A consistent approach for the treatment of Fermi acceleration in time-dependent billiards

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by  A. K. Karlis

A consistent approach for the treatment of Fermi acceleration in
time-dependent billiards
A. K. Karlis,1,2,a) F. K. Diakonos,1,b) and V. Constantoudis3,c)
1)Department of Physics, University of Athens, GR-15771 Athens, Greece
2)Physikalisches Institut, Universita¨t Heidelberg, Philosophenweg 12, 69120 Heidelberg,
Germany
3)Institute of Microelectronics, NCSR Demokritos, P.O. Box 60228, Attiki, Greece
(Dated: 12 January 2013)
The standard description of Fermi acceleration, developing in a class of time-dependent billiards, is given in
terms of a diffusion process taking place in momentum space. Within this framework the evolution of the
probability density function (PDF) of the magnitude of particle velocities as a function of the number of
collisions n is determined by the Fokker-Planckequation (FPE). In the literature the FPE is constructed by
2
1 identifying the transport coefficients with the ensemble averages of the change of the magnitude of particle
0 velocityanditssquareinthecourseofonecollision. Althoughthistreatmentleadstothecorrectsolutionafter
2 asufficientlylargenumberofcollisionshasbeenreached,thetransientpartoftheevolutionofthePDFisnot
n described. Moreover, in the case of the Fermi-Ulam model (FUM), if a stadanrd simplification is employed,
a thesolutionoftheFPEiseveninconsistentwiththevaluesofthetransportcoefficientsusedforitsderivation.
J The goal of our work is to provide a self-consistent methodology for the treatment of Fermi acceleration in

7 time-dependent billiards. The proposedapproachobviates any assumptions for the continuity of the random
processandthe existenceofthelimits formallydefining thetransportcoefficientsofthe FPE.Specifically,we

]
suggest, instead of the calculation of ensemble averages,the derivation of the one-step transition probability
D
function and the use of the Chapman-Kolmogorov forward equation. This approach is generic and can be
C
applied to any time-dependent billiard for the treatment of Fermi-acceleration. As a first step, we apply this
.
n methodology to the FUM, being the archetype of time-dependent billiards to exhibit Fermi acceleration.
i
l
n PACS numbers: 05.60.Cd,05.45.Ac,05.45.Pq
[

2
Fermi acceleration, that is the increase of the the increase of the mean energy of particles as a result
v
mean energy of an ensemble of particles due to of random collisions with moving scatterers. Soon after
9
random collisions off moving scatterers, is clearly his seminal paper, his co-workerUlam introduced a sim-
4
5 one of the most interesting physical mechanisms plemechanicalmodelfortestingFermi’sidea2,knownas
1 linked to time-dependent billiards. Despite this the Fermi-Ulam model (FUM), linking for the first time
. fact, the standard approach in the literature Fermiaccelerationwith the study oftime-dependent bil-
2
0 for its analytical treatment can at best describe liards.
1 Fermi acceleration in the asymptotic time limit. Since the introduction of the FUM, the standard de-
1 Herein, we propose a methodology, which de-
scription of Fermi acceleration developing in a class of
: scribes the evolution of Fermi acceleration at all
v time-dependent billiards is given in terms of a diffusion
i times and, even more, obviates any unclear or ad processtakingplaceinmomentumspace3–5. Withinthis
X hoc assumptions, which can lead to inconstencies
framework the evolution of the probability density func-
r or unreliable results. We exemplify the proposed
a tion (PDF) of the magnitude of particle velocities as a
approach in the prototype of billiards exhibiting
function of the number of collisions n is determined by
Fermi acceleration; The Fermi-Ulam model.
the Fokker-Planck equation (FPE). In the literature the
FPE is constructed by identifying the transport coeffi-
cients with the ensemble averages of the change of the
magnitudeofparticlevelocityanditssquareinthecourse
I. INTRODUCTION
of one collision3–6. Although this treatment leads to the
correctsolution after a sufficiently large number of colli-
More than 60 years ago, Fermi1 proposed an intuitive sionshasbeenreached,thetransientpartoftheevolution
mechanismfortheexplanationoftheoriginofthehighly ofthePDFisnotdescribed. Moreover,inthecaseofthe
energetic cosmic ray particles and ever since it has been FUM, if a standard simplification is employed —known
a subject of intense study. The mechanism consists in asthe staticwallapproximation(SWA) orthe simplified
FermiUlamModel(SFUM)— the solutionofthe FPEis
even inconsistent with the values of the transport coeffi-
cients used for its derivation.
a)Electronicmail: [email protected]
b)Electronicmail: [email protected] The aim of the work presented, is to provide a self-
c)Electronicmail: [email protected] consistent methodology for the derivation of the PDF
2
of particle velocities for all times. The proposed ap- A. The Fokker-Planck approximation
proachobviatesanyassumptionsforthecontinuityofthe
random process and the existence of the limits formally
The standard approach in the literature for the de-
defining the transport coefficients of the FPE. Specifi-
termination of the asymptotic behaviour of the PDF of
cally, we suggest, instead of the calculation of ensem-
particlevelocities,is the approximationof the jump pro-
ble averages, the derivation of the one-step transition cess with a diffusion process, described by the FPE3–5.
probabilityfunction(TPF)andtheuseoftheChapman-
This approximation has also been used for the analyti-
Kolmogorov (forward) equation (CKE). This approach
caltreatmentofFermiaccelerationdevelopinginhigher-
is generic and can be applied to any time-dependent bil-
dimensionalbilliards,likethe simplifiedperiodicLorentz
liard for the treatment of Fermi-acceleration. As a first gas6, i.e. the oscillating hard circular scatterers oscillate
step, we apply this methodology to the FUM, being the
only in the velocity space. An equation of the form of
archetype of time-dependent billiards to exhibit Fermi the FPE can be derived from Eq. (1) as follows7:
acceleration. In this context, we show that the FPE re-
If we introduce ∆v v z, then the integrand in
ported in the literature5 describing the evolution of the ≡ −
Eq. (1) can be rewritten as,
PDF of the magnitude of particle velocities is not valid,
and that the observed agreement for n 1 between the
analytical and numerical results, in this≫case, should be ρ(v,nv′,n′)= d(∆v)ρ(v ∆v,n 1v′,n′)
| − − | (2)
regarded as accidental, i.e. due to the validity of the Z
W (v ∆v+∆v,v ∆v).
central limit theorem (CLT). × − −
′ ′
Expanding the distribution function ρ(v,∆v,v ,n) and
the transition probability function (TPF) W(v;∆v) in a
II. STATISTICALDESCRIPTION OF FERMI
ACCELERATION Taylor series yields,
∞
( 1)m
Fermiaccelerationdevelopinginatime-dependentbil- ρ(v,nv′,n′)= d(∆v) − (∆v)m
liard can be described in terms of a stochastic process | m!
Z m=0
taking place in the velocity space. Let W(v,z) denote ∂m X
′ ′
the probability of a particle being at the velocity z to ρ(v,n 1v ,n)W (v+∆v,v)
× ∂vm − |
perform a jump to velocity v in the course of a single
(3)
′ ′
collision and ρ(v,nv ,n) the probability of a particle to
| ′
possessvelocityv afterncollisionsgiventhatatthenth
Integrating now Eq. (3) over ∆v we obtain,
′
collision it had velocity v . This jump process can be
described by the following equation: ∞
( 1)m ∂m
′ ′ ′ ′
ρ(v,nv ,n)= − M (v)ρ(v,n 1v ,n),
| m! ∂vm m − |
′ ′ ′ ′
ρ(v,nv ,n)= dzρ(z,n 1v ,n)W (v,z) (1) m=0
X
| − | (4)
Z
where M (v) stands for the mth moment of the TPF,
m
Eq. (1) is exact, on the condition that the process is
i.e.
Markovian. From a physical point of view, this means
thatthe probabilityofaparticle toexperience avelocity
M (v)= (∆v)mW(v+∆v,z)d(∆v).
jumpequalto∆v uponthenthcollisiondependsonlyon m
Z
the velocity it had at the previous step, i.e at the nth-1
collision. Therefore,
∞
( 1)m ∂m
′ ′ ′ ′ ′ ′
ρ(v,nv ,n) ρ(v,n 1v ,n)= − M (v)ρ(v,nv ,n) (5)
| − − | m! ∂vm m |
m=1
X
By truncating the above series to the second order, and for n 1 one obtains an equation resembling the FPE.
≫
further by approximating the discrete derivative
∂ ∂
′ ′ ′ ′
ρ(v,nv ,n)= [Bρ(v,nv ,n)]
∂n | − ∂v |
′ ′ ′ ′ ′ ′ (6)
∆kρ(v,n|v ,n)=[ρ(v,n+k|v ,n)−ρ(v,n|v ,n)]/k, + 1 ∂2 [Dρ(v,nv′,n′)],
2∂v2 |
′ ′
(k =1)withthecontinuousderivative∂ρ(v,nv ,n)/∂n, wherethe coefficientB andD isthe ensembleaverageof
|
3
the change of particle velocities and its square, respec- Moreover,in this case an approximate solution can be
tively, in one mapping period. obtained directly in the velocity space, using the sad-
Theapproximationsappliedabovefortheconstruction dlepoint approximation technique9. Specifically, from
of the FPE are valid on the condition that only very Eq. (8), one can derive the moment generating function
small jumps are probable and further that the solution
∞
′ ′
ρ(v,nv ,n)variesslowlywithvsothatonecanperform φ(t,n)= etxρ(v,nz,0)dv
the ex|pansion in a Taylor series. More formally8, we −∞ |
Z (11)
demand that there exists a δ >0, ∞ n
= etvW(v)dv etz
−∞
W(z+∆z,z) 0, for ∆z >δ (7a) (cid:18)Z (cid:19)
≈ | |
ρ(v+∆v,nv′,n′) ρ(v,nv′,n′), for ∆v <δ. (7b) of the velocity PDF. To find the saddlepoint tˆ(v,n),
| ≈ | | | ′
we solve the equation κ(t,n) = v, where κ(t.n) =
Intheliterature5thederivationofanFPEfromEq.(1) log(φ(t,n)). Then, the PDF is approximately,
forthestatisticaldescriptionofFermiaccelerationiscar-
ried out on an ad hoc basis. As a consequence, as shown 1
ρ(v,nz,0)
in the following, it has produced contradictory results. | ≈s2πκ′′(tˆ(v,n)) (12)
Moreover, by construction, the description of Fermi ac-
exp κ(tˆ(v,n)) tˆ(v,n)v .
celeration with a continuous stochastic process, can at × −
best describe the statistics only for n 1. Hence, a full
In the following sectio(cid:2)ns we implement th(cid:3)e proposed
≫
descriptionofFAinatime-dependentbilliardcanonlybe
methodologyintheprototypeoftime-dependentbilliards
given in the context of a jump process and consequently
exhibiting Fermi acceleration; The Fermi-Ulam model
by Eq. (1).
(FUM).
B. A complete description: The Chapman-Kolmogorov III. FERMI ACCELERATION IN THE STOCHASTIC
equation SIMPLIFIED FUM
Thestudyofthetransientstatisticscanonlybeaccom- The Fermi-Ulam model, originally proposed for test-
plishedbymeansoftheChapman-Kolmogorovequation, ing the feasibility of gaining energy through scatter-
i.e. Eq.(1). Assumingthatinitiallyparticlevelocitiesare ing off moving targets, i.e. Fermi acceleration, consists
distributed according to ρ(v,0) = δ(v z), Eq. (1) can of one harmonically oscillating and one fixed infinitely
− ′
be rewritten in respect with the one-step TPF. W(v,v ) heavy hardwalland anensemble of non-interactingpar-
as, ticles bouncing between them. Ever since, many dif-
ferent versions of the original model have been sug-
ρ(vn,nz,0)= W(vn,vn−1) W(v1,z)dv, (8) gested and investigated, such as variants of the FUM
| ··· ··· with dissipation10–14, different deterministic or random
Z Z
drivingsofthemovingwall15,16 the quantum-mechanical
n−1 version17–22 and the, so called, bouncer model23, where
where dv = dv . The derivation of the one-step
i a particle performs elastic24 or inelastic25–31 collisions
i=1
TPF can be acQhieved by determining the PDF p(q) of with an oscillating infinitely heavy platform under the
the variables q xi appearing in the dynamicalequa- influence of a gravitationalfield. Recently, a hybrid ver-
≡{ }
tion defining the velocity of a particle after a collision sion of the FUM and the bouncer model has also been
withthemovingboundaryofthetime-dependentbilliard, investigated32,33.
vn =f(vn−1,q). Then, the TPF is The equations defining the dynamics of the FUM are
of implicit form with respect to the collisiontime, which
complicatesnumericalsimulationsandhindersananalyt-
W(vn,vn−1)= p(q)δ[vn−f(q,vn−1)]dq. (9) ical treatment. A simplification5 —known as the static
Z
wall approximation (SWA)15,34– consists in treating the
IftheresultingTPFisafunctionofthedifferenceofve-
oscillating wall as immobile, located at its equilibrium
locitiesatsuccessivesteps W(vn,vn−1)=W(vn vn−1), position, yet allowing the transfer of momentum upon
−
Eq. (8) can be easily solved in the Fourier space. Specif-
impact with a particle as if the wall were harmonically
ically,ifthis conditionis met,thenbytakingtheFourier
oscillating. This simplificationhas become overthe time
transform of Eq. (8) we find, the standard approximation for studying the FUM35.
The SWA speeds-up numerical simulations and facili-
[ρ˜(v,nz,0)]=(2π)n−21 e−ikz [W(v)] n, (10) tates the analytical treatment of the problem, while it
F | {F }
hasbeengeneralizedtohigher-dimensionalbilliardswith
∞
time-dependent boundaries, such as the time-dependent
where =1/(√2π) exp[ ikv]dv.
F −∞ − Lorentz Gas6,15.
R
4
n
Letusconsider,withoutlossofgenerality,aFUMcon-
enced up to this point, i.e. v = ∆v +v . Fur-
n n 0
sisting of a fixed wall on the right and a moving wall m=1
on the left, oscillating with frequency ω. If we further thermore, due to Fermi-accelerationPdeveloping in the
assume that the positive direction of particle velocities SFUM, after n 1 collisions, the vast majority of the
≫
is towards the right, then the dynamics of the billiard particles has acquired velocities much greater than the
within the framework of the SWA is defined by the fol- maximum wall velocity, irrespective of the initial distri-
lowing set of dimensionless difference equations: bution. Therefore, most of the collisions, after a suf-
ficiently large “time”, take place in the high velocity
2 regime. In this limit, the absolute value function can be
tn =tn−1+ vn−1 (13a) neglected and we immediately obtain3,36 ∆v = 0 and
vn = vn−1+2un (13b) (∆v)2 = 2ǫ2. Therefore, the velocity juhmpsiare com-
| | pletely uncorrelated, i.e. do not depend on the velocity
un =ǫcos(tn+ηn), (13c) (cid:10) (cid:11)
the particle had at the previous step. Thus, for n 1
≫
whereu isthevelocityofthe“oscillating”wall,v isthe and v ǫ the centrallimit theorem (CLT) dictates that
n n ≫
algebraic value of the particle velocity immediately after the PDF ofparticle velocitiestends to aGaussiandistri-
n
the nth collision with the “oscillating” wall measured in bution,withameanvalueequalto ∆v andvariance
i
units of ωw (w denoting the spacing between the walls), i=1h i
tn the time when the nth collision occurs measured in σ2 = n (∆v )2 ∆v 2 . P
i i
units of1/ω,η a randomvariableuniformlydistributed −h i
n i=1
in the interval [0,2π) updated immediately after each HenPce,htDhe PDFEof particleivelocities for n 1 is
≫
collision between a particle and the fixed wall and ǫ the
1 v2
dimensionless ratio of the amplitude of oscillation to the ρ(v,n)= exp . (14)
ǫ√πn −4ǫ2n
spacing between the “oscillating” and the fixed wall. It (cid:20) (cid:21)
is noted that in all numerical simulations ǫ was fixed at
InFig.1, Eq.(14)is plotted alongwiththe histogramof
1/10.
particle velocities obtained from the simulation of 1.2
The absolute value in Eq. (13b) is introduced in order 106 trajectoriesforn=105 collisions. The ensemblewa×s
toavoidtheoccurrenceofpositiveparticlevelocitiesafter initiallydistributedaccordingtothedeltafunctionδ(v
a collision with the “oscillating” wall, which would lead ǫ). The analytical result obtained from the applicatio−n
to the escape of the particle from the area between the of the CLT is in perfect agreement with the numerically
walls. It should be stressed that such a collision, within computed PDF.
theframeworkoftheexactmodel,correspondstoaparti-
cle experiencing at least one second consecutive collision
with the “oscillating” wall. Therefore, if Vn−1 < 2un 2. FPE equation in the SFUM
| | | |
and u 0, in order to prevent the particle from escap-
n
≤
ing the region between the walls the velocity is reversed
As mentioned in the previous section, assuming that
artificially. Thepresenceoftheabsolutevaluefunctionin
forn 1theprobabilitymeasureoftheeventsoccurring
Eq.(13b), nevertheless,complicatesthe analyticaltreat- in the≫low-velocity regime is negligible, B ∆v ⋍ 0,
ment of the accelerationproblem. For this reason,it has D (∆v)2 ⋍2ǫ2. InthislimitEq.(6)obta≡inhsthieform
becomeastandardpracticeinthetreatmentoftheFUM ≡h i
ofastandarddiffusionequation. whichforadeltainitial
to remove the absolute value function, thereby neglect-
distribution of velocities v = z together with reflecting
ing the set of collision events upon which the particle
conditions at v = 0 has as a solution the sum of two
direction is not reversed after its collision with the “os-
spreading Gaussians
cillating” wall. However,this further simplification gives
rise to a fundamental inconsistency: the ensemble mean 1 (v z)2
ρ(v,nz,0)= exp −
of the absolute velocity obtained analytically does not | 2√πnǫ2 − 4nǫ2
changethroughcollisions with the “oscillating” wall, de- (cid:26) (cid:20) (cid:21) (15)
(v+z)2
spitethewell-establishednumericalresultthatFermiac- +exp ,
celerationdoestakeplaceinthephase-randomizedFUM. − 4nǫ2
(cid:20) (cid:21)(cid:27)
which for n z2/(4ǫ2ln2) transforms to Eq. (14).
≥
A. The asymptotics of the PDF of particle velocities
3. Remarks
1. Application of the central limit theorem
Although the solution derived by means of the FPE
Inthissectionwewilldiscusstheasymptoticbehaviour is in agreement with the one obtained from the applica-
ofthePDFofparticlevelocitiesintheSFUM.Evidently, tionoftheCLT,themethodologyusedforthederivation
afterncollisionsthe velocityofaparticleevolvinginthe of Eq. (6) stands on very shaky ground, since the ter-
SFUM is the sum of the velocity jumps it has experi- mination of the series at the second term in Eq. (5) is
5
0.018 sion is small, then successive collisions are likely to oc-
cur, the exact dynamics result to higher exit velocities.
Consequently, successive collisions render small particle
0.016 velocitiesimprobable,asopposedtotheSFUM,whereas
showninSec.IIIA1,v =0isthemostprobablevelocity.
Summarizing, the application of the CLT for the de-
0.014 termination of the long-time statistics is much more
straightforwardand renders the solution of a differential
equation redundant. More importantly, the assumption
0.012 of continuity of the stochastic process describing Fermi
acceleration,whichis essentialfor the constructionofan
)
0 FPE, is not required.
0
0 0.01
0
0
1 B. Short-time statistics in the SFUM
=
0.008
n From Eqs. (13) the particle velocity after the nth col-
v, lision given that it had velocity z is,
(
ρ
0.006
v = z 2u 2(z+2u)Θ(2u+z), (16)
− − −
where Θ(x) is the Heaviside unit-step function.
0.004 According to Eq. (13b) the wall velocity u is deter-
n
mined by the phase ξ t +η of oscillation at the
n n n
≡
instant of the nth collision. Due to the fact that in the
0.002 stochastic SFUM the phase is randomly shifted through
the addition of a random number η —distributed uni-
n
formly in the interval (0,2π)— after each collision, the
oscillation phase ξ is completely uncorrelated between
0 n
0 100 200 300 collisions,followingauniformdistribution. Furthermore,
v given that in the context of the SFUM the wall remains
fixed in the configuration space, the wall velocity upon
collision does not depend on the velocity of the particle,
FIG. 1. Histogram —diagonal crosses— of particle velocities therefore, un and vn−1 are also uncorrelated. From the
aftern=105collisions,obtainedbytheiterationofEqs.(13), fundamentaltransformationlawofprobabilitiesthePDF
on the basis of an ensemble of 1.2 ×106 particles initially of the wall velocity upon collision is,
distributedasρ(v,0)=δ(v−ǫ). Theanalyticalresultderived
1
through the application of the CLT [Eq. (14)] is also plotted p(u)= . (17)
—solid (red) line. π√ǫ2 u2
−
For the single-step TPF W(v,z) we can write,
completely arbitrary8. In general, a jump process can
ǫ
be approximatedbyadiffusionprocess,onthe condition W(v,z)= p(u)δ[v v(u,z)]du. (18)
that a scaling assumption for the transition probability Z−ǫ −
holds. Namely, in the limit of infinitely small time in-
Substituting Eqs. (17) and (16) into Eq. (18) we obtain
tervals, the jumps should become smaller and more fre-
after integrating over u,
quent, such that the random process can be viewed as a
continuous one37. An intuitive way to examine this is to
1 Θ(2ǫ v z,2ǫ z)
consider the average square of the jump size (∆v)2 a W(v,z)= − − −
h i π 4ǫ2 (v+z)2
particle makes having a velocity v prior to the collision. (cid:20) − (19)
Given that the SWA treats the moving wall as fixed +pΘ(2ǫ−v+z,2ǫ+v−z)
in the configuration space, all phases upon collision are 4ǫ2 (v z)2
possible, independently of the velocity v of particles be- − − (cid:21)
fore a collision. As a result, the averagejump size is not In Fig. 2 the analyticalpresult of Eq. (19) is compared
reducedasv 0. Incontrast,withintheexactmodel,as with the histogram of particle velocities after a single
→
the velocity of the particle prior to a collision decreases, collision, obtained numerically using Eqs. (13) and an
it becomes increasingly probable to collide with the wall ensemble of 1.2 106 particles, with initial velocity z =
×
at the turning points, where its velocity is close to zero. 0.1. Clearly, the numerical and analytical results are in
Moreover, if the velocity of the particle before a colli- agreement.
6
fore, changes in velocity in the high-velocity regime are
completely uncorrelated.
120 In more detail, Eq. (19), consists of two parts, one of
which does indeed depend only on the jump size. How-
ever,theotherbranchoftheTPF,takingeffectforv <2ǫ
—relating to the set of rare events36— depends also on
100 the velocity at the last step. Nevertheless, the action of
both branchesofW allowsofa simple geometricalinter-
pretation: At each step, the second branch of the TPF
stretches the PDF ρ(v,nz,0), resulting to a probability
|
flux towards negative values of velocity. This unphysi-
80
calresultcausedby the stretching is negatedby the first
branch, which folds the part of the ρ density over the
1) vertical line at v = 0. Therefore, the solution of Eq. (8)
.0
= can be obtained by extending the domain of ρ(v,nz,0)
60 |
z tothewholereallineandapplyingthemethodofimages.
v,
( Thus, for any number of collisions, we have
W
ρ(v,nz,0)=ρ˜(v,nz,0)+ρ˜(v,n z,0), (20)
| | |−
40
whereρ˜isthe solutionoftheunrestrictedproblem. Sub-
stituting Eq. (19) into Eq. (10) we obtain,
1
20 ρ˜(k,nz,0)= exp( ikz)J0(2ǫk )n (21)
| √2π − | |
Eq. (21) cannot be inverted analytically. To obtain an
analytical result into the velocity space, we use the sad-
0 dlepoint approximation [Eq. (12)]. The moment gener-
0 0.1 0.2 0.3 0.4
v ating function of ρ(v,nz,0) is,
|
φ(t,n)=(I (2tǫ))netz, (22)
0
FIG. 2. Histogram —diagonal crosses— of particle velocities whereI0 is the modifiedBesselfunctionofthe firstkind.
afterasinglecollisionwiththe“moving”wall,obtainedusing Consequently, the characteristic function is κ(t,n) =
Eqs. (13) and an ensemble of 1.2×106 particles with initial logφ(t,n) = nlog(I (2tǫ))+tz. The saddlepoint is the
0
velocity z = 0.1. The analytical result [Eq. (19)] for the point tˆ(v,n) that satisfies
one-step transition probability is also plotted for the sake of
comparison —solid line 2nǫI (2tǫ)
′ 1
κ(t,n)=v +z =v. (23)
⇒ I (2tǫ)
0
Eq.(23)isimplicitandcannotbesolvedanalytically. To
The analyticalresult of Eq. (19) reveals that the TPF
′
derive an explicit equation we expand κ(t,n) in powers
depends only on the most immediate history of a parti-
of ǫ to second order. Doing so we get,
cle, that is on the velocity it had at the previous step.
Consequently, the stochastic process is indeed Marko- v z
vian. Even more, if the particle before a collision has tˆ(v,n)= − . (24)
2nǫ2
velocityz >2ǫ,thenthe velocityjump ∆v =v z itun-
−
dergoes is completely independent on its history. There- Substituting Eq. (24) into Eq. (12) we have,
ρ(v,n|z,0)≈ 21ǫe−(v2−nǫz2)2I0(cid:18)vn−ǫz(cid:19)n"πnI0 vn−ǫz 2+πnI2 Iv0n−(cid:0)ǫzvn−ǫIz0(cid:1)2vn−ǫz −2πnI1 vn−ǫz 2#1/2. (25)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
In Fig. 3 we present the exact numerical solution of given by Eq. (25) [blue solid line] for n = 3,5,10,31 ,
{ }
Eq. (8) [red solid line] as well as the approximate one using only the second branch of the one-step TPF
7
[uprightcrosses],foralltimes. Evenmore,weseethatthe
saddlepoint approximative solution describes very accu-
ratelythe evolutionofthe PDFfor n 5. Ascanbe ob-
4 ≥
) served,the PDF ofparticle velocitiesquicklyapproaches
3
to a Gaussian distribution, in accordance with the pre-
=
dictionoftheCLT.Thisisattributedtothefactthatthe
2
n TPFcanbereducedtoadifferencekernel. Consequently,
v, theadditionalassumptionwemadefortheapplicationof
(
ρ 0 theCLTinSec.IIIA1,namelythatthestatisticalweight
0 0.2 0.4 0.6 of the rare events36 is negligible, is redundant. This can
v be circumvented, as aforementioned, by extending the
domain of particle velocities. Thus, if one applies the
4 CLT on the whole real line, then all the conditions for
)
5 its applications are met exactly. As a final remark, we
= wouldliketostressthatthesuccessoftheFokker-Planck
2
n type of equationreportedinthe literature for evenshort
, timesisattributedtothevalidityoftheCLT,guarantee-
v
( ing that the PDF will convergeto a normaldistribution,
ρ 0
allowing for the use of a diffusion equation. If however,
0 0.5 1
thereductionoftheTPFtoadifferencekernelisnotfea-
v
sible, then the transient can be arbitrarily long, a point
2 demonstrated via an example in the following section.
)
0
1
=
1
n C. Long Transients
,
v
(
ρ 0 In the last section we showed that the specific choice
0 0.5 1 1.5
made fortreating negativevelocities aftera collision,i.e.
v
reflection with respect to the v =0, reduces the TPF to
2 anevenfunctionofthe jumpsize. Asa consequence,the
)
1 PDF of particle velocities approaches rapidly to a sum
3
of two spreading Gaussians. Clearly, after a number of
=
1 collisions the system will “forget” its initial distribution,
n and the sum will converge to a single half-Gaussian cen-
v, tered at v = 0+. Therefore, the most probable velocity
(
ρ 0 for a particle evolving in the phase-randomized SFUM
0 1 2 will eventually be v = 0+, in clear contrast with the
p
v results given by the numerical simulation and analytical
results derived using the exact dynamical mapping15,34,
according to which as v 0, ρ(v,n) 0. From a phys-
→ →
ical point of view this happens because if the motion of
the wallin the configurationspace is takeninto account,
as v 0 collisions resulting in an energy loss can occur
FIG. 3. Histogram —upright crosses— of particle velocities
→
after n={3,5,10,31} collisions, obtained by the iteration of only in a small neighborhood around the wall’s extreme
Eqs. (13), on the basis of an ensemble of 1.2×106 particles positions, where its velocity is zero, resulting to a min-
initiallydistributedasρ(v,0)=δ(v−ǫ). Theexactnumerical imal energy loss. Furthermore, if the particle velocity
solution of Eq. (8) [red solid line] as well as the approximate is comparable to the wall velocity, consecutive collisions
one given by Eq. (25) [blue solid line] for n = {3,5,10,31}, can take place, resulting in a higher exit velocity from
usingonly thesecond branch oftheone-stepTPF [Eq.(19)], the interaction region within the exact model.
followedbytheapplicationofthemethodofimages[Eq.(20)]
On account of these properties of the collision process
are also plotted for the sakeof comparison. line.
inthe exactmodel, the reflectionofnegativevelocitiesis
notrealistic. Togapthedifferencebetweentheresultsof
the simplifiedandthe exactFUM,we proposeinsteadof
[Eq. (19)], followed by the application of the method the inversionofnegativeparticlevelocities,the inversion
of images [Eq. (20)]. The numerical solution is in total ofthedirectionofthewall’svelocity,ifthecollisionwould
agreement with the histogram of particle velocities ob- lead to a negative particle velocity. This would lead in a
tained by the iteration of the dynamical equations (13) greaterenergy gain in comparisonwith the reflection, as
8
v+u v + u. Therefore, Eqs (13) change to,
| |≤| | | |
2
tn =tn−1+ (26a)
Vn−1
Vn =Vn−1+2un (26b)
| |
u =ǫcos(t +η ). (26c)
n n n
Let us now derive the TPF. From Eqs. (17), (18) and
(26b) we obtain,
Θ(2ǫ+z v)
W(v,z)= − Θ(2ǫ+z v)+Θ(ǫ z/2)[Θ(z v 2ǫ)+Θ(v 2z)] (27)
π 4ǫ2 (v z)2 − − − − −
− −
(cid:8) (cid:9)
p
AcomparisonoftheanalyticalresultgivenbyEq.(27)
and the histogram of velocities obtained on the basis of
an ensemble of 1.2 106 particles after 1 iteration of 20
×
Eqs. (26) is presented in Fig .4, proving the validity of
the derived result.
18
As expected, the TPF has two branches, one taking
effect only in the low velocity regime, i.e. z < 2ǫ and
another which is nonzero for any velocity z prior to a 16
collision. As was also the case in the SFUM with re-
flection of negative velocities, the part of the TPF that
14
is relevant to the low-velocity regime, depends on the
jump size, as well as, on the velocity of a particle prior
to a collision. However, due to the fact that this branch ) 12
1
of W does not have a simple geometrical interpretation, .
0
the single-step transition function cannot be reduced to =
a difference propagatorby anextensionofthe domainof 10
z
W to the whole real line. Thus, the conditions for the ,
v
application of the CLT are not met exactly. However, (
W 8
due to the acceleration of the particles, as n , the
→ ∞
probability measure of particles having velocity z < 2ǫ
becomes negligible. Therefore, for n 1 and v ǫ the 6
≫ ≫
PDF of particle velocities tends to a Gaussian distribu-
tion [Eq. (14)].
4
The study of the transient behaviour of the PDF re-
quires the solution of the CKE [Eq. (8)]. The numerical
solution of Eq. (8) at times n = 3,5,17,316 is pre- 2
{ }
sentedinFig.5. The histogramsofparticlevelocitiesfor
the same times, calculated by iterating an ensemble of
1.2 106 particles for up to n = 105 collisions, are also 0
× 0 0.1 0.2 0.3 0.4
plotted for the sake of comparison. It can be seen, that
v
the solution of the CKE is in agreement with the results
of the simulation for all times presented. In Fig. 6 the
histogram of velocities for n = 105 collisions is plotted.
The solution obtained from the application of the CLT FIG. 4. Histogram —diagonal crosses— of particle velocities
—on the assumption that the statistical weight of col- afterasinglecollisionwiththe“moving”wall,obtainedusing
lisions happening in the region v < 2ǫ is negligible, is Eqs. (26) and an ensemble of 1.2×106 particles with initial
also plotted, and is full agreement with the PDF in this velocity z = 0.1. The analytical result [Eq. (27)] for the
one-step transition probability is also plotted for the sake of
velocity region. However, a blow-up of the low-velocity
region shows that even after 105 collisions, the PDF di- comparison —solid line
verges from the Gaussian profile. This is clear evidence
thatevenafteraverylargenumberofcollisions,thePDF
in the whole velocity domain cannot be described by an FPE, in contrast to the standard version of the SFUM
9
0.03
4
0.02
)
3
=
2
n 0.025 0.015
,
v
(
ρ
0
0 0.2 0.4 0.6 0.01
0.02
v
)
4 0
0
) 0
5 0 0.005
0
= 1 0 0.5 1
0.015
n 2 =
, n
v
( v,
ρ (
0 ρ
0 0.5 1 0.01
v
2
)
7
1
0.005
=
1
n
,
v
(
ρ 0 0
0 1 2 0 50 100 150
v v
0.4
)
6
1
3
FIG. 6. Histogram —upright crosses— of particle velocities
= 0.2 aftern=105collisions,obtainedbytheiterationofEqs.(26),
n onthebasisofanensembleof1.2×106 particlesinitiallydis-
, tributed as ρ(v,0)=δ(v−ǫ). The asymptotic Gaussian dis-
v
( tribution[Eq.(14)]predictedbytheCLTisalsoplotted–solid
ρ 0
(red) line. A blow-up of the numerically obtained histogram
0 2 4 6
at thelow-velocity region is illustrated in the inset.
v
D. Fermi acceleration in the exact FUM
FIG. 5. Histogram —upright crosses— of particle velocities
after n = {3,5,17,316} collisions, obtained by the iteration Thesimplificationemployedtothe treatmentofFermi
ofEqs.(26),onthebasisofanensembleof1.2×106 particles acceleration in the FUM —treating the wall as fixed in
initially distributed as ρ(v,0) = δ(v−ǫ). The solution ob- realspace—howeverwidespreadandthoroughlystudied,
tained by numerically solving the forward CKE [Eq.8] using prohibits the study of the details of Fermi acceleration.
as the TPF of the modified version of the SFUM [Eq. 27] is InRefs.15,34,36 itwasprovedthatsmalladditionalfluctu-
also plotted —solid (red) line. ations of the time of collision due to dynamical correla-
tions induced by the displacement of the scatterer upon
impact,quantitativelyaswellasqualitativelychangethe
evolution of the PDF of velocities, increasing the effi-
[Eqs. (13)] even though the argumentation used in both ciencyofFermiacceleration. Moreover,the development
cases was the same, i.e. the particles are accelerated. of correlations causes the CLT to break down and the
This exemplifies the potential pitfalls of a diffusion ap- asymptotic PDF ceases to be a normal distribution.
proximationofFermi-accelerationintime-dependentbil- In Ref.34 utilizing a novel simplification, the so-called
liards. hoppingapproximation,whichsucceedsintoretainingall
10
the essentials of the exact dynamics, an analytical solu-
tion for the asymptotic behaviour ofthe PDF of particle
velocitiesintheexactFUM,whichwasinexcellentagree-
2
ment with the numerical simulation of the exact FUM.
)
Specifically, it was shown by means of a Fokker-Planck 3
type of equation that, in contrast with the SFUM, the =
attractorofthePDFofvelocitiesinthefunctionspaceis n 1
aMaxwell-Boltzmanlikedistribution,i.e. independently ,
v
oftheinitialdistributionofvelocities,thePDFconverges (
ρ
to a Maxwell-Boltzman like distribution. Therefore, in 0
the case of the exact FUM for v 0+, ρ(v,nz,0) 0, 0 2 4 6
→ | →
in contrast to the SFUM where ρ(v,nz,0) attains its v
maximumvalue for v 0+. This differe|nce between the 2
→
simplified and the exact FUM can be understood as fol- )
5
lows: Ifthevelocityofaparticleafteracollisionwiththe
=
moving wall is small, then multiple successive collisions 1
n
are likely to occur within the exact FUM, resulting into
,
higherexitvelocities,asopposedtothesimplifiedmodel, v
(
within which successive collisions cannot be realized. ρ
0
Another subtle difference between the simplified and
0 2 4 6
the exact FUM [see Sec. IIIA3] is that within the exact v
FUMparticleswithlowvelocityaremorelikelytocollide 1
with the oscillating wall near its turning points, where )
5
the wall velocity is close to 0. Hence, the velocity jump 1
performed by a particle due to a collision with the wall =
0.5
∆v 0 as v 0. As a result, Fermi acceleration when n
→ →
usingthe exactdynamicscanbebetterapproximatedby ,
v
a continuous stochastic process, or equivalently, by the (
ρ
FPE.Still,thetransientstatisticsinthesystemcanonly 0
0 2 4 6
be studied by means of the CKE.
v
As aforementioned, the movement of the wall in the
1
configuration space described by the exact dynamics re-
)
1
sults into a more efficient energy transfer from the mov- 3
ingwalltotheparticlesuponcollision,whencomparedto =
the SFUM. In mathematical terms, this causes the PDF 0.5
n
of the oscillation phase on collisions to deviate from the
,
v
uniformdistribution,reflectingthefactthathead-oncol- (
ρ
lisionsaremorepreferablethanhead-tailcollisions. How- 0
ever, the phase of oscillation of the moving wall when a 0 2 4 6
particles collides with the fixed wall —or when it passes v
throughanyfixed pointwithin the areabetweenthe two
walls comprising the FUM— is uniformly distributed.
The map describing the exact dynamics is,
FIG. 7. Histogram —upright crosses— of particle velocities
dn, =ǫsin(δtn+tn−1+ηn) (28a) after n = {3,5,17,316} collisions, obtained by the iteration
un =ǫcos(δtn+tn−1+ηn) (28b) ofEqs.(28),onthebasisofanensembleof1.2×106 particles
vn =vn−1+2un, (28c) initially distributed as ρ(v,0) = δ(v−10). The solution ob-
tained by numerically solving the forward CKE [Eq.8] using
astheTPFoftheexactmodel[Eq.32]isalsoplotted—solid
where d stands for the position of the moving wall in
n,
(red) line.
theinstantofthenthcollision,u forthewallvelocity,η
n n
for the random phase component and v for the particle
n
velocity after the nth collision. The time of free flight
Ifwedenotethephaseofoscillationofthemovingwall
δt is obtained by solving the implicit equation
n when a particle collides with the fixed wall with ψ, then
xn−1+vn−1δtn =dn, (29) ψ =cos−1 un + 1 1+ ǫ2 u2 (30)
n ǫ z − n
where xn stands for the position of the particle in the (cid:16) (cid:17) (cid:16) p (cid:17)
instant of the nth collision. For Eq. (30) we obtain for the distribution of the wall