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Particle transport at arbitrary timescales with Poisson-distributed collisions.

22 Nov 2019-Physical Review E (American Physical Society (APS))-Vol. 100, Iss: 5, pp 052134-052134
TL;DR: It is observed that transport can change character over time and that early times show features that, in general, depend on the initial conditions of the walker.
Abstract: We develop a model to investigate the time evolution of the mean location and variance of a random walker subject to Poisson-distributed collisions at constant rate. The collisions are instantaneous velocity changes where a new value of velocity is generated from a model probability function. The walker is persistent, which means that it moves at constant velocity between collisions. We study three different cases of velocity transition functions and compute the transport properties from the evolution of the variance. We observe that transport can change character over time and that early times show features that, in general, depend on the initial conditions of the walker.

Summary (1 min read)

Introduction

  • The collisions are instantaneous velocity changes where a new value of velocity is generated from a model probability function.
  • The authors study three different cases of velocity transition functions and compute the transport properties from the evolution of the variance.
  • Many results on random walks rely on the assumption of long timescales.
  • The authors study the time evolution of the variance of the walker position and show that the character of transport, understood here as the scaling of the variance as a function of time, changes over time.

II. THEORETICAL FRAMEWORK

  • The walker moves with constant velocity except at collisions, when the velocity transitions instantaneously to a new value (see Fig. 1). (3) We use the transform variable κ , instead of x, to distinguish the transformed function from the original one.the authors.the authors.
  • The transformed function is written as the original one but uses the transform variable s. Equation (4) means that the initial location and velocity are assumed to be statistically independent [28].
  • For what follows, the authors only consider the case where these initial PDFs have finite first and second moments and can be Fourier-transformed.
  • This form of the equation is very useful once one realizes that the Fourier transform of a PDF is its characteristic function [28].

VI. CONCLUSIONS

  • Borrowing some ideas from PRW [9,10,15] and biological dispersion [7], the authors have developed a technique to find exact analytic expressions for the mean, variance and transport exponent of a 1D random walker subject to Poisson-distributed collisions (with constant rate) which moves at constant velocity between collisions.
  • Transport properties, as established with the value of the transport exponent λ [Eq. (13)], have in general been shown to be time-varying and dependent on the possibly random initial conditions of location and velocity.
  • Three different types of velocity transitions are studied.
  • In particular, the authors follow some basic steps in the approach described in Refs. [7,9,10,15].
  • In that case, the walker only has two possible motion states (towards increasing x or decreasing x, respectively).

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PHYSICAL REVIEW E 100, 052134 (2019)
Particle transport at arbitrary timescales with Poisson-distributed collisions
M. Baquero-Ruiz , F. Manke, I. Furno , A. Fasoli, and P. Ricci
École Polytechnique Fédérale de Lausanne (EPFL), Swiss Plasma Center (SPC), CH-1015 Lausanne, Switzerland
(Received 4 July 2019; published 22 November 2019)
We develop a model to investigate the time evolution of the mean location and variance of a random walker
subject to Poisson-distributed collisions at constant rate. The collisions are instantaneous velocity changes where
a new value of velocity is generated from a model probability function. The walker is persistent, which means that
it moves at constant velocity between collisions. We study three different cases of velocity transition functions
and compute the transport properties from the evolution of the variance. We observe that transport can change
character over time and that early times show features that, in general, depend on the initial conditions of the
walker.
DOI: 10.1103/PhysRevE.100.052134
I. INTRODUCTION
Many results on random walks rely on the assumption
of long timescales. This is particularly true in the case of
continuous time random walks (CTRW), where the limit t
is widely used to simplify the Montrol-Weiss equation
[13] and cast it in a form that ultimately leads to the diffusion
equation or, more in general, to a type of fractional diffusion
equation [14].
In some situations, however, one may be interested in the
evolution of the walker at early times, i.e., when t τ , where
τ is a typical collision time. In those cases, the results obtained
for long timescales may not apply. This is exemplified by
measurements of particle motion in a rarefied gas [5]ora
liquid [6], the motion of organisms in biology [7,8], and trans-
port of particles in semiconductors [9,10], where the walker
dynamics between collisions plays an important role. This
was already recognized in 1930 by Uhlenbeck and Ornstein
in their model of Brownian motion [11,12], which allowed
them to compute explicit results for all t 0.
One simple model for dynamics between collisions is to
assume that the walker moves with constant velocity. This
persistent motion is a useful model in biology, where cor-
relations at short timescales have been seen to be relevant
to modeling of dispersal in biological systems [7]. It is also
useful in physics, as it models the situation of no intercollision
forces. The collisions are then understood to be instanta-
neous changes in velocity occurring randomly in time. In
1D persistent random walks (PRW) [9,1315], for example,
one usually describes the evolution of a walker moving at
constant speed (modulus of velocity) but subject to random
reversals of the direction of motion. The simplicity of PRW
makes it amenable to analytical investigations that lead to
time-changing transport properties and a close relationship to
the Telegrapher’s equation (TE) [9,14].
In addition to the changes in direction, one can also
consider changes in speed [2,16]. This has proven useful in
studies of first-passage times of a persistent random walker
[17] and, more recently, in some generalizations of the TE
[18]. We pursue the idea by studying situations in which
collisions lead to new (random) values of velocity arising
from model probability density functions (PDFs). We study
the time evolution of the variance of the walker position and
show that the character of transport, understood here as the
scaling of the variance as a function of time, changes over
time. The results are valid for all times t 0, in contrast to
Refs. [2,16
], where attention is paid to establishing asymptotic
properties by assuming large spatial and temporal scales. In
contrast to Ref. [19], where a Langevin approach is used, we
develop a method that allows us to study different types of
collisions and the effect of initial conditions of position and
velocity.
We start with a theoretical framework (Sec. II). We develop
a procedure to compute the time evolution of statistics of
the walker’s location (including the mean and variance) when
subject to Poisson-distributed collisions. Then, we apply this
method to three different types of velocity transition PDFs.
First we consider the case when, upon a collision, the
walker transitions to a new random velocity independent of
the precollision one (Sec. III). In this way we can model a
tracer particle moving in a medium at constant temperature,
a situation that may, for example, be of interest in studies
of scattering of neutrons in a moderator [20]orscattering
of electrons in a metal [10]. It can also find applications in
studies of particle transport in plasmas [21], for example, in
the context of suprathermal ions [3,22,23], or in studies of
particle dynamics in plasma turbulence [24]. We show that
our approach yields exact expressions that can be compared
to numerical simulations and give interesting insights of trans-
port at short timescales. Notably, in certain circumstances, we
observe superballistic behavior characterized by a t
3
scaling
of the variance [19]. This constitutes an interesting analogy to
a similar result observed in the formally different problem of
relative separation of particles [24,25].
In Sec. IV we explore the case of addition of a random ve-
locity. We show that this choice typically yields a t
3
scaling (at
long enough t), providing a possible basis to the superballistic
results observed in Sec. III.
In Sec. V, we study scaled velocity changes. This is the
case when the walker velocity is multiplied by a constant
2470-0045/2019/100(5)/052134(14) 052134-1 ©2019 American Physical Society

M. BAQUERO-RUIZ et al. PHYSICAL REVIEW E 100, 052134 (2019)
FIG. 1. Depiction of the system dynamics. A walker moves along
x with velocity u (a) until a random event (a collision) makes the
velocity change instantaneously to the new value v (b). The walker
then moves with this velocity (c) until a new collision forces a
change.
factor in each collision, a more artificial situation which
nevertheless shares some similarities with exponential Fermi
acceleration [26]. We show that transport can then have an
exponential behavior. Also, we obtain a generalized version
of the TE, thus showing that the formalism can recover results
found in PRW while also enabling studies of other situations.
A summary of these results, as well as an outlook of future
studies, is given in Sec. VI.
II. THEORETICAL FRAMEWORK
Let p(x, t ) be the PDF of the random 1D location χ (t)
of a walker moving along x at time t”. This means that
p(x, t ) dx is the probability of finding the walker in the
interval (x, x + dx) at time t. The walker moves with constant
velocity except at collisions, when the velocity transitions
instantaneously to a new value (see Fig. 1). The resulting
randomness in walker velocity leads us to consider p(x, v , t ),
the PDF of walker location and velocity. For given x, v, and
t, p(x, v, t ) dxdv is the probability of finding the walker with
some velocity in (v, v + dv ) at the location (x, x + dx) at time
t. We note that
p(x, t ) =
p(x, v, t) dv, (1)
as p(x, t ) is just the PDF of location irrespective of velocity.
Then, as shown in Appendix A, the evolution of p(x, v , t )is
determined by the transport equation [27]
t
+ v
x
+ γ
p(x, v, t) = γ
f (u, v ) p(x, u, t ) du,
(2)
where γ 1 is a constant collision rate (which for finite
τ satisfies γ>0) and f (u, v ) is a kernel giving the proba-
bility that, upon a collision, a walker moving with velocity
u transitions to a velocity v. The function f needs to fulfill
f (u, v ) dv = 1forall u (see Appendix A).
Equation (2) is a partial integrodifferential equation for
p(x, v, t), with known γ and f (u, v ), which is in general
difficult to solve. In transport studies, however, one is typ-
ically interested [2,15] in statistics of χ (t), such as the
mean walker location
χ (t )=
xp(x, t ) dx μ
χ
(t) and
the mean square displacement χ
2
(t)=
x
2
p(x, t ) dx
σ
2
χ
(t) + (μ
χ
(t))
2
, where σ
2
χ
(t)isthevariance. In other words,
one wants to find moments [28]ofχ (t ). In that case, as
discussed next, one does not need to find the full solution
p(x, v, t).
We assume that the walker moves in an unbounded space
and that the Fourier transform of p(x, v, t ) on the spatial
coordinate x exists for t 0 and all values of v .Westate
clearly the definition of the transform to avoid confusion with
references that may use a different convention [29]:
p(x, v, t)
F
p(κ,v, t )
−∞
p(x, v, t) e
ı κ x
dx.(3)
We use the transform variable κ, instead of x, to distinguish
the transformed function from the original one. Similarly, we
assume that the Laplace transform of p(x, v, t)int exists for
all x and v and the definition is
p(x, v, t)
L
p(x, v, s)
0
p(x, v, t) e
st
dt .
In this case, the transformed function is written as the original
one but uses the transform variable s. The use of the Laplace
transform in time is needed to include initial conditions at t =
0, which we assume to be of the form
p(x, v, t)|
t=0
= g
0
(x) f
0
(v ). (4)
In this expression g
0
(x) is the PDF of the (in general)
random initial location χ
0
, and f
0
(v ) is the PDF of the
initial velocity V
0
. Then
g
0
(x) dx = 1 and
f
0
(v ) dv = 1
so that

p(x, v, t)|
t=0
dxdv = 1 (i.e., the probability of
finding the walker anywhere moving at any speed when
t = 0isone). Equation (4) means that the initial location
and velocity are assumed to be statistically independent [28].
For what follows, we only consider the case where these
initial PDFs have finite first and second moments and can be
Fourier-transformed.
Taking Eq. (2) and applying a Fourier transform in x
followed by a Laplace transform in t we obtain
(s + ı κv + γ )p(κ,v, s) g
0
(κ ) f
0
(v )
= γ
f (u, v ) p(κ, u, s) du,(5)
where g
0
(κ ) is the Fourier transform of g
0
(x). This form
of the equation is very useful once one realizes that the
Fourier transform of a PDF is its characteristic function [28].
Then, results similar to those used in probability theory [28]
can be employed to compute the mean μ
χ
(t) and variance
σ
2
χ
(t) directly from p(κ,t), without inverse-transforming to
the original x-space [15]. For example,
∂κ
p(κ,t )|
κ=0
=
∂κ
−∞
p(x, t ) e
ı κ x
dx
κ=0
=
−∞
∂κ
(p(x, t ) e
ı κ x
)|
κ=0
dx
=−ı
−∞
xp(x, t ) dx =−ı μ
χ
(t), (6)
and, following a similar procedure,
2
∂κ
2
p(κ,t )|
κ=0
=−σ
2
χ
(t) [μ
χ
(t)]
2
.
052134-2

PARTICLE TRANSPORT AT ARBITRARY TIMESCALES PHYSICAL REVIEW E 100, 052134 (2019)
In terms of p(κ,s) we then have
μ
χ
(t) = L
1
ı
∂κ
p(κ,s)|
κ=0
σ
2
χ
(t) = L
1
2
∂κ
2
p(κ,s)|
κ=0
[μ
χ
(t)]
2
.(7)
In the next sections we show that Eqs. (7) together with
Eq. (5) can be used to find exact analytic expressions for μ
χ
(t)
and σ
2
χ
(t). Higher-order moments of p(x, t) can be computed
in a similar fashion.
III. RANDOM VELOCITY UPON A COLLISION
A. Model
As a first example of the applicability of the theory of
Sec. II, we consider
f (u, v ) = f
R
(v ), (8)
where f
R
is a PDF with finite first- and second-order moments.
This is the case when, upon a collision, the walker transitions
to a new random velocity V distributed f
R
(v ), and the new
velocity is independent of the one prior to the collision. This
choice can, for example, model situations where a tracer
particle moves through a medium at constant temperature.
In that case, collisions make the particle jump to random
velocities allowed by a velocity distribution associated to the
medium temperature.
Since
f
R
(v ) p(κ,u, s) du = f
R
(v )
p(κ,u , s) du =
f
R
(v ) p(κ,s)[seeEq.(1)], replacing f
R
in Eq. (5)gives
(s + ı κv + γ )p(κ,v , s) g
0
(κ ) f
0
(v ) = γ f
R
(v ) p(κ,s),
which leads to
p(κ,v , s) =
g
0
(κ ) f
0
(v )
s + ı κv + γ
+
γ f
R
(v )
s + ı κv + γ
p(κ,s).
Then, integrating on both sides with respect to v, the left-hand
side (LHS) becomes
p(κ,v , s) dv = p(κ,s), which allows
us to solve for p(κ,s) and obtain
p(κ,s)
=
g
0
(κ ) f
0
(v )
s + ı κv + γ
dv

1 γ
f
R
(v )
s + ı κv + γ
dv
1
.
(9)
This expression gives the complete time evolution of p(x, t )
in the transformed spaces.
Since p(κ,s) = L{F {p(x, t )}} = L{
−∞
p(x, t ) e
ı κx
dx},
we have
−∞
p(x, t ) dx = L
1
{p(κ,s)|
κ=0
}. Evaluating the
expression in Eq. (9)atκ = 0, we have
p(κ,s)|
κ=0
=
f
0
(v )
s + γ
dv

1 γ
f
R
(v )
s + γ
dv
1
=
1
s + γ

1 γ
1
s + γ
1
=
1
s
,
since g
0
(κ )|
κ=0
= 1,
f
0
(v ) dv = 1, and
f
R
(v ) dv = 1.
Therefore,
−∞
p(x, t ) dx = L
1
{
1
s
}=1 as expected, since
probability needs to be conserved at all times t 0.
FIG. 2. Evolution of μ
χ
(t) for different ratios μ
f
0
f
R
of mean
initial and mean post-collision velocities. We assume here μ
f
R
> 0.
As Eq. (10) shows, the functional form of μ
χ
(t) only depends on
μ
f
0
f
R
and γ t (the mean number of collisions at time t), once μ
g
0
has been subtracted.
Inserting Eq. (9) into Eqs. (7) allows us to compute the
mean walker location (see Appendix B):
μ
χ
(t) = L
1
μ
g
0
s
+
μ
f
0
s (s + γ )
+
γμ
f
R
s
2
(s + γ )
= μ
g
0
+
1
γ
[μ
f
0
(1 e
γ t
) + μ
f
R
(1 + γ t + e
γ t
)].
(10)
Here, μ
g
0
and μ
f
0
are the means of the initial conditions χ
0
and V
0
with PDFs g
0
and f
0
, respectively, and μ
f
R
is the mean
of V f
R
(v ). It is interesting to note that μ
χ
(t) only depends
on first-order moments of the other distributions. No other
additional information on g
0
(x), f
0
(v ), and f
R
(v ) is needed
(such as higher-order moments) to establish the complete time
evolution of μ
χ
(t). In passing, we note that making γ = 0
in the first line of Eq. (10) leads to the expected result for a
particle moving always at constant velocity, even though γ is
nonzero by definition (see Sec. II).
If μ
f
0
= μ
f
R
= 0, then Eq. (10) implies constant μ
χ
(t) =
μ
g
0
for all t 0. If μ
f
R
= 0, then the time evolution of the
mean is more interesting. Figure 2 shows μ
χ
(t) for sev-
eral choices of μ
f
0
f
R
. The initial behavior is determined
by a competition between initial conditions and mean post-
collision velocities. Then, when collisions have had enough
time (t γ
1
) to randomize the motion, μ
f
R
starts dominat-
ing. Finally, μ
χ
(t) μ
f
R
t when γ t 1 (irrespective of μ
f
0
).
We focus now on the variance σ
2
χ
(t). For simplicity, we
consider the case μ
f 0
= μ
f
R
= 0. The general situation for
nonzero values is discussed in Appendix C. Following a
similar procedure as for the mean, we obtain
σ
2
χ
(t) = σ
2
g
0
+
2 σ
2
f
0
γ
2
[1 (1 + γ t ) e
γ t
]
+
2 σ
2
f
R
γ
2
[2 + γ t + (2 + γ t ) e
γ t
]. (11)
This expression only depends on second-order moments of g
0
,
f
0
, and f
R
. Therefore, PDFs of very different form but similar
mean and variance should lead to the same time evolution
052134-3

M. BAQUERO-RUIZ et al. PHYSICAL REVIEW E 100, 052134 (2019)
FIG. 3. (a) Evolution of σ
2
χ
(t)forσ
g
0
= 0, μ
f
0
= μ
f
R
= 0and
different values of σ
f
0
f
R
as indicated in the legend. (b) Transport
exponent λ for the same cases (same color legend).
of σ
2
χ
. This is indeed verified using numerical simulations
in Sec. III B. The case σ
f
R
= 0 corresponds to a Dirac-delta
[30] distribution modeling only one possible post-collision
velocity (in this case v = 0, as we have chosen μ
f
R
= 0).
Equation (11) shows that, in that situation, σ
2
χ
(t) tends to a
constant when γ t 1. If we consider now nonzero σ
f
R
, then
we can rewrite Eq. (11)as
γ
2
2 σ
2
f
R
σ
2
χ
(t) =
γ
2
2 σ
2
f
R
σ
2
g
0
+
σ
2
f
0
σ
2
f
R
2 + γ t
+ e
γ t
2
σ
2
f
0
σ
2
f
R
+ γ t
1
σ
2
f
0
σ
2
f
R

.
(12)
The RHS of this expression only depends on three parameters.
The first one is γσ
g
0
f
R
, a normalized initial width. The
second one is γ t, the average number of collisions by time
t. The third one is σ
f
0
f
R
, the ratio of the standard deviations
of V
0
and V . The individual values of σ
f
0
and σ
f
R
do not play
a role in the functional form of the time evolution of σ
2
χ
(it is
only their ratio that matters). We note that making σ
f
0
f
R
= 1
in Eq. (12) leads to a result with the same functional form
as the known solution for arbitrary times using the Langevin
approach [5,11].
Figure 3(a) shows the evolution of σ
2
χ
when σ
g
0
= 0. This is
the case when the initial location is known with complete cer-
tainty [g
0
(x) is a Dirac-delta]. The behavior at long timescales
(i.e., γ t 1) is similar for all different values of σ
f
0
f
R
.Itis
in fact diffusive,asσ
2
χ
is seen to scale linearly with time, i.e.,
σ
2
χ
t
λ
with λ = 1. We know this from computing the slope
of the curves and verifying that in all cases it is equivalent to 1
for γ t 1. In the log-log plot, the slope is in fact equivalent
to λ. The number λ, which we call [3] transport exponent
(or transport scaling [2]), may, however, take on other values
FIG. 4. (a) Evolution of σ
2
χ
(t)forμ
f
0
= μ
f
R
= 0 and different
values of σ
g
0
as indicated in the legend. We use σ
f
0
= σ
f
R
in all cases.
(b) Transport exponent λ of the same curves. Changes in normalized
initial width (γσ
g
0
)
f
R
have a significant effect on the slope at early
times and could lead to misinterpretations of transport features if not
properly considered. The case σ
g
0
= 0 is similar to the σ
f
0
f
R
= 1
curve in Fig. 3.
when γ t 1. Figure 3(a) shows that transport at early times
can indeed be different from diffusive.
Figure 3(b) shows the time evolution of λ for the same
values of σ
f
0
f
R
as in Fig. 3(a).Weuse
λ(t) =
t
σ
2
χ
(t)
d
dt
σ
2
χ
(t)
(13)
to compute λ from Eq. (12) (valid for t > 0) and then plot
it. We note that this definition of λ uses the variance (some
references may use a different convention [2,3]). There are
several interesting observations. First, if σ
f
0
f
R
1, then the
initial behavior is ballistic; i.e., λ = 2. However, transport is
significantly reduced at γ t 1 and can become subdiffusive.
In fact, we see that λ can approach the value zero when
σ
f
0
f
R
> 10. The case σ
f
0
= σ
f
R
models a situation similar
[14] to PRW and indeed shows the same smooth transition
from ballistic to diffusive around γ t = 1. Probably more
interesting is the case σ
f
0
f
R
< 1 when γ t 1. In that
situation, λ can attain values larger than 2 including λ = 3.
This superballistic transport, already observed in Ref. [19], is
seen in Sec. IV A to be characteristic of velocity diffusion.
Figure 4 shows the evolution of σ
2
χ
for fixed σ
f
0
f
R
=
1 and different values of σ
g
0
. Also shown is λ which, for
large normalized initial widths γσ
g
0
f
R
> 1, can completely
change character from initially ballistic to no transport (λ =
0). This exemplifies the importance of correctly accounting
for σ
g
0
in transport studies at early times. In practice, it is not
possible to know σ
g
0
exactly, so finite-width uncertainties are
foreseen to always be present in practical applications which
will mostly impact observations when t
1
.
052134-4

PARTICLE TRANSPORT AT ARBITRARY TIMESCALES PHYSICAL REVIEW E 100, 052134 (2019)
The methods presented can be used to compute higher-
order moments of p(x, t ) and find symbolic expressions for,
for example, the skewness and kurtosis [28]ofχ (t). As we are
concerned with determining transport features, which involve
the first and second moments, we do not undertake higher-
order moment calculations here.
B. Comparison of theory and numerical simulations
We developed a simple code in MATLAB [31] (see Ap-
pendix D) that creates numerical estimates m
χ
(t), s
2
χ
(t)of
μ
χ
(t), σ
2
χ
(t), respectively.
We consider three possible choices of PDFs,
Gaussian(x; μ, σ
2
) =
1
2 πσ
2
e
(xμ)
2
2 σ
2
,
Discrete(x; μ, σ
2
) =
1
2
[δ(x μ σ ) + δ(x μ + σ )],
Uni form(x; μ, σ
2
) =
(2
3 σ )
1
if μ
3 σ x μ +
3 σ ,
0 otherwise,
(14)
for the initial conditions (g
0
and f
0
) and post-collision velocity
distribution (f
R
).
In a given simulation, we choose one of three possible
sets of values of γ and σ
f
R
, and one choice of PDFs. This
means that we take g
0
, f
0
and f
R
to be all Gaussian,all
Uni form,orallDiscrete. The particular selection is indi-
cated with different markers in Fig. 5. Circles correspond
to γ = 3s
1
, σ
f
R
= 2m/s and Discrete PDFs. Triangles are
γ = 1s
1
, σ
f
R
= 1m/s and Gaussian PDFs. Crosses are γ =
2s
1
, σ
f
R
= 3m/s and Uni form distributions. We use in all
cases μ
f
0
= μ
f
R
= 0 and set σ
g
0
such that (γσ
g
0
)
f
R
= 10
4
.
The choice of a nonzero initial width is made to avoid
otherwise large rounding errors in the variance, and conse-
quently the transport exponent, if σ
g
0
is too close to zero near
t = 0.
The simulation results (see Fig. 5) show good agreement
with the theory of Sec. III A. The evolution, in units of
γ t, of the normalized variance [Fig. 5(b)] and the transport
exponent [Fig. 5(c)] only depends on the ratio σ
f
0
f
R
and
is independent of the particular choices of γ , σ
f
R
, and PDF
shape, as expected. The increased variability of the markers in
Fig. 5(a) when γ t 10 is due to the statistical error expected
for the computation of the mean m
χ
(t). This is illustrated
in the case σ
f
0
f
R
= 1 for which the markers lie within
or near the shaded area representing the 1σ region around
μ
χ
(t) = 0.
IV. ADDITIVE VELOCITY JUMPS
A. Model
We now consider the case
f (u, v ) = f
A
(v u), (15)
where f
A
is a PDF with finite first and second-order moments
and can be Fourier-transformed. Upon a collision, the walker
transitions to a new random velocity u + V where V is dis-
tributed f
A
. Note that the new velocity depends on the velocity
prior to the collision, and the dependency is additive. This
choice models a generalized diffusion of velocities, as dis-
cussed in Appendix E, with a priori no limit on the maximum
attainable speed. Relativistic effects are not included in the
model, so nothing precludes reaching artificial supraluminal
regimes after long enough times.
The analysis requires a different approach from the one
used in Sec. III A.WetakeEq.(5) and perform a Fourier
transform with respect to the velocity variable v,usinga
convention similar to Eq. (3). We call β the variable conjugate
FIG. 5. Using (γσ
g
0
)
f
R
= 10
4
, μ
f
0
= μ
f
R
= 0 we compute
(a) the mean m
χ
(t), (b) the variance s
2
χ
(t) and (c) the transport
exponent of χ (t) at some sample times (markers) during simulations
performed with different values of σ
f
0
f
R
[colored according to the
label in (b)]. Different markers (circles, triangles, crosses) are used
to indicate one of three sets of parameter values of γ , σ
f
R
and choice
of PDFs used in a particular simulation, as explained in the text.
The solid lines are the theoretical values μ
χ
(t), σ
2
χ
(t)andλ(t )from
Sec. III A corresponding to the different σ
f
0
f
R
(same color coding).
There is good agreement between simulations and theory.
052134-5

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References
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TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.

7,412 citations

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TL;DR: In this paper, the mean values of all the powers of the velocity $u$ and the displacement $s$ of a free particle in Brownian motion are calculated and the exact expressions for the square of the deviation of a harmonically bound particle in the Fokker-Planck partial differential equation as a function of the time and the initial deviation are obtained.
Abstract: With a method first indicated by Ornstein the mean values of all the powers of the velocity $u$ and the displacement $s$ of a free particle in Brownian motion are calculated It is shown that $u\ensuremath{-}{u}_{0}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}t)$ and $s\ensuremath{-}\frac{{u}_{0}}{\ensuremath{\beta}[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}t)]}$ where ${u}_{0}$ is the initial velocity and $\ensuremath{\beta}$ the friction coefficient divided by the mass of the particle, follow the normal Gaussian distribution law For $s$ this gives the exact frequency distribution corresponding to the exact formula for ${s}^{2}$ of Ornstein and F\"urth Discussion is given of the connection with the Fokker-Planck partial differential equation By the same method exact expressions are obtained for the square of the deviation of a harmonically bound particle in Brownian motion as a function of the time and the initial deviation Here the periodic, aperiodic and overdamped cases have to be treated separately In the last case, when $\ensuremath{\beta}$ is much larger than the frequency and for values of $t\ensuremath{\gg}{\ensuremath{\beta}}^{\ensuremath{-}1}$, the formula takes the form of that previously given by Smoluchowski

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  • ...The case σ f0 = σ fR models a situation similar [14] to PRW and indeed shows the same smooth transition from ballistic to diffusive around γ t = 1....

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  • ...Many other methods have been developed to investigate similar situations in PRW [13,14,32,33] but are not pursued here....

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  • ...Borrowing some ideas from PRW [9,10,15] and biological dispersion [7], we have developed a technique to find exact analytic expressions for the mean, variance and transport exponent of a 1D random walker subject to Poisson-distributed collisions (with constant rate) which moves at constant velocity between collisions....

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Abstract: If the diffusivity K of a substance whose mass per volume of atmosphere is χ be defined by an equation of Fick’s type ū ∂ χ /∂ x + v - ∂ χ /∂ y + w - ∂ χ /∂ z + ∂ χ /∂ t = ∂/∂ x (K ∂ χ /∂ x ) + ∂/∂ y (K ∂ χ /∂ y ) ∂/∂ z (K ∂ χ /∂ z ), (1) x , y , z , t being Cartesian co-ordinates and time, ū , v -, w - being the components of mean velocity, then the measured values of K have been found to be 0·2 cm.2 sec.-1 in capillary tubes (Kaye and Laby’s Tables), 105 cm.2 sec.-1 when gusts are smoothed out of the mean wind (Akerblom, G. I. Taylor, Hesselberg, etc.), 108 cm.2 sec.-1 when the means extend over a time comparable with 4 hours (L. F. Richardson and D. Proctor), 1011 cm.2 sec.-1 when the mean wind is taken to be the general circulation characteristic of the latitude (Defant). Thus the so-called constant K varies in a ratio of 2 to a billion. The present paper records an attempt to comprehend all this range of diffusivity in one coherent scheme. Lest the method which I shall adopt should strike the reader as queer and roundabout, I wish to justify it by showing first why some known methods are in difficulties.

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Frequently Asked Questions (2)
Q1. What contributions have the authors mentioned in the paper "Particle transport at arbitrary timescales with poisson-distributed collisions" ?

The authors develop a model to investigate the time evolution of the mean location and variance of a random walker subject to Poisson-distributed collisions at constant rate. The authors study three different cases of velocity transition functions and compute the transport properties from the evolution of the variance. The authors observe that transport can change character over time and that early times show features that, in general, depend on the initial conditions of the walker. 

Although the authors focus on first and second-order moments ( relevant to transport studies ), the procedures presented here can straightforwardly be extended to higher-order statistics of the random walker location ( as a function of time ), for example, the skewness and kurtosis. These will be the subject of future studies. Indeed, obtaining t/τ collisions by time t can be seen as successes in t/dt Bernoulli trials [ 33 ] with probability of success ( t/τ ) / ( t/dt ) = dt/τ. The authors can then rewrite Eq. ( A2 ) as ( ∂ ∂t + v ∂ ∂x + γ ) p ( x, v, t ) = γ ∫ f ( u, v ) p ( x, u, t ) du.