + x or x 4 =x 4(x 2 x 3 + x 1 x 3 + x 1 x 2 + x 3 + x 2 + x 1 + 1) + x 1 x 2 x 3 + x 2 x 3 + x 1 x 3 + x 1 x or x 4 =x 4(x 2 x 3 + x 1 x 3 + x 1 x 2 + x 3 + x 2 + x 1 + 1) + x 1 x 2 x 3 + x 2 x 3 + x 1 x 3 + x 1 x or x 4 =x 3 + x 2 + x or x 4 =x 4(x 2 x 3 +x 1 x 3 +x 1 x 2 +x 3 +x 2 +x 1 )+x 1 x 2 x 3 +x 2 x 3 +x 1 x 3 +x 1 x 2 +x 3 +x 2 +x 1 Cube, 2 additional reversible rules 51 or x 4 =x 3 + x 2 + x or x 4 =x 3 + x 2 + x 1 Dodecahedron and Graphene 3 4, no additional reversible rules V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

25 Lattice Models On Symmetric Lattices Distribution function via summation of group orbit sizes in statistical physics Ising model: s i Q = { 1, 1} H = s i s k e H kt i,k Dodecahedron Graphene 3 4 o Torus + Klein bottle V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26

26 Main Results C program for symmetry analysis of finite discrete dynamical systems has been created. Trajectories of dynamical systems with non-trivial symmetry go always in the direction of non-increasing sizes of group orbits. Cyclic traectories lie within orbits of the same size. Reversibility is rare property. Taking into account lattice symmetries facilitates work with physical lattice models. V. V. Kornyak (LIT, JINR) Symmetries of Discrete Systems February 23, / 26