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

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Study of Systems with Variable Length using Processes Without Collisions

C. S. Sousa∗, A. D. Ramos∗∗ and A. Toom∗

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.

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Study of Systems with Variable Length using Processes Without Collisions

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 = ZZ

n₊

,

where

q

k

∈ ZZ

+ denotes the number of

k

-particles. For any vector

c ∈ C

n we denote the

norm of

c

by

kck =

n

X

i=1

|c

i

|.

Distance in

IR

n:

∀v, v

0

∈ IR

n

:

dist(v, v

0

) = kv − v

0

k.

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Study of Systems with Variable Length using Processes Without Collisions

We call a

n

-dimensional vector

v =







v

₁ .. .

v

n







∈ IR

n positive if

v

₁

≥ 0, . . . , v

n

≥ 0

and

v

1

+ . . . + v

n

> 0.

For any vector

v ∈ IR

n we call

v

normalized if

kvk = 1

. We denote

D =

















v

1 .. .

v

n







: v

₁

≥ 0, . . . , v

n

≥ 0, v

1

+ . . . + v

n

= 1











,

D

+

=

















v

₁ .. .

v

n







: v

₁

> 0, . . . , v

n

> 0, v

1

+ . . . + v

n

= 1











.

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Study of Systems with Variable Length using Processes Without Collisions

The Probability Operator

We denote the operator of norming by

N orm(v) = v/kvk,

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

0 the set of in fact finite probability distributions on

Ω.

Π

0 is the space where we shall work.

Π

and

Π

0 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

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Study of Systems with Variable Length using Processes Without Collisions

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,

δ

k

=







0

.. .

1

.. .

0 





∈ Ω

and

δ

₀

=







0

.. .

0 





∈ Ω.

Type of particles Resulting vector Transformation

k

δ

0 Death

k

δ

k no change

k

2 · δ

k Mitosis

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Study of Systems with Variable Length using Processes Without Collisions

Of course,

∀k :

X

q∈Ω

θ

k,q

= 1.

We assume that there is a constant

C

such that

(q

₁

+ . . . + 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.

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Study of Systems with Variable Length using Processes Without Collisions

Step 1:

For every

q ∈ Ω

we denote by

∆

q

∈ Π

0

the distribution concentrated in one vector

q ∈ Ω.

We define

M ∆

q

∈ Π

0

as the result of application of

M

to

∆

q for every q:

M ∆

q def

=

n

X

k=1 qk

X

j=1

V

_kj

.

Here

V

_kj are jointly independent random vectors. Every random vector

V

_kj 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

0

,

assuming linearity:

M P =

X

q∈Ω

(M ∆

q

) · P

q

,

where

P

q is the value of distribution

P

on vector

q.

Thus the operator

M : Π

0

→ Π

0 is defined.

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Study of Systems with Variable Length using Processes Without Collisions

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

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Study of Systems with Variable Length using Processes Without Collisions

Definition 2 For all

d ∈ D

we define the operator

M : D −→ D

˜

by the formula

˜

M d = N orm (M

0

d) ,

where

M

₀

=







X

r∈Ω

r

1

θ

1,r

· · ·

X

r∈Ω

r

1

θ

n,r .. . ...

X

r∈Ω

r

n

θ

1,r

· · ·

X

r∈Ω

r

n

θ

n,r







.

Properties of

M :

˜

•

The map

M : D → D

˜

has at least one fixed point.

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Study of Systems with Variable Length using Processes Without Collisions

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

)







.

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Study of Systems with Variable Length using Processes Without Collisions

•

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

N orm

³

_{M ∆round}

_(L·d)

´

distributed in

D.

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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

M d,

˜

when

L −→ ∞.

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Study of Systems with Variable Length using Processes Without Collisions

Theorem 1

∀d ∈ D, ∀ε > 0,

P rob

Ã

dist

Ã M ∆round

(L·d)

L

, M

0

· d

!

> ε

!

L→∞

−→ 0.

Theorem 2

∀d ∈ D, ∀ε > 0,

P rob





dist

µ

N orm

³

_{M ∆round}

_(L·d)

´

,

M d

˜

¶

> ε





L→∞

−→ 0.

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Study of Systems with Variable Length using Processes Without Collisions

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.

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

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Study of Systems with Variable Length using Processes Without Collisions

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 Mortα

,

Flipβ and

Mit_γ respectively.

•

Mort_α

:

one component in the state

1

disappears with probability

α

independently from the other components.

•

Flipβ

:

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.

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Study of Systems with Variable Length using Processes Without Collisions

Our process is the sequence of measures

µ

t

= µQ

t

,

(2)

where

Q =

MortαFlipβMitγ

.

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

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Study of Systems with Variable Length using Processes Without Collisions

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Study of Systems with Variable Length using Processes Without Collisions

In this case,

M

₀

=







1 (1 − α) β

0 1 − (1 − α) β + (1 − α) (1 − β) γ − α







.

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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





v

1

0 



.

The eigenvectors associated

with eigenvalue

λ

₂ are







(1 − α) β

− (1 − α) β + (1 − α) (1 − β) γ − α

v

2

v

2







.

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Study of Systems with Variable Length using Processes Without Collisions

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;

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Study of Systems with Variable Length using Processes Without Collisions

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 Part´ıculas sem Colis˜oes. Tese de Doutorado, Universidade Federal de Pernambuco, Pernambuco, (2007).

Veja http://www.de.ufpe.br/˜ 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. 895-924.

[3] A.D. Ramos. Processos de Part´ıculas com Comprimento Vari ´avel. Tese de Doutorado, Universidade Federal de Pernambuco, Pernambuco, (2007).