Study of Systems with Variable Length using Processes Without Collisions C. S. Sousa, A. D. Ramos and A. Toom

Transcrição

1 12th Brazilian School of Probability, Minas Gerais, 08/ Introduction We consider a class of particle processes with a finite number of types of particles, which we call Processes Without Collisions or PWC for short. As the discrete time goes on, any particle may die or transform itself into one or several particles of any types with certain probabilities, but there are no collisions, that is every transformation applies to only one particle and probabilities of its transformations do not depend on other particles. We pay special attention to the limit, when the number of particles tends to infinity and the quantities of particles of all types may be treated as real rather than integer numbers. In fact, this approximation is often used in various sciences, sometimes without a proper foundation. In [1], we prove that under mild conditions the resulting dynamical systems have at least one fixed point and under additional conditions tend to it when time tends to infinity.

2 12th Brazilian School of Probability, Minas Gerais, 08/ Definitions and Description of Process Particles of type k = 1,..., n will be called k-particles. A generic state of our process is a vector q = q 1.. q n Ω = ZZn +, where q k ZZ + denotes the number of k-particles. For any vector c C n we denote the n norm of c by c = c i. Distance in IR n : i=1 v, v IR n : dist(v, v ) = v v.

3 12th Brazilian School of Probability, Minas Gerais, 08/ We call a n-dimensional vector v = v 1. v n v 1 0,..., v n 0 and v v n > 0. v n IRn positive if For any vector v IR n we call v normalized if v = 1. We denote v 1 D =. : v 1 0,..., v n 0, v v n = 1, D + = v 1. v n Properties of set D : compact and convex. : v 1 > 0,..., v n > 0, v v n = 1.

4 The Probability Operator We denote the operator of norming by Norm(v) = v/ v, for any non-null vector v. We denote by Π the set of probability distributions on Ω = ZZ n +. Definition 1 Let P Π. We denote by P q the value of P at vector q Ω. We say that the distribution P is in fact finite, if the set {q Ω : P q > 0} is finite. We denote by Π the set of in fact finite probability distributions on Ω. Π is the space where we shall work. Π and Π are convex. At every step of discrete time all particles decide independently into what they are going to transform themselves. Every k-particle turns into a vector q with a probability θ k,q. Type of particles Transitional probabilities Resulting vector k θ k,q q 12th Brazilian School of Probability, Minas Gerais, 08/2008 4

5 12th Brazilian School of Probability, Minas Gerais, 08/ We denote by δ k Ω the vector having one at the k-th place and zeros at all the other places and by δ 0 Ω the vector having zeros at all the places, that is, 0. 0 δ k = 1 Ω and δ 0 =. Ω Type of particles Resulting vector Transformation k δ 0 Death k δ k no change k 2 δ k Mitosis k δ i Conversion

6 12th Brazilian School of Probability, Minas Gerais, 08/ Of course, k : θ k,q = 1. q Ω We assume that there is a constant C such that (q q n > C) = (θ k,q = 0). that is, no particle may transform into more than C particles at once. Now, we define an operator M : Π Π in two steps.

7 Step 1: For every q Ω we denote by q Π the distribution concentrated in one vector q Ω. We define M q Π as the result of application of M to q for every q: Here V j k def M q = n q k k=1 j=1 V j k. are jointly independent random vectors. Every random vector V j k equals p with a probability θ k,p for all p Ω. Here q k is the k-th component of q. Thus we have defined how M acts on q. Step 2: Now we define how M acts on any probability distribution P Π, assuming linearity: MP = q Ω(M q ) P q, where P q is the value of distribution P on vector q. Thus the operator M : Π defined. Π is 12th Brazilian School of Probability, Minas Gerais, 08/2008 7

8 The Deterministic Operator In practice we deal with macro-amounts, that is large amounts of particles, so large that we use continuous macro-units instead of discrete micro-units. For example, talking of a chemical or nuclear reaction, we measure the amounts of substances in macro-units, for example grams or kilograms rather than in micro-units like the number of molecules or atoms. In this connection, we consider the non-integer analog of our operator, in which instead of non-negative integer numbers of particles of each type, we have non-negative real numbers, called densities d 1,..., d n. We are interested only in proportion of different particles rather than in the total amount. For this reason, we use the operator of normalization which acts on any probability distribution P in Ω as follows: Norm(P ) is a distribution in D induced by P with the map Norm. 12th Brazilian School of Probability, Minas Gerais, 08/2008 8

9 12th Brazilian School of Probability, Minas Gerais, 08/ Definition 2 For all d D we define the operator M : D D by the formula Md = Norm (M 0 d), where M 0 = r 1 θ 1,r r Ω. r n θ 1,r r 1 θ n,r r Ω. r n θ n,r. r Ω r Ω Properties of M : The map M : D D has at least one fixed point. The map M is Lipschitz, with constant 4 C.

10 12th Brazilian School of Probability, Minas Gerais, 08/ Relation between M, M 0 and M Now let us describe the relation between operators M, M 0 and M. For any real number x 0 we denote by round(x) the smallest integer number which is not less than x and call this operator rounding. For any positive vector v we denote by round(v) the vector, whose components are rounded components of v: round(v) = round(v 1 ). round(v n ). For every d D and every value of a real parameter L > 0, we define two distributions.

11 12th Brazilian School of Probability, Minas Gerais, 08/ First, given d D and L > 0, we transform it into a micro-state round(l d). Then, since round(l d) Ω, we can apply M to the distribution round(l d), which is concentrated in round(l d), and obtain M round(l d), which is a random vector distributed in Ω. Normalizing M round(l d), we obtain a random vector ) Norm (M round(l d) distributed in D. (1) Now we can say how M, M 0 and M are related: the distribution M round(l d) L concentrates to M 0 d, when L ; and the distribution (1) concentrates to Md, when L. The following theorems show what do we mean by the word concentrates :

12 12th Brazilian School of Probability, Minas Gerais, 08/ Theorem 1 d D, ε > 0, ( ) P rob dist ( M round(l d) L, M 0 d > ε ) L 0. Theorem 2 d D, ε > 0, ( ) ) P rob dist Norm (M round(l d), Md > ε L 0.

13 12th Brazilian School of Probability, Minas Gerais, 08/ Definition 3 Let M be a square matrix of order n. We call M reducible if there is a permutation of indices which reduces the matrix M to A 0 B C where A and C are square matrix and 0 is null matrix. Otherwise we call the matrix M irreducible. Definition 4 Let M be a non-negative and irreducible square matrix of order n and let λ dom be its dominant eigenvalue. Let h be the number of eigenvalues of the matrix M, whose moduli are equal to λ dom. If h = 1, we call M primitive. If h > 1, the matrix is called imprimitive, and h is called the index of imprimitivity. Theorem 3 Let the matrix M 0 be primitive. Then the sequence of vectors M t d has a limit when t and this limit is one and the same for all initial vectors d D +.,

14 12th Brazilian School of Probability, Minas Gerais, 08/ Application: Systems with Variable Length Systems with Variable Length or SVL is the name of a new class of 1-D particle systems, whose components can appear and disappear in the process of interaction. We consider SVL with discrete time, with only two types of particles 1 and 2. This process includes three operators death, flip and mitosis, denoted by Mort α, Flip β and Mit γ respectively. Mort α : one component in the state 1 disappears with probability α independently from the other components. Flip β : changes the state of one component from 1 to 2 with probability β independently from the other components. Mit γ : one component in the state 1 duplicates itself with probability γ independently from the other components. Also we included normalization in our processes in order to apply our results to Systems with Variable Length.

15 12th Brazilian School of Probability, Minas Gerais, 08/ Our process is the sequence of measures where µ t = µq t, (2) Q = Mort α Flip β Mit γ. At every time step first operates the operator death, then the operator flip and finally the operator mitosis. Here we write operators after measures (as in the references[2] e [3]). Let us represent the process (2) in our terms. We use r = 1 0, s = 0 1, u = 0 2, v = 0 0 and the following transitional probabilities θ 1,r = 1, θ 2,r = (1 α) β, θ 2,u = (1 α) (1 β) γ, θ 2,v = α and θ 2,s = 1 (1 α) β (1 α) (1 β) γ α.

16 12th Brazilian School of Probability, Minas Gerais, 08/

17 12th Brazilian School of Probability, Minas Gerais, 08/ In this case, M 0 = 1 (1 α) β 0 1 (1 α) β + (1 α) (1 β) γ α. (3) M 0 is reducible, and therefore not primitive. The eigenvalues of the matrix M 0 are: λ 1 = 1 and λ 2 = 1 (1 α) β + (1 α) (1 β) γ α. The eigenvectors associated with eigenvalue λ 1 are with eigenvalue λ 2 are v 1 0 (1 α) β (1 α) β + (1 α) (1 β) γ α v 2 v 2. The eigenvectors associated.

18 12th Brazilian School of Probability, Minas Gerais, 08/ Results: Let γ > β (1 α) + α (1 β) (1 α). Then the stochastic process (2) has at least two different invariant measures; its deterministic approximation with matrix M 0 (3) has at least two different fixed points. Let γ β (1 α) + α (1 β) (1 α). Then the stochastic process (2) has only one invariant measure; its deterministic approximation with matrix M 0 (3) has only one fixed point.

19 12th Brazilian School of Probability, Minas Gerais, 08/ Acknowledgements C. S. Sousa acknowledge financial support from Facepe/Brazil. A. D. Ramos and A. Toom were supported by CNPq. References [1] C.S. Sousa. Processos de Partículas sem Colisões. Tese de Doutorado, Universidade Federal de Pernambuco, Pernambuco, (2007). Veja toom/ensino/doutorado/alunos/caliteia/calitese.pdf. [2] A. Toom. Non-ergodicity in a 1-D particle process with variable length. Journal of Statistical Physics, v. 115, n. 3/4, (2004), pp [3] A.D. Ramos. Processos de Partículas com Comprimento Variável. Tese de Doutorado, Universidade Federal de Pernambuco, Pernambuco, (2007). Veja toom/ensino/doutorado/alunos/alex/alex.pdf.