# Ballistic annihilation in one dimension : A critical review

Soham Biswas  Departamento de Física, Universidad de Guadalajara, Guadalajara,
Jalisco, Mexico<EMAIL_ADDRESS>Francois Leyvraz  Instituto de
Ciencias Físicas, Universidad Nacional Autónoma de México, Mexico
<EMAIL_ADDRESS>Centro de Ciencias Físicas, Cuernavaca, Morelos,
Mexico

###### Abstract

In this article we review the problem of reaction annihilation
$A+A\rightarrow\emptyset$ on a real lattice in one dimension, where $A$
particles move ballistically in one direction with a discrete set of possible
velocities. We first discuss the case of pure ballistic annihilation, that is
a model in which each particle moves simultaneously at constant speed. We then
review ballistic annihilation with superimposed diffusion in one dimension.
This model consists of diffusing particles each of which diffuses with a fixed
bias, which can be either positive or negative with probability $1/2$, and
annihilate upon contact. When the initial concentration of left and right
moving particles is same, the concentration $c(t)$ decays as $t^{-1/2}$ with
time, for pure ballistic annihilation. However when the diffusion is
superimposed decay is faster and the concentration $c(t)\sim t^{-3/4}$. We
also discuss the nearest-neighbor distance distribution as well as crossover
behavior.

##  I Introduction

The dependence of chemical kinetics on the dynamics of the reacting particles
has attracted considerable attention review ; book . In all these works, one
of the most important objective is to determine how the laws of motion
influence the space-time evolution of the species concentrations. For
irreversible diffusion-controlled reactions, when reactants diffuse and react
on contact, the reaction creates spatial correlations that invalidate the
mean-field approximation and hence the applicability of rate equations. For
instance, consider the case of one-species annihilation (or aggregation) where
one species reacts with itself via the bi-molecular reaction,

$A+A\mathop{\longrightarrow}_{K}C,$ (1)

where $C$ is an inert species. The rate equation reads :

$\dot{c}_{A}=-Kc_{A}^{2},$ (2)

According to the rate equation the $A$ concentration $c_{A}(t)$ decays as
$1/t$ at long times. However for the same process where $A$ walkers diffuse
randomly in one dimension, the concentration decay is given by $(Dt)^{-1/2}$
spouge . Also the amplitude in Eq.(2) depends on the initial concentration,
whereas the the amplitude of the correct $1/\sqrt{t}$ decay only depends on
the diffusion constant.

Beyond the immediate interest of such models for chemical reactions, the
dynamical behaviour of particle annihilation ($A+A\rightarrow\emptyset$) and
particle aggregation ($A+A\rightarrow A$) have a direct one-to-one
correspondence with the kinetic Ising and Potts model in one dimension. In
that case, the zero-temperature Glauber dynamics can be mapped to a random
walk problem for both the $q$ state nearest neighbour Potts model and Ising
model ($q=2$). In both cases domain walls perform random walks, and whenever
the walkers meet, they either coagulate ($A+A\rightarrow A$) or annihilate
($A+A\rightarrow\emptyset$). Two domain walls will annihilate each other by
contact, if the two spins that surround that domain wall are in same state.
Specifically, for the Ising model ($q=2$ state Potts model) the domain walls
always annihilate. For the $q$-state Potts model with random initial
conditions, they will annihilate with a probability $\dfrac{1}{q-1}$ and will
coagulate with probability $\dfrac{q-2}{q-1}$. In either case the domain walls
perform pure random walks. The exponent for the concentration decay is
identical to that of the domain growth exponent Francois  or ordering exponent
bnnni . However for several binary opinion dynamic models (which can be
directly mapped to the Ising spin system) in one dimension, the corresponding
walker pictures are much more complicated than a simple random walk. Sometimes
the motion of the $A$ walkers can also have ballistic components opn . Hence
understanding the ballistic annihilation with and without superimposed
diffusion could also be useful for understanding these more complex dynamics.

Here in this article we review the problem of ballistic annihilation in one
dimension. The main aim is to determine the decay law for the concentration of
the particles for this single species reaction-annihilation problem. First, in
Section II we discuss the ballistic annihilation ($A+A\rightarrow\emptyset$)
where the $A$ particles move in a purely ballistic way on the real line with
velocities which can take only two values, $v$ and $-v$. We assume the
particles to have always one of a discrete set of velocities. This model can
also be simulated on a lattice, if we assume that all particles always move in
one direction and if their positions are updated synchronously. We shall thus
refer to this model as the synchronous update model. In section III, we talk
about the ballistic annihilation with superimposed diffusion in one dimension.
This model can be obtained, for instance, by modifying the update rule of the
previous model to being asynchronous. The conclusion and discussions are
stated in the final section.

##  II Synchronous Ballistic annihilation

###  II.1 Particles with only two velocities $+v$ and $-v$

The purely ballistic model, that is the Ballistic annihilation under the
synchronous update rule was first discussed by Elskens and Frisch balsyn . In
this model, particles are initially distributed randomly on a one dimensional
lattice and move freely with independent initial velocities. In this single
species annihilation problem whenever two particle meet they annihilate
($A+A\rightarrow\emptyset$). The velocity distribution function, that is, the
probability that the velocity is $v$ or $-v$,can be written as

$D(w)=q\delta(w+v)+p\delta(w-v),\qquad p+q=1$ (3)

The main interest is to determine the decay law for the concentration decay of
the particles.

We will now look at the problem in a little more detail. Initially let the
position and velocity of the $k$th particle be $x_{k}$ and $v_{k}$. The
particle positions are initially random and the velocities are independent.
What we want to calculate is the average survival probability $S(t)$ of the
particles at time $t$, as the concentration decay is identical to the decay of
survival probability with time.

Figure 1: This shows a schematic picture for the space–time evolution of the
system of particles going through the process of pure ballistic annihilation,
where the distribution of the velocities are given by equation (3), that is,
where the particles can have only two velocities $+v$ or $-v$. Vertical
direction is the direction of time.

In this case two particles which are moving in the same direction move with
the same velocity and thus cannot annihilate (Figure. 1a). Since, because of
the annihilation process, particles cannot cross each other’s trajectories ,
it follows that the entire distribution of velocities of surviving particles
to the right of a given surviving particle (let us denote the index by $k$) is
independent of the whole distribution to its left.

The time at which any given particle, say a right moving particle, will
annihilate is readily determined as follows, if the initial conditions are
given: let us denote the given particle by the index 0 and place it at the
origin. We further assign the index $k$ to the $k$’th particle to the right
which is located at $x(k)$, and compute

$S(k)=\sum_{l=0}^{k}\rm{sign}(v_{k})$ (4)

Denote by $k_{0}$ the smallest value of $k$ such that $S(k_{0})=0$. Note that
$k_{0}$ is necessarily even. We then find that the annihilation time for the
particle 0 is given by

$T_{0}=x(k_{0})/(2c)$ (5)

The probability density $p(t)dt$ for $T_{0}$ is thus given by

$p(t)\,dt=\sum_{k=1}^{\infty}\mbox{\rm Prob}\left[S(k)=0\mbox{\rm\ for the
first time}\right]\cdot\mbox{\rm
Prob}\left[2vt\leq\sum_{l=1}^{k}\left(x_{l+1}-x_{l}\right)\leq
2c(t+dt)\right]$ (6)

This sum is readily evaluated as follows: the first factor in the product is a
well-known problem in probability, it is the so-called gambler’s ruin problem,
whereas the second is simply the expression for the distribution of the
distance of the $k$’th neighbour to the origin, given the nearest neighbor
distance distribution. By the law of large numbers, if this distribution has a
finite mean $\langle l\rangle$ and variance $\sigma^{2}$, the second factor,
for large $k$, is a Gaussian of average $k\langle l\rangle$ and variance
$k\sigma^{2}$.

The behaviour of the first factor is somewhat non-trivial but well-known, see
for instance feller . When $p=q=1/2$, the asymptotic behaviour of this
quantity goes a a power law $k^{-3/2}$. It corresponds to the probability that
an unbiased random walk remains to the right of the origin for $k$ steps. On
the other hand, if $p>q$, the random walk is biased and with probability one
eventually remains forever on the right side of the origin, in other words,
right-going particles have a finite probability of surviving indefinitely.

For $p=1/2$ and an initial exponential nearest neighbor distance distribution,
the expression [6] can be evaluated without any approximation balsyn . The
survival probability is given by

$S(t)=\int_{t}^{\infty}p(t^{\prime})\,dt^{\prime}=e^{-2\rho vt}[I_{0}(2\rho
vt)+I_{1}(2\rho vt)]$ (7)

where $I_{0}$ and $I_{1}$ are the modified Bessel’s functions and
$\rho=\langle l\rangle^{-1}$. Asymptotically for $t\rightarrow\infty$, $S(t)$
decays in a power law fashion:

$S(t)=\dfrac{1}{\sqrt{\pi\rho vt}}\left[1+O(t^{-1})\right]$ (8)

For $p>q$, on the other hand, the expressions for the probability of a random
walk remaining for $k$ steps to the right of the origin become significantly
more intricate and an explicit expression cannot be found, The asymptotic
behaviour, however, is readily evaluated: it is an exponential decay to zero
of the concentration of the minority species (in this case, the left-going
particles) and a corresponding exponential saturation of the majority species
concentration to an asymptotic non-zero value of $p-q$. This exponential
approach is further corrected by a power-law factor:

$S(t)-S_{\infty}\simeq(pq)^{1/4}(\rho vt)^{-3/2}\dfrac{\exp(-2r\rho
vt)}{r\sqrt{8\pi}}$ (9)

where $r=1-2\sqrt{pq}$ is a quantity that measures the strength of the bias.

For the case $p=q=1/2$, one can physically argue the decay law at long time as
follows: $S(t)$ is mostly determined by the difference between the positive
and negative velocity particles in an interval of time $2vt$. As the particles
are initially distributed over the lattice independently with density $\rho$,
the difference between the positive and negative velocity particles is of the
order of $(2\rho vt)^{-1/2}$ for the population of order $2\rho vt$. Hence the
concentration of the particles at time $t$ (for $t\rightarrow\infty$) is given
by

$c(t)=\rho S(t)\simeq\sqrt{\dfrac{c(0)}{\pi vt}},$ (10)

where $c(0)=\rho$ is the initial concentration of the particles.

We can expand the argument given in the previous paragraph to give a simple
derivation for the concentration decay using scaling analysis baldif . If the
size of the one dimensional lattice is $L$, the initial number of particles
are $c(0)L$. The difference between the number of left-moving and right-moving
particles is of the order of

$\Delta N=|N_{+}-N_{-}|\sim\sqrt{N}.$ (11)

At long time ($t\rightarrow\infty$) all the particles which belong to the
minority-velocity species are annihilated and hence
$c(L,t\rightarrow\infty)\sim\Delta N/L\sim(c(0)/L)^{1/2}$. We can assume a
scaling form for the concentration

$c(L,t)\sim(c(0)/L)^{1/2}g(z),$ (12)

with $z=L/vt$. Following the above argument, we have

$g(z)\rightarrow\mbox{\rm constant}\qquad(t\to\infty,z\to 0).$ (13)

On the other hand in the short time limit, that is for $z\rightarrow\infty$,
the concentration cannot depend on the size of the lattice, which implies

$g(z)\propto z^{1/2}$ (14)

Thus we find

$c(t)\sim\sqrt{\dfrac{c(0)}{vt}}$ (15)

an expression similar to that of the solution given by equation (10).

###  II.2 The case of three velocities

Matters become rather more intricate in the case of three velocities. If we
have particles moving with velocities $v_{1}$, $v_{2}$, and $v_{3}$ we may
suppose, using the Galilean invariance of the process, that one of the
velocities is equal to zero, say the intermediate velocity. To limit the
number of parameters, let us assume that the two extreme velocities are
opposite, that is, we have velocities $\pm v$ and $0$. Let the probability
distribution be

$D(w)=p_{-}\delta(w+v)+p_{0}\delta(w)+p_{+}\delta(w-v)\qquad(p_{-}+p_{0}+p_{+}=1).$
(16)

This system had been studied numerically in ball3v . Qualitatively, the
behaviour is the following: whenever the concentrations of one type of
particles clearly dominates all others, only those particles that have higher
concentrations will eventually survive. The phase diagram is therefore divided
in three phases, in which either the $+$ particles, the $-$ particles or the
immobile particles survive. Each phase has a boundary with the other two, and
the three have only one point in common, which is numerically found to be very
close to $p_{\pm}=3/8$ and $p_{0}=1/4$.

We now describe the behaviour of each species in different regions of the
phase diagram. Within the regions in which only $\pm$ or immobile particles
survive, the approach to equilibrium is exponential. In the boundary
separating the $+$ from the $-$ region, we have

$c_{\pm}(t)\simeq t^{-1/2}\qquad c_{0}(t)\simeq t^{-1}.$ (17)

In other words, all particles disappear, but the immobile particles do so much
faster than the moving particles. If this is assumed, it is clear that the
decay of the $\pm$ particles will asymptotically not be affected by the
immobile ones, so that the problem is reduced to the case of two velocities
treated in the previous section. On the boundaries separating the $+$ or the
$-$ phase from the immobile phase, the behaviour is unclear. Finally, at the
tricritical point where all regions meet, namely $p_{0}=1/4$ and
$p_{\pm}=3/8$, all particles again disappear, but all with the same exponent,
which is found to be $2/3$ with high numerical accuracy.

These same results were later confirmed analytically ball3v1 ; ball3v2 . The
essential argument described in ball3v1  involves the factorization of the
distribution of velocities to the left and to the right of any given particle.
This implies that the nearest neighbour distance distributions obey a closed
hierarchy of equations. Note that this is similar, but by no means identical,
to the usual hierarchies involving correlation functions hierarchy . The
distribution of nearest-neighbour distances involve correlation functions of
arbitrarily high order. The quantities used in this approach are the nearest-
neighbour distributions, and these factor for reasons related to those we have
already discussed.

The approach is not straightforward, however. In principle, it appears that
any ballistic aggregation system can be in principle analyzed in this manner.
However, even the detailed analysis of the three-velocity ballistic
annihilation problem studied numerically in ball3v  is an amazingly involved
tour de force. An altogether different problem is the case of ballistic
aggregation in the case of continuous velocity distributions. This was treated
within a Boltzmann equation approximation in ballcontinuous . The results fit
closely the numerics.

##  III Ballistic Annihilation with superimposed diffusion

###  III.1 Description of the model

In this section, we review the problem of ballistic annihilation with
superimposed diffusion for a the two-velocity case, in which particles can
have only the velocities, $\pm v$ Initially the fraction of positive velocity
particles are $p=1/2+\varepsilon$ and the negative velocity particles are
$q=1/2-\varepsilon$. The particles are randomly distributed over the one
dimensional lattice as usual. Most importantly, additionally to the ballistic
particle motion we have a diffusive motion for the particle. This may, for
instance, arise from an asynchronous update for the Monte Carlo Simulation. At
each time step one particle is chosen at random and moved in the direction of
the corresponding velocity of that particle. After $N$ such update one Monte
Carlo time step is over, if the total number of particles over the lattice is
$N$. We consider the one species annihilation, that is whenever two particles
meet they annihilate ($A+A\rightarrow\emptyset$) baldif ; shf .

The crucial difference between this model and the purely ballistic model is
the possible annihilation between particles having the same velocity. In other
words, the purely ballistic model involves two particle species (left-going
and right-going) which cannot react. In the model with diffusion, however, the
particles of the same species have another mechanism, namely diffusion, that
allows them to annihilate. Since the two mechanisms scale differently,
however, the resulting model is in a different universality class from either
the ballistic or the purely diffusive model. A trivial example of the
difference is given by the case in which $p$ is significantly larger than $q$:
in a time of order one, the purely ballistic model reaches a limiting
concentration, whereas the model involving diffusion decays as $t^{-1/2}$.

For $\varepsilon=0$, that is, for $p=q$, at the beginning, the numbers of
positive and negative velocity particles are equal on an average. When
$\varepsilon\neq 0$ ($-1/2\leq\varepsilon\leq 1/2$) introduces inequality in
the initial numbers of positive and negative velocity particles, and
$\varepsilon=\pm 1/2$ means there is only one kind of particle.

Unlike in the previous section, we study this problem in discrete time and at
each time step we choose a particle at random and move it one step to the
right (or left) if the velocity of the particle is positive (or negative). In
the asynchronous update, the random choice of particle leads to an effective
diffusive motion of the particles with respect to their neighbouring particles
that have the same speed. This asynchronous update rule introduces diffusion
so that, as remarked above, two particles with the same velocity can
annihilate.

Figure 2: A schematic picture with two crossovers for the decay of the
concentration $c(t)$ for the total number of particles with time $t$.

Let us here summarize the effects of this apparently minor change in the
update rule. These are non-trivial and profound. For $\varepsilon>0$
($\varepsilon\ll 1$), there are three different regimes [Figure. 2]. In the
first stage we can neglect the concentration difference between left- and
right-going particles. The common concentration $c(t)$ then decays, as we will
show, as $t^{-3/4}$. This regime ends when the minority ( that is left-going)
particles begin to have as significantly different concentration fro the
majority. This is followed in short order by their complete disappearance.
This happens at a time $t_{1}(\varepsilon)$ of the order of $\varepsilon^{-2}$
[Fig 2]. At this point, starts the second stage which is characterized by the
decay of concentration as $t^{-1/4}$ [Fig 2]. This is due to the fact that the
distribution of the positions of the surviving right going particles over the
one dimensional lattice are far from being uniform, leading to an anomalous
decay alemany . Finally, a third stage set in, when the distribution of the
particles over the lattice becomes uniform due to the presence of diffusion in
the system, which happens at a time $t_{2}(\varepsilon)$ that scales as
$\varepsilon^{-4}$ [Fig 2]. In this third stage, the system contains only the
right-moving particles moving ballistically, with superimposed diffusion. Due
to Galilean invariance, this is equivalent to pure diffusive dynamics. Hence
the concentration decay as $(Dt)^{-1/2}$ asymptotically.

Even for $\varepsilon=0$, there will be an inequality in the right and left
going particle due to the random distribution of the particles at the
beginning. For $\varepsilon=0$,the concentration of an excess number of
particles, $c_{ex}(0)\simeq 1/\sqrt{N}$ (up to an $O(1)$ factor) where $N$ is
the initial number of particles. Hence we can see both the first and second
stage there, but the crossover time diverges with the system size. In the next
subsection we will discuss the initial $t^{-3/4}$ decay using dimensional
analysis mostly for $\varepsilon=0$.

###  III.2 Dimensional analysis for $\varepsilon=0$ : Infinite system size

Due to the convection, after some time, particles organize themselves into
domains of right and left-moving particles. However because of asynchronous
update rule, inside each domain of same-velocity particles, diffusive
annihilation takes place. One could therefore with some plausibility argue
that the diffusive annihilation mechanism leads to an effective time-dependent
initial concentration $c_{0}(t)\sim(Dt)^{-1/2}$, which plays the role of
$c(0)$ in Equation (15), leading to the $t^{-3/4}$ decay of the concentration
with time.

However in this subsection we present a more thorough approach to this system.
Specifically, we shall use dimensional analysis, or scaling, to derive the
$t^{-3/4}$ decay law, for $\varepsilon=0$. We follow the argument presented in
shf . We define two dimensionless parameters:

$\displaystyle x=\dfrac{v}{Dc(0)}$ (18a) $\displaystyle\tau=\dfrac{v^{2}t}{D}$
(18b)

where $c(0)$ is the initial concentration, $D$ is the diffusion constant and
$v$ is the velocity of the particles. The meaning of these quantities is as
follows: $x$ is the ratio of two times: the time needed to cross the initial
inter-particle separation ballistically, and that needed to cross the same
distance diffusively. O the other hand, $\tau$ is a dimensionless time
(Equation 18b), given by the ratio of $t$ to the time scale $\tau_{v}$ above
which ballistic motion dominates diffusion. Hence at times corresponding to
$\tau\ll 1$, diffusion dominates, whereas for $\tau\gg 1$, drift does.

Now the concentration of the particles as a function of $x$ and $\tau$, at
time $t$ can be written as :

$c(t)\simeq c(0)\Phi(x,\tau)$ (19)

If $x\ll 1$, $c(t)$ cannot depend on $v$ at small times, since then $\tau\ll
1$ and everything is dominated by diffusion. That implies that
$c(t)\sim(Dt)^{-1/2}$ for small $t$. Hence using Equation (18a) and (18b) we
get

$\Phi(x,\tau)=\dfrac{x}{\sqrt{\tau}}$ (20)

This should be valid until $\tau\sim 1$, as this is the crossover time between
drift and diffusion.

Now consider the case $x\gg 1$. In that case, since time is discrete, $\tau\gg
1$ for all times. It may therefore appear that the dynamics is purely
ballistic since $\tau\gg 1$; $c(t)$ is thus independent of $D$ in this regime,
and it is given by $\sqrt{c(0)/vt}$, as discussed in the previous section. In
this case, again using Equation (18a) and (18b), we get

$\Phi(x,\tau)=\sqrt{\dfrac{x}{\tau}}.$ (21)

The structure resulting from this dynamics is therefore that of the ballistic
aggregation without diffusion studied in the previous Section. I this model,
however, there form large domains of particles moving at the same velocity,
which do not react in the pure ballistic model.

Now, due to the presence of the diffusion, the particles forming such domains
can annihilate via diffusion. Since the particles in a domain have never
interacted, we may roughly assume their separation to be given, in terms of
the initial concentration, by $1/c(0)$. Particles having the same speed
therefore react on a time scale $1/(Dc(0)^{2})$, which in terms of the
dimensionless quantities, is $\tau\sim x^{2}$. Hence Equation (21) will be
valid until time $\tau\sim x^{2}$. The validity of (21) is therefore limited
to finite, though large, values of $\tau$.

As explained above, the previous equations (20) and (21) only hold for finite
values of $\tau$. Now let us describe the large $\tau$ behaviour for the full
dynamics. In that case we can write

$\Phi(x,\tau)\simeq x^{\alpha}\tau^{\beta}$ (22)

When $\tau\sim 1$ and $x\ll 1$, we can use both equation (20) and (22), which
leads to

$x\sim x^{\alpha}$

That gives $\alpha=1$. Similarly, if $x\gg 1$ and $\tau\sim x^{2}$, we can use
the equation (21) as well as (22), which lead us to

$\sqrt{\dfrac{x}{\tau}}\sim x^{-1/2}\sim x^{\alpha+2\beta}=x^{1+2\beta},$

and that give us

$\beta=-\dfrac{3}{4}.$

Putting the value of $\alpha$ and $\beta$ in Equation 22 we get

$\Phi(x,\tau)\simeq x\tau^{-3/4}.$

From this we finally get that for large time ($t\rightarrow\infty$) the
concentration of the particles $c(t)$ will have the following form :

$c(t)\simeq v^{-1/2}D^{-1/4}t^{-3/4}.$ (23)

Now we will see an alternative way to determine the decay law using
dimensional analysis in a little different way. As we have already discussed
the system can be completely characterized by the initial concentration
$c(0)$, velocity $v$ and diffusion constant $D$. From these parameters, one
can get different combinations with the dimensions of concentrations, which
are $c(0)$, $1/vt$ and $1/\sqrt{Dt}$. From this one can anticipate that these
three concentration scales could be written multiplicatively to express the
time-dependent concentration in a conventional scaling form. Hence we can
write the time-dependent concentration as follows :

$c(t)\sim
c(0)^{\rho}\left(\dfrac{1}{vt}\right)^{\sigma}\left(\dfrac{1}{\sqrt{Dt}}\right)^{1-\rho-\sigma}$
(24)

The exponents $\rho$ and $\sigma$ can be obtained by the following argument:
first, let $t<\tau_{v}\simeq D/v^{2}$. Here $\tau_{v}$ is the characteristic
time below which the diffusion dominates over drift. The particle then
undergoes (weakly) biased diffusion, which leads to

$c(t)\sim(Dt)^{-1/2}.$ (25)

independently of $c(0)$. At $t=\tau_{v}$, putting the expression of $c(t)$ and
$\tau_{v}$ in equation (24), we obtain $\rho=0$. Secondly, consider
$\tau_{v}<t<\tau_{D}\simeq 1/(Dc(0)^{2}$, where $\tau_{D}$ is the time needed
for two neighbouring same velocity particles to meet by diffusion. In this
case the dynamics is mostly ballistic and hence

$c(t)\sim\sqrt{c(0)/(vt)}.$ (26)

Again at $t=\tau_{D}$, putting the expression of $c(t)$ and $\tau_{D}$ in
equation (24), we obtain $\sigma=1/2$. Using the value of $\rho$ and $\sigma$
in equation (24), one can get back the same expression for $c(t)$ at large
time which is given by equation (23).

Note that so far we have not at all taken into account the possibility of
unequal concentrations of left-going and right-going particles. Similarly, we
have not considered the effect of a finite system size. We now turn to these
issues.

###  III.3 The case of finite system size : Second stage of dynamics

In an infinite system, it is readily seen that the arguments of Subsection
III.2 hold for all times. For finite systems, another regime appears, which we
now describe.

When all the minority particles have been annihilated, the only reaction
mechanism remaining is diffusive annihilation. One might thus be tempted to
conclude that, for finite systems, we merely have ab additional final stage in
which ordinary diffusive annihilation occurs, with a decay of $1/\sqrt{Dt}$.
As we show, matters are a bit less straightforward.

At the beginning of this stage, that is, when all minority particles have
disappeared, the distribution of all the surviving particles (which are a
fraction of the majority particles) over the one dimensional lattice is far
from being uniform. This leads to the next decay law. In this subsection we
describe by simple arguments, the spatial distribution of the particles over
the one dimensional lattice, and argue that it leads to $t^{-1/4}$ decay. We
follow the arguments given in shf , to which the reader is referred for
details.

Let us start with the behaviour of the surviving particles in ballistic
annihilation with synchronous updating. Once the initial condition is set at
random, each particle has a unique annihilation partner. Otherwise the
particle survives forever. Initially the distance between the particle
(positive velocity) situated at the $k^{th}$ lattice cite and the unique
annihilation partner of that particle be $\pi_{+}(k)$. $k$ runs from $0$ to
$L$, where $L$ is the length of the one dimensional lattice. Once the initial
configuration is fixed each particle survives until it encounters its reaction
partner. Hence the collision time is

$\tau_{+}(k)=\dfrac{\pi_{+}(k)}{2}$ (27)

Here we have considered the positive velocity particles. The entire
description is also valid for the negative velocity particles. In that case we
would say that the initial distance of the unique annihilation partner be
$\pi_{-}(k)$.

We can now try to determine the structure of the set $\Sigma_{t}$, which is
defined as

$\Sigma_{t}=\left\\{k:\tau_{+}(k)>t\right\\}$ (28)

Let us consider that $k$ is the distance between two positive velocity
particles. Without loss of generality we may assume that we have two
particles, one at $0$ and the other one at $k>0$.

If $k>t$, then the two intervals $[0,t]$ and $[k,k+t]$ are disjoint. The
probability that both sites belong to $\Sigma_{t}$ is simply the product of
either site belonging to $\Sigma_{t}$ and hence no dependence on $k$ exists.
This is independent of the fact whether we consider $\varepsilon=0$ for the
initial configuration or not.

On the other hand, for $k<t$ the situation is little different. Let us assume
a virtual random walk that takes a step to the right whenever there is
positive velocity particle in the initial configuration and takes a step to
the left if the velocity of the particle is negative. For $\varepsilon=0$, the
walk is symmetric. Hence for $\varepsilon=0$, the probability of having two
particles separated by a distance $k<t$ and both surviving a time $t$ is
equivalent to the the probability that a symmetric random walk, which starts
at the origin and takes a step to the right does not return to the origin
before time $k$. This probability scales as $k^{-1/2}$ for $k\gg 1$, if the
walk is symmetric Weiss . This description is compatible with the idea that
the set $\Sigma_{t}$ forms a fractal set—below the cutoff value $t$ and with
fractal dimension $1/2$.

For synchronous update we see that the set $\Sigma_{t}$ is a fractal with a
lower cutoff at length $1$ (lattice spacing) and an upper cutoff $t$ with
fractal dimension $1/2$. In case of asynchronous update, annihilation of the
same velocity particles due to the the presence of diffusion, eliminates all
(well, actually, most of) the particles with distance less than $\sqrt{t}$.
Hence the corresponding set of surviving particles becomes a fractal set of
dimension $1/2$ with lower cutoff $\sqrt{t}$ and upper cutoff $t$.

For $\varepsilon>0$, the argument expanded so far is still valid. The main
difference is that now the virtual random walk is biased. That virtual walker
now takes a step to the right with probability $1/2+\varepsilon$, whereas a
step to the left has probability $1/2-\varepsilon$. For $k<t$, the probability
that two particles, one at $0$ and another at $k$ both belong to $\Sigma_{t}$,
is still given by the probability (say $p_{\varepsilon}(k)$), that a random
walk (biased now), which starts at the origin and takes a step to the right,
does not return to the origin before time $k$.

The probability $p_{\varepsilon}(k)$ saturates to a positive value
$p_{\varepsilon}(\infty)$ as $k\rightarrow\infty$. This saturation happens
when $k\sim k_{c}(\varepsilon)\sim\varepsilon^{-2}$ as $\varepsilon\to 0$
Weiss ; Hughes  and when $\varepsilon\rightarrow 0$ we have
$p_{\infty}(\varepsilon)\simeq\varepsilon$. The set $\Sigma_{t}$ is therefore
a fractal set with a cutoff which is either at $t$ or at $\varepsilon^{-2}$,
whichever is smaller.

From the arguments given above we see that the particles surviving at time
$t_{1}$ (Fig. 2), at which only one species (either $+ve$ or $-ve$ velocity
particles) survives, lie on a fractal of dimension $d_{f}=1/2$. This is also
true for both $\varepsilon=0$ and $\varepsilon>0$ in the initial condition.

The fact that $\Sigma_{t_{1}}$ is a fractal further implies that the second
stage of the dynamics is purely diffusive (as only one species survives),
starting from an initial condition where the particles are distributed on a
fractal of dimension $d_{f}$ ($0<d_{f}<1$). The probability distribution
function for this initial inter-particle distances for two nearest particles
has, for $x\gg 1$, a behavior similar to that of the distribution

$P(x)=\dfrac{1}{\zeta(\lambda)}\sum_{l=1}^{\infty}l^{-\lambda}\delta(x-l)$
(29)

where $\lambda=d_{f}+1$. For the purposes of the asymptotic estimations we
shall be dealing with, $P(x)$ and the exact probability distribution can be
used interchangeably. Considering this discrete distribution given by equation
(29), and following the formalism developed in alemany , one can obtain the
number of particles $n(t)$ at time $t$ (normalized by initial number of
particles $n(0)$) which is given by

$\displaystyle
n(t)=-\dfrac{\Gamma(1-\lambda)}{2\zeta(\lambda)\Gamma(\dfrac{3-\lambda}{2})}\tau^{-\dfrac{\lambda-1}{2}}+\dfrac{\Gamma(1-\lambda)}{4(1-\lambda)[\zeta(\lambda)]^{2}}~{}\tau^{-(\lambda-1)}$
$\displaystyle+\dfrac{\zeta(\lambda-1)}{2\zeta(\lambda)\sqrt{\pi}}~{}\tau^{-1/2}+O(\tau^{-\lambda/2})~{}~{}~{}~{}~{}~{}$
(30)

where $\tau=D_{eff}~{}t$. See Appendix in shf  for details. In our case the
fractal dimension $d_{f}=1/2$, and hence $\lambda=3/2$. The leading term of
the decay law is $t^{-1/4}$ and the coefficient of this leading term is
positive, as $\Gamma(1-\lambda)$ is negative for all $\lambda>1$. Putting the
value of $\lambda$ in equation (30) we get

$\displaystyle n(t)=\dfrac{\sqrt{\pi}}{\zeta(3/2)\Gamma(3/4)}\tau^{-1/4}+$
$\displaystyle\left(\dfrac{\sqrt{\pi}}{\zeta(3/2)^{2}}+\dfrac{\zeta(1/2)}{2\sqrt{\pi}\,\zeta(3/2)}\right)\tau^{-1/2}+O(\tau^{-3/4})$
$\displaystyle\approx 0.5537\,\tau^{-1/4}+0.102\,\tau^{-1/2}+O(\tau^{-3/4})$
(31)

Now let us progress to determine the crossover times $t_{1}(\varepsilon)$ and
$t_{1}(\varepsilon)$. $t_{1}(\varepsilon)$ is the time at which the dynamics
have a crossover from $t^{-3/4}$ initial behaviour, to the long-time
$t^{-1/4}$ behaviour. Hence at $t_{1}(\varepsilon)$ we can write

$at_{1}(\varepsilon)^{-3/4}=b(\varepsilon)t_{1}(\varepsilon)^{-1/4}$ (32)

where the coefficient $a$ does not depend on $\varepsilon$, as the decay at
the beginning does not depend on the initial concentration. However
$b(\varepsilon)\sim\varepsilon$, as $p_{\infty}(\varepsilon)\simeq\varepsilon$
for $\varepsilon\rightarrow 0$. Simplifying equation (32), we obtain

$t_{1}(\varepsilon)\sim\varepsilon^{-2}$ (33)

For $\varepsilon\rightarrow 0$, this crossover time $t_{c}$ will diverge and
hence will scale with system size $L$ for finite $L$. We will still be able to
see the $t^{-1/4}$ decay due to the concentration of an excess number of
particles, $c_{ex}(0)\simeq 1/\sqrt{N}$ for setting the initial configuration
at random.

Though it is possible to see the $t^{-1/4}$ decay for $\varepsilon=0$ it is
difficult as there remain very few particles at this late stage. However
$\varepsilon\neq 0$, it is respectively easier to detect the $t^{-1/4}$ (which
is the leading term) decay of concentration on the fractal, as the remaining
particles are more in this situation. As the dynamics is entirely diffusive at
this stage, the fractal structure will eventually die out to uniform
distribution. So there will be another crossover after which the concentration
will decay as $t^{-1/2}$ at very long time. If we consider the crossover time
for the second crossover as $t_{2}(\varepsilon)$ we can write

$b(\varepsilon)t_{2}(\varepsilon)^{-1/4}=c\,t_{2}(\varepsilon)^{-1/2}$

where the coefficient $c$ is not a function of $\varepsilon$ as it does not
depend on the initial concentration. Simplifying the above expression we get

$t_{2}(\varepsilon)\sim\varepsilon^{-4}$ (34)

It is indeed rather challenging to measure this second crossover time
$t_{2}(\varepsilon)$, as the number of remaining particles is very low and the
time very large.

In Subsection III.4 we will show some of the numerical evidence supporting the
theoretical claims described in this subsection and in Subsection III.2.

###  III.4 Numerical Evidences and scaling

If initially there are $N$ number of particles, then at time
$t_{1}(\varepsilon)$ there remain $N\varepsilon^{3/2}$ particles and at
$t_{2}(\varepsilon)$ the number of particles remain only $N\varepsilon^{5/2}$.
This makes some numerically challenging features of the model: if we wish to
have a clean separation of time scales, we need at least $\varepsilon\sim
10^{-2}$. In a different language, if we wish to have more or less hundred
particles at time $t_{2}(\varepsilon)$ (for being able to observe the final
exponent reliably), we need to start with $N\sim 10^{7}$, which is
unrealistic. Hence to drag out the features of the system explained in the
previous subsection one have to rely on various simulations with different
values of the parameters and the corresponding scaling laws.

First we see the decay of concentration for $\varepsilon=0$ which shows the
$t^{-3/4}$ decay law at the beginning and then a crossover to $t^{-1/4}$ decay
(Fig. 3).

Figure 3:  The concentration $c(t)$ (red points) for the total number of
particles decay as $t^{-3/4}$, (the slope is given by blue dotted line) before
the crossover and following the equation (31) after the crossover. Decay for
the concentration for the excess number of particles $c_{ex}(t)$ is plotted in
green points. $n(t)/\sqrt{N}$, with $\lambda=1.5$ is plotted as the
theoretical curve (black dotted line), where the expression of $n(t)$ is given
by equation (30). The decay of the concentrations is plotted starting with
initially $N=80000$ particles.

As we know, initial concentration for the excess particles
$c_{ex}(0)=1/\sqrt{N}$. If the excess particles, which decay due to diffusive
annihilation lie on a fractal of dimension $1/2$ from the very beginning of
the dynamics, then they will decay following the equation (31). The plot of
Equ. 31 with the numerical data for $c_{ex}(t)$ (Fig. 3) shows perfect
agreement, supporting this assumption.

Next we will show the distribution of inter-particle distances between two
neighbouring particles at late times, when the particles decay as $t^{-1/4}$.
$P_{s}(l)$ and $P_{d}(l)$ are the distribution functions of inter-particle
distances between two neighbouring particles with same and different
velocities [Fig. 4]. $P_{s}(l)\sim l^{-3/2}$ for large $l$ [Inset at the
bottom of Fig.4], which is evidence that inside a domain of same velocity
particles, the particles are distributed over a fractal of dimension $1/2$.
For small $l$, $P_{s}(l)\sim l$ due to the diffusion-limited annihilation DBA
. The distribution $P_{d}(l)$ is almost flat at the beginning and then has an
exponential decay [Top right inset of Fig. 4].

Figure 4: Main plot shows the general distribution $P(l)$ for the inter-
particle distances between two neighbouring particles. Inset at the bottom
shows the distribution $P_{s}(l)$ and the top right inset shows the
distribution $P_{d}(l)$. Saffron line have slope $l^{-3/2}$ for both the main
plot and at the bottom inset. Black dashed line shows the linear increase.

At the part of the exponential decay the value of $P_{d}(l)$ suddenly
increases as two domains or fractals of same velocity particles are moving
apart from each other making the probability of having some large value of $l$
very high. $P(l)$ is the general distribution function for inter-particle
distances between two neighbouring particles, independent of their velocity
[Main plot of 4].

Next we will show the numerics for $0<\varepsilon<1/2$. Though for
$\varepsilon=0$, there exist three different dynamical regimes, due to the
finiteness of the system, the regimes are not that well separated and hence
the crossover times are not very clean. We will mostly discuss the scaling
behaviours involving $\varepsilon$ and $t$, which hold for $|\varepsilon|\ll
1$, for two crossovers in this article. For details of the numerics one should
consult shf .

The scaling function for the first two dynamical regimes can be written as

$c(\varepsilon,t)\sim\varepsilon^{2\delta}f(\varepsilon^{2}t),\qquad\hbox{\rm
for }\varepsilon\ll 1$ (35)

where $f(x)\rightarrow x^{-\delta}$ with $\delta=3/4$, for $x\ll 1$ and
$f(x)\rightarrow x^{-1/4}$ when $x\gg 1$. The raw data as well as the scaled
data using $\delta=3/4$ are shown in Fig. 5. The collapse is good for the
first two dynamical regimes where the exponent values are $3/4$ and $1/4$
respectively.

Figure 5: The main plot shows the collapse of scaled data of concentration of
particles with $\delta=3/4$ for different $\varepsilon\neq 0$. Inset shows the
raw data.

The scaling law for the second and third dynamical regime (where
$\varepsilon^{4}t$ is the relevant dimensionless quantity) can be written in
the following form

$c(\varepsilon,t)\sim\varepsilon^{4\eta}g(\varepsilon^{4}t),\qquad\hbox{\rm
for all }\varepsilon\ll 1$ (36)

where $g(x)\rightarrow x^{-\eta}$ with $\eta=1/2$, for $x\rightarrow\infty$
and $g(x)\rightarrow x^{-1/4}$ when $x\ll 1$.

Figure 6: (Collapsed plot of scaled data for the concentration of particles
with $\eta=1/2$ for different $\varepsilon\neq 0$. Inset shows the raw data.
The average number of particles is again considerably less than $one$ in this
exponentially decaying region.

The quality of collapse of the scaled data is good for second and third
dynamical region where the exponent values are $1/4$ and $1/2$ respectively
[Figure. 6]. Data collapse can not be obtained for the exponentially decaying
part, as the scaling theory applies only for the power law regions.

##  IV Conclusions

In this article we have reviewed the problem of ballistic annihilation with
discrete velocities in one dimension. First we discussed the decay of the
concentration of the particles under the synchronous update rule. After that
we described the properties and characteristics of the dynamics when diffusion
is superimposed, for instance by making the update rule asynchronous. There
are several open questions on ballistic annihilation which could be addressed
in future. One of the important issue that could be to studied is the
consequence of initial particle distributions on the decay law for the
ballistic annihilation with superimposed diffusion. If initially the particles
are distributed over a fractal of fractal dimension $d\neq 1$, then the
concentration of the particles should still decay in a power law fashion, but
with a different exponent depending on the value of $d$, the dimension of the
system. Following the argument presented in the section III.3, the
concentration decay should again go through a crossover in time, before it
finally start decaying with exponent $1/2$. Checking this hypothesis could be
a part of a future project.

The problems of two species ballistic annihilation frnRchd  with superimposed
diffusion in one and higher dimension are also open for the future study. On
the other hand, the annihilation problem for the persistent random walkers
persrw  is also very interesting that could be studied in future. One can view
the problem of ballistic annihilation as a limiting case for
$A+A\rightarrow\emptyset$, where $A$ walkers are the persistent walkers. This
problem can be studied under both the synchronous and asynchronous update rule
in future. The generalization of the above results to three or more velocities
is also a challenging problem.

##  V Acknowledgement

Financial support from the project of CONACyT Ciencia de Frontera 2019, Number
10872, is gratefully acknowledged. FL acknowledges the financial support from
the project of CONACyT Ciencias Basicas, Number 254515.

Author Contribution Statement: Both the authors contributed equally to the
paper.

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