All Particle Processes Without Collisions Have A

Transcrição

All Particle Processes Without Collisions Have A Fixed Point
C. S. Sousa*, A. D. Ramos*, A. Toom*
Federal University of Pernambuco
Abstract
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 its probability does not depend on other particles. This excludes predator-prey and sexual
relations, but includes death and asexual reproduction with possible mutations. Due to this
restriction, positions of particles are irrelevant and we do not mention them.
If the number of particles tends to infinity, the behavior of the process becomes
deterministic after normalization. In fact, this approximation is often used without a proper
foundation. Our intention is to prove that the resulting dynamic system has a fixed point and
that under additional conditions the fixed point is unique and the system tends to it from any
initial condition. Besides, this approach raises some questions about the manner in which
some books on populational dynamics have been written.
IMS Annual Meeting & X Brazilian School of Probability, Rio de Janeiro, 08/2006
*Departament of Statistics; e-mail: [email protected];alex dias [email protected];[email protected]
–1–
Introduction
The theory of stochastic processes is an area of modern mathematics which has a lot of
applications. It is important to study ergodic vs. non-ergodic processes, which tend vs. do not
tend to a unique limit when t
→ ∞.
We consider one class of random processes with discrete time. We have n types of particles.
Any particle may die or transform itself into one or several particles of any types with certain
probabilities. Our main assumption is that there are no collisions, that is every transformation
applies to only one particle and its probability does not depend on the other particles. Due to
this restriction, positions of particles are irrelevant and we do not mention them. We call such
processes Processes Without Collisions or PWC for short. We study PWC, with the objective
of proving that all of them are ergodic under reasonable conditions.
–2–
Absence of collisions implies absence of any analogy with predator-prey or sexual relations.
However, our methods may be applied to problems of survival, proliferation, mutation and
evolution of species with asexual reproduction, e.g. viruses. Also our processes may be
useful to model social dynamics, which includes both social growth and mobility.
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.
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.
–3–
Definitions and Description of Process
Particles of type k
= 1, . . . , n will be called k -particles. A generic state of our process is a
n
vector q = (q1 , . . . , qn ) ∈ Ω = ZZ+ , where qk ∈ ZZ+ denotes the number of k -particles.
∈ Ω the vector having one at the k -th place and zeros at all the other
places and by δ0 ∈ Ω the vector having zeros at all the places.
We denote by δk
We call a n-dimensional vector v
= (v1 , . . . , vn ) ∈ IRn positive if
v1 ≥ 0, . . . , vn ≥ 0 and v1 + . . . + vn > 0.
–4–
For any positive vector v
= (v1 , . . . , vn ) we denote |v| = v1 + . . . + vn , call |v| norm of v
and call v normed if |v| = 1. We denote by D the set of normed vectors, that is,
D = {(v1 , . . . , vn ) : v1 ≥ 0, . . . , vn ≥ 0, v1 + . . . + vn = 1}.
We denote by N orm the operator of norming, that is, N orm(v)
= v/|v|, for any positive
vector v .
We also use an operator of norming which acts on any distribution P in Ω as follows:
N orm(P ) is a distribution in D induced by P with map N orm.
–5–
The Probability Operator
We denote by Π the set of probability distributions on
Ω.
At every step of discrete time all particles decide independently into what they are going to
turn themselves. Every k -particle turns into a vector q
= (q1 , . . . , qn ) with a probability
θk,q . This includes death if q = δ0 and remaining unchanged if q = δk . Of course,
X
∀k :
θk,q = 1.
q∈Ω
We assume that there is a constant C such that
q1 + . . . + qn > C =⇒ θk,q = 0,
(1)
that is, no particle may transform into more than C particles at once.
Now, we define an operator M
: Π → Π in two steps.
–6–
First, for every q
∈ Ω, we denote by ∆q the distribution concentrated in one vector q ∈ Ω.
We define M ∆q for every q as the following sum of distributions in Ω, which are jointly
j
independent random vectors Vk :
def
M ∆q =
qk
n X
X
Vkj ,
(2)
k=1 j=1
j
where every Vk equals q with a probability θk,q for all q
∈ Ω. Here qk is the k -th component
of q = (q1 , . . . , qn ). Thus we have defined how M acts on ∆q .
Now, we define how M acts on any P
∈ Π assuming its linearity:
X
MP =
(M ∆q ) · Pq .
(3)
q∈Ω
Thus the operator M
: Π → Π is defined.
–7–
The Deterministic Operator
In practice we deal with macro-amounts, that is large amounts of particles, so large that we
use continuous macro-measures instead of discrete micro-measures. For example, talking of
a chemical or nuclear reaction, we measure the amounts of substances in macro-measures,
for example grams or kilograms rather than in micro-measures like the number of molecules
or atoms. We have the habit of treating macro-amounts as continuous in spite of the discrete
micro-nature of these substances.
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 densities d1 , . . . , dn of each type. In this work we are interested only in proportions of
substances. For this reason we assume that vector of densities is always normed, that is
d1 + . . . + dn = 1, so (d1 , . . . , dn ) ∈ D. We shall show how to obtain the behavior of
densities as a limit of the original process when the number of particles tends to infinity.
–8–
Let us define a deterministic operator M̃
macro-state M̃ d
: D → D which transforms d into another
:
M̃ d = N orm
Ã
n
X
k=1
where dk is the k -th component of d
dk
X
rθk,r
r∈Ω
!
,
(4)
= (d1 , . . . , dn ).
Now let us explain how this operator is related to our random operator.
–9–
≥ 0 we denote by round(x) the smallest integer number, which is not
n
less than x and call this operation rounding. For any positive vector v ∈ IR+ we denote by
round(v) the vector, whose components are rounded components of v :
For any number x
round(v) = ( round(v1 ), . . . , round(vn ) ).
We start with a macro-state d
= (d1 , . . . , dn ) ∈ D. First we transform it into a micro-state
as follows: We choose a large number L, multiply all the components of d by L and take their
integer approximations. Thus we obtain an integer vector round(L · d). After that we apply
one step of our random process, thus obtaining a random vector M ∆round(L·d) , where M
was defined in (3).
–10–
After that we come back to the macro-state applying the operator N orm. Thus we obtain a
random vector
³
´
N orm M ∆round(L·d) .
(5)
In practice L is finite (for example, the Avogadro number), but in our study we suppose that
L −→ ∞. We want to substantiate the practice of dealing with macro-level. For this we show
that the distribution (5) concentrates in the vicinity M̃ d when L is large.
–11–
Theorems
n
Let us introduce the following distance in IR :
∀q, q 0 ∈ IRn : dist(q, q 0 ) =
n
X
|qi −qi0 |, where q = (q1 , . . . , qn ) and q 0 = (q10 , . . . , qn0 ).
i=1
Theorem 1
∀d ∈ D and ∀ε > 0 :


¶
µ
´
³
L→∞
P rob dist N orm M ∆round(L·d) , M̃ d > ε −→ 0.
³
´
Theorem 1 means that the measure N orm M ∆round(L·d) concentrates in the
neighbourhood of M̃ d, when L
−→ ∞.
–12–
Theorem 1 shows that if the number of particles tends to infinity, after norming the behavior of
the process becomes deterministic governed by the formula (4). Thus the deterministic
process can be used as an approximation of our original stochastic process, when the
number of particles is large.
Theorem 2
∀d, d0 ∈ D : dist(M̃ d0 , M̃ d) < const · dist(d0 , d).
So, the map M̃ is Lipschitz.
Theorem 3 The map M̃ has at least one fixed point.
Due to theorem 2 the map M̃
: D → D is continuous. Then theorem 3 follows from the
Brouwer’s Fixed Point Theorem. In our case it is better to use the Schauder’s version[4]. Thus
M̃ has at least one fixed point d ∈ D.
–13–
Example
Let us consider a special case, in which n
= 2, that is there are only two types of particles
and the following transitional probabilities are positive:
θ1,δ1 , θ2,δ1 , θ1,δ2 , θ2,δ2 .
Due to formula (4) we obtain ∀d
∈ D = {(d1 , d2 ) : d1 ≥ 0, d2 ≥ 0, d1 + d2 = 1}
M̃ d = (d1 θ1,δ1 + d2 θ2,δ1 , d1 θ1,δ2 + d2 θ2,δ2 ) .
In this case
µ
θ2,δ1
,
θ2,δ1 + θ1,δ2
θ1,δ2
θ2,δ1 + θ1,δ2
¶
is the only fixed point of M̃ .
–14–
Acknowledgements
CS Sousa and AD Ramos took part at this project under the Computational Mathematics PhD
Programme of the Universidade Federal de Pernambuco and they acknowledge financial
support from Fapesb/Brazil and Propesq/Brazil. A. Toom was supported by CNPq.
–15–
References
[1] A. Okubo, and S. A. Levin. Diffusion and Ecological Problems-Modern Perspectives. New
York, Springer-Verlag, 2nd. ed., (1980).
[2] A.Toom. Non-ergodicity in a 1-D particle process with variable length. Journal of
Statistical Physics, v.115, nn. 3/4, (2004), pp. 895-924.
[3] A.Toom. Particle systems with variable length. Bulletin of the Brazilian Mathematical
Society, v.33, n.3, (2002), pp. 419-425.
[4] C. S. Hönig. Aplicações da Topologia à Análise. Rio de Janeiro, Instituto de Matemática
Pura e Aplicada, CNPq, (1976).
[5] L.D.Landau and E.M.Lifshitz. Statistical Physics. (V.5 of Course of Theorical Physics.)
2-d Edition. Pergamon Press, (1969).
[6] N. T. J. Barley. The Elements of Stochastic Processes-with application to the natural
sciences. New York, John Wiley & Sons, Inc, (1976).
–16–

All Particle Processes Without Collisions Have A

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