Velocity Distribution of Driven Inelastic One-component Maxwell gas

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by  V. V. Prasad

Velocity Distribution of Driven Inelastic One-component Maxwell gas
V. V. Prasad,1,2 Dibyendu Das,3 Sanjib Sabhapandit,4 and R.Rajesh1,2
1The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai-600113, India
2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai-400094, India
3Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai-400076,India
4Raman Research Institute, Bangalore - 560080, India
(Dated: January 16, 2017)
Thenatureofthevelocitydistributionofadrivengranulargas,thoughwellstudied,isunknown
astowhetheritisuniversalornot,andifuniversalwhatitis. Wedeterminethetailsofthesteady
7 state velocity distribution of a driven inelastic Maxwell gas, which is a simple model of a granular
1 gas where the rate of collision between particles is independent of the separation as well as the
0 relative velocity. We show that the steady state velocity distribution is non-universal and depends
2
strongly on the nature of driving. The asymptotic behavior of the velocity distribution are shown

n tobeidenticaltothatofanon-interactingmodelwherethecollisionsbetweenparticlesareignored.
a For diffusive driving, where collisions with the wall are modelled by an additive noise, the tails of
J the velocity distribution is universal only if the noise distribution decays faster than exponential.

3
1
I. INTRODUCTION θ =d/(d+2) [9, 12, 15].

] In dilute driven granular gases, the focus of this pa-
h
per, the system reaches a steady state where the energy
c Granular matter, constituted of particles that interact
e lost in collisions is balanced by external driving. Several
through inelastic collisions, exhibit diverse phenomena
m experiments, simulationsandtheoreticalstudieshavefo-
such as cluster formation, jamming, phase separation,
cused on determining the steady state velocity distribu-
-
pattern formation, static piles with intricate stress net-
t tion P(v). In experiments, driving is done either me-
a works, etc. [1–5]. Itsubiquity innature andin industrial
t chanically [17–24] through collision of the particles with
s applications makes it important to understand how the
vibratingwallofthecontainerorbyapplyingelectric[25]
.
t macroscopically observed behavior of granular systems
a or magnetic fields [26] on the granular beads. Almost
arises from the microscopic dynamics. A well studied
m all the experiments find the tails of P(v) to be non-
macroscopic property is the velocity distribution of a di-
Maxwellian, and described by a stretched exponential
- lutegranulargas. Whileseveralstudies(seebelow)have
d formP(v) exp( avβ)forlargev. Someoftheseexper-
n shown that the inherent non-equilibrium nature of the iments find∼P(v) t−o be universal with β =3/2 for a wide
o system, induced by inelasticity, could result in a non-
range of parameters [21, 24]. In contrast, other experi-
c Maxwellian velocity distribution, they fail to pinpoint
ments [20, 23] find P(v) to be non-universal with the ex-
[ whether the velocity distribution is universal, and if yes,
ponentβ varyingwiththesystemparameters,sometimes
1 what its form is. In this paper, we focus on the role of even approaching a Gaussian distribution (β =2) [20].
v driving in determining the velocity distribution within a
In numerical simulations, driving is done either from
0 simplified model for a granular gas, namely the inelastic
the boundaries [8, 27] which leads to clustering, or ho-
0 Maxwell model.
6 mogeneously [28–31] within the bulk. In simulations
3 Dilute granular gases are of two kinds: freely cooling of a granular gas in three dimensions, driven homoge-
0 in which there is no input of energy [6–16], or driven, neously by addition of white noise to the velocity (dif-
1. in which energy is injected at a constant rate. In the fusive driving), it was observed that β = 3/2 for large
0 freely cooling granular gas, the velocity distribution at enough inelasticity [29]. However, similar simulations of
7 different times t has the form P(v,t) (cid:39) vr−m1sf(v/vrms), a bounded two dimensional granular gases with diffusive
1 where v is any of the velocity components, vrms(t) is drivingfoundarangeofdistributionsinthesteadystate,
v: the time dependent root mean square velocity and f withβ rangingfrom0.7to2astheparametersinthesys-
i is a scaling function. vrms(t) decreases in time as a tem are varied [30, 31].
X power law vrms(t) t−θ. To determine the behavior Theoretical approaches have been of two kinds: ki-
∼
r of f for large argument, it was argued that the con- netic theory, or by studying simple models which cap-
a
tributions to the tails of the velocity distributions are ture essential physics but are analytically tractable. In
from particles that do not undergo any collisions, im- kinetic theory [32], the Boltzmann equation describing
plying an exponential decay of P(v,t) with time t [12]. the evolution of the distribution function is obtained by
Thus,f(x) exp( ax1/θ),orP(v,t) e av1/θtforlarge truncating the BBGKY hierarchy by assuming product
−
∼ − ∼
v. It is known that at initial times, the granular parti- measure for joint distribution functions. While it is dif-
cles remain homogeneously distributed with θ = 1 [6], ficult to solve this non-linear equation exactly, the de-
leading to P(v,t) having an exponential decay in all di- viation of the velocity distribution from Gaussian can
mensions. At late times they tend to cluster resulting in be expressed as a perturbation expansion using Sonine
densityinhomogeneitieswithcurrentevidencesuggesting polynomials[11,32–34]. Thisapproachdescribestheve-
2
locity distribution near the typical velocities. The tails tribution for dissipative driving. For the pseudo steady
of the distribution can be obtained by linearizing the stateindiffusivedriving, wefindthatthevelocitydistri-
Boltzmann equation [11, 35, 36]. Notably, for granular bution is universal if the noise distribution decays faster
gases with diffusive driving, this leads to the prediction thanexponentialanddeterminedbynoisestatisticsifthe
P(v) exp( bv β) with β = 3/2 for large velocities, noise distribution decays slower than exponential.
(cid:39) C − | |
independentofthecoefficientofrestitution,stronglysug- Therestofthepaperisorganizedasfollows. InSec.II
gesting that the velocity distribution is universal [11]. wedefinetheMaxwellmodelanditsdynamicsmorepre-
cisely. InSec.IIIthesteadystatevelocitydistributionof
The alternate theoretical approach is to study simpler
the system are determined by studying its characteristic
modelliketheinelasticMaxwellgas, inwhichspatialco-
function as well as the asymptotic behavior of ratios of
ordinates of the particles are ignored and each pair of
successivemoments. Inparticular,weobtainthevelocity
particles collide at constant rate [10]. In the freely cool-
distribution for a family of stretched exponential distri-
ing Maxwell gas, the velocity distribution decays as a
butions for the noise. The results for dissipative driving
power law with an exponent that depends on dimension
maybefoundinSec.IIIAandthosefordiffusivedriving
and coefficient of restitution [37–40]. In contrast, for a
in Sec. IIIB. In Sec. IV, the exact solution of the non-
diffusively driven Maxwell gas, in which collisions with
interacting problem is presented. Section V contains a
the wall and modelled by velocities being modified by
summary and discussion of results.
an additive noise, it was shown that P(v) has a univer-
sal exponential tail (β =1) for all coefficients of restitu-
tion[41,42]. However,ithasbeenrecentlyshown[43,44]
II. DRIVEN MAXWELL GAS
thatwhenthedrivingisdiffusive,thevelocityofthecen-
terofmassdoesaBrownianmotion,andthetotalenergy
increases linearly with time at large times. Thus, the Consider N particles of unit mass. Each particle i
system fails to reach a time-independent steady state, has a one-component velocity vi, i = 1,2,...,N. The
making the results for diffusive driving valid only for in- particles undergo two-body collisions that conserve mo-
termediate times when a pesudo-steady state might be mentum but dissipate energy, such that when particles i
assumed. This drawback may be overcome by model- and j collide, the post-collision velocities vi(cid:48) and vj(cid:48) are
ingdrivingthroughcollisionswithawall, wherethenew given in terms of the pre-collision velocities vi and vj as:
velocity v(cid:48) of a particle colliding with a wall is given by (1 r) (1+r)
vp(cid:48)ar=tic−ler-wwva+llηc,owllhiseiorensr,wainsdthηeicsouenfficcoirernetlaotferdesntoitiusetiroenpfroer- vi(cid:48) = −2 vi+ 2 vj, (1)
(1+r) (1 r)
senting the momentum transfer due to the wall [43] (dif- vj(cid:48) = 2 vi+ −2 vj,
fusive driving corresponds to r = 1). For this dissipa-
w
tivedriving(r <1),thesystemr−eachesasteadystate, where r [0,1] is the coefficient of restitution. For
and the veloc|itwy|distribution was shown to be Gaussian energy-con∈serving elastic collisions, r = 1. In the
when η is taken from a normalized Gaussian distribu- Maxwell gas, the rate of collision of a pair of particles
tion [43]. If η is described by a Cauchy distribution, the is assumed to be independent of their spatial separation
steadystateP(v)isalsoaCauchydistribution, butwith as well as their relative velocity. These simplifying as-
a different parameter [43]. sumptions make the model more tractable as the spatial
coordinates of the particles may now be ignored.
Thus,whilethevelocitydistributionforthefreelycool-
The system is driven by input of energy, modeled by
ing granular gas is universal and reasonably well under-
particles colliding with a vibrating wall [43]. If particle i
stood, it has remained unclear whether the velocity dis-
withvelocityv collideswiththewallhavingvelocityV ,
tribution of a driven granular gas is universal. Also, if i w
thenewvelocitiesv ,V respectively,satisfytherelation
thevelocitydistributionisnon-Maxwellian,aclearphys- i(cid:48) w(cid:48)
v V = r (v V ), where the parameter r is the
ical picture for its origin is missing. Intuitively, it would i(cid:48) − w(cid:48) − w i− w w
coefficientofrestitutionforparticle-wallcollisions. Since
appear that the tails of the velocity distribution would
the wall is much heavier than the particles, V V ,
bedominatedbyparticlesthathavebeenrecentlydriven w(cid:48) ≈ w
and hence v = r v +(1+r )V . Since the motion of
and not undergone any collision henceforth. This would i(cid:48) − w i w w
the wall is independent of the particles and the particle-
mean the P(v) cannot decay faster than the distribution
wallcollisiontimesarerandom,itisreasonabletoreplace
of the noise associated with the driving. If this reason-
(1+r )V by a random noise η and the new velocity v
ing is right, the noise statistics should play a crucial role w w i(cid:48)
is now given by [43],
in determining the velocity distribution, making it non-
universal. How sensitive is P(v) to the details of the vi(cid:48) =−rwvi+ηi. (2)
driving? In particular, how does P(v) behave for large
Inthispaper,weconsideraclassofnormalizedstretched
v for different noise distributions Φ(η)? We answer this
exponential distributions for the noise η,
question within the Maxwell model, both for dissipative
1
dforrivdiniffgu(s0iv≤e drrwiv<ing1)(raws w=el−l 1a)s.thIne ppsaerutidcoulsatre,awdye sshtaotwe Φ(η)= 2Γ(cid:16)a1γ+ 1(cid:17)exp(−a|η|γ) a,γ >0, (3)
that the tail statistics are determined by the noise dis- γ
3
characterized by the exponent γ. Note that there is no is also interesting. When r = 1, the structure of the
w
apriori reason to assume that the noise is Gaussian as equations obeyed by the steady state velocity distribu-
the noise is not averaged over many random kicks. tion is identical to those obeyed by the distribution in
The system is evolved in discrete time steps. At each the pseudo-steady state of the Maxwell gas with diffu-
step, a pair of particles are chosen at random and with sive driving (r = 1) [43].
w
−
probability p they collide according to Eq. (1), and with
probability(1 p),theycollidewiththewallaccordingto
−
Eq. (2). We note that evolving the system in continuous III. STEADY STATE VELOCITY
time does not change the results obtained for the steady DISTRIBUTION
state.
We also note that though the physical range of r is We use two diagnostic tools to obtain the tail of the
w
[0,1], it is useful to mathematically extend its range to steady state velocity distribution: (1) by directly study-
[ 1,1]. This makes it convenient to treat special limit- ingthecharacteristicfunctionofthevelocitydistribution
−
ing cases in one general framework. For instance, when and(2)bydeterminingtheratiosoflargemomentsofthe
r = 1, the driving reduces to a random noise being velocity distribution.
w
−
added to the velocities, corresponding to diffusive driv- In the steady state, due to collisions being random,
ing. In this case, the system reaches a pseudo-steady there are no correlations between velocities of two differ-
state before energy starts increasing linearly with time ent particles in the thermodynamic limit. We note that
for large times [43, 44]. When r = 1, the system for finite systems, there are correlations that are propor-
w
(cid:54) −
reaches a steady state that is independent of the initial tional to N 1 [43]. The two point joint probability dis-
−
conditions. In the limit r 1, and rate of collisions tributions can thus be written as a product of one-point
w
→ −
with the wall going to infinity, the problem reduces to probability distributions. It is then straightforward to
an Ornstein-Uhlenbeck process [44]. The case r = 1 write
w
(cid:90)(cid:90) (cid:20) (cid:21) (cid:90)(cid:90)
1 r 1+r
P(v,t+1)=p dv dv P(v ,t)P(v ,t)δ − v + v v +(1 p) dηdv Φ(η)P(v ,t)δ[η r v v],(4)
1 2 1 2 1 2 1 1 w 1
2 2 − − − −
where the first term on the right hand side describes the Z(λ) = 1 λ2 v2 /2 for small λ, one can use this re-
− (cid:104) (cid:105)
evolution due to collisions between particles and the sec- cursion relation to calculate characteristic function for
ondtermdescribestheevolutionduetocollisionbetween any value of λ. Here v2 may be calculated exactly [see
(cid:104) (cid:105)
particles and wall. In the steady state, the velocity dis- Eq.(9)]. Thevelocitydistributionmaybeobtainedfrom
tributions become time independent and we use the no- the inverse Fourier transform of Z(λ).
tation limt P(v,t) = P(v). Equation (4) is best ana- When rw =1, Eq. (6) allows the tail statistics of P(v)
lyzedinthe→F∞ourierspace. Letthecharacteristicfunction to be determined exactly. In this case, the characteristic
of the velocity distribution be defined as function satisfies the relation
Z(λ)= exp( iλv) . (5) pZ([1 r]λ/2)Z([1+r]λ/2)
(cid:104) − (cid:105) Z(λ)= − , r =1. (8)
w
It can be shown from Eq. (4) that Z(λ) satisfies the re- [1 (1 p)f(λ)]
− −
lation [43]
Equation (8) may be iteratively solved to obtain an infi-
(cid:18) (cid:19) (cid:18) (cid:19)
[1 r]λ [1+r]λ nite product involving simple poles. The behavior of the
Z(λ)=pZ − Z +(1 p)Z(r λ)f(λ),
2 2 − w velocitydistributionforasymptoticallylargevelocitiesis
(6) determined by the pole closest to the origin, and has the
where f(λ) exp( iλη) . Equation (6) is non-linear form P(v) exp( λ v ), where λ is determined from
≡ (cid:104) − (cid:105)η ∼ − ∗| | ∗
and non-local (in the argument of Z) and is not solvable 1 (1 p)f(λ) = 0 [43]. When r = 1/2, the iterative
− −
in general. But it is possible to numerically obtain the numerical scheme discussed above for dissipative driving
probability distribution for certain choices of the param- may be followed for determining the characteristic func-
eters. tion for the diffusive case.
When r =0 and r =1/2, Eq. (6) takes the form, Thedynamics[Eqs.(1,2)]alsoallowsthecalculationof
w
the moments of the steady state distribution. For the
(cid:20) (cid:18) (cid:19)(cid:21)2 (cid:18) (cid:19)
λ λ 1 Maxwell model, it was shown that the equations for the
Z(λ)=p Z +(1 p)Z f(λ), r =0,r = .
2 − 2 w 2 twopointcorrelationfunctionsclose[43,44]. Theclosure
(7) canbealsoextendedtoonedimensionalpseudoMaxwell
Thus, Z(λ) is determined if Z(λ/2) is known. By it- models where particles collide only with nearest neigh-
erating to smaller λ, and considering the initial value borparticleswithequalrates[45]. Usingthissimplifying
4
property, the variance of the steady state velocity dis-
tribution in the thermodynamic limit was determined to
be:
2κσ2
v2 = , (9)
(cid:104) (cid:105) 1 r2+2κ(1 r2)
− − w
where κ = (1 p)/p and σ2 is the variance of the noise
−
distribution. On the other hand, the two-point velocity
correlations in the steady state vanishes in the thermo-
dynamic limit.
Among the higher moments, the odd moments van-
ish as the velocity distributions is even. Define 2n-th
moment of the distribution to be v2n =M . The evo-
2n
(cid:104) (cid:105)
lution equation for M may be obtained by multiplying
2n
Eq. (4) by v2n, and integrating over the velocities. It is
then straightforward to show that they satisfy a recur-
rence relation, FIG.1. (coloronline)Thenumericallycalculatedvelocitydis-
tribution P(v), obtained from the inverse Fourier Transform
(cid:104)1 (cid:15)2n (1 (cid:15))2n+κ(cid:0)1 r2n(cid:1)(cid:105)M = of the characteristic function Z(λ), for different noise distri-
− − − − w 2n butions as described in Eq. (3) with (a) γ = 1/2, (b) γ = 1,
(c) γ = 2, and (d) γ = 3 for a = 3. P(v) is computed for
n 1(cid:18) (cid:19)
(cid:88)− 2n (cid:15)2m(1 (cid:15))2n−2mM2mM2n 2m rw =1/2 (dissipative driving) and rw =1 (diffusive driving)
2m − − and compared with the noise distribution.
m=1
n 1(cid:18) (cid:19)
(cid:88)− 2n
+κ r2mM N , (10)
2m w 2m 2n−2m A. Dissipative Driving (rw <1)
m=0
where (cid:15) = (1 r)/2 and Ni is the i-th moment of the We first evaluate the velocity distribution numerically
−
noisedistribution. Equation(10)expressesM2n interms by inverting the characteristic function Z(λ). For this
oflowerordermoments. SinceP(v)isanormalizabledis- calculation, f(λ), the Fourier transform of the noise dis-
tribution,M0 =1. AlsoM2 isgivenbyEq.(9). Knowing tribution in Eq. (3), is determined numerically using
thesetwomoments,allhigherordermomentsmaybede- Eq. (7). Figure 1 shows the velocity distributions ob-
rived recursively using Eq. (10). tained for γ = 1/2,1,2,3 for fixed a = 3 (see Eq. (3)
The ratios of moments may be used for determining for definition of a). For the case r = 1/2, correspond-
w
thetailofthevelocitydistribution. Supposethevelocity ing to dissipative driving, the velocity distribution P(v)
distribution is a stretched exponential: approaches the noise distribution for large velocities for
all values of γ. This suggests that the tail of the distri-
b1/β
P(v)= exp( bv β), b,β >0, (11) bution is determined by the characteristics of the noise.
2Γ(1+β−1) − | | However, using this method, it is not possible to extend
the range of v to larger values so that the large v behav-
where Γ is the Gamma function. For this distribution
iormaybedeterminedunambiguously. Therangeofv is
the 2nth moment is
limitedbytheprecisiontowhichf(λ)canbedetermined
Γ(2n+1) numerically.
M2n =b−2n/ββΓ(1β+ 1), (12) The ratios of moments [see Eq. (13)] is a more robust
β method for determining the tail of the velocity distribu-
tion. The moments are calculated from the recurrence
such that that the ratios for large n is
relation Eq. (10) where the moments of the noise distri-
M (cid:18)2n(cid:19)2/β bution described in Eq. (3) is given by
2n
, n 1. (13)
M2n 2 ≈ bβ (cid:29) Γ(2n+1)
− N =a 2n/γ γ . (14)
Though Eq. (13) has been derived for the specific distri- 2n − γΓ(1+ 1)
γ
bution given in Eq. (11), the moment ratios will asymp-
totically obey Eq. (13) even if only the tail of the distri- The numerically obtained moment ratios of the steady
bution is a stretched exponential. This is because large statevelocitydistributionfordissipativedrivingisshown
moments are determined only by the tail of the distri- inFig.2,fordifferentnoisedistributionscharacterizedby
bution. Thus, the exponent β can be obtained unam- γ. Themomentratiosincreasewithnasapowerlawwith
biguously from the asymptotic behavior of the moment an exponent 2/γ, independent of the value of r and the
w
ratios. coefficient of restitution r. Comparing with Eq. (13), we
5
(a) γ=1/2 (b) γ=1 3.006 (a) γ=1/2 r =1/4,r=1/4
1010 n4 n2 104 (b) γ=2 rw=1/4,r=3/4
/MM2n2n-2 rrrrnwwwwo====n1111-i////n4422t,,,, rrrrr====1111=////14242/4100 bn()33..0000243 rrrrwwwww====3313////4444,, rrnn==oo13nn//--44iinntt 2
100 non-int rww=1/2 2 0.000 0.007 0.014 0 0.007 0.014
102 (c) γ=2 n (d) γ=3 n2/3 10 1/n 1/n
2
M2n- 6 (c) γ=1/2 (d) γ=1 rr==13//44 6
/n non-int
M2100 100 b b3(=13-r )2
w
3 3
0 1 2 3 1 2 3
10 10 10 10 10 10 10
n n
(e) γ=2 (f) γ=3
FIG. 2. (Color Online) The moment ratios [see Eq. (13)] for 3 3
different noise distributions as described in Eq. (3) with (a)
b
γ = 1/2, (b) γ = 1, (c) γ = 2, and (d) γ = 3 for a = 3.
In each figure the ratios are plotted for r = 1/4, 1/2, as
well as r = 1/4, 1/2, corresponding to dissipative driving.
w 0 0
These are is compared with the moment ratios of the non- 0 0.2 0.4r0.6 0.8 0 0.2 0.4r0.6 0.8 1
interacting system in which collisions are ignored, as well as w w
the noise distribution (dashed green line).
FIG. 3. (Color Online) The coefficient b(n) obtained from
Eq. (15) for (a) γ = 1/2, (b) γ = 2 varies linearly with n−1
obtainβ =γ,andthatthetailofthevelocitydistribution for dissipative driving rw < 1. The choice of r and rw are
isdeterminedbythenoisedistribution. Wealsocompare thesameinbothplotsandlabeledin(b). Thecorresponding
theresultswiththosefordrivennon-interactingparticles. b(n) obtained for the non-interacting system are also shown.
The variation of b=b(∞) with r is shown for (c) γ =1/2,
Here, collisions between particles are completely ignored w
(d) γ =1, (e) γ =2 and (f) γ =3.
sothatthetimeevolutionofparticlesareindependentof
eachother,andeachparticleisdrivenindependently. For
the range of parameters, considered, the moment ratios
system coincide. In addition, for γ 1, we find that the
oftheinteractingsystemisasymptoticallyindistinguish- ≤
valueofbapproachesthevalueacharacterizingthenoise
able from that of the non-interacting system, showing
distribution.
that for dissipative driving collisions between particles
do not affect the tails of the velocity distribution. The
moment ratios are also compared with those of the noise
B. Velocity distributions for diffusive driving
distribution. Here, we observe that while the ratios have
the same power law exponent, the prefactor is different.
Wenowdeterminetheconstantbintheexponentialin TheMaxwellgaswithdiffusivedriving(rw = 1)does
−
Eq. (11). It may be determined from Eq. (13) once β is not have a steady state in the long time limit, when the
determined. Rearranging Eq. (13), we obtain total energy diverges. However, it has a pseudo steady
statesolutionthatisvalidatintermediatetimes. Onthe
2n(cid:18) M2n (cid:19)−β/2 other hand when rw = 1 the system reaches a steady
b(n) , n 1. (15)
state at large time. It has been shown that the veloc-
≈ β M (cid:29)
2n 2
− ity distribution in the pseudo steady state for the case
Figures 3 (a) and(b) show the variation of b(n) with n r = 1 is the same as the velocity distribution in the
w
−
for different γ. We find that for large n, b(n) is indepen- steady state of the system with r = 1 [43]. For r = 1
w w
dent of coefficient of restitution r, but may depend on and η taken from a Gaussian distribution, the velocity
r . Also, we find that b b(n) n 1 for all values of distribution was shown to have an exponential distribu-
w −
− ∝
γ. Figures 3 (c) and(f) show the variation of b with r tion [43]. In this section, we determine this steady state
w
for different γ. For γ = 1/2 and 1, b is independent of for other noise distributions.
r , while for γ = 2 and 3, it depends on r . We have In Fig. 1, the numerically obtained P(v) is shown for
w w
checked that b is independent of r for γ up to 1. In different values of γ. We find that for γ = 1/2,1 the
w
Figures 3 (c) and(f), the values of b are also compared velocity distribution approaches the noise distribution.
withthatobtainedforanon-interactingsysteminwhich Interestingly, when γ =2,3 the velocity distribution de-
collisionsbetweenparticlesareignored. Wefindthatthe viatessignificantlyfromthenoisedistribution. Whilethe
values of b for both the interacting and non-interacting dataforlnP(v)appearstovarylinearlywithv,therange
6
3.010 2.13
1M/M0nn22-210 (a) γ=1/~2n4 (b) γ=1~n2 rrww==11,,rr==11//42 105 )b(n3.005 (a) γ=1/2 (b) γ=1 λrr*==13//44
Noise
0 3.000
Analytic 10
0 2.9 2.12
10
(c) γ=2 ~n2 (d) γ=3 106 (c) γ=2 (d) γ=3 2.88
104 ~n2
)
/Mnn2-2 ~n ~n2/3 (bn2.89 2.87
M2
100 100 0 1000 20000 1000 2000
0 1 2 3 1 2 3 n n
10 10 10 10 10 10 10
n n
FIG. 5. (Color Online) The variation of the coefficient b(n)
FIG. 4. (color online) The moment ratios [see Eq. (13)] for with n obtained from Eq. (15) for diffusive driving rw = 1
different noise distributions as described in Eq. (3) with (a) and for different values of r, for different noise distribution
γ = 1/2, (b) γ = 1, (c) γ = 2, and (d) γ = 3 for a = 3. characterized by (a) γ = 1/2, (b) γ = 1, (c) γ = 2, (d) γ =
Thedataareforr =1(diffusivedriving)andtheratiosare 3. The dashed line corresponds to the analytically obtained
w
plotted for r = 1/4,and 1/2. These are compared with the asymptotic value λ∗ [See Eqs. (16), (17)].
noisedistribution(dashedgreenline). In(b),(c)and(d),we
alsoplotmomentratiosfortheexponentialdistributionwith
analytically obtained value of λ∗ [See Eqs. (16), (17)]. that when γ <1, the coefficient b(n) approaches that of
the noise distribution a = 3. For γ 1, b is calculated
≥
by substituting β =1 in Eq. (15). One finds in this case
is limited and it is not possible to unambiguously con- that b approaches λ which is obtained analytically.
∗
clude that P(v) is exponential independent of the noise
distribution.
Asforthedissipativecase,thebettertooltoprobethe IV. NON-INTERACTING SYSTEM
tail of the distributions is the moment ratios Eq. (13).
Figure 4 shows that moment ratios increase with n as a We showed in Sec. IIIA that, for dissipative driving,
power law. The power law exponent is 2/γ for γ < 1 the tail of the velocity distribution P(v) is identical to
[see Fig. 4(a)] and equal to 2 for γ 1 [see Fig. 4((b)- that of a non-interacting system in which collisions be-
≥
(d)]. Thus, we conclude that β = min[γ,1]. Thus, P(v) tween particles may be ignored. In this section, we de-
is universal, and has an exponential tail for γ 1. termine the velocity distribution of the non-interacting
≥
The exact form of the universal exponential tail can system in terms of the noise distribution. In the non-
be analytically obtained as follows. If the velocity distri- interacting system, the particle is driven at each time
bution has the form P(v)=(λ∗/2)exp(−λ∗|v|), the mo- step. If vn is the velocity after the nth collision, then
mentratiosinthelargenlimitbehavesasM /M
2n 2n 2
(4n2 2n)/(λ∗)2. But we have seen in Sec. III that−, fo≈r vn =−rwvn−1+ηn−1. (18)
−
diffusive driving Eq. (8)satisfies a solution such that the For a particle that is initially at rest (v =0),
0
velocitydistributionisdeterminedbythepolenearestto
n 1 n 1
the origin ±iλ∗ obtained from relation 1 = (1−p)f(λ). v = (cid:88)− rmη = (cid:88)− rmη , (19)
When γ =1, the pole has the form given by n w n m 1 w m
− −
m=0 m=0
λ∗ = a√p, γ =1, (16) where the second equality is in the statistical sense, and
±
(cid:112)
2ln(1 p) followsfromthefactthatnoiseisuncorrelatedandthere-
λ∗ = − − , γ =2. (17) fore the order is irrelevant.
± σ
Now, consider the moment generating function of the
When γ = 3, we obtain complicated Hypergeometric
noisedistribution, exp( λη) exp[µ(λ)]whereµ(λ)is
function for f(λ) from which λ may be determined nu- (cid:104) − (cid:105)≡
∗ the cumulant generating function,
merically. The moment ratios thus obtained are plotted
in Fig. 4(b), (c), and (d) which matches with the nu- (cid:88)∞ λ2n
µ(λ) C , (20)
merically calculated moment ratio. It can be seen that ≡ 2n! 2n
i=1
when γ < 1, there is no λ which satisfies the relation
∗
1=(1 p)f(λ). whereC isthe2nth cumulantofthenoisedistribution.
2n
−
From Eq. (15), we obtain the coefficient b for the dif- It has been assumed that the noise distribution is sym-
fusively driven system and is shown in Fig. 5. It is seen metric such that only even cumulants are non-zero. The
7
moment generating function of the velocity after infinite
time-steps is,
(cid:42) (cid:34) (cid:35)(cid:43) 3
(cid:88)∞
exp( λv ) = exp λ rmη ,
(cid:104) − ∞ (cid:105)η − w m
m=0 η P(v) e b(rw) v �
(cid:34) (cid:35) � | |
⇠
(cid:88)∞
= exp µ(rmλ) . (21) �
− w
m=0
1
From the definition of µ(λ) [see Eq. (20)], we obtain
P(v) e av �
µ(rwmλ)= (cid:88)∞ (rwm2nλ!)2nC2n. (22) 0 ⇠ � | |
n=1 1 r 1
Summing over m, No s�teady state w P(v) e�v|v⇤|
⇠
(cid:88)∞ µ(r2mλ)= (cid:88)∞ (cid:88)∞ (rwmλ)2nC , FIG. 6. (color online) Schematic diagram summarizing the
w 2n! 2n results obtained in paper. The parameters r ∈ [−1,1] is
w
m=0 m=0n=1
the coefficient of restitution of wall-particle collisions and γ
(cid:88)∞ λ2n (cid:18) 1 (cid:19) characterizesthenoisedistribution[seeEq.(3)]. Whenr =
= C . (23) w
2n! 1 r2n 2n −1, the system does not reach a time independent steady
n=1 − w state. When r = 1, P(v) is universal when γ ≥ 1, and has
w
the same asymptotic behavior as the noise distribution when
But, exp( λv ) = exp[ξ(λ)] where ξ(λ) is the cumu-
lant g(cid:104)enera−ting∞fu(cid:105)nction of the velocity distribution at γ < 1. When the driving is dissipative (|rw| < 1), P(v) has
the same asymptotic behavior as the noise distribution for
large times,
γ < 1. When γ ≥ 1, the coefficient in the exponential gets
modified.
(cid:88)∞ λ2n
ξ(λ)= D , (24)
2n
2n!
n=1
These results generalize the results in Ref. [43], where
where D2n is the 2nth cumulant of the velocity distribu- it was shown that for dissipative driving that when the
tion. Comparing with Eq. (23), we obtain noise distribution is gaussian or Cauchy, the tails of the
velocitydistributionaresimilartothatofthenoisedistri-
C
D = 2n . (25) bution. The results are also consistent with the intuitive
2n 1−rw2n understanding that the tails of velocity distribution are
boundedfrombelowbythenoisedistribution. Thisisbe-
Forlargen,behaviorofthecumulantsofthevelocitydis-
cause the tails are populated by particles that have been
tributionapproachesthatofthenoisedistribution. Thus,
recently driven and then do not undergo any collision.
byknowingallcumulants,thevelocitydistributionofthe
Weexpectthatmorecomplicatedkernelsofcollisionwill
non-interacting system is completely determined.
notchangetheresult. Thiscouldbethereasonwhymany
of the experimental results [23] see non-universal behav-
ior. However, there are experiments that see universal
V. DISCUSSION AND CONCLUSION
behavior [21, 24]. In these experiments the P(v) is mea-
suredindirectionsperpendiculartothedrivingdirection.
Insummary,weconsideredaninelasticonecomponent
It may be that the details of the driving are lost when
Maxwell gas in which particles are driven through colli-
energy is transferred to other directions. Transferring
sionswithawall. Wedeterminedpreciselythetailofthe
energy in other directions ensures that collisions cannot
velocity distribution P(v) by analyzing the asymptotic
be ignored, unlike the case of one-component Maxwell
behavior of the ratio of consecutive moments. Our main
gas studied in this paper. The two component Maxwell
results are: (1) For dissipative driving, the tail of P(v)
model is a good starting point to answer this question.
is identical to that of the corresponding non-interacting
Methodsdevelopedinthepaperwillbeusefultoanalyze
system where collisions are ignored. By solving the non-
the same. This is a promising area for future study.
interacting problem, the cumulants of the velocity dis-
tribution may be expressed in terms of the noise distri-
bution and. Thus, P(v) is highly non-universal. (2) For
ACKNOWLEDGMENTS
diffusive driving, P(v) is universal and decays exponen-
tially when the noise distribution decays faster than ex-
ponential. If Φ(η) decays slower than exponential, then This research was supported in part by the Interna-
P(v) is non-universal and the tails are similar to the tail tional Centre for Theoretical Sciences (ICTS) during a
of Φ(η). These results are summarized in Fig. 6. visit for participating in the program -Indian Statistical
8
Physics Community Meeting 2016 (Code: ICTS/Prog- ISPC/2016/02)
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