Sand Piles. From Physics to Cellular Automata.

Transcription

1 FRAC 2009, December, Nice p. 1/53 UNIVERSITÀ DEGLI STUDI DI MILANO BICOCCA DIP. DI INFORMATICA, SISTEMISTICA E COMUNICAZIONE (DISCO) Sand Piles From Physics to Cellular Automata G. CATTANEO, M. COMITO, AND D. BIANUCCI cattang@disco.unimib.it FRAC 2009, December 2, Nice, France

2 FRAC 2009, December, Nice p. 2/53 Talk outline Sand Piles Introduction

3 FRAC 2009, December, Nice p. 2/53 Talk outline Sand Piles Introduction The Sand Pile Lagrangian

4 FRAC 2009, December, Nice p. 2/53 Talk outline Sand Piles Introduction The Sand Pile Lagrangian The Sand Pile Entropy

5 FRAC 2009, December, Nice p. 2/53 Talk outline Sand Piles Introduction The Sand Pile Lagrangian The Sand Pile Entropy The Sand Pile Cellular Automata

6 FRAC 2009, December, Nice p. 3/53 Part I Introduction to Sand Piles Unidirectional Wind Flow from Left-to-Right

7 FRAC 2009, December, Nice p. 4/53 Sand Pile model Formally, a Sand Pile Model (SPM) is based on the basic notion of integer partition. Precisely, an integer partition of the integer N, with N N fixed positive integer number, is mathematically described by a N length finite sequence of nonnegative integer numbers n =(n 1 n 2... n N ) with n i N for every i = 1,2,..., N under the condition N n i = N i=1 and the constraint n 1 0 and n N = 0,1 Trivially, the condition N i=1 n i = N implies i, n i N

8 FRAC 2009, December, Nice p. 5/53 Sand Pile Configurations We consider sand piles according to the two conditions: the total mass N condition: N i=1 n i = N (integer partitions of N) the 0 finiteness condition: k {1,...,N} s.t. n i > 0 for every i k, and n j = 0 for every j > k: n = (n 1,...,n i,...,n k, 0,...,0) The collection of all such sand piles of total mass N, also called N length configurations, will be denoted by Ω(N)

9 FRAC 2009, December, Nice p. 6/53 Horizontal rule The unidirectional sand pile model (SPM) we treat is based on the dynamical evolution formalized by the movement of a sand grain with respect to the following rule: Horizontal (H) Rule. If n i n i+1 2, i.e., a jump is located at cell i, then (n 1,...,n i, n i+1,..., n N ) (n 1,...,n i 1, n i+1 +1,..., n N ) The typical left-to-right granule movement of a sand pile is drawn as The total number N of granules is conserved: Principle of total mass conservation

10 FRAC 2009, December, Nice p. 7/53 Equilibrium Configuration A configuration n = (n 1 n 2...n N ) is said to be of equilibrium iff n i+1 n i 1 for every i = 1,...,N 1, In this case this configuration is a fixed point with respect to the application of the horizontal rule H: (n 1,...,n i,n i+1,...,n N ) (n 1,...,n i,n i+1,...,n N )

11 FRAC 2009, December, Nice p. 8/53 Possible or Admissible Paths We can define a possible path (also admissible path) generated by the H evolution rule starting from a given initial configuration n 0 as a Ω(N) valued finite sequence γ n0 ( n(0), n(1), n(2),..., n(t f ) ) satisfying the following conditions: (P1) The initial configuration at time t = 0 is n(0) = n 0 ; (P2) the configuration n(t + 1) γ n0 at time t + 1 is obtained from the configuration n(t) γ n0 at time t by the application of the H evolution rule, i.e., n(t + 1) is a successor of n(t) (formally, n(t) n(t + 1)); (P3) the final configuration n(t f ) γ n0 is an equilibrium configuration.

12 Possible Motions: An Example Thus, as time elapses, the configuration may change and the motion of the sand system can be described by discrete time orbits in configuration space Ω(N), each of which must be a possible path of the system. Example. Hasse diagram of the two possible dynamical evolutions with initial configuration (5, 1), both reaching the equilibrium (3, 2, 1): (5, 1) (4, 2) (3, 3) (4, 1, 1) (3, 2, 1) FRAC 2009, December, Nice p. 9/53

13 FRAC 2009, December, Nice p. 10/53 The Lattice of All Possible Motions Let us consider the following algorithmic construction (with graphical representation) Level 0: Let us choose an initial configuration n 0 at time t = 0 Level 1: Let us solve any jump of n 0 by the horizontal rule H from the left to the right, connecting by an arrow the configuration n 0 with any of such new configuration at time t = 1: n (1), n (1), Level t f : This procedure leads (theorem) to a unique final equilibrium configuration n(t f ) at time t f N The so obtained diagram is the Hasse diagram of a (finite) bounded (by n 0 and n(t f )) lattice!!!

14 FRAC 2009, December, Nice p. 11/53 The Lattice of All Possible Motions (2) Figure 1: Example of the 0 and 1 levels construction starting from the initial configuration (6, 4, 2, 0), solving jumps from left to right.

15 FRAC 2009, December, Nice p. 12/53 The Lattice of All Possible Motions (3) Figure 2: Example of the first three levels construction starting from the initial configuration (6,4,2,0). Note that at the second level the same configuration is obtained by two different configurations of the first level.

16 FRAC 2009, December, Nice p. 13/53 Black Hole n (t 1) n (t 1)... n k 1 (t 1) n(t) n k (t 1) n (t + 1) n (t + 1)... n h 1 (t + 1) n h (t + 1) Black Hole A black hole configuration n(t) absorbs any (t 1)-configurations, and generates at least one (t + 1)-configuration

17 FRAC 2009, December, Nice p. 14/53 Una analogia Una interessante analogia tra il diagramma spazio-temporale (discreto) dei sand piles a quello dei Sistemi Dinamici a spazio-tempo continuo Questo mi ha portato ad una congettura/euristica in cui la usuale teoria del Calcolo Variazionale, con i capitoli finali sull approccia Hamiltoniano alla meccanica classica, ha dato lo spunto per una applicazione al caso dei Sand Piles. Ovviamente il caso Meccanica Classica si basa su un asse temporale continuo I = R (o un suo sotto-intervallo finito) uno spazio degli stati continuo Ω = R 2f (o un suo sotto-insieme aperto)

18 FRAC 2009, December, Nice p. 15/53 Meccanica Classica Nel caso contiuo la procedura del Calcolo Variazionale si articola nei seguenti punti Si introduce una grandezza osservabile particolare L : Ω R, chiamata Lagrangiana (di cui ora non é interessante dare la formalizzazione) Si valuta tale osservabile rispetto a tutte le possibili traiettorie congiungenti 2 stati prefissati γ : [t i,t f ] Ω, con γ(t i ) = s 0 e γ(t f ) = s f preassegnati: (L γ) : [t i,t f ] Ω R L azione é l integrale, dipendente dalla traiettoria γ, S(γ) = tf t i (L γ)(t)dt

19 FRAC 2009, December, Nice p. 16/53 Principio Hamiltoniano di Minima Azione Formulazione usuale in lettratura fisica-matematica Tra tutte le traiettorie possibili congiungenti s i con s f, la traiettoria attuale é quella che minimizza l azione Formulazione matematicamente (formalmente) corretta Tra tutte le traiettorie possibilicongiungenti s i con s f, la traiettoria attuale é quella che rende l azione stazionaria Lagrangiana per campi conservativi: L = Energia Cinetica - Energia Potenziale Estensione del Principio di Azione Stazionaria al caso discreto sia nello spazio che nel tempo.

20 FRAC 2009, December, Nice p. 17/53 Part II The Sand Pile Lagrangian In the Conservative Gravitational Field

21 FRAC 2009, December, Nice p. 18/53 Energy considerations m U = mgh h g In classical mechanics a massive particle of mass m, at high h with respect to the ground level (as reference system) under the action of the uniform gravitational field with constant gravitational acceleration g is characterized by the Potential Energy U = mgh

22 FRAC 2009, December, Nice p. 19/53 Energy considerations (2) δ m U = (mgδ)n i h = n i δ g In a pile column of n i granules located at cell i, each of linear dimension δ the higher granule is at a distance h i = n i δ from the ground reference level, and so the corresponding potential energy of this granule is U i = m g n i δ

23 FRAC 2009, December, Nice p. 20/53 Energy considerations (3) The column of n i granules at cell i of a configuration n = (n 1,...,n i,...,n N ) can be considered as a vertical pile of massive particles, each of them of mass m and linear dimension δ n i δ i m

24 FRAC 2009, December, Nice p. 20/53 Energy considerations (3) The column of n i granules at cell i of a configuration n = (n 1,...,n i,...,n N ) can be considered as a vertical pile of massive particles, each of them of mass m and linear dimension δ The gravitational potential energy of the i-column is U i ( n) = m g δ ( n i 1 ) (n i k) k=0 n i δ i m

25 FRAC 2009, December, Nice p. 20/53 Energy considerations (3) The column of n i granules at cell i of a configuration n = (n 1,...,n i,...,n N ) can be considered as a vertical pile of massive particles, each of them of mass m and linear dimension δ The gravitational potential energy of the i-column is U i ( n) = m g δ ( n i 1 ) (n i k) k=0 n i δ i m Putting, as usual in physics, m g δ = 1, the total potential energy of the configuration n is then U( n) = N i=1 U i( n), i.e.,

26 FRAC 2009, December, Nice p. 20/53 Energy considerations (3) The column of n i granules at cell i of a configuration n = (n 1,...,n i,...,n N ) can be considered as a vertical pile of massive particles, each of them of mass m and linear dimension δ The gravitational potential energy of the i-column is U i ( n) = m g δ ( n i 1 ) (n i k) k=0 n i δ i m Putting, as usual in physics, m g δ = 1, the total potential energy of the configuration n is then U( n) = N i=1 U i( n), i.e., U( n) = N n i 1 i=1 k=0 (n i k)

27 FRAC 2009, December, Nice p. 21/53 Energy Conservation During the Dynamics Fixing at time t = 0 an initial sand pile configuration of N granules, n(0) Ω(N), considered at rest,

28 FRAC 2009, December, Nice p. 21/53 Energy Conservation During the Dynamics Fixing at time t = 0 an initial sand pile configuration of N granules, n(0) Ω(N), considered at rest, the potential energy of this configuration coincides with the total energy of the whole pile E 0 = U( n(0))

29 FRAC 2009, December, Nice p. 21/53 Energy Conservation During the Dynamics Fixing at time t = 0 an initial sand pile configuration of N granules, n(0) Ω(N), considered at rest, the potential energy of this configuration coincides with the total energy of the whole pile E 0 = U( n(0)) The application of the H rule may lead to a new configuration n(1) at time t = 1 withe invariant total number of granules N (mass conservation) and whose potential energy is now (proof!) U( n(1)) < U( n(0))

30 FRAC 2009, December, Nice p. 21/53 Energy Conservation During the Dynamics Fixing at time t = 0 an initial sand pile configuration of N granules, n(0) Ω(N), considered at rest, the potential energy of this configuration coincides with the total energy of the whole pile E 0 = U( n(0)) The application of the H rule may lead to a new configuration n(1) at time t = 1 withe invariant total number of granules N (mass conservation) and whose potential energy is now (proof!) U( n(1)) < U( n(0)) Owing to the fact that the gravitational field is conservative, this decrease of potential energy corresponds to the rise of kinetic energy T( n(1)) = U( n(0)) U( n(1)) = E 0 U( n(1))

31 Energy Conservation During the Dynamics Fixing at time t = 0 an initial sand pile configuration of N granules, n(0) Ω(N), considered at rest, the potential energy of this configuration coincides with the total energy of the whole pile E 0 = U( n(0)) The application of the H rule may lead to a new configuration n(1) at time t = 1 withe invariant total number of granules N (mass conservation) and whose potential energy is now (proof!) U( n(1)) < U( n(0)) Owing to the fact that the gravitational field is conservative, this decrease of potential energy corresponds to the rise of kinetic energy T( n(1)) = U( n(0)) U( n(1)) = E 0 U( n(1)) in such a way that the total energy is conserved T( n(1)) + U( n(1)) = E 0 FRAC 2009, December, Nice p. 21/53

32 FRAC 2009, December, Nice p. 22/53 Energy Conservation During the Dynamics (2) Let us now consider a possible path γ n(0) from the initial configuration n(0) to the (unique) equilibrium configuration n(t f ) = n f, reached for the first time at the instant t f : γ n(0) ( n(0), n(1), n(2),..., n(t), n(t + 1),..., n(t f ) )

33 FRAC 2009, December, Nice p. 22/53 Energy Conservation During the Dynamics (2) Let us now consider a possible path γ n(0) from the initial configuration n(0) to the (unique) equilibrium configuration n(t f ) = n f, reached for the first time at the instant t f : γ n(0) ( n(0), n(1), n(2),..., n(t), n(t + 1),..., n(t f ) ) We obtain the two ordered chains for the (decreasing) potential energy and the (increasing) kinetic energy, respectively U( n(0)) > U( n(1)) >... > U( n(t)) > U( n(t + 1)) > T( n(1)) <... < T( n(t)) < T( n(t + 1)) <... E 0

34 FRAC 2009, December, Nice p. 22/53 Energy Conservation During the Dynamics (2) Let us now consider a possible path γ n(0) from the initial configuration n(0) to the (unique) equilibrium configuration n(t f ) = n f, reached for the first time at the instant t f : γ n(0) ( n(0), n(1), n(2),..., n(t), n(t + 1),..., n(t f ) ) We obtain the two ordered chains for the (decreasing) potential energy and the (increasing) kinetic energy, respectively U( n(0)) > U( n(1)) >... > U( n(t)) > U( n(t + 1)) > T( n(1)) <... < T( n(t)) < T( n(t + 1)) <... E 0 under the total energy conservation (the system is conservative): t N, T( n(t)) + U( n(t)) = E 0 In conservative systems the total energy is the same along each possible path

35 FRAC 2009, December, Nice p. 23/53 The Lagrangian The Lagrangian at time t of this conservative dynamical system is L( n(t)) = T( n(t)) U( n(t)) = E 0 2U( n(t)) = 2T( n(t)) E 0

36 FRAC 2009, December, Nice p. 23/53 The Lagrangian The Lagrangian at time t of this conservative dynamical system is L( n(t)) = T( n(t)) U( n(t)) = E 0 2U( n(t)) = 2T( n(t)) E 0 which is an increasing quantity during the dynamical evolution: E 0 = L( n(0)) < L( n(1)) <... < L( n(t)) < L( n(t + 1)) <... < E 0

37 FRAC 2009, December, Nice p. 23/53 The Lagrangian The Lagrangian at time t of this conservative dynamical system is L( n(t)) = T( n(t)) U( n(t)) = E 0 2U( n(t)) = 2T( n(t)) E 0 which is an increasing quantity during the dynamical evolution: E 0 = L( n(0)) < L( n(1)) <... < L( n(t)) < L( n(t + 1)) <... < E 0 The action associated to the possible path γ n0 is defined as S(γ n0 ) := t f L( n(t)) = 2 t f t=0 t=0 with associated reduced action A(γ n0 ) := 2 t f t=0 T( n(t)) t f E 0 T( n(t))

38 FRAC 2009, December, Nice p. 24/53 Stationary action principle Now, we can formulate the discrete time version of The Stationary Action or Hamilton s Principle. The actual path connecting the initial configuration n 0 to the final one n f, is characterized by an extremum (either minimum or maximum) of the associated action (i.e., is the one along which the action is stationary) with respect to all the other possible paths from n 0 to n f for which the total energy is constant and equal to E 0.

39 FRAC 2009, December, Nice p. 24/53 Stationary action principle Now, we can formulate the discrete time version of The Stationary Action or Hamilton s Principle. The actual path connecting the initial configuration n 0 to the final one n f, is characterized by an extremum (either minimum or maximum) of the associated action (i.e., is the one along which the action is stationary) with respect to all the other possible paths from n 0 to n f for which the total energy is constant and equal to E 0. From the computational point of view, in order to reach this result one must know all the possible paths (which a priori is not so easy to obtain if the total number of granules N is very large),

40 FRAC 2009, December, Nice p. 24/53 Stationary action principle Now, we can formulate the discrete time version of The Stationary Action or Hamilton s Principle. The actual path connecting the initial configuration n 0 to the final one n f, is characterized by an extremum (either minimum or maximum) of the associated action (i.e., is the one along which the action is stationary) with respect to all the other possible paths from n 0 to n f for which the total energy is constant and equal to E 0. From the computational point of view, in order to reach this result one must know all the possible paths (which a priori is not so easy to obtain if the total number of granules N is very large), then he must compute the action of each of these paths in order to select the one producing the Lagrangian extremum value.

41 FRAC 2009, December, Nice p. 25/53 Local Stationary action principle (2) On the contrary, we propose a local procedure which consists in evaluating, step by step, the Lagrangian of all the possible configurations at time t + 1 with respect to the Lagrangian of the actual configuration at time t.

42 FRAC 2009, December, Nice p. 25/53 Local Stationary action principle (2) On the contrary, we propose a local procedure which consists in evaluating, step by step, the Lagrangian of all the possible configurations at time t + 1 with respect to the Lagrangian of the actual configuration at time t. This in agreement with Landau Lifshitz, Mechanics : it should be mentioned that the formulation of the principle of stationary action is not always valid for the entire path of the system, but only for a sufficiently short segment of the path

43 FRAC 2009, December, Nice p. 26/53 Local least action principle In order to select the dynamical evolutions characterized by the local least action principle (local minimization of the action), one must apply the following steps: (LL1) Let n(t) be the configuration reached at time t starting from the initial configuration n(0) making use of the H rule (LL2) Let n (t + 1), n (t + 1),..., n k (t + 1) be all the possible admissible next configurations at time t + 1 obtained from n(t) by the H rule (LL3) Let us compute the corresponding Lagrangian values L( n (t + 1)), L( n (t + 1)),...,L( n k (t + 1)) (LL4) Then one chooses the next configuration of point (LL2) which maximizes the potential energy, and so minimizes the corresponding local action.

44 FRAC 2009, December, Nice p. 27/53 An example (1) (5, 5, 2) (5, 4, 3) (6, 4, 2) (6, 3, 3) (5, 5, 1, 1) (5, 4, 2, 1) (6, 4, 1, 1) (6, 3, 2, 1) Level t = 0 Level t = 1 Level t = 2 Level t = 3. The t = 0,1,2,3 time steps possible paths with initial configuration (6,4,2)

45 FRAC 2009, December, Nice p. 28/53 An example (2) U(5, 5, 2) = 33 U(5, 4, 3) = 31 U(6, 4, 2) = 34 U(6, 3, 3) = 33 U(5, 5, 1, 1) = 32 U(5, 4, 2, 1) = 29 U(6, 4, 1, 1) = 33 U(6, 3, 2, 1) = 31. The corresponding Potential Energy valuation

46 FRAC 2009, December, Nice p. 29/53 An example (3) (5, 5, 2) (5, 4, 3) (6, 4, 2) (6, 3, 3) (6, 4, 1, 1) (6, 3, 2, 1) (5, 4, 2, 1) Minimal Potential Energy, i.e., SbS Maximal Action: U(5,4,3) = U(6,3,2,1) = 31 (5, 5, 2) (6, 4, 2) (6, 3, 3) (5, 5, 1, 1) (5, 4, 2, 1) (6, 4, 1, 1) Maximal Potential Energy, i.e., SbS Minimal Action: U(5,5,1,1) = 32

47 FRAC 2009, December, Nice p. 30/53 Some conclusions It is clear from this example that the local action, either maximization or minimization, procedure is not able to select a unique path Also if we continue the possible paths construction after the black hole configuration (5, 4, 2, 1), these local paths turn out to remain distinct among them, whatever be the subsequent time step configurations obtained by applications of the H rule

48 FRAC 2009, December, Nice p. 31/53 Part III The Entropy of Sand Piles Shannon Information Entropy and Granulation Entropy

49 FRAC 2009, December, Nice p. 32/53 Integer partitions: the Information Entropies In discrete measure theory an integer partition n = (n 1,n 2,...,n l ) of total mass N = l i=1 n i, can be considered as a measure distribution on the finite space of l abstract elementary events E l := {1, 2,...,l}. The normalized finite sequence (depending from n) ( p( n) = p 1 = n 1 N, p 2 = n 2 N,...,p l = n ) l N is a probability distribution on the same set of elementary events E l : l i=1 p 1 = 1 and p i 0 for every i.

50 FRAC 2009, December, Nice p. 33/53 The Information and Granulation Entropies Let us define the following two random variables (RV): (RV1) The Granularity RV based on n G( n) := ( G 1 = log n 1, G 2 = log n 2,..., G l = log n l ) where the single quantity G i = log n i furnishes a measure of the granularity of the column i of the pile. (RV2) The Information (also uncertainty) RV based on p I( n) := ( I 1 = log p 1, I 2 = log p 2,..., I l = log p l ) where the single quantity I i = log p i measures the uncertainty linked to the probability p i (probability 0 means the maximum of uncertainty, probability 1 means no uncertainty, and for p < q the uncertainty measure is strictly decreasing I(p) < I(q)). Trivially, for any index i one has I i + G i = log N.

51 FRAC 2009, December, Nice p. 34/53 The Information and Granulation Entropies (2) The average uncertainty and the average granularity of the integer partition n are expressed by the quantities H( n) := l i=1 I i p i and G( n) := l i=1 G i p i They are called the information or Shannon uncertainty entropy and the information granularity entropy, respectively.

52 FRAC 2009, December, Nice p. 34/53 The Information and Granulation Entropies (2) The average uncertainty and the average granularity of the integer partition n are expressed by the quantities H( n) := l i=1 I i p i and G( n) := l i=1 G i p i They are called the information or Shannon uncertainty entropy and the information granularity entropy, respectively. Trivially, one has the conservation of the total information log N for any configuration n of total mass N, i.e., for any n Ω(N): H( n) + G( n) = log N

53 FRAC 2009, December, Nice p. 35/53 The Information and Granulation Entropies (3) Let us consider the two extreme cases: E1) The certain configuration n c = (N, 0,...,0), in which all the N }{{} N times granules are located in a unique column, corresponds to the maximum of certainty (probability distribution p( n c ) = (1, 0,...,0)) and can be considered as a situation of maximum order of the pile. E2) The uniform configuration n u = (1, 1,...,1), in which we have N }{{} N times columns each containing a unique granule, corresponds to a maximum of uncertainty (uniform probability distribution) and can be considered as a situation of maximum disorder. For any configuration n of total mass N we have the boundaries H( n c ) = 0 H( n) log N = H( n u )

54 FRAC 2009, December, Nice p. 36/53 The Information (Shannon) Entropy Proposition: The information (Shannon) entropy of the configuration n(t + 1) at time t + 1, obtained by the application of the H evolution rule to a jump centered at the cell i of the configuration n(t) at time t, i.e., during the dynamical transition is strictly increasing n(t) H-rule n(t + 1) H( n(t)) < H( n(t + 1))

55 FRAC 2009, December, Nice p. 37/53 The Information (Shannon) Entropy (2) Proposition: Let n(t) be a configuration and let n i (t + 1) and n j (t + 1), with i < j, be two configurations obtained by the application of the H evolution rule to the jumps centered, respectively, at the cell i and at the cell j of the same father configuration n(t). Then, the two corresponding information entropies are different. In particular, we have that the latter is strictly greater than the former, i.e. H( n i (t + 1)) < H( n j (t + 1))

56 FRAC 2009, December, Nice p. 38/53 The Information (Shannon) Entropy (3) n(t) H( n(t)) n 1 (t + 1)... n k (t + 1) H( n 1 (t + 1)) <... < H( n k (t + 1)) The information entropy at time t is bigger than all the information entropies at time t + 1 but at time t + 1 the information entropy of the solution by H-rule of the first jump is the smallest one and the information entropy of the solution by H-rule of the last jump is the greatest one

57 FRAC 2009, December, Nice p. 39/53 Entropy Conservation During the Dynamics Let us now consider a possible path γ n(0) from the initial configuration n(0) to the (unique) equilibrium configuration n(t f ) = n f, reached at the time t f : γ n(0) ( n(0), n(1), n(2),..., n(t), n(t + 1),..., n(t f ) ) The information uncertainty entropy increases and the granularity entropy decreases according to the following chains: H( n(0)) < H( n(1)) <... < H( n(t)) < H( n(t + 1)) <... log N G( n(0)) > G( n(1)) >... > G( n(t)) > G( n(t + 1)) >... 0 under the total entropy conservation (the system is isolated from the information point of view): t N, H( n(t)) + G( n(t)) = log N

58 FRAC 2009, December, Nice p. 40/53 The Entropy Lagrangian and Action We can define the entropy Lagrangian at time t along a possible path L( n(t)) = H( n(t)) G( n(t)) = log N 2G( n(t)) = 2H( n(t)) log N which is an increasing quantity during the dynamical evolution: L( n(0)) < L( n(1)) <... < L( n(t)) < L( n(t + 1)) <... log N We can also introduce the entropy action along a possible path γ n0 S(γ n0 ) := t f t=0 L( n(t)) = 2 t f t=0 H( n(t)) t f log N with associated entropy reduced action A(γ n0 ) := 2 t f t=0 H( n(t))

59 FRAC 2009, December, Nice p. 41/53 The Entropy Stationary Action Principle Local Adiabatic Evolution: The actual path connecting the initial configuration n 0 to the final configuration n f is the one whose configuration at any time step t minimizes the information (Shannon) entropy, among all the possible ones i.e., minimizes the disorder degree. Local Anti Adiabatic Evolution: The actual path connecting the initial configuration n 0 to the final configuration n f is the one whose configuration at any time step t maximizes the information (Shannon) entropy, among all the possible ones i.e., maximizes the disorder degree.

60 FRAC 2009, December, Nice p. 42/53 The Shannon Information Entropy: Drawbacks Drawbacks of the Shannon Information entropy: The adiabatic path has been obtained by a step-by-step minimization of the information (Shannon) entropy But in general it does not correspond to the path which minimizes the global entropy action, i.e., to the path for which the sum of all the information entropies of its configurations is the minimum among all the possible paths. The step-by-step anti adiabatic path in general does not correspond to the path which maximizes the global entropy action, i.e., to the path for which the sum of all the information entropies of its configurations is the maximum among all the possible paths.

61 FRAC 2009, December, Nice p. 43/53 Part IV Sand Cellular Automata

62 FRAC 2009, December, Nice p. 44/53 Sand CA lattice and phase space Formally, the CA version of the total mass N sand pile model is based on:

63 FRAC 2009, December, Nice p. 44/53 Sand CA lattice and phase space Formally, the CA version of the total mass N sand pile model is based on: a one dimensional (1D) lattice of N cells L(N) := {1, 2,...,N} where every integer i L(N) represents the cell of position i

64 FRAC 2009, December, Nice p. 44/53 Sand CA lattice and phase space Formally, the CA version of the total mass N sand pile model is based on: a one dimensional (1D) lattice of N cells L(N) := {1, 2,...,N} where every integer i L(N) represents the cell of position i a set of states of the CA S(N) := {0, 1,...,N}

65 FRAC 2009, December, Nice p. 44/53 Sand CA lattice and phase space Formally, the CA version of the total mass N sand pile model is based on: a one dimensional (1D) lattice of N cells L(N) := {1, 2,...,N} where every integer i L(N) represents the cell of position i a set of states of the CA S(N) := {0, 1,...,N} a configuration is any map n : L S(N) assigning to any cell of the lattice i L(N) the number n i S(n) of granules in the cell of position i of the lattice

66 FRAC 2009, December, Nice p. 44/53 Sand CA lattice and phase space Formally, the CA version of the total mass N sand pile model is based on: a one dimensional (1D) lattice of N cells L(N) := {1, 2,...,N} where every integer i L(N) represents the cell of position i a set of states of the CA S(N) := {0, 1,...,N} a configuration is any map n : L S(N) assigning to any cell of the lattice i L(N) the number n i S(n) of granules in the cell of position i of the lattice under the two conditions: the N total mass condition i L(N) n i = N the 0 finiteness condition k L(N) s.t. n i > 0 for every i k and n j = 0 for every j > k:

67 FRAC 2009, December, Nice p. 44/53 Sand CA lattice and phase space Formally, the CA version of the total mass N sand pile model is based on: a one dimensional (1D) lattice of N cells L(N) := {1, 2,...,N} where every integer i L(N) represents the cell of position i a set of states of the CA S(N) := {0, 1,...,N} a configuration is any map n : L S(N) assigning to any cell of the lattice i L(N) the number n i S(n) of granules in the cell of position i of the lattice under the two conditions: the N total mass condition i L(N) n i = N the 0 finiteness condition k L(N) s.t. n i > 0 for every i k and n j = 0 for every j > k: n 1... n i... n k

68 FRAC 2009, December, Nice p. 44/53 Sand CA lattice and phase space Formally, the CA version of the total mass N sand pile model is based on: a one dimensional (1D) lattice of N cells L(N) := {1, 2,...,N} where every integer i L(N) represents the cell of position i a set of states of the CA S(N) := {0, 1,...,N} a configuration is any map n : L S(N) assigning to any cell of the lattice i L(N) the number n i S(n) of granules in the cell of position i of the lattice under the two conditions: the N total mass condition i L(N) n i = N the 0 finiteness condition k L(N) s.t. n i > 0 for every i k and n j = 0 for every j > k: n 1... n i... n k The collection of all such mass N configurations as usual will be denoted by Ω(N) = S(N) L(N) and it is the CA state space

69 FRAC 2009, December, Nice p. 45/53 (From left-to-right) Local Rule CA LR Local Rule x if x y + 1 and (x u or x = u + 1) x if x u 2 and x y + 2 f LR (u, x, y) = x 1 if (x u or x = u + 1) and x y + 2 x + 1 if x u 2 and (x = y or x = y + 1)

70 FRAC 2009, December, Nice p. 45/53 (From left-to-right) Local Rule CA LR Local Rule x if x y + 1 and (x u or x = u + 1) x if x u 2 and x y + 2 f LR (u, x, y) = x 1 if (x u or x = u + 1) and x y + 2 x + 1 if x u 2 and (x = y or x = y + 1) Global behavior (6,4,2) (5,4,2,1) (5,3,3,1) (4,4,2,2) (4,3,3,1,1) (4,3,2,2,1)

71 FRAC 2009, December, Nice p. 46/53 The CA dynamics is Sand CA and Shannon entropy (8, 3, 1) (7, 3, 2) (6,4,1,1) (5, 4, 2, 1) (5, 3, 3, 1) (4,4,2,2) (4, 3, 3, 1, 1) (4, 3, 2, 2, 1) which does not fit any of the two entropy dynamics.

72 FRAC 2009, December, Nice p. 46/53 Sand CA and Shannon entropy The CA dynamics is (8, 3, 1) (7, 3, 2) (6,4,1,1) (5, 4, 2, 1) (5, 3, 3, 1) (4,4,2,2) (4, 3, 3, 1, 1) (4, 3, 2, 2, 1) which does not fit any of the two entropy dynamics. The adiabatic evolution is (no (6, 4, 1, 1)!!!) (8, 3, 1) (8, 2, 2) (7, 3, 2) (6, 4, 2) (5, 5, 2) (5, 4, 3) (5, 4, 2, 1) (5, 3, 3, 1) (4, 4, 3, 1) (4,4,2,2) (4, 3, 3, 2) (4, 3, 3, 1, 1) (4, 3, 2, 2, 1)

73 FRAC 2009, December, Nice p. 46/53 Sand CA and Shannon entropy The CA dynamics is (8, 3, 1) (7, 3, 2) (6,4,1,1) (5, 4, 2, 1) (5, 3, 3, 1) (4,4,2,2) (4, 3, 3, 1, 1) (4, 3, 2, 2, 1) which does not fit any of the two entropy dynamics. The adiabatic evolution is (no (6, 4, 1, 1)!!!) (8, 3, 1) (8, 2, 2) (7, 3, 2) (6, 4, 2) (5, 5, 2) (5, 4, 3) (5, 4, 2, 1) (5, 3, 3, 1) (4, 4, 3, 1) (4,4,2,2) (4, 3, 3, 2) (4, 3, 3, 1, 1) (4, 3, 2, 2, 1) The anti adiabatic evolution is (no (4, 4, 2, 2)!!!) (8, 3, 1) (7, 4, 1) (7, 3, 2) (7, 3, 1, 1) (6,4,1,1) (6, 3, 2, 1) (5, 4, 2, 1) (5, 3, 3, 1) (5, 3, 2, 2) (5, 3, 2, 1, 1) (4, 4, 2, 1, 1) (4, 3, 3, 1, 1) (4, 3, 2, 2, 1)

74 FRAC 2009, December, Nice p. 47/53 CA and Shannon entropy: Comments In particular, one can observe that the adiabatic path excludes the configuration (6,4,1,1) of the CA dynamics while the anti adiabatic one does not admit the configuration (4,4,2,2) of the same dynamics. But, anyway, this CA deterministic dynamics fits with one of the paths of global minimal (mechanical) action. Conclusions: There may be CA dynamics that do not fit either the Shannon entropy adiabatic (single step) evolution, or the anti adiabatic (single step) one.

75 FRAC 2009, December, Nice p. 48/53 Part V Conclusion and Further Developments Bi-Directional Sand CA

76 FRAC 2009, December, Nice p. 49/53 Bi-directional SA Another interesting situation is the one in which the direction of the wind is not the same during the time evolution. This can be formalized by introducing the corresponding symmetrical right-to-left local rule f RL (u, x, y) = and then introducing x if x u + 1 and (x y or x = y + 1) x if x y 2 and x u + 2 x 1 if (x y or x = y + 1) and x u + 2 x + 1 if x y 2 and (x = u or x = u + 1) one of the two symbols, either RL or LR. This choice, denoted by j 0 {RL, LR}, determines one of the two alternate sequences, either σ 1 := (RL, LR, RL,...) if j 0 = RL or σ 2 := (LR, RL, LR,...) if j 0 = LR,respectively; (1)

77 FRAC 2009, December, Nice p. 50/53 Towards equilibrium Left-to-Right equilibrium Non Equilibrium LR-Equilibrium Bi-directional equilibrium Non Equilibrium Non Equilibrium Equilibrium

78 FRAC 2009, December, Nice p. 51/53 Bidirectional Sand CA Equilibrium (Instable) Equilibrium

79 THE END FRAC 2009, December, Nice p. 52/53

80 FRAC 2009, December, Nice p. 53/53 Integer partitions: the (Boltzmann) Thermodynamics Entropy Given an integer partition n = (n 1, n 2,...,n N ) of total mass N, the Boltzmann entropy is defined as: S( n) = log W N ( n) where W N ( n), also called thermodynamics probability, is the following quantity W N ( n) = N! n 1!n 2!...n N!