Page 1

arXiv:1001.3526v2 [cond-mat.stat-mech] 9 Apr 2010

Exact correlations in the one-dimensional

coagulation-diffusion process

by the empty-interval method

Xavier Duranga, Jean-Yves Fortina, Diego Del Biondo‡a,

Malte Henkelaand Jean Richertb

aGroupe de Physique Statistique, D´ epartement de Physique de la Mati` ere et des

Mat´ eriaux, Institut Jean Lamour§, CNRS – Nancy-Universit´ e – UPVM,

B.P. 70239, F - 54506 Vandœuvre les Nancy Cedex, France

bInstitut de Physique, Universit´ e de Strasbourg, 3 rue de l’Universit´ e, F - 67084

Strasbourg Cedex

E-mail:

durang@lpm.u-nancy.fr,fortin@lpm.u-nancy.fr,delbiond@lpmcn.univ-lyon1.fr,

henkel@lpm.u-nancy.fr,richert@fresnel.u-strasbg.fr

Abstract.

dimensions is dominated by fluctuation effects. The one-dimensional coagulation-

diffusion process describes the kinetics of particles which freely hop between

the sites of a chain and where upon encounter of two particles, one of them

disappears with probability one. The empty-interval method has, since a long

time, been a convenient tool for the exact calculation of time-dependent particle

densities in this model. We generalize the empty-interval method by considering

the probability distributions of two simultaneous empty intervals at a given

distance.While the equations of motion of these probabilities reduce for the

coagulation-diffusion process to a simple diffusion equation in the continuum

limit, consistency with the single-interval distribution introduces several non-

trivial boundary conditions which are solved for the first time for arbitrary initial

configurations. In this way, exact space-time-dependent correlation functions can

be directly obtained and their dynamic scaling behaviour is analysed for large

classes of initial conditions.

The long-time dynamics of reaction-diffusion processes in low

PACS numbers: 05.20-y, 64.60.Ht

Keywords:

correlation functions

exact results, phase transitions into absorbing states, diffusion,

‡ adresse actuelle: Laboratoire de Physique de la Mati` ere Condens´ ee et Nanostructures, Universit´ e

Claude Bernard Lyon 1 and CNRS, Domaine Scientifique de la Doua, Bˆ atiment L´ eon Brillouin, 43

Boulevard du 11 Novembre 1918, F - 69622 Villeurbanne, France

§ Laboratoire associ´ e au CNRS UMR 7198

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Coagulation-diffusion process in 1D by the empty-interval method2

1. Introduction

The precise description of cooperative effects in strongly interacting many-body

systems continues to pose many challenges.

which may be described in terms of diffusion-limited reaction-diffusion processes.

Applications of these systems and their non-equilibrium phase transitions have arisen

in fields as different as solid-state physics, physical chemistry, physical and chemical

ageing, cosmology, biology, financial markets or population evolution in social sciences.

If the spatial dimension of these systems is low enough, that is d ≤ d∗where d∗is the

upper critical dimension, fluctuation effects dominate the long-time kinetics of these

systems and their behaviour is different from the one expected from the solutions of

(mean-field) reaction-diffusion equations, which attempt to describe the interactions

of the elementary constituents in terms of the macroscopic law of mass-action, see e.g.

[37, 26, 42, 6, 34, 18].

One of the motivations for this work is the continuing practical interest in systems

with reduced dimensionality, and such that homogenisation through stirring is not

possible.Specifically, we shall consider the one-dimensional coagulation-diffusion

process, which is defined as follows. Consider a single species A of indistinguishable

particles, such that each site of an infinitely long chain can either be empty or else

be occupied by a single particle. The dynamics of the system is described in terms

of a Markov process, where allowed two-site microscopic reactions A + ∅ ↔ ∅ + A

and A + A → A + ∅ or ∅ + A are implemented as follows: at each microscopic time

step, a randomly selected single particle hops to a nearest-neighbour site, with a rate

Γ := Da2, where a is the lattice constant. If that site was empty, the particle is placed

there. On the other hand, if the site was already occupied, one of the two particles is

removed from the system with probability one. This model is one of the best-studied

examples of a diffusion-limited process and at least since the work of Toussaint and

Wilczek [46] it is known that the mean particle concentration c(t) ∼ t−1/2for large

times and with an amplitude which is thought to be universal as confirmed by the field-

theoretical renormalisation group [24, 7]; in contrast a mean-field treatment would

have predicted c(t) ∼ t−1. These theoretically predicted fluctuation effects have been

confirmed experimentally, for example using the kinetics of excitons on long chains of

the polymer TMMC = (CH3)4N(MnCl3) [23], but also in other polymers confined to

quasi-one-dimensional geometries [36, 21], see also the reviews in [37]. Another recent

application of diffusion-limited reactions concerns carbon nanotubes, for example the

relaxation of photoexcitations [41] or the photoluminescence saturation [44]. On the

other hand, the 1D coagulation-diffusion process has also received attention from

mathematicians [9, 31] and is simple enough that it can be related to integrable

quantum chains, see [3, 42].Hence, by a consideration of the quantum chain

Hamiltonians, which can be derived from the master equation, the time-dependence

of its observables could in principle be found via a Bethe ansatz, see [40]. In practice,

however, it has turned out to be easier to find the time-dependent densities from

the empty-interval method, which considers the time-dependent probabilities En(t)

that n ≥ 1 consecutive sites of the chain are empty [5, 11, 4, 6, 27], see also [43].

The En(t) satisfy a closed set of differential-difference equations, subject to the

boundary condition E0(t) = 1 and the average particle concentration is obtained

as c(t) =?1−E1(t)?/a. The scaling behaviour of the averages can be directly studied

diffusion equation?∂t− 2D∂2

Paradigmatic examples are systems

in the continuum limit a → 0, when En(t) −→ E(x,t) which in turn satisfies the

x

?E(x,t) = 0 with the boundary condition E(0,t) = 1

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Coagulation-diffusion process in 1D by the empty-interval method

such that the concentration now becomes c(t) = −∂xE(x,t)??

consider instead ρ(x,t) := ∂2

such that standard Green’s functions of the diffusion equation can be used, see [6] and

references therein.

Remarkably, the empty-interval method can be applied to a large class of

coagulation-diffusion models, where several additional reactions can be added, see

e.g.[5, 11, 4, 6, 27, 29, 17, 20, 2, 31, 32, 28].

hamiltonian/Liouvillian of coagulation-diffusion models with the (reversible) reaction

2A ↔ A can, by a stochastic similarity transformation [22, 15, 10], be transformed to

the one of pair annihilation/creation 2A ↔ ∅ which in one dimension can be solved

by free-fermion methods, see e.g. [9, 45, 39, 25, 3, 14, 43, 30]. Those one-dimensional

reaction-diffusion systems which can be treated with free-fermion methods have been

classified [16, 42], but the empty-interval technique has the advantage that further

reactions can be treated, such as ∅ −→ A or A∅A −→ AAA, which have no known

analogue in a free-fermion description. In particular, Peschel et al. [35] suggested a

systematic way to identify observables for which closed systems of equations of motion

can be derived from the reformulation of the master equation in terms of a Hamiltonian

matrix in a controllable way. Their approach includes the method of empty intervals as

the most simple special case. In principle, their method can be extended to include the

probabilities of having several empty intervals of sizes n1,n2,... at certain distances

which allows to find correlation functions as well. Their study is the main subject of

this paper.

In particular, our approach allows to consider arbitrary initial configurations

of particules and hence our results will include many of the existing results in the

literature as special cases. As we shall see, there exists a natural decomposition of

the time-dependent observables which may be arranged in terms of the information

required on the initial state. This can be formulated through single-interval or two-

interval probabilities for those quantities which we consider explicitly. We shall give

examples which suggest a clear order to relevance in the long-time limit. On the other

hand, we shall assume spatial translation-invariance from the outset, which simplifies

the equations to be analysed. However, if one were to investigate the effects of disorder,

one would have to revert to a formalism [11, 4] where translation-invariance is not

required.

The study of correlation functions of reaction-diffusion systems is also motivated

by the recent interest in ageing phenomena: having begun in the study of slow

relaxation in glassy systems brought out of equilibrium after a rapid change in the

thermodynamic parameters, it was later realised that the three main characteristics

of physical ageing, namely (i) slow, non-exponential relaxation, (ii) breaking of time-

translation-invariance and (iii) dynamical scaling also occur in many-body systems

which in contrast to glasses are neither disordered nor frustrated, see [13] for a brief

review and a forthcoming book [19]. Furthermore, these characteristics have also been

found in several many-particle systems with absorbing stationary states, such as the

contact process [12, 38, 8], the non-equilibrium kinetic Ising model [33] or kinetically

constrained systems such as the Frederikson-Andersen model [28]. One particular

point of interest in these ageing systems is the relation between two-time correlations

and responses and a study of the coagulation-diffusion process (along with its exactly

solved extensions) should be useful, since exact results can be expected, at least in

one dimension. However, while such an analysis is readily formulated in terms of

3

x=0. Still, the direct

solution of the problem is usually considered to be complicated enough to prefer to

xE(x,t) where the boundary condition becomes ρ(0,t) = 0

Furthermore, the quantum

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Coagulation-diffusion process in 1D by the empty-interval method4

the empty-interval method at two different times, the explicit calculation requires

the knowledge of the exact equal-time two-interval probabilities. In this paper, we

shall provide this information, which will become an initial condition for the two-time

correlator, and is going to be used in a sequel paper where the ageing behaviour in

exactly solvable reaction-diffusion processes will be addressed.

This paper is organised as follows. In order to make the presentation more self-

contained, we recall in section 2 the derivation of the equation of motion for the

empty single-interval probability En(t) before we proceed to show that the boundary

condition E0(t) = 1 can be fixed through an analytic continuation to negative values

of n. The techniques thereby developed are to be generalized to the two-interval

probability in the remainder of the paper. The passage from the initial state towards

the scaling long-time regime as a function of the initial distrbution is analysed and

we also compare between the discrete model and its continuum limit. In section 3,

the equations of motion and the formal solution is given, to be followed by the

derivation of the consistency conditions with the single-interval probabilities. The

general two-interval probability for arbitrary initial conditions is derived in section 4

and in section 5 we use the results for the derivation of the equal-time correlators.

We conclude in section 6. Several appendices (A-G) contain technical details of the

calculations.

2. Single-interval probability

2.1. Equations of motion

Using the definition of the coagulation-diffusion process as given in the introduction,

we begin by recalling the derivation of the equation of the empty-interval probabilities

[5]. The same equations can also be found within a quantum Hamiltonian formalism

[35], but this will not be repeated here. We denote by En(t) the time-dependent

probability of having an interval of n consecutive empty sites at time t. Since the

system is assumed to be homogeneous, En(t) is site-independent and will depend only

on the interval size n and time t. The time evolution of this quantity is governed by

the rate at which particles move on adjacent intervals of size n or n−1. In an interval

of length n, which will be denoted by

n , a particle (•) can enter from the left or

the right between the time period t and t+dt, and En(t) decreases during this period

of time by the amount

?

The probability Pr(•?

the left of the interval, or Pr(•?

the relation

−

Pr(•?

n ) + Pr( n

?•)

?

n ) is proportional to the probability that a particle lies on

n ) = Pr(• n )Γdt, which can be evaluated using

Pr(• n ) + Pr(◦ n ) = Pr( n ) = En(t)

where the symbol (◦) refers to an empty site. Since by definition Pr(◦ n ) = En+1(t)

we obtain directly

Pr(• n ) = Pr( n •) = En(t) − En+1(t).

En(t) may also increase, if we consider the possibility that a particle sitting next to

an interval of size n − 1 moves away from this interval,

+Pr(?• n-1 ) + Pr( n-1 •?)

(1)

?

?

.

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Coagulation-diffusion process in 1D by the empty-interval method5

This is possible because the process A + A → A constrains each site to contain at

most one particle. Hence there is no need to consider the case when the particle

encounters another particle when it moves away from the interval. As before, we have

Pr(?• n-1 ) = Pr(• n-1 )Γdt =?En−1(t) − En(t)?Γdt. Summing the contributions,

∂tEn(t) = 2Γ

− {En(t) − En+1(t)} + {En−1(t) − En(t)}

= 2Γ(En−1− 2En+ En+1).

This equation is valid only for a positive index n > 1. For n = 1 the rate of change

for E1(t) is given as previously by the equation

?

We also have Pr(•?) = Pr(•)Γdt and Pr(•?◦) = Pr(•◦)Γdt. The solutions for each

of these quantities can be found by considering the probability conditions

the rate of change for En(t) is given by

??

(2)

∂tE1(t)dt = Pr(•?) + Pr(?•) − Pr(•?◦) − Pr(◦?•)

?

.

Pr(•) + Pr(◦) = 1 ⇒ Pr(•) = 1 − E1(t)

Pr(•◦) + Pr(◦◦) = Pr(◦) ⇒ Pr(•◦) = E1(t) − E2(t).

Therefore, the equation for n = 1 is given by

?

In order to be able to write this as the extension of eq. (2) for n = 1, it appears

convenient and is, indeed, common, to introduce the constraint E0(t) = 1. We shall

do the same, but return to this condition below. However, the boundary conditions,

including E(0,t) = 1 were considered to be sufficiently complicated so that an explicit

solution of (5) is usually avoided. Ingenious ways have been developed to extract

physically interesting information, such as the particle-density c(t). We shall require

the explicit form of E(x,t) below when looking for correlation functions and shall now

give it. In the continuum limit, when a is small, we set x = na and E(x,t) = En(t).

The previous relation (2) can be expanded with respect to a and a rescaled hopping

rate D = Γ/a2, which leads to a simple diffusion equation, together with a boundary

condition

(3)

∂tE1(t) = 2Γ1 − 2E1(t) + E2(t)

?

.(4)

∂tE(x,t) = 2D∂xxE(x,t), and E(0,t) = 1.

If we could use a spatially infinite Fourier transform E(x,t) =?+∞

E(x,t) =

−∞

where the integrals over the real axis are unrestricted. In the above expression, a

diffusion length

√8Dt

(5)

−∞

dk

2πexp(ikx)? E(k,t)

E(x′,0),

to solve the previous equation, we would obtain in the standard fashion

?∞

dx′

√π ℓ0

exp

?

−

1

ℓ02(x − x′)2?

ℓ0:=

(6)

acts as the scaling length of the function E(x,t) = E(x/ℓ0).

2.2. Effect of the boundary condition: continuum limit

The simplistic approach outlined at the end of the previous subsection must evidently

be modified in order to take the boundary condition E(0,t) = 1 into account. This

amounts to define in eq. (6) the meaning of the probability E(x′,0) for negative x′

and is achieved by the following result.

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Coagulation-diffusion process in 1D by the empty-interval method6

Lemma 2.1 If one extends the validity of eq. (2) to all n ∈ Z, together with the

boundary condition E0(t) = 1, one has

E−n(t) = 2 − En(t).

In the continuum limit, this leads to E(−x,t) = 2 − E(x,t).

Proof: This is proven by induction. First, we consider the case n = 0. Using eq. (2)

and E0(t) = 1, we obtain

∂tE0(t) = 2Γ(E−1− 2E0+ E1),

which implies E−1(t) = 2E0(t)−E1(t) = 2−E1(t). In the general case, let us consider

the equation of motion for the index −n−1 and use the assumption (7) for the indices

−n and −n + 1:

E−n−1= 2E−n− E−n+1+

(7)

1

2Γ∂tE−n

= 2(2 − En) − (2 − En−1) +

= 2 − En+1

1

2Γ∂t(2 − En)

where the equations of motion (2) were used again. This completes the proof. q.e.d.

In the continuum limit, this relation allows us to rewrite the integral (6) over the

positive axis only

?∞

and where erfc is the complementary error function [1].

Eq. (8) is the general solution for the probability E(x,t) of having an empty

interval, at least of length x and at time t, where the initial state is described by the

function E(x,0). The particle concentration c(t) = Pr(•)/a can be obtained in the

continuum limit from the relation (3):

Pr(•) + Pr(◦) = 1 ⇒ Pr(•) = ac(t) = 1 − E1(t),

where ac(t) = 1 − E1(t) ≃ 1 − E(0,t) − a∂xE(x = 0,t), and therefore

c(t) = − ∂xE(x,t)|x=0.

The function E(x,t) can by definition be written as a cumulative sum of the

probabilities for having bounded on the left, of size at least equal to x′or P(x′,t) =

Pr(• x′

E(x,t) =

x

This imposes two boundary conditions: first, we have E(0,t) =?∞

We can express E(x,t) as function of P(x,0) by performing an integration by

parts of (8):

E(x,t) = 1 −1

2

0

By differentiation with respect to x, we obtain the expression for the concentration

2

√πℓ0

0

From this, all initial conditions, characterised by P(x,0), lead to the long-time

behaviour of the concentration:

E(x,t) = erfc(x/ℓ0) +

0

dx′

√π ℓ0

E(x′,0)

?

e−(x−x′)2/ℓ02− e−(x+x′)2/ℓ02?

. (8)

(9)

):

?∞

dx′P(x′,t).(10)

0dxP(x,t) = 1 by

normalisation. Then, in the limit x → ∞, one must have E(x,t) → 0.

?∞

dx′P(x′,0)

?

erf

?x′+ x

ℓ0

?

− erf

?x′− x

ℓ0

??

. (11)

c(t) =

?∞

dx′P(x′,0)exp

?

−x′2

ℓ02

?

(12)

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Coagulation-diffusion process in 1D by the empty-interval method7

Lemma 2.2 For sufficiently long times and any initial distribution P(x,0), the

concentration decreases as

2

√πℓ0

Proof: Since P(x,0) is a normalised probability distribution,?∞

2

√π

ℓ0

0

?

where the last estimate follows from the large-x behaviour of P(x,0).

c(t) ≃

1

+ o?ℓ0−1?

(13)

0dxP(x,0) = 1, we

must have P(x,0) = o(1/x) for x → ∞. We rewrite eq. (12) as follows

1

−

c(t) =

2

√π

?∞

dxP(xℓ0,0)

?

1 − e−x2?

???

o(1/ℓ0)

(14)

q.e.d.

In a more explicit way, this may be obtained from eq. (12) by a formal expansion of

the exponential. If the second moment of P is well-defined, this leads to the long-time

behaviour

2

√πℓ0

ℓ02

On the other hand, the case of a diverging second moment is illustrated by the example

a0

1 + x1+α,

where 0 < α < 1 and a0= (1+α)sin[π/(1+α)]/π is the normalisation factor. In this

case, a calculation analogous to the proof of lemma 2.2 gives

?

1 + x1+α

?

0

x1+α

which gives the leading correction in the long-time limit as function of the

exponent α. We also notice that if particles occupy each site with probability p, the

concentration is defined by c0= p/a. When the system is filled with a concentration

c0of particles, the function En(0) is proportional to

c(t) ≃

?

1 −?x2?

?

+ o(ℓ0−2). (15)

P(x,0) =

(16)

c(t) =

2

√π ℓ0

?∞

1 −a0

0

dxa0

exp

−x2

ℓ02

?

=

2

√π ℓ0

ℓ0α

?∞

dx1 − e−x2

+ ···

?

,(17)

En(0) ∼ (1 − p)n= (1 − ac0)x/aa→0

However, if the system is entirely filled with particles, p = 1 and En(0) = 0 for

n ?= 0, then, from (8), E(x,t) is simply given by erfc(x/ℓ0), and c(t) = 2/√πℓ0. In

the general case of a given concentration c0 where E0(x) = e−c0x, we simply have

P(x,0) = c0e−c0xand from (12)

?1

In figure 1 we illustrate the effect of several initial empty-interval distributions

E(x,0). Clearly, as expected from the above discussion, all initial distributions lead

to the same long-time asymptotics c(t) ∼ t−1/2but the way this asymptotic regime

is reached depends on the initial state. This can be better understood when plotting

the interparticle distribution function (IPDF)

∂2E(x,t)

∂x2

−→ E(x,0) = exp(−c0x) (18)

c(t) = c0exp

4c2

0ℓ02

?

erfc

?c0ℓ0

2

?

.(19)

p(x,t) :=

1

c(t)

, (20)

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Coagulation-diffusion process in 1D by the empty-interval method8

0.01

1

t

100

0.01

0.1

1

c(t)

exp(-c0x)

(1+c0x)-1

erfc(√ πc0x/2)

012

x

34

0

0.2

0.4

0.6

0.8

1

IPDF (t=0)

Figure 1. Time-evolution of the concentration c(t), for several initial conditions

expressed in terms of E(x,0) and parameter c0= 1/2. For any initial distribution,

the particle concentration shows the same asymptotic behaviour. The passage

between the initial and the asymptotic regime can be qualitatively explained in

terms of the interparticle distribution function (IPDF) defined in eq. (20) and

shown for t = 0 in the inset. See text for details.

which gives the probability density that the next neighbour of a particle is at distance

x at time t [5, 6]. This function is shown for t = 0 in the inset of figure 1. One

observes that in those cases when p(x,0) decays monotonously with x, the transition

to the asymptotic regime is more gradual. On the other hand, in the third case there

is an initial non-vanishing distance the particles must overcome before they can react.

This leads to a very sharp transition between the initial and the asymptotic regimes.

2.3. The discrete case

We now give the solution of the discrete case, without performing a continuum limit.

First we recall the solution of the differential equation (2).

function F(z,t) =

∂tF(z,t) = 2D(z + 1/z − 2)F(z,t), with the solution

The generating

?+∞

n=−∞znEn(t) satisfies as usual the differential equation

F(z,t) = F(z,0)e2D(z+1/z−2)t= e−4Dt

+∞

?

n=−∞

zn

+∞

?

m=−∞

En−m(0)Im(4Dt)(21)

Identifying En(t) in the previous expression, we write

En(t) = e−4Dt

+∞

?

m=−∞

Em(0)In−m(4Dt).(22)

We now must take the boundary condition E0(t) = 1 into account.

continuum case, we replace the summation over negative values of the index m by

As in the

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Coagulation-diffusion process in 1D by the empty-interval method9

0.01

1

t

100

0.01

0.1

1

c(t)

discrete, c0=1

discrete, c0=0.5

discrete, c0=0.1

continuum limit, c0=1

continuum limit, c0=0.5

continuum limit, c0=0.1

Figure 2. Time evolution of the concentration in the discrete case (full curves)

and the continuous limit (dashed lines). The initial concentration c0is [1,0.5,0.1]

from top to bottom.

using the discrete relation E−n(t) = 2 − En(t) and find

?+∞

m=1

In the discrete case, the particle concentration is given by c(t) = 1 − E1(t). Using

summation and recurrence relations over modified Bessel functions, we obtain

?

En(t) = e−4Dt

?

Em(0)

?

In−m(4Dt) − In+m(4Dt)

?

+

+∞

?

m=1

2In+m(4Dt) + I1(4Dt)

?

.(23)

c(t) = e−4Dt

I0(4Dt) + I1(4Dt) −

+∞

?

m=1

m

2DtEm(0)Im(4Dt)

?

, (24)

which generalises earlier results of Spouge [43]. In figure 2, we compare the particle

concentration according to the discrete case eq. (24) with the previously obtained

solution eq. (19) in the continuum limit, for three values of the initial concentration

c0. We used, respectively, the initial distributions En(0) = (1 − c0/a)nand E(x,0) =

e−c0x. As expected, the same asymptotics is found in all cases, independently of

the initial concentration. We observe that the passage between the initial and the

asymptotic regimes is more gradual in the continuum limit. As above, we interpret

this as coming from the fact that in the discrete case the particles must first overcome

a finite distance before then can react.

3. Two-interval probability

In this section, we generalize the previous result and evaluate the probability

En1,n2(d,t) to have two empty intervals, at least of sizes n1 and n2 and separated

by the distance d: we denote it by En1,n2(d,t) = Pr?

En1,n2(d,t) = En2,n1(d,t)

n1

dn2

?. This function is

expected to have the following symmetries

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Coagulation-diffusion process in 1D by the empty-interval method10

En1,0(d,t) = En1(t)

En1,n2(0,t) = En1+n2(t).

andE0,n2(d,t) = En2(t)

(25)

3.1. Equations of motion

As before and using the notations of previous section, we consider the different

possibilities for the variation of En1,n2(d,t) between the time t and t + dt:

?

+ Pr( n1

?

+ Pr( n1− 1 •?d

The probability rates are given by considering the sum rules for static probabilities.

First, we consider the negative contributions for which we obtain the relations

∂tEn1,n2(d,t)dt = −

Pr(•?

n1

dn2 ) + Pr( n1

dn2

?•)(26)

?

?•d − 1

n2 ) + Pr( n1

d − 1•?

n2− 1 •?)

d?• n2− 1 )

n2 )

+Pr(?• n1− 1dn2 ) + Pr( n1

d

n2 ) + Pr( n1

?

.

Pr(•?

Pr(• n1 d

⇒ Pr(• n1

n1

dn2 ) = Pr(• n1

n2 ) + Pr(◦ n1 d

dn2 ) = En1,n2(d,t) − En1+1,n2(d,t)

dn2 )Γdt,

n2 ) = Pr( n1

dn2 )

and

Pr( n1

?•d − 1

• d − 1

• d − 1

n2 ) = Pr( n1

• d − 1

◦ d − 1

n2 )Γdt,

Pr( n1

n2 ) + Pr( n1

n2 ) = Pr( n1 dn2 )

⇒ Pr( n1

n2 ) = En1,n2(d,t) − En1+1,n2(d − 1,t).

For the positive contibutions, we have

Pr(?• n1− 1

Pr(• n1− 1

Pr(• n1− 1

and

dn2 ) = Pr(• n1− 1

n2 ) + Pr(◦ n1− 1

dn2 ) = En1−1,n2(d,t) − En1,n2(d,t),

dn2 )Γdt,

ddn2 ) = Pr( n1− 1dn2 )

⇒

Pr( n1− 1 •?d

Pr( n1− 1 • d

⇒ Pr( n1− 1 • d

(similarly for the other terms which are symmetric).

contributions, we finally find (the time variable is from now on suppressed)

n2 ) = Pr( n1− 1 • d

n2 ) + Pr( n1− 1 ◦ d

n2 ) = En1−1,n2(d + 1,t) − En1,n2(d,t),

n2 )Γdt,

n2 ) = Pr( n1− 1d + 1n2 )

After gathering all the

∂tEn1,n2(d) = Γ[−8En1,n2(d)

+ En1+1,n2(d) + En1,n2+1(d) + En1−1,n2(d) + En1,n2−1(d)

+ En1+1,n2(d − 1) + En1,n2+1(d − 1) + En1−1,n2(d + 1) + En1,n2−1(d + 1)].

We have checked that the same closed system of equations of motion is also obtained

when the master equation is rewritten in terms of a quantum Hamiltonian [35].

(27)

Page 11

Coagulation-diffusion process in 1D by the empty-interval method 11

The continuum limit of this diffusion equation is obtained by expanding the terms

up to the second order in the lattice step a (the a2will be absorbed in Γ). Setting

x = n1a, y = n2a and z = da we obtain the following linear differential equation:

?

∂tE(x,y,z) = 2D∂2

x+ ∂2

y+ ∂2

z−

?

∂x∂z+ ∂y∂z

??

E(x,y,z).(28)

3.2. General solution

The general solution for (28) is obtained by diagonalising the quadratic form associated

with the differential operator?P := ∂2

X = α(x + y +√2z) , Y = β(x + y −√2z) , Z = γ(x − y),

with the positive constants

√2)]−1/2, β = [2(2 +

such that α2+ β2= 1 and α2− β2= 1/√2.

In the new variables, the operator?P = ∂2

the boundary conditions are ignored for a moment, the function E(x,y,z) can be found

via a Fourier transformation, and explicitly expressed as a kernel integral depending

on the initial conditions E0(x,y,z) := E(x,y,z,0) :

√2

(√π ℓ0)3

−∞

where the Gaussian kernel W(u,v,w) is given by

1

ℓ02

1

ℓ02

the Jacobian of the transformation (29) being equal to 4√2αβγ =√2.

x+∂2

y+∂2

z−∂x∂z−∂y∂z. We find that the following

change of variables (x,y,z) → (X,Y,Z) diagonalisesˆP:

(29)

α = [2(2 −

√2)]−1/2, γ = 1/√2,

X+ ∂2

Y+ ∂2

Zis diagonal. As in the

previous section (see (6)) for the one-interval problem, and in the continuum limit, if

E(x,y,z) =

?∞

dx′dy′dz′W(x − x′,y − y′,z − z′)E0(x′,y′,z′)(30)

W(u,v,w)= exp

?

?

− α2(u + v +√2w)2− β2(u + v −√2w)2− γ2(u − v)2?

− (u + v + w)2− w2−1

= exp

2(u − v)2?

,(31)

3.3. Compatibility conditions

As expressed before by equation (7), it is important to make a correspondence

between intervals of formally negative and positive lengths.

in the formal solution (30) which requires real variables, whereas the probability

E(x,y,z) has an obvious physical meaning depends for positive distances only. We

have to consider 3 cases, depending on whether x, y or z are negative or positive. By

symmetry considerations (25), it is only necessary to consider the case where x or z

are individually negative (the case y < 0 being deduced by means of the first equation

(25)), the other variables being positive. The explicit evaluation of eq. (30) in the

following sections requires several identities, stated as lemmata for clarity and proven

in appendix A and B, respectively. The first one treats the case of formally negative

interval lengths.

This will be needed

Page 12

Coagulation-diffusion process in 1D by the empty-interval method12

Lemma 3.1 The probability of two-empty-intervals of negative lengths is related to

the probability of positive lengths as follows.

E−n1,n2(d) = 2En2− En1,n2(d − n1),

En1,−n2(d) = 2En1− En1,n2(d − n2),

E−n1,−n2(d) = 4 − 2En1− 2En2+ En1,n2(d − n1− n2).

A further relation connects the negative separations −d between two intervals to the

positive ones.

(32)

(33)

Lemma 3.2 We have

En1,n2(−d) = 2En1+n2−d− En1−d,n2−d(d).

Later on, we shall require these results in the continuum limit, where the

expressions (7), (32), (33) and (34) take the following form

(34)

E(−x) = 2 − E(x),

E(−x,y,z) = 2E(y) − E(x,y,z − x),

E(x,−y,z) = 2E(x) − E(x,y,z − y),

E(−x,−y,z) = 4 − 2E(x) − 2E(y) + E(x,y,z − x − y),

E(x,y,−z) = 2E(x + y − z) − E(x − z,y − z,z).

This allows us to rewrite (30) in the restricted domain where (x′,y′,z′) are all positive,

and where E0(x′,y′,z′) is physically well-defined.

(35)

4. General solution for E(x,y,z,t)

From the general equation (30), we separate the 8 different domains of integration

around the origin for example (x′> 0,y′> 0,z′> 0), (x′< 0,y′> 0,z′> 0) etc..,

and use relations (7), (32), (33) and (34) to map all domains into the single domain

(x′> 0,y′> 0,z′> 0). This calculation is done in the appendix C and, here, we just

summarize the results. The general solution can be decomposed as follows:

E(x,y,z,t) = E(0)(x,y,z,t) + E(1)(x,y,z,t) + E(2)(x,y,z,t),(36)

where E(0)(x,y,z,t) is obtained from the terms independent of the initial conditions,

E(1)(x,y,z,t) from the initial one-interval probability E0(x′) and E(2)(x,y,z,t) from

the initial two-interval probability E0(x,y,z), respectively. Note that E(1)(x,y,z,t)

and E(2)(x,y,z,t) depend on E0(x′), E0(x′,y′,z′) with arguments positive, hence

this gives us the physical answer to the diffusion process in the coagulation problem

starting from arbitrary initial conditions and constraints on the differential equation.

We now analyse these three terms one by one.

4.1. Special case of a system initially entirely filled with particles

We notice that (C6), (C7), and (C8) contain initial conditions for the single-interval

distribution E0(x′), some constants independent of the initial conditions, and initial

conditions for the two-interval distribution E0(x′,y′,z′). To simplify notations, we

shall re-scale all lengths by ℓ0such that Eℓ0(x,y,z) = E(xℓ0,yℓ0,zℓ0,t). In (C8), we

can isolate from E0(−x′,−y′,z′) two terms independent of the initial conditions,

4 − 4θ(y′− z′)θ(x′− z′)θ(x′+ y′− z′).

Page 13

Coagulation-diffusion process in 1D by the empty-interval method13

It is obvious that θ(y′− z′)θ(x′− z′)θ(x′+ y′− z′) = θ(y′− z′)θ(x′− z′). These two

terms, plus the first term in (C3), give a contribution to the general function equal to

E(0)

ℓ0(x,y,z) = erfc(z)erfc(x + y + z)

?

+

Performing the translations x′− z′→ x′and y′−z′→ y′in the last contribution, we

obtain

E(0)

ℓ0(x,y,z) = erfc(z)erfc(x + y + z)

?

+

Using the relation (C4), and the identity

+

2

π3

?

R3

dx′dy′dz′ ˜ Wℓ0(−x′,−y′,z′){4 − 4θ(y′− z′)θ(x′− z′)}.

+

2

π3

?

R3

dx′dy′dz′?

˜

Wℓ0(−x′,−y′,z′) −˜

Wℓ0(−x′− z′,−y′− z′,z′)

?

.

Wℓ0(x − x′− z′,y − y′− z′,z + z′) = Wℓ0(x − x′,y − y′,z − z′)e−4zz′

⇒˜

we can rewrite E as

E(0)

ℓ0(x,y,z) = erfc(z)erfc(x + y + z)

?

+

= erfc(z)erfc(x + y + z)

?

+

The integral over z′now gives a gaussian exponential.

integrals can also be carried out explicitly (see appendix G). Introducing again the

diffusion length ℓ0, we find

?x

− erfc

ℓ0

This is the exact two-interval probability in the case of an initially fully filled lattice

(where both E0(x) and E0(x,y,z) vanish). In particular, the solution (38) satisfies

the symmetry conditions (25).In the limit of z large, one has the factorisation

E(x,y,z,t) ≃ E(x,t)E(y,t).

The remaining terms of the full solution depend on the initial conditions. They

are of two kinds, and involve either the single-interval or else the two-interval initial

probabilities. We turn to them now.

Wℓ0(x′+ z′,y′+ z′,−z′) = −˜ Wℓ0(x′,y′,z′)(37)

+

32

π3

?

R3

dx′dy′dz′?

˜

Wℓ0(−x′,−y′,z′) +˜ Wℓ0(−x′,−y′,−z′)

?

+

32

π3

?

R2

dx′dy′

?

R

dz′ ˜

Wℓ0(−x′,−y′,z′)

Then the two remaining

E(0)(x,y,z) = erfc

ℓ0

?x + z

?

erfc

?y

?

ℓ0

?

+ erfc

?y + z

?z

?

ℓ0

?

erfc

?x + y + z

ℓ0

?

erfc

ℓ0

.(38)

4.2. Contributions to E(x,y,z,t) from terms with a single-interval initial distribution

The contributions to E(x,y,z,t) of single-interval distributions come from the previous

relations (C6), (C7) and (C8), where we can isolate the following individual terms

• the second term of equation (C3)

• the first 3 terms in (C6) and (C7)

Page 14

Coagulation-diffusion process in 1D by the empty-interval method14

• terms 2, 3, 4 in (C8).

On the whole, there are 10 terms contributing to the initial conditions given for a

given choice of E0(x′). Gathering these terms and performing successive translations

in x′, y′or z′when necessary, we obtain

?∞

?

R3

+

?

?

+˜ Wℓ0(−x′− z′,−y′− z′,z′) −˜ Wℓ0(−x′,−y′,z′)

?

+˜ Wℓ0(−y′− z′,−x′− z′,z′) −˜ Wℓ0(−y′,−x′,z′)

?

+θ(−x′− y′+ z′)[˜

?

+θ(−x′+ y′+ z′)˜

?

By performing partial translations and integrations and simplifying all terms from I1

to I6(see appendix D for details), we obtain

?∞

+erfc(x)

− erfc(x + z)

+erfc(y)

− erfc(y + z)

+erfc(x + y + z)

?∞

where the kernel K1,ℓ0is positive. When z = 0, we recover the result (8) for a single-

interval distribution of size x + y. For some functions E0(x′), the previous integrals

can be performed exactly since K1is gaussian in the variable x′. For example, if we

take as initial function E0(x) = e−c0x, where c0is an initial concentration of particles,

we find

?z

+erfc

ℓ0

ℓ0

with the following abbreviation

?∞

E(1)

ℓ0(x,y,z) = erfc(z)

?

0

dx′

√πE0,ℓ0(x′)

?

e−(x+y+z−x′)2− e−(x+y+z+x′)2?

Wℓ0(x′,−y′,z′) −˜ Wℓ0(x′− z′,−y′− z′,z′)

??

?

?

?

Wℓ0(y′,−z′,x′) +˜

??

Wℓ0(−z′,−y′,x′)

??

+

8

π3

dx′dy′dz′E0(x′)

?

˜

?

I1

+˜ Wℓ0(−y′,x′,z′) −˜ Wℓ0(−y′− z′,x′− z′,z′)

??

I2

??

??

I3

I4

Wℓ0(−z′,y′,x′)]

?

I5

?

I6

?

(39)

E(1)

ℓ0(x,y,z) =

0

dx′

√πE0,ℓ0(x′)

?

erfc(z)

?

e−(x′−x−y−z)2− e−(x′+x+y+z)2?

?

?

?

?

e−(x′−y)2− e−(x′+y)2?

e−(x′−x)2− e−(x′+x)2?

?

dx′

√πE0,ℓ0(x′)K1,ℓ0(x′;x,y,z),

e−(x′−y−z)2− e−(x′+y+z)2?

e−(x′−x−z)2− e−(x′+x+z)2?

e−(x′−z)2− e−(x′+z)2??

=:

0

(40)

E(1)(x,y,z) = erfc

?y

ℓ0

?

Fc0(x + y + z) + erfc

?y + z

?x

ℓ0

?

Fc0(y) − erfc

?x + y + z

?x + z

?

ℓ0

?

Fc0(y + z)

?

Fc0(x) − erfc

?

Fc0(x + z) + erfc

ℓ0

Fc0(z),(41)

Fc0(x) :=

0

dx′

ℓ0√π

e−c0x′?

e−(x′−x)2/ℓ02− e−(x′+x)2/ℓ02?

(42)

Page 15

Coagulation-diffusion process in 1D by the empty-interval method

?

The limiting values of this function read

15

=1

2eℓ02c2

0/4

e−c0xerf

?x

ℓ0

−ℓ0c0

2

?

+ ec0xerf

?x

ℓ0

+ℓ0c0

2

?

− 2sinh(c0x)

?

.

Fc0(x)

x→∞

≃

ec2

0ℓ02/4−c0x, Fc0(x)

x→0

≃

?

2

ℓ0√π− c0ec2

0ℓ02/4erfc(ℓ0c0/2)

?

x, (43)

and F0(x) = erf(x/ℓ0).

In the long time limit, the above expression goes to zero like 1/ℓ03

4x

√πc2

and the dominant part of contribution E(1)(x,y,z,t) in the same limit behaves like

E(1)(x,y,z) ≃4(x + y)

It is interesting to compare this expansion with the long-time limit expansion of

E(0)(x,y,z), which is independent of c0

Fc0(x) ≃

0ℓ03

?

1 −

6

c2

0ℓ02−x2

ℓ02+ ...

?

(44)

√πc2

0ℓ03.(45)

E(0)(x,y,z) ≃ 1 −2(x + y)

√πℓ02+

2

?

(x + y)3+ 6xyz

3√πℓ03

?

.(46)

The first term (45) tends to increase the two-interval probability by a factor

independent of the distance, since there are less particles in the system for a finite

concentration of particles.

4.3. Contributions to E(x,y,z) of two-interval initial distributions

As noticed previously, we can isolate from equations (C6), (C7), and (C8) terms

involving E(x′,y′,z′,t = 0) = E0(x′,y′,z′). In particular

• 1 term E0(x′,y′,z′) when all variables are positive, in combination with

˜

Wℓ0(x′,y′,z′)

• 2×3 terms in (C6) and (C7)

• 5 terms in (C8).

When combining and simplifying these terms (see appendix E for details), we obtain

a reduced integral form for E(2)

ℓ0(x,y,z), as function of a kernel K2,ℓ0(x′,y′,z′;x,y,z)

?

+

with

K2,ℓ0(x′,y′,z′;x,y,z) = [1 − e−4(x′+y′+z′)(x+y+z)](1 − e−4z′z)

+e−4x′x−4y′y[1 − e−4z′(x+y+z)][1 − e−4(x′+y′+z′)z]

−e−4x′x[1 − e−4(y′+z′)(x+y+z)][1 − e−4(x′+z′)z]

−e−4y′y[1 − e−4(x′+z′)(x+y+z)][1 − e−4(y′+z′)z]

+e−4x′x−4z′(x+z)[1 − e−4y′(x+y+z)](1 − e−4x′z)

+e−4y′y−4z′(y+z)[1 − e−4x′(x+y+z)](1 − e−4y′z).

It is important to notice for the following sections that K2(x′,y′,z′,0,0,z) =

∂xK2(x′,y′,z′,x,y,z)|x=y=0= ∂yK2(x′,y′,z′,x,y,z)|x=y=0= 0.

E(2)

ℓ0(x,y,z) =

2

π3

?

R3

dx′dy′dz′E0,ℓ0(x′,y′,z′)Wℓ0(x − x′,y − y′,z − z′)K2,ℓ0(x′,y′,z′;x,y,z)

(47)