Physica 133A (1985) 551-560

North-Holland, Amsterdam

METHODS OF HILBERT SPACES IN THE BIRTH

AND DEATH PROCESSES FORMALISM

Krzysztof KOWALSKI

Department of Biophysics, Institute of Physiology and Biochemistry, Medical School of Lddi,

3 Lindley St., 90-131 Lddi, Poland

Received 22 April 1985

It is shown that classical birth and death processes can be described by a Hilbert space,

Schriidinger-like equation with the Hamiltonian (a non-Hermitian one) expressed in terms of Bose

creation and annihilation operators. A new algebraic method of solving the master equations of

birth and death type is introduced.

1. Introduction

In spite of its limitations the stochastic birth and death processes approach

still remains an important tool in the study of both chemical and biological

kinetics and of some problems of theoretical ecology’-5). The fundamental

equation in the birth and death processes formalism is a differential-difference

equation (i.e. an infinite system of ordinary differential equations called the

Kolmogorov equations) called a master equation. The appearance of infinite

dimension in the master equation suggests that it can be related with some

problem in a Hilbert space. Moreover, the nature of the birth and death processes

indicates that this Hilbert space coincides with the Fock space.

The aim of the present paper is to show that the solution of a birth and death

master equation is equivalent to the solution of an abstract, Hilbert space,

Schrodinger-like equation with the Hamiltonian (a non-Hermitian one)

expressed in terms of Bose creation and annihilation operators. It is demon-

strated that in the particular case of the Bargmann representation the solution

of this Schriidinger-like equation coincides with the generating function of the

probability. Using the coherent states representation the new formula on the

expectation value of an arbitrary Bore1 function of a birth and death process is

obtained. As a non-trivial application of the presented canonical Hilbert space

0378-4371/85/$03.30 @ Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

552 K. KOWALSKI

formalism, the master equation describing a bimolecular chemical reaction is

solved.

2. Schriidinger equation

Consider the birth and death process g(t), where 5: R, + 2:; Z(Z+) is the set

of integers (non-negative integers). The direct Kolmogorov equations can be

written as

P(n; t)= -w(n)P(n; t)+ C w(n- r, n)P(n- r; t), GS

0)

where P(n; t)= P@(t)= n), n = (n,, . . . , n,)E Z:, S is a bounded subset of

Zk such that {O}@ S, w(n, m), n, m E 2: are the transition probability in-

tensities, w(n) = CIES w(n, n + r) and overdot denotes differentiation with

respect to the time.

We assume that the master equation (1) is subject to the initial data

0 ; 0) = ir %;“(), . (2) i=l

This assumption is the usual one made in applications of birth and death

processes.

Notice that if there exists a subscript i E (1, . . . , k} and an element ni such

that ni < r,, ri >O, then the following condition has to hold:

w(n - r, n) = 0.

We assume that w(n - r, n) is of the form

w(n-r,n)=

I 6r(n)fifi(n,-ri-j+1), vislriJ, vi>O, i= l,..., k, r=l j=l

G,,(n) fi fi(ni-r,--j+l), vi 2 jril , vs = 0, s E A C (1, . . . , k},

(3)

where G,,(n) is analytic in n.

BIRTH AND DEATH PROCESSES 553

The last assumption is not a too restrictive one. The condition (3) is valid in

every practical application of birth and death processes such as for example

chemical kinetics (the summation extended over elementary reactions, in the

case of many-stage ones, is omitted for brevity from the relations (1) (3)).

Consider now the Bose creation and annihilation operators a:, uj, i, j =

1 , . * . , k. These operators satisfy the standard commutation relations

[ Ui, Uf ] = 6ij 7

Suppose that we are given an operator

it4 = -w(N) + C ‘rlN) ~ ( e(ri)(ut)” (NT li)! 1ES i=l I

Ni ! ‘e(-ri’(~i-ri_~i)!ui”

where Ni = at a,, i = 1, . . . , k, are the number operators and O(x) is the step

function

If we put

!P(n; t) = (fi ni!)l’zP(n; t), (5)

then it follows from (l)-(4) that P(n; t) can be written in the form

P(n; t) = (n 1 ?P; t) , (6)

where the vectors In), it E 2: span the occupation number representation (see

appendix A) and the vector IP; t) satisfies the equation

(7)

It thus appears that solutions of very general birth and death master equations

554 K. KOWALSKI

embracing everything that arises in practice is equivalent to the solution of an

abstract Schrodinger-like equation in Hilbert space.

Consider the Bargmann representation (see appendix B). The “Schrodinger

equation” (7) written in this representation is the linear partial differential

equation

a,!&*, t) = M (z*, $1 S(z*, t) ) @(z*, 0) = h z; ““1 1 ,=,

where the differential operator A4 is M(z*, a/az*) = MIO+=LI,(I=a,3Z*.

The Bargmann amplitude $(z*, t) and the probability P(n; t) are related by

P(z*, t) = c P(n; t) n z; n1 ) “EZ: i=l

i.e. the Bargmann amplitude $(z*, t) is the generating function of the multi-

index sequence P(n; t).

We have thus shown that the method of the generating function which is

frequently used for the construction of the solutions of master equations

corresponds to the particular, Bargmann representation in the presented

canonical, Hilbert space formalism.

The Bargmann amplitude (generating function) ?&(z*, t) contains the expec-

tation value of an arbitrary Bore1 function F of the random variable 5. In fact,

by definition

(F;[g(t)]> = c F;(n)P(n; t), i = 1,. . ) k. “EZf

Hence, using the relation (6) written in the coherent states representation (see

appendix B), we obtain

(F,M41) = /R2k dp(z) e-‘L’2@(z*, t)gi(z), i = 1, . , k, (8)

where p,(z) is the exponential generating function of c(n):

i=l,...,k.

The author does not know any other formula like (8) which allows to compute

the expectation value of an arbitrary Bore1 function of a birth and death

process from the given generating function.

BIRTH AND DEATH PROCESSES 555

In the case of F polynomial in 5 well-known relations can be obtained from the formula (8), such as for example

and

(5itt)) = a!tqz*, t)

az*

I ?=(I, 1.. , 1)

i=l,...,k.

(g(4) = [& (6 I& @fi(z*, o)] / 1 I r’=(l, 1,. , 1)

Example. Consider the radioactive decay process. The probability to have n surviving nuclei at time t changes in time according to the master equation

i)(n; t)= -AnP(n; t)+A(n+ l)P(n+l; t), P(n;O)= anno. (9)

The corresponding “Schrodinger equation” is of the form

- )p,t)=MIp;t), )?P;O)=Vn,!~n,),

where M = A(a -N); N = a+a is the number operator. Passing to the inter- action picture

jpTt)= eAN’IF; t), (10)

the following equation is obtained:

I?FYt) = ti(t)J!Er), IFTO) = IW; 0) = V/n,!ln,),

where the “Hamiltonian” in the interaction picture is given by

a(t) = A eANra emAN’ = A e-“‘a .

Taking into account the obvious relation

one

[a(t), &f(f)] = 0 , t, t’ E R,

gets

I!&)= -

eJ:fii(ddT1qio) = dn,! e”-“-h”“ln,) , (11)

556 K. KOWALSKI

Hence, by virtue of (6), (lo), (11) the solution P(n; t) of the master equation (9)

can be obtained easily:

P(n; t) = &(n 1 P; t) = +yNf$y,) n.

n0 = ( ) e-A”‘(l _ e-At)q-” .

n

Example. Consider now the nontrivial case of the chemical reaction

2A:B.

The probability distribution for the particle number of the molecules A satisfies

the following equation’):

P(n;t)=i(n+2)(n+l)P(n+2 ; t) - i n(n - l)P(n ; f), P(n

The corresponding “Schrodinger equation” can be written as

- \1VI1)=Mj?J$r), ~?P;O)=dno!~no),

where the “Hamiltonian” M is given by

M=;[a2-N(N-I)].

Passing to the interaction picture

1 ,j--Tt) = ,WWWl)~ q; Q ,

the following equation is obtained:

- I?f?I) =ti(t)lWTt), Ilu;O)= I!JJ;O)= Vno!Ino),

where the “Hamiltonian” &f(t) is of the form

Q(Q = ! eWW’W1) 2 a e-WWWU = ‘e -n(2N+1)uz . 2 2

0) = 8,“” .

(12)

(13)

(14)

BIRTH AND DEATH PROCESSES 551

Now, using the easily checked relation

e ~t(2h'+l)~2= e4rta2 e-rf(2N+1)

the solution of eq. (14) can be written as

where the evolution operator V(t) is given by

V(~)=l+i:jdr,jdr,_,...~dr,~(~j)...A(r,) j=l

0 0 0

= 1 + 5 (i)ja2j j dfj / dfj_l . . . / dt, e4rzc,I iti e-r(2N+1)& ti .

j=l 0 0 0

It follows from (6), (13) and (15) that the solution of the master equation (12) can be expressed by

P(n; t) = h(n) !P; t)= --&(n1 e-cm)WcNml)l?i~t)

?Z,! -(n/Z)n(n-I) L Kwwl-~ll

z--e

n. I 0 2 G,WWzo-n,](t) ’

where [x] designates the biggest integer in x and the function G,,,(t) is determined by the following recursion relation:

G,,,(t)= j e(4m-2no-1)rsG,,_1(r) dr, G,(t) = 1 ; WI = 1,2,3, . . . .

0

3. Conclusion

It appears that the semantic similarity between the denominations “birth and death” of stochastic processes and “creation and annihilation” of the operators is not an accidental one. Namely, it is shown in this paper that the general master equation of a birth and death type can be transformed so that the

558 K. KOWALSKI

resulting equation is the Fock space realisation of an abstract Hilbert space,

Schrodinger-like one with the Hamiltonian expressed in terms of Bose creation and annihilation operators. In this way the new canonical Hilbert space description of birth and death processes is found.

The introduced approach generalizes the classical methods which are usually employed in the study of birth and death master equations. For example the method of generating function corresponds to the particular Bargmann representation in the presented “quantum mechanical” Hilbert space for- malism. It is demonstrated that the introduced technique enables us to deter- mine some interesting quantities which are related to the master equation. Moreover, the algorithm of solving the birth and death master equations described herein is in fact a new algebraic method of obtaining the solutions of these equations in a systematic manner. The presented canonical Hilbert space formalism will undoubtedly be a useful tool in the study of birth and death processes.

Acknowledgements

I would like to thank Dr. P. Kosinski and Dr. S. Giler for helpful comments.

Appendix A

Occupation number representation

We recall the basic properties of the occupation number representation. Let us assume that there exists in the Hilbert space of states X, a unique,

normed vector IO) (vacuum vector) which satisfies

a,lO)=O, i= l,..., k. (A.11

We also assume that there is no nontrivial closed subspace of X, which is invariant under the action of operators

ai, al, i,j=l,..., k.

The state vectors In), II E Zt, defined as follows:

In>= @~),U), 1’

(A.2)

BIRTH AND DEATH PROCESSES 559

are the common eigenvectors of the number operators i.e.

Ni = a:q, i = 1,. . . , k,

Niln)= nJn>, i = 1,. . . , k.

These vectors satisfy the following relations:

(A-3)

(n I n’> = ii s,,,; 9 orthogonality ,

i=l

c In>(4= 1, completeness .

"EZ:

The action of the Bose operators on following form:

+1, *. * 7 rzk) = XGgn,, . . . ) ni - 1, , I-,

the vectors In) = lizI,

,..) nk), i= 1,.

(A.41

’ > nk) has the

,k. (A.51 a;ln,, . . .) n,)=Vn,+lln, ,...) ni+l )...) ?Q>,

Appendix B

Coherent states representation

We recall now the basic properties of coherent states. The coherent states lz), where z E Ck, i.e. the eigenvectors of the annihilation operators ui,

a,lz) = z,lz), i = 1,. . . , k, (B.1)

fulfill the following relations:

(z I w) = exp[-i(Jz12 + )w12 - 2z* - w)] , (B-2)

where z* = (zr, . . . , z*,), the asterisk over zi, i = 1,. . . , k, denotes the complex conjugation, z - w = Efcl ziwi and lz12 = z - z* = CF=, jzij2,

I dp(z) Iz)(zl = I, completeness,

R=

(B.3)

where

560 K. KOWALSKI

k 1 dp((z)=n--d(Rezi)d(Imzi).

;=* n

The passage from the occupation number representation to the coherent states

representation is given by the kernel

(B.4)

Suppose now that we are given an arbitrary state IF’). It is easily shown that the

function P(z*) = (z 1 P) has the following form:

P(z*) = S(z*) exp(-i)z12), 03.5)

where @(z*) is an analytic function (entire function). Hence, we get

d/l.(z) exp(-)zl*)$*(z*)@z*). 03.6)

RZk

The representation (B.6) is called the Bargmann representation’). The Bose

operators ai, at, i, j = 1, . . , k, act in this representation as follows:

a,@(z*) = & @(z*) )

I i=l,...,k.

u;s(z*) = 2; $(z*),

(B.7)

References

1) D.A. McQuarrie, in: Suppl. Rev. Ser. Appl. Probab. 8 (Methuen, London, 1967).

2) N.S. Goel and N. Richter-Dyn, Stochastic Models in Biology (Academic Press, New York, 1974).

3) N.G. van Kampen, in: Proc. Int. Summer School in Stat. Mech. IV, Jadwisin, Poland, 1977.

4) I. Oppenheim, K.E. Shuler and G.H. Weiss, Stochastic Processes in Chemical Physics (MIT Press, Cambridge, 1977).

5) C.W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1983).

6) V. Bargmann, Commun. Pure Appl. Math. 14 (l%l) 187.