Table Of ContentBrownian motion under annihilation dynamics
Mar´ıa Isabel Garc´ıa de Soria,1 Pablo Maynar,2,3 and Emmanuel Trizac1
1Universit´e Paris-Sud, LPTMS, UMR 8626, Orsay Cedex, F-91405 and CNRS, Orsay, F-91405
2Laboratoire de Physique Th´eorique (CNRS UMR 8627),
Bˆatiment 210, Universit´e Paris-Sud, 91405 Orsay cedex, France
3F´ısica Te´orica, Universidad de Sevilla, Apartado de Correos 1065, E-41080, Sevilla, Spain
(Dated: January 6, 2009)
The behavior of a heavy tagged intruderimmersed in a bath of particles evolving underballistic
9
annihilation dynamics is investigated. The Fokker-Planck equation for this system is derived and
0
0 thepeculiarities of thecorresponding diffusivebehaviorare worked out. Inthelong timelimit, the
2 intrudervelocitydistributionfunctionapproachesaGaussianform,butwithadifferenttemperature
from its bath counterpart. As a consequence of the continuous decay of particles in the bath, the
n
meansquareddisplacementincreasesexponentiallyinthecollisionperparticletimescale. Analytical
a
results are finally successfully tested against Monte Carlo numerical simulations.
J
6 PACSnumbers: 51.10.+y,05.20.Dd,82.20.Nk
]
h
c I. INTRODUCTION
e
m
Inrecentyears,therehasbeensomeinterestforsystemswhereparticlesannihilateballistically[1,2,3,4,5,6,7,8].
- In these studies, the model considered consists of an ensemble of hard particles which evolve freely until a binary
t
a encounter, which leads either to the annihilation of the colliding partners with probability p, or to an elastic collision
t
s with probability 1 p. For this probabilistic annihilation model, there are no collisional invariants, and numerical
. simulationshavesh−ownthatfor abroadclassofinitialconditions,the systemreachesanhomogeneousstate inwhich
t
a allthetimedependenceoftheoneparticledistributionfunctionisencodedinthedensityandtemperature(definedas
m
thesecondvelocitymomentofthedistributionfunction)[4,5]. Thisistheso-called“HomogeneousDecayState”. Such
- abehaviorresemblesthe oneofgranularfluids (see[9]andreferencestherein)where,ifthesystemisstable,itevolves
d
intoanhomogeneouscoolingstate,inwhichallthetimedependenceisbornebythegranulartemperature(inthiscase
n
o the density is conserved) [10]. For the annihilation model, the hydrodynamic equations have been derived using the
c Chapmann-Enskogmethod[11]bytheusualassumptionoftheexistenceofa“normalsolution”,whosespaceandtime
[ dependenceoccursonlythroughthehydrodynamicfields[6]. Recently,thehydrodynamicequationslinearizedaround
the homogeneous decay state have been derived relaxing such an assumption [12]. Nevertheless, it must be assumed
1
v that there is scale separation, i.e that the spectrum of the linearized Boltzmann collision operator is such that the
4 eigenvaluesassociatedto the hydrodynamicexcitationsareseparatedfromthe faster“kinetic eigenvalues”. Although
5 this propertyis valid forelastic collisions[13], ithas notbeenprovenfor the probabilisticballistic annihilationmodel
6 in general, but only for Maxwell molecules [14] and for p smaller than a given threshold [12].
0
The objective in this paper is to study the simplest transport process in this system in which we can rigorously
.
1 prove that there is scale separation. We will consider a tagged particle in a fluid in the homogeneous decay state,
0 but collisions between the tagged particle and the particles of the fluid will be always elastic. The equation for the
9 tagged particle is the Boltzmann-Lorentz equation [13, 15] which depends on the one particle distribution function
0
of the bath. In the limit of asymptotically large relative mass for the tagged particle, this equation reduces to a
:
v Fokker-Planck equation which depends on the time-dependent density and temperature of the bath. Due to the
i structure of this equation, we can prove that there is scale separation and that, in the long-time limit, the velocity
X
distribution function of the tagged particle approaches a Gaussian distribution but with a temperature that differs
r
a fromthat ofthe bath. Asimilarbreakdownofequipartitionhasbeenreportedforaheavyparticleina granularbath
[17, 18, 19, 20, 21], a problemthat canbe mapped onto an elastic situation [22], atvariance with the situation under
scrutiny here. We alsostudy the diffusion of the heavyparticle and identify the diffusion coefficientas a Green-Kubo
formula in terms of the velocity autocorrelation function. Finally, we perform Monte Carlo numerical simulations in
order to test our theoretical results.
II. FOKKER-PLANCK EQUATION
We consider a tagged particle of mass m and diameter σ immersed in a low-density gas. This gas is composed of
hardspheres or disks of mass m and diameter σ which moveballistically until one particle meets another one; such
g g
2
binary encounters lead to the annihilation of the colliding partners with probability p or to an elastic collision with
probability1 p[1,2,3,4,5,8]. Collisionsbetweentheparticlesofthegasandthetaggedparticlearealwayselastic.
−
A. From Boltzmann-Lorentz to Fokker-Planck
TheevolutionequationfortheprobabilitydensityF(r,v,t)ofthetaggedparticleistheBoltzmann-Lorentzequation
[13, 15]
∂
+v F(r,v,t)=J[r,v,tF,f], (1)
(cid:18)∂t ·∇(cid:19) |
where the collision operator is given by
J[r,v,tF,f] = σd−1 dv dσˆΘ(g σˆ)(g σˆ) F(r,v∗,t)f(r,v∗,t)
| 0 Z 1Z · · { 1
F(r,v,t)f(r,v ,t) . (2)
1
− }
Here f(r,v,t) is the distribution function of the particles in the gas, d is the space dimension, g = v v is the
1
relative velocity, Θ is the Heaviside step function, σˆ is a unit vector pointing from the centre of the gas−particle to
the centre of the tagged particle at contact, and σ = σ+σg. The precollisionalvelocities v∗ and v∗ are given by
0 2 1
2∆
v∗ =v (g σˆ)σˆ, (3)
− 1+∆ ·
2
v∗ =v + (g σˆ)σˆ, (4)
1 1 1+∆ ·
with ∆=m /m the (gas/taggedparticle) mass ratio.
g
We shall consider that the gas is in the homogeneous decay state, so its distribution function has the scaling form
[5]
n (t) v
f (v ,t)= g χ (c ), c = 1 , (5)
H 1 vd(t) H 1 1 v (t)
g g
1/2
where n (t) is the number density of the gas, v (t) = 2Tg(t) is the thermal velocity of the particles in the gas
g g (cid:16) mg (cid:17)
and χ is an isotropic function depending only on the modulus c = c of the rescaled velocity. It can be seen that
H
| |
the homogeneous density and temperature obey the following equations [6]
∂n (t)
g
= pν (t)ζ n (t), (6)
g n g
∂t −
∂T (t)
g
= pν (t)ζ T (t), (7)
g T g
∂t −
where we have introduced the collision frequency of the corresponding hard sphere fluid in equilibrium (with same
temperature and density)
ν (t)= ng(t)Tg1/2(t)σgd−1 8πd−21 . (8)
g m1/2 (d+2)Γ(d/2)
Here the dimensionless decay rates ζ and ζ are functionals of the distribution function and are approximately
n T
known in the first Sonine approximation [4, 7], see Appendix A. Equations (6) and (7) can be integrated to obtain
the following power laws for the decay of the density and temperature
ng(t) = ng(0)[1+νg(0)p(ζn+ζT/2)t]−2ζn2ζ+nζT , (9)
Tg(t) = Tg(0)[1+νg(0)p(ζn+ζT/2)t]−2ζ2nζ+TζT . (10)
As a consequence, we get n T1/2 t−1, a simplified form of a scaling relation common to all ballistically controlled
g g
∝
processes [16].
3
We next study the evolution equation for the tagged particle in the limit of large relative mass for the tagged
particle. Inthe limit ∆ 1, it is possible to expandthe collisionoperatorJ[r,v,tF,f] inpowersof∆. InAppendix
≪ |
B it is shown that the leading order is
∂ 1 ∂ ∂
J[r,v,tF,f] [A(v)F(r,v,t)]+ :[N(v)F(r,v,t)], (11)
| ≃ ∂v · 2∂v∂v
where
A(v,t)=γ(t)v, N(v,t)=2γ¯(t)I, (12)
and I is the second order unit tensor. The definitions of γ and γ¯ are respectively
γ(t) = γ [n (t),T (t)]a(p), (13)
e g g
T (t)
g
γ¯(t) = γ [n (t),T (t)]a(p)b(p) , (14)
e g g
m
The friction coefficient γ (t) is the same as for elastic bodies, and appears here as a function of the time-dependent
e
density n (t) and temperature T (t)
g g
γ [n (t),T (t)]= 4πd−21 ∆1/2n (t) 2Tg(t) 1/2σd−1, (15)
e g g dΓ(d/2) g (cid:18) m (cid:19) 0
with a(p) and b(p) functionals of the distribution function of the bath which depend only on the parameter p
Γ(d/2)
a(p) = dc χ (c )c , (16)
1 H 1 1
Γ((d+1)/2)Z
2 dc χ (c )c3
b(p) = 1 H 1 1. (17)
d+1Rdc χ (c )c
1 H 1 1
R
In Appendix B, it is shown that the two terms on the right hand side of equation (11) are both of order n v σd−1∆,
g g 0
while the other contributions in the Kramers-Moyal expansion are at least of order n v σd−1∆3/2. In the same
g g 0
Appendix, the expressions for a(p) and b(p) are evaluated to first order in a Sonine expansion.
Taking into account the approximate expression for the collision operator, Eq. (11), it is possible to write the
Boltzmann-Lorentz equation as a Fokker-Planckequation for asymptotically small ∆
∂ ∂ T (t) ∂
+v F(r,v,t)=γ (t)a(p) v+b(p) g F(r,v,t). (18)
(cid:20)∂t ·∇(cid:21) e ∂v ·(cid:20) m ∂v(cid:21)
As in the inelastic case, the Einstein relation is violated due to the fact that the distribution function of the bath
is not Maxwellian [18, 23, 24, 25], which in turn implies that b(p) = 1. On the other hand, if we suppose that the
6
velocity of the taggedparticle obeys a Markovprocessand write the correspondingFokker-Planckequation,in terms
ofthejumpmoments,lim ∆v /∆tandlim ∆v2 /∆t,weobtainexactlyEq. (18). Here ... meansaverage
∆t→0 ∆t→0
h i h i h i
over different noise (bath) realizations.
B. Coarse grained fields and relevant scales
We now focus on the study of the hydrodynamic fields of the tagged particle with the aid of the Fokker-Planck
equation, Eq. (18). We define the mean velocity and the temperature of the Brownian particle as
u(t) = dr dvvF(r,v,t), (19)
Z Z
d 1
T(t) = dr dv m(v u)2F(r,v,t). (20)
2 Z Z 2 −
Taking moments in the Fokker-Planckequation, we obtain (see Appendix C)
∂u(t)
= γ (t)a(p)u(t), (21)
e
∂t −
∂T(t)
= 2γ (t)a(p)[T(t) b(p)T (t)]. (22)
e g
∂t − −
4
As the function γ is a known functional of the gas density n and temperature T , equations (21) and (22) can be
e g g
integrated, which yields
u(t)=u(0)[1+νg(0)p(ζn+ζT/2)t]−(2ζan(p+)ζζTT)ǫ , (23)
and
T(t) = b(p)Tg(0)[1+νg(0)p(ζn+ζT/2)t]−2ζ2nζ+TζT
1 ǫ
−
+ T(0) b(p)Tg(0) [1+νg(0)p(ζn+ζT/2)t]−ǫ(2ζ2nζ+TζT) . (24)
(cid:20) − 1 ǫ (cid:21)
−
In the above equation, we have introduced the dimensionless coefficient ǫ
pζ ν (t) √2dζ σ d−1 p
T g T g
ǫ= = , (25)
2a(p)γ (t) 2(d+2)a(p)(cid:18)σ (cid:19) ∆
e 0
that is not necessarily a small quantity.
As can be seen in Eq. (24), the behavior of the temperature depends strongly on the value of ǫ. If ǫ < 1 the first
term of equation (24) dominates in the long time limit and the temperature of the tagged particle asymptotically
decays with the same power as the temperature of the gas (see equation (10)). As a consequence of (24) we have
T(t) b(p)
lim = , ǫ<1. (26)
t→∞Tg(t) 1 ǫ
−
On the other hand, if ǫ > 1 the second term of equation (24) dominates in the long time limit and the temperature
decays slower than the gas temperature. One can understand this behavior as follows; the parameter ǫ is essentially
the quotient between the cooling rate of the gas and the relaxation rate of the tagged particle’s temperature. If the
former is smaller than the latter, the taggedparticle’s temperature is eventually slavedby T due to the secondterm
g
of the Eq. (22). In the reversed case, the tagged particle’s temperature evolves independently in the long time limit
with a cooling rate slower than that of the gas. It should be emphasized here that in the expansion made in the
previous section, we implicitly assumed that T/T remains finite, because the coefficients A and N were expanded
g
in powers of ∆1/2 T (see Appendix B). Hence, the Fokker-Planck equation for ǫ > 1 is restricted to a time window
Tg
in which T is small enough. In the following, we will only consider the case in which the Fokker-Planck equation is
Tg
valid for all times, i.e. the double limit
d−1
σ p
g
∆ 0, p 0, ǫ<1, ǫ , (27)
→ → ∝(cid:18)σ (cid:19) ∆
0
wherethe requirementofsmallp stemsfrom∆ 1andǫboundedfromabove. Hence,inorderto be consistentwith
≪
this limit, we can substitute in the Fokker-Planck equation, Eq. (18), the values of the coefficients, a(p) and b(p) by
their elastic limits
lima(p)=1, limb(p)=1, (28)
p→0 p→0
so that
∂ ∂ T (t) ∂
+v F(r,v,t)=γ (t) v+ g F(r,v,t). (29)
(cid:20)∂t ·∇(cid:21) e ∂v ·(cid:20) m ∂v(cid:21)
This equation is formally identical to the one obtained for an elastic gas [15], except for the fact that the density
and temperature of the gas depend on time. This is a consequence of the elastic limit to which we are restricted. In
general, the coefficients a(p) and b(p), Eqs. (16) and (17), differ from unity due to the non Maxwellian character of
the distribution function of the bath, and Einstein relation is violated.
In order to analyze this equation, it is convenientto introduce the dimensionless time scale, t∗, proportionalto the
number of collisions of the tagged particle
t
t∗ =(1 ǫ ) dt′γ (t′), (30)
0 e
− Z
0
5
where ǫ is defined by substituting ζ (p)/a(p) in (25) by its p 0 limit:
0 T
→
√2 σ d−1 p
g
ǫ = . (31)
0
16 (cid:18)σ (cid:19) ∆
0
The dimensionless time scale t∗ is related to the real time t by
γ 2(1 ǫ ) 1
t∗ = e − 0 log[1+ν (0)p(ζ +ζ /2)t]. (32)
g n T
ν p(2ζ +ζ )
g n T
In this time scale, the evolution of the mean velocity and temperature of the tagged particle are particularly simple
u(t∗)=u(0)e−1−t∗ǫ0, (33)
and
T(t∗) = T(0)e−2t∗ + 1 (1 e−2t∗). (34)
T (t∗) T (0) 1 ǫ −
g g 0
−
Such predictions will be compared against numerical simulations in section IV. Let us also introduce the scaled
distribution
1 v
F(r,v,t)= F∗(r,v∗,t∗), v∗ = , (35)
vd(t) v (t)
ǫ ǫ
where
1 1/2 2T (t) 1/2
g
v (t)= . (36)
ǫ
(cid:18)1 ǫ (cid:19) (cid:18) m (cid:19)
0
−
The function v (t) is introduced because with these definitions we have
ǫ
2T(t) 1/2
v (t) , (37)
ǫ
→(cid:20) m (cid:21)
in the long time limit. In these variables the Fokker-Planckequation (29) reduces to
∂
+ℓ (t∗)v∗ F∗(r,v∗,t∗)=Λ (v∗)F∗(r,v∗,t∗), (38)
(cid:18)∂t∗ 0 ·∇(cid:19) FP
where we have introduced the standard homogeneous Fokker-Planck operator
∂ 1 ∂
Λ (v∗)= v∗+ , (39)
FP ∂v∗ ·(cid:18) 2∂v∗(cid:19)
and the function proportionalto the mean free path
d−1
v (t) dΓ(d/2) σ
ℓ (t∗)= ǫ = ∆−1/2 g [n (t∗)σd−1]−1. (40)
0 (1−ǫ0)γe(t) (1−ǫ0)3/24πd−21 (cid:18)σ0(cid:19) g g
Taking into account the definition of ǫ, Eq. (25), we can write explicitly ℓ as a function of the t∗ variable as
0
ℓ (t∗)=ℓ (0)eǫ∗t∗, ǫ∗ = 2ζnǫ0 . (41)
0 0
ζ (1 ǫ )
T 0
−
To sum up, we have obtained the evolution equation for the distribution function of a tagged particle in a bath of
particleswhichannihilate,inthelimitwherethetaggedparticleismuchheavierthantheparticlesofthebath. There
are some points in common with the elastic case, but also some important differences. The homogeneous operator,
whose spectral properties are well-known [13, 15], is exactly the same, but the flux term is weighted by a function
depending on time and that diverges in the long time limit. This will have important consequences in the study of
diffusion as we will see in the following section. Moreover,the equation is not valid for all values of the probability p
of annihilationin the bathand massesof the taggedparticlebut, as alreadymentioned, is limited to the double limit
of Eq. (27), in which ǫ <1.
0
6
III. LONG-TIME LIMIT SOLUTION OF THE FOKKER-PLANCK EQUATION
In this section we investigate the long time behavior of the solution of the Fokker-Planck equation, Eq. (38),
startingwith anarbitraryinitialcondition. The objectiveis to study if the taggedparticle reachessome scalingstate
in the long time limit and also to analyze how the particle diffuses.
A. Evolution towards a scaling form
As the Fokker-Planck equation is linear, it is convenient to work in the Fourier space. The Fourier component of
the tagged particle distribution function is defined as
Fk(v∗,t∗)= dre−ik·rF∗(r,v∗,t∗), (42)
Z
so that Eq. (38) yields Fk(v∗,t∗)
∂
∂t∗Fk(v∗,t∗)=[ΛFP(v∗)−iℓ0(t∗)k·v∗]Fk(v∗,t∗). (43)
The spectrum of the operator Λ (v∗) iℓ (t∗)k v∗ is known [13, 15]. The eigenvalues are
FP 0
− ·
d
1
λn(k,t)= [kℓ0(t)]2 nj, (44)
−2 −
Xj=1
where we have introduced the vector label n = (n ,...,n ), with possible coordinate values n = 0,1,2,... .
1 d i
∞
Hence, for any initial condition, all the k-Fourier components decay and only the k = 0 remains. Moreover, as the
eigenfunction associated to the vanishing eigenvalue is the Maxwellian distribution [13, 15]
χ (v∗)= 1 e−v∗2, (45)
M πd/2
we obtain
1
F(r,v,t) χ (v∗), (46)
→ vd(t) M
ǫ
in the long time limit.
As aconsequence,the taggedparticledistributionfunctionapproachesascalingformsimilarto(5)forthe gas,but
with a different temperature (see (26)). In this regime the cooling rates of the bath and of the tagged particle are
the same and the temperatures are proportional. The situation is similar to that of an elastic particle in a bath of
inelastic grains [18, 19]. Nevertheless, there is an important difference: in the inelastic case it has been proved that
there exists an exact mapping with an elastic system. On the other hand, in our problem such a mapping fails due
to the flux term which explicitly depends on time.
B. Characteristics of diffusive motion
Our objective is to study the evolution equation for the density of tagged particles, n(r,t) = dvF(r,v,t) in a
“macroscopic” scale, i.e. in a long time and length scale compared to the microscopic ones. TheRmicroscopic time
scale is defined by the slowest kinetic modes of Λ , i.e. the modes with a single non vanishing component, labeled
FP
by n =δ for a givenvalue ofj in[1,d]. The microscopiclengthscaleis defined bythe meanfree pathofthe tagged
i ij
particlewhichisproportionaltoℓ0(t∗). ThestartingpointwillbetheFokker-PlanckequationforFk,Eq. (43). Asthe
generator of the dynamics, the operator Λ (v∗) iℓ (t∗)k v∗, does not commute with its time derivative, it is not
FP 0
− ·
possible to write the general solution of equation (43) in terms of the initial condition in a simple way. Nevertheless,
it is shown in Appendix D that in the hydrodynamic limit, i.e. kℓ (t∗) 1 for all the time evolution and t∗ 1, a
0
closed equation for the Fourier component of the density, nk = dv∗Fk(≪v∗,t∗), is obtained ≫
R
∂n∂kt(∗t∗) =−D0[kℓ0(t∗)]2nk(t∗), (47)
7
where the diffusion coefficient is
1
D = . (48)
0 2(1+ǫ∗)
This asymptotic behavior can be evaluated, taking advantage of the scale separation (i.e. the mode with n = 0 is
isolated from the other modes). From equation (47) we can derive the evolution equation for the density
∂n(r,t∗)
=D ℓ2(t∗) 2n(r,t∗). (49)
∂t∗ 0 0 ∇
This is the equation we were looking for and it is only valid in the “macroscopic” time and length scale. If we
transform this equation to real time with the aid of formula (30) we obtain
∂n(r,t) ζ
=D(t) 2n(r,t), D(t)= T D (t), (50)
e
∂t ∇ (1 ǫ )[ζ +(2ζ ζ )ǫ ]
0 T n T 0
− −
where D (t)=2v2(t)/γ (t) is the same as the diffusion coefficient for elastic collisions except that it appears here as
e g e
afunctionofthe time-dependentgastemperatureanddensity. As canbeseen,foroursystemthe diffusioncoefficient
D(t) is far from being a trivial generalization of the elastic diffusion coefficient.
Nowletusfocusonthepredictionsofourdiffusionequation. Tothisend,weintroducethemeansquaredisplacement
r2(t∗) = drr2n(r,t∗). (51)
h i Z
If we consider an infinite system, we obtain from equation (49)
∂
r2(t∗) =2dD ℓ (t∗)2, (52)
∂t∗h i 0 0
that will only be valid in the long time limit. It is straightforwardto integrate equation(52), taking into accountthe
explicit formula for ℓ (t∗), Eq. (41). This gives
0
e2ε∗t∗ 1
r2(t∗) =dD ℓ2(0) − , (53)
h i 0 0 ε∗
or in real time
hr2(t)i= dD0ǫℓ∗20(0)n[1+νg(0)p(ζn+ζT/2)t]2ζn4ζ+nζT −1o. (54)
AscanbeseeninEq. (53),thediffusivebehavioriscompletelydifferentfromitselasticoreveninelasticcounterparts,
where it was found that the meansquare displacementis proportionalto the number ofcollisionper particle [18, 19].
The fact that the bath is loosing particles significantly affects this dynamics, and the mean square displacement
increases exponentially in the collision per particle scale. As we will see in the following section, simulation results
agree well with our theoretical prediction. Roughly speaking, the exponent 4ζ /(2ζ +ζ ) is close to 8d/(4d+1)
n n T
(approximately 1.77 in two dimensions, and 1.84 in three dimensions).
C. Diffusive behavior : an alternative derivation
Inthe remainderofthis section,we showthat, under plausiblehypothesis,itis possibleto re-derive“a`la Einstein”
the formula for the mean square displacement, Eq. (53). This derivation has the merit of leading to a Green-Kubo
like expression for the diffusion coefficient. We start by writing the mean-squared displacement as
t t
r2(t) = dt′ dt′′ V(t′) V(t′′) . (55)
h i Z Z h · i
0 0
Here, the position and the velocity of the tagged particle, r(t) and V(t), are considered as a stochastic process and
... denotes an ensemble averageover different trajectories. Let us change the variables from t t∗ and let us also
h i →
introduce the scaled velocity
V
w(t∗) , (56)
≡ v (t)
ǫ
8
1.5
1
g
T
T/
0.5
m=20m
g
m=200m
g
0
0 100 200
τ
Figure1: (Coloronline)Evolutionofthetemperatureratioasafunctionofthenumberofcollisionsperparticleτ forasystem
with p=0.01 and two values of thetagged particle’s mass, m=20mg (∆=1/20) and m=200mg (∆=1/200). The dashed
line is the theoretical prediction given by Eq. (34).
where v (t) is defined in (36). With these definitions we have
ǫ
1 t∗ t∗
r2(t∗) = ds γ−1(s ) ds γ−1(s ) V(s ) V(s )
h i (1 ǫ )2 Z 1 e 1 Z 2 e 2 h 1 · 2 i
− 0 0 0
t∗ t∗
= ℓ2(0) ds ds eǫ∗(s1+s2) w(s ) w(s ) , (57)
0 Z 1Z 2 h 1 · 2 i
0 0
wherewehaveusedthe definitions oft∗ andℓ (0), Eq. (30)andEq. (41). Now,ifwe assumethatthe taggedparticle
0
is in the scaled regime, i.e. the temperature is T (t∗)/(1 ǫ ) and that the correlation function w(s ) w(s ) is
g 0 1 2
− h · i
a function of s s , by integrating in the new variables S = (s +s )/2 and s = s s , the following relation is
1 2 1 2 1 2
− −
obtained
hr2(t∗)i ǫ∗ 1 t∗ds w(s) w(0) e−ǫ∗s, (58)
e2ǫ∗t∗ dℓ2(0) → dZ h · i
0 0
inthelongtimelimit. ThisformulaisthegeneralizationoftheEinsteinformulaforthediffusioncoefficientofaheavy
particleinafluidinthehomogeneousdecaystate. Itrelatestheasymptoticbehaviorofthemean-squaredisplacement
with the time integral of the autocorrelation function of the velocity weighted by the exponential e−ǫ∗s. So far we
have considered the Fokker-Planck equation as the equation for the one-time probability distribution function. If
we assume that the velocity of the tagged particle is a Markov process, then the Fokker-Planck equation is also the
equation for the conditional probability and we can evaluate easily the correlation function w(s ) w(s ) . Taking
1 2
h · i
into account Eqs. (33) and (36), we have
d
w(s ) w(s ) = e−|s1−s2|. (59)
1 2
h · i 2
By substituting this formula into Eq. (58) we re-derive Eq. (53) with the same diffusion coefficient D , that can be
0
written in the Green-Kubo form
D = 1 ∞ds w(s) w(0) e−ǫ∗s. (60)
0
dZ h · i
0
IV. DIRECT SIMULATION MONTE CARLO RESULTS
The objective of this section is to put to the test the main results of the previous sections by means of the direct
simulationMonteCarlomethod(DSMC).Moreprecisely,wewillanalyzethetemperatureandmeanvelocityevolution,
together with the tagged particle diffusion. We have performed DSMC simulations of a system of N hard disks of
g
mass m and diameter σ which annihilate with probability p or collide elastically with probability 1 p everytime
g g
−
9
1.5
1
Φ
p=0.01, m=38m
0.5 g
p=0.01, m=150m
g
p=0.1, m=20m
g
p=0.1, m=60m
g
0
0 1 2 3 4 5
t*
∗
Figure2: (Coloronline)ScalingfunctionΦdefinedinthemaintextasafunctionofreducedtimet fordifferentsystemswith
p = 0.1 and p = 0.01 and different values of the tagged particle’s mass. The dashed line is the theoretical prediction of Eq.
(62).
1.4
st
)
g
T 1.2
T/
(
1
0 0.02 0.04 0.06 0.08
∆
Figure 3: (Color online) Stationary values of the temperatures ratio for a system with p = 0.01 and different values of the
tagged particle’smass ∆=mg/m. SymbolsarefortheMonteCarlo dataandthedashedlineshowsthelongtimelimit ofEq.
(34).
two particles meet. Bird’s algorithm [26] has been used. The parameters in all the simulations were m =1, σ =1,
g g
N (0)=105 and T (0)=1. We have considered only one tagged particle in each simulation, that collides elastically
g g
with the surrounding bath. The diameter of this particle has been set to unity (σ =1) and we have varied the value
of its mass m. The values of the probability of annihilation p of the particles in the bath have been p = 0.1 and
p=0.01, and the results have been averagedover 2 104 and 4 103 trajectories respectively. For a given value of p,
· ·
we have performed a series of simulations for different values of the mass of the tagged particle. Taking due account
of the constraint ǫ < 1, the value of the tagged particle’s mass must be smaller than 102 for p = 0.1 and 103 for
0
p=0.01.
Figure1showstheevolutionoftheratioT/T forasystemwithp=0.01andfortwovaluesofthetaggedparticle’s
g
mass, that is, m=20m and m=200m , as a function of the number of collisions per particle, τ, defined as
g g
1 t 1 ν
τ = dt′ν (t′)= gt∗. (61)
g
2Z 2(1 ǫ )γ
0 − 0 e
The initial value of temperature of the taggedparticle is T(0)=0 for all the trajectories, since at t=0, the intruder
has a prescribed velocity. As we can see, in this scale, the evolution to the stationary value of the ratio of the
temperaturesis slowerasweincreasethe massofthe taggedparticle. This implies thattherearevaluesofthe tagged
particle’smassforwhichthe ratioofthe temperatureswillnotreachitsstationaryvalueinthetime ofthe simulation
(forinstance,thenumberofparticlesinthebathforτ =200isN 1800,whichhinderscorrectstatisticalsampling).
g
≃
10
1.2
m=15m
g
1.1 m=20mg
2> m=60mg
2
v
<
2 1
>/
4
v
<
0.9
0.8
0 5 10 15 20
τ
Figure 4: (Color online) Reduced fourth moment of the tagged particle velocity as a function of the dimensionless time τ for
p=0.1 and different values of the tagged particle’s mass.
1
0
0)] -1
0.8 u(x-2
0) n[u(t*)/x--43
( 0.6 l
x -5
u 0 1 2 3 4 5
*)/ t*/(1-ε0)
u(tx0.4 mm==1250mmgg
m=38m
g
m=60m
0.2 g
m=80m
g
m=100m
g
0
0 1 2 3 4 5
t*/(1-ε )
0
Figure5: (Color online)Meanvelocityofthetagged particleasafunctionoft∗/(1−ǫ0)forasystemwithp=0.1andseveral
values of the tagged particle’s mass. The dashed line is the theoretical prediction given by Eq. (33). Inset: Mean velocity of
thetagged particle as a function of t∗/(1−ǫ0) for a system with p=0.1 on a logarithmic scale.
The theoretical prediction in this time scale (dashed line) is obtained directly from Eq. (34), taking into account Eq.
(61). The agreement between theory and simulations is good.
Similar simulations were performed with different values of the taggedparticle mass (m ranging from 15 to 100 for
p=0.1 and from 15 to 900 for p=10−2). Since T (0)=0, Eq. (34) predicts
g
Φ (1 ǫ )T = 1 e−2t∗. (62)
0
≡ − T −
g
AscanbeobservedinFig. 2,allsimulationdataforΦcollapseontoasinglecurve. Thetimescalet∗ canbecalculated
fromthe scale τ definedin (61). In the same vein,we canobtainthe stationaryvalue ofthe temperature ratio,which
is plotted as a function of ∆=m /m in Fig. 3 ; this validates our theoretical analysis.
g
In order to probe –atleast partially–the Gaussiannature of the time dependent taggedparticle velocity statistics,
we havemeasuredthe reducedfourthmoment4 v4 /(d(d+2) v2 2). As canbe seeninFig. 4, wherewe haveplotted
h i h i
the results for p = 0.1 and several values of ∆ as a function of the number of collisions per particle, τ, the value of
this quantity is in agreement with the Gaussian prediction (that is unity) within the statistical uncertainties.
Considernext the meanvelocityof the taggedparticle,u(t∗). Inorderto study the decayofthis quantity,we have
performed a set of simulations starting with a component of the velocity field in the x direction when it is immersed
in a bath in the homogeneous decay state, u (0). In Fig. 5, we plot u (t∗)/u (0) as a function of t∗/(1 ǫ ) for
x x x 0
−
p=0.1 and severalvalues of m. In this time scale, the data for all values of m collapse due to Eq. (33). In the inset,
the same quantity is plotted on a logarithmic scale. If the theoretical prediction in Eq. (33) is verified, the above
plot must lead to a straightline with slope ζu =1 (dashed line), where ζu is the decay rate of the mean velocity. On
