1]Physics Department, Bar-Ilan University, Ramat-Gan, 52900, Israel
2]Institute of Nanotechnology and Advanced Materials, Bar-Ilan University,
Ramat-Gan, 52900, Israel

# Laplace’s first law of errors applied to diffusive motion

Omer Hamdi  Stanislav Burov  Eli Barkai  [  [

###### Abstract

In biological, glassy, and active systems, various tracers exhibit Laplace-
like, i.e., exponential, spreading of the diffusing packet of particles. The
limitations of the central limit theorem in fully capturing the behaviors of
such diffusive processes, especially in the tails, have been studied using the
continuous time random walk model. For cases when the jump length distribution
is super-exponential, e.g., a Gaussian, we use large deviations theory and
relate it to the appearance of exponential tails. When the jump length
distribution is sub-exponential the packet of spreading particles is described
by the big jump principle. We demonstrate the applicability of our approach
for finite time, indicating that rare events and the asymptotics of the large
deviations rate function can be sampled for large length scales within a
reasonably short measurement time.

##  1 Introduction

††footnotetext: Invited paper for the topical issue in the European Physical
Journal B: New Trends in Statistical Physics of Complex Systems: Theoretical
and Experimental Approaches.

Laplace’s two laws of error are milestones in statistics. The first was
published in $1774$ [1] and states that the frequency of an error could be
expressed as an exponential of the magnitude of the error, in absolute value.
The second law of errors, from $1778$ [2], states that the frequency of the
error is an exponential of a quadratic function of the error. In the context
of diffusing particles in one dimension, and for a packet starting at the
origin, the probability density of the particles is $P(x,t)$. The first law
states $P(x,t)\propto\exp(-\mbox{const}\ |x|)$ and the second is the more
familiar law, $P(x,t)\propto\exp(-\mbox{const}\ x^{2})$. In $1923$ Wilson [3]
discussed some of the history of the problem. He noted that the second law is
typically called the normal or Gauss law; however, despite Gauss’s well-known
precocity, he probably did not make this discovery before he was two years
old† †††Gauss utilized the least squares method of error estimation for the
discovery of the lost dwarf planet, Ceres. [4, 5].. Indeed, it was Laplace who
promoted the central limit theorem, putting a firm mathematical basis for the
second law. The normal distribution has since been used in all fields of
science, while the first law, as far as we know, was not in the spotlight for
an extended period.

However, using single-molecule tracking data and computer-generated
trajectories of a variety of tracers diffusing in disordered media, in more
recent years, the first law experienced a revival, as it was used to fit a
large body of experimental data [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24]. These observations are related to the diffusing
diffusivity models [25, 26, 21, 27] and a phenomena known as Brownian yet non-
Gaussian diffusion [14, 28, 15, 29, 30, 31], and Fickian yet non-Gaussian
diffusion [16, 17, 18, 19, 20]. By now, the exponential decay of
$P(x,t)\propto\exp(-\mbox{const}\ |x|)$ is well documented. In some cases,
only the large $x$ limit is described by this law, while in others, the
Laplace law, as a fitting procedure, holds for all $x$ [14]. A large body of
phenomenological models was used to describe these behaviors, for example, by
assuming that the diffusion constant $D(t)$ is a stochastic process.

Chaudhuri, Berthier, and Kob [22] analyzed four systems, focusing on particle
displacements near glass and jamming transitions, highlighting behaviors like
sticking (caging) and rapid jumps between basins. This type of dynamics is
described by the continuous time random walk (CTRW) [32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45], which will also be analyzed in this paper.
Using specific waiting time distributions, they showed how the basic CTRW
model can be used to predict exponential tails, in accordance with Laplace’s
first law. This theory was later advanced by Wang, Burov, and Barkai [46, 47],
showing that it holds for very general settings. They have shown that, for any
distribution of waiting times, which is analytical for short waiting times,
and for any jump length distribution, which decays faster than exponential,
Laplace’s first law holds at the tails of $P(x,t)$. The two above-mentioned
conditions are expected to hold in many systems. The analysis was based on
large deviations [48, 49, 50, 51, 52, 53, 54, 55] arguments, and a saddle
point approximation [56], namely to put things in a historical context, the
technique used is an extension of the Laplace method for solving integrals
[1].

Schramma et al. [6] discovered that chloroplasts, which are components of
plant cells, adapt to dim light by moving in a manner closely resembling the
caged dynamics observed in supercooled liquids or colloidal suspensions near
the glass transition. This movement demonstrates exponential tails, fitting
both the CTRW and the diffusing diffusivity models.

The connection between Laplace diffusion and CTRW in polymer nanocomposites
was further explored by Hu et al [57]. They demonstrated the ability to
manipulate the emergence of exponential tails in contrast to Gaussian
diffusion by modifying the strength of the disorder.

In this manuscript, we first provide a more detailed analysis of CTRW dynamics
using three tools. We will then conclude the paper with a broader perspective
on the Laplace-like behavior, i.e., exponential decay of spreading particles.

We start with a study of the CTRW model for the case when the jump length
distribution is either sub or super-exponential. In the former case large
deviations theory holds, while for the latter the big jump principle is valid
(see below). This transition is related to the fact that for sub-exponential
jump length distribution, the cumulant generating function diverges, and thus
the standard Laplace-Cramers-Daniles tool of saddle point approximation of
large deviations theory does not hold. We show how the big-jump principle is
related to an extension of the Laplace method of solving integrals. In
Laplace’s method, close to the saddle point, an analytic i.e., quadratic
function is used, and hence, eventually, a Gaussian integral is computed,
while the original integral is non-Gaussian. For the sub-exponential
distribution of jump lengths, we find a similar extremum, but with non-
analytical features. Namely, the quadratic expansion close to the extremum is
invalid. Finally, we present the Edgeworth expansion to approximate the CTRW
probability density [58]. This approximation provides corrections for the
central limit theorem (CLT). It deals with a long time limit, and not very
large $x$, while the large deviations theory and the big-jump principle cover
the behavior of the tails of the density. To study these effects, we
constructed a numerical tool, to sample finite time and finite $x$ propagators
$P(x,t)$ using CTRW.

##  2  Model

In the CTRW model, waiting times between jump events are independent
identically distributed (IID) random variables with a probability density
function (PDF) $\psi(\tau)$. The process starts at time $t=0$, and then the
particle waits at its initial position $x=0$, for a random time $\tau$ drawn
from $\psi(\tau)$. At time $t=\tau$, the random walker jumps to a new
position, the duration of the jump being negligible. The jump lengths are also
IID random variables, with $f(\chi)$ as the PDF. The process is then renewed,
namely, a second waiting time is drawn, followed by a spatial jump, etc. The
goal is to find the PDF $P(x,t)$ of finding the particle at $x$ at time $t$.
The CTRW process is called a semi-Markovian process and the statistics for the
number of jumps can be analyzed utilizing the renewal theory [59, 60].

We focus on exponentially distributed waiting times, $\psi(\tau)=\exp(-\tau)$,
with the mean time between jumps set to unity. It follows that the number of
jumps in the time interval $(0,t)$ is described by Poisson statistics. The
jump length PDF $f\left(\chi\right)$ is:

$f\left(\chi\right)=\widetilde{N}\exp\left(-\alpha^{\beta}|\chi|^{\beta}\right).$
(1)

The mean jump length is zero, and the symmetry of $f\left(\chi\right)$ sets
the symmetry of $P(x,t)$, i.e., $P(x,t)=P(-x,t)$. The variance of the jump
length is set to be $1$. Here we have
$\alpha=\sqrt{\Gamma(3/\beta)/\Gamma(1/\beta)}$, and the normalization
$\widetilde{N}=\beta\sqrt{\Gamma(3/\beta)}/2\Gamma^{3/2}(1/\beta)$, with
$\Gamma(\dots)$ being the Gamma function. The exponent $\beta>0$ is an
important parameter in this study. For $\beta<1$ $(\beta>1)$ we have sub (or
super) exponential decay of the jump length distribution. The case $\beta=2$
corresponds to Gaussian statistics for the jump length PDF.

The probability of jumping $n$ times during time $t$ is $t^{n}\exp(-t)/n!$
hence

$P(x,t)=\sum_{n=0}^{\infty}\frac{{\rm e}^{-t}t^{n}}{n!}\phi(x|n),$ (2)

$\phi(x|n)$ is the PDF of finding the particle at $x$, conditioned on
performing exactly $n$ jumps. In the previous equation, the summation is
performed over all the possible number of jumps that can occur during the
process. Using the fact that jump lengths are IID random variables, the
characteristic function of $\phi(x|n)$

$\int_{-\infty}^{\infty}{\rm e}^{ikx}\phi(x|n){\rm d}x=\langle{\rm
e}^{ik(\chi_{1}+...\chi_{n})}\rangle=\widetilde{f}^{n}(k),$ (3)

is expressed in terms of the Fourier transform [61] of $f(\chi)$, namely
$\widetilde{f}(k)=\int_{-\infty}^{\infty}\exp(ik\chi)f(\chi){\rm d}\chi$. The
Fourier transform of $P(x,t)$, i.e., $\widetilde{P}(k,t)$, is obtained from
Eq.(2) and Eq.(3)

$\widetilde{P}(k,t)=\exp\left[-t\left(1-\widetilde{f}(k)\right)\right].$ (4)

The goal is to find the inverse Fourier transform of the expression above. The
CLT is valid in the limit $t\to\infty$ and $x\to\infty$ while the ratio
$x/\sqrt{t}$ is finite. We utilize the expansion of $\widetilde{f}(k)$ around
$k\to 0$. Since the jump length distribution is an even function,
$\widetilde{f}(k)\sim 1-\sigma^{2}k^{2}/2$, and according to our notation
$\sigma^{2}=1$. Therefore Eq. (4) yields

$P(x,t)\sim\frac{\exp\left(-\frac{x^{2}}{2t}\right)}{\sqrt{2\pi t}}.$ (5)

Namely, the central limit theorem holds.

##  3 Large Deviations $\beta\ >\ 1$

Capturing the essence of the limiting scaling law in Eq. (5) is but one aspect
of the large deviations Theory. This theory also delves into analyzing a
different yet significant limit. The inverse Fourier transform of
$\widetilde{P}(k,t)$ in Eq. (4), while changing variable $ik=u$, reads

$\displaystyle P(x,t)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int_{-i\infty}^{i\infty}\exp{\left[-xK(u)\right]}{\rm d}u,$ (6)

where

$\displaystyle K(u)$ $\displaystyle=$ $\displaystyle
u+\frac{1}{q}[1-\hat{f}(u)],$ (7)

$q\equiv x/t$, and $\hat{f}(u)=\int_{-\infty}^{\infty}\exp(u\chi)f(\chi){\rm
d}\chi$ is the moment generating function. Note that $\hat{f}(u)$ diverges for
$\beta<1$, leading to the divergence of the integral in Eq. (6). Therefore, in
this section, we focus on super-exponential PDFs, i.e., $\beta>1$.

Large deviations theory is based on the saddle point approximation and
considers the case of large $x$ limit of $P(x,t)$ in Eq. (6). Strictly
speaking, here the scaling is $x\to\infty$ and $t\to\infty$ while the ratio
$q=x/t$ is kept finite. Nevertheless, we show that the approximate solution
for $P(x,t)$ works reasonably well for large $x$ while $t$ is kept finite.

The problem of finding $P(x,t)$ is solved by using the following steps: First,
find the $u$ that satisfies $K^{\prime}(u)=0$ and term this $u$ as $u_{0}$.
Then, expand $K(u)$ in Eq. (6), in the vicinity of $u_{0}$, up to a quadratic
term. The obtained Gaussian integral yields

$P(x,t)\sim\frac{\exp\left[-xK(u_{0})\right]}{\sqrt{2\pi|xK^{\prime\prime}(u_{0})|}}.$
(8)

The approximation for large x can be obtained through numerical methods.
Namely, we numerically find $u$ that satisfy $K^{\prime}(u)=0$, and insert it
into Eq. (8). We call this solution method the numerical large deviations
method (Numerical-LDT). Below, we discuss how to obtain the exact form of
$P(x,t)$ and compare it to the solution obtained via Numerical-LDT.

Using $K^{\prime}(u)=0$, and the definition of $\hat{f}(u)$, the equation for
$u_{0}$ reads

$1-\frac{1}{q}\int_{-\infty}^{\infty}\chi f(\chi)\exp(u_{0}\chi){\rm
d}\chi=0.$ (9)

We treat this equation in two limits, small and large $q=x/t$. For small $q$
the value of integral in Eq. (9), must be small. Since the integral in Eq. (9)
is a growing function of $u$, the limit of small $q$ corresponds to small
values of $u$. Expanding $\exp(u\chi)$ and using the fact that the variance of
jump lengths equals $1$ we find $u_{0}\sim q\ll 1$, and $K(u_{0})\sim
u_{0}+\frac{1}{q}[1-\hat{f}(u_{0})]\sim q/2$. It then follows that for $q\ll
1$, $P(x,t)\propto\exp[-x^{2}/(2t)]$. This is the standard prediction of the
CLT, as delineated in Eq. (5), hence the more interesting case is $q\gg 1$.

We use Eq. (9) in the large $q$ limit, considering large $u$, to find $u_{0}$.
For that aim, we need to find $\hat{f}(u)$ when $u$ is large. With a change of
variable $\chi=u^{\frac{1}{\beta-1}}\xi$, we obtain

$\hat{f}(u)=\widetilde{N}u^{\frac{1}{\beta-1}}\int_{-\infty}^{\infty}\exp\left[u^{\frac{\beta}{\beta-1}}\left(\xi-\alpha^{\beta}|\xi|^{\beta}\right)\right]{\rm
d}\xi.$ (10)

Since $u$ is large this integral is solved using Laplace’s method

$\hat{f}(u)\sim\widetilde{N}C_{2}u^{\frac{1}{2}\frac{2-\beta}{\beta-1}}\exp\left(C_{1}u^{\frac{\beta}{\beta-1}}\right),$
(11)

where $C_{1}=(\beta-1)(\beta\alpha)^{\frac{\beta}{1-\beta}}$, and
$C_{2}=\sqrt{2\pi/(\beta-1)(\beta\alpha^{\beta})^{\frac{1}{\beta-1}}}$. This
expression is exact for the Gaussian case $\beta=2$. Eq. (9) for $u_{0}$ is
written as $0=1-\frac{1}{q}\hat{f^{\prime}}(u)$, and using Eq. (11) we find
that $u_{0}$ satisfies

$\displaystyle
0=1-\frac{1}{q}\frac{\beta\widetilde{N}C_{1}C_{2}}{\beta-1}u_{0}^{\frac{4-\beta}{2\beta-2}}\exp\left(C_{1}u_{0}^{\frac{\beta}{\beta-1}}\right).$
(12)

Note that in the case of $\beta=4$, this approximation breaks down. For that
reason, in this work, we focused on the range $1<\beta<4$ and the transition
to $\beta<1$, where the big jump principle holds. To solve Eq. (12) we utilize
the Lambert function $W(x)$ [62] which satisfies
$W(x)\exp\left[W(x)\right]=x$. The branch of the Lambert function that is
relevant to our study is the well-documented principal branch $W_{0}(x)$.
Solving Eq. (12) we obtain

$u_{0}\sim
C_{3}W_{0}\left[C_{4}q^{\frac{2\beta}{4-\beta}}\right]^{\frac{\beta-1}{\beta}}.$
(13)

For large values of the argument, the Lambert function is expressed in terms
of logarithmic functions, i.e.,
$W_{0}(z)\underset{z\to\infty}{\sim}\ln(z)-\ln(\ln(z))$, hence $u_{0}$ is
growing with $x$, however very slowly. This holds true only when $\beta<4$,
while the opposite case demands further study. The constants are:

$\displaystyle
C_{3}=\beta\alpha\left(\frac{2\beta(\beta-1)}{4-\beta}\right)^{\frac{1-\beta}{\beta}}\quad,\quad$
$\displaystyle
C_{4}=\frac{2(\beta(\beta-1))^{\frac{4}{4-\beta}}}{4-\beta}\left(\frac{\alpha^{2}}{\sqrt{2\pi}\widetilde{N}(\beta)}\right)^{\frac{2\beta}{4-\beta}}.$
(14)

Using $\exp[aW(x)]=(x)^{a}/[W(x)]^{a}$, we obtain
$K(u_{0})=u_{0}+\frac{1}{q}[1-\hat{f}(u_{0})]$. Therefore, according to Eq.
(8), we have that

$P(x,t)\sim\frac{\exp\left\\{-t\left(\frac{|x|}{t}Z\left(\frac{|x|}{t}\right)+1\right)\right\\}}{\sqrt{2\pi
K^{{}^{\prime\prime}}(u_{0})}},$ (15)

where

$Z(q)=\frac{c_{3}W_{0}\left[c_{4}q^{\frac{2\beta}{4-\beta}}\right]-\alpha\left(\frac{2\beta(\beta-1)}{4-\beta}\right)^{\frac{1}{\beta}}}{W_{0}\left[c_{4}q^{\frac{2\beta}{4-\beta}}\right]^{\frac{1}{\beta}}}.$
(16)

Using $W_{0}(x)\sim\ln(x)$ we find

$P(x,t)\sim\frac{\exp\left\\{-t\left[\kappa\frac{|x|}{t}\log\left(\frac{|x|}{t}\right)^{1-1/\beta}+1\right]\right\\}}{\sqrt{2\pi
K^{{}^{\prime\prime}}(u_{0})}},$ (17)

where $\kappa=\beta\alpha/(\beta-1)^{1-1/\beta}$. Thus, for large $x$, the
packet is decaying according to Laplace’s first law, with logarithmic
corrections which are typically hard to detect in experiments. The result in
Eq. (17) was found in a more general setting in [46]. We also note a mistake
in Eq. (9) in [46] the later equation corresponds to Eq. (16). Finally, we do
not advocate the practical use of Eq. (17) beyond an asymptotic result. The
Lambert functions in Eq. (16) are needed unless, $x/t$ is astronomically
large. To see this consider the asymptotic expansion
$W_{0}(x)\sim\ln(x)-\ln(\ln(x))$ the second term is one percent of the first
when $x\sim 10^{280}$. In other words, the leading term, in applications not
sampling extremely rare events is insufficient. We now show how the theory is
applicable for finite times, using numerically exact $P(x,t)$.

Figure 1: Comparative analysis of the CTRW exact solution (circles), our
approximation derived from Eq. (15) (blue line), numerical LDT derived using
Eq. (8) (dashed black), and CLT predictions (orange line), log-plotted for two
time scales (0.3,3), and various $\beta$ values. Notably, the nearly
exponential tails are a dominant feature in both time scales, with an
observable convergence to the CLT regime over time. The numerical LDT shows
remarkable fitting in the tails and also exhibits a strong performance in the
central part as time progresses. The approximation we propose in Eq.(15)
demonstrates a close match with the numerical data, however, deviations become
more pronounced for large time values as $\beta\rightarrow 1$. This trend,
indicative of slower convergence at smaller $\beta$ values, is further
analyzed and discussed in the following section.

###  3.1 CTRW Propagator

Sampling the “rare events” of the CTRW, particularly the tails of the PDF,
poses significant challenges in trajectory simulations. To address this, we
utilize the convolution theorem of the Fourier transform to compute
$\phi(x|n)$ in Eq (2). More specifically, in Fourier space, the equation
$\hat{\phi}(k|n)=\widetilde{f}^{n}(k)$ holds true. This leads to the recursive
relation: $\phi(x|n)=f\ast f\dots\ast f=\phi(x|n-1)\ast f$, where $*$ is the
convolution operator. For the case of $N=2$, this translates to
$\phi(x|2)=f\ast
f=\int_{-\infty}^{\infty}f(\chi^{\prime})f(x-\chi^{\prime})d\chi^{\prime}$.
This equation for $\phi(x|2)$ represents the integration over the probability
of making a first step to a certain point $\chi^{\prime}$ and then a
subsequent step to $x$.

In Fig. 1, we conduct a comparison between this numerically exact solution
(CTRW), our derived approximation in Eq. (15), and the numerical LDT
calculated using Eq. (8), juxtaposed with the predictions of the CLT. For
short time scales, our derived approximation and the numerical LDT demonstrate
excellent agreement in the tails, where the CLT fails. As the time scale
increases, the numerical LDT consistently aligns for all values of $x$ and
$\beta$, whereas our derived approximation, though precise for $\beta=2$
(Gaussian), and closely fitting for $\beta=3/2$, converges more gradually when
$\beta=1.1$, nearing the critical transition for $\beta=1$ that was studied in
[63]. For this case, when $\beta=1$, exponential-like tails are still
exhibited but are not described by Eq. (17).

Upon further examination of Fig. 1, it becomes apparent that for $t\gg 1$, the
application of the CLT or the Edgeworth expansion is also viable, a topic we
will discuss shortly.

Figure 2: Convergence of the CTRW results (dashed-dotted) for increasing time
scales, $t=1,5,15$, towards the numerical LDT, as derived in Eq. (8) (blue
stars). The theory demonstrates exceptional accuracy, with the CTRW aligning
with our numerical LDT across all values of $\beta$ for long times.
Additionally, the linear-like nature of the tails is observed, aligning with
the expected Laplace’s exponential tails. We also plot the asymptotical rate
function (ARF) from Eq. (19) (shown as a black line). Notably, for $\beta=2$,
the ARF aligns perfectly with the predictions, and for $\beta=1.5$, we achieve
a close fit. However, as we approach the regime of the big jump principle, for
$\beta=1.1$, convergence is slow, as we later describe.

###  3.2 Rate Function

The rate function is an important concept in the large deviations literature
as it holds the main characteristics of the propagator, describing both the
typical and rare events. It is obtained in the limit of $t\rightarrow\infty$,
but $x/t$ finite:

$\displaystyle\mathcal{I}(x/t)=\lim_{t\to\infty}-\frac{1}{t}\ln{\left[P(x,t)\right]}.$
(18)

In our model, we can extract the rate function from Eq. (8). Taking into
consideration the fact that $u_{0}$ is a function of $q=x/t$, see Eq. (9),
i.e. $q$ is the scaling parameter of the rate function:

$\displaystyle\mathcal{I}(q)=qK(u_{0})\approx\begin{dcases*}q^{2}/2&, $q\ll
1$\\\ |q|Z(|q|)+1&, $q\gg 1$ .\end{dcases*}$ (19)

The rate function in the small $q$ limit exhibits the quadratic CLT behavior,
whereas the large $q$ behavior is what we call the asymptotic rate function
(ARF). In Fig. 2, we demonstrate how our model effectively captures both the
central part, corresponding to the CLT, and the Laplace(exponential)-like
tails, corresponding to the large $q$ limit in Eq. (19), as $Z(q)$ is a slowly
varying function. It becomes evident that, as time progresses, the numerical
solution (CTRW) converges to our numerical LDT rate function. Moreover,
although the ARF theory is valid, as expected, we observe an issue of slow
convergence for values of $\beta$ near 1.

Figure 3: Slow convergence of the asymptotical rate function (ARF) as
formulated in Eq. (19), to the numerical LDT obtained using Eq. (8), for the
case $\beta=1.1$. As we approach $\beta=1$ from above, the convergence of our
approximation (ARF) becomes increasingly gradual. This slowing of convergence
is attributed to the proximity to the transition point. At this critical
juncture, the moment-generating function becomes non-existent, and the system
transitions into the big jump principle (BJP) regime.

###  3.3 Slow Convergence of the ARF

The issue of slow convergence of the ARF, as observed in Fig. 2, becomes
increasingly pronounced as we mentioned before. More specifically, we focus on
the scenario where our approximation, formulated in Eq. (19), under the
umbrella of LDT $(\beta>1)$, is nearing the critical transition point that
occurs for $\beta=1$. For this value of $\beta$, the moment-generating
function ceases to exist and our approximation becomes invalid.

In Fig. 3, we highlight how, for $\beta=1.1$, the ARF converges slowly to the
numerical LDT, as $q=x/t\rightarrow\infty$. This gradual convergence is a
direct consequence of our approximation for $u_{0}$, and the numerical
solution for $u_{0}$ eliminates this problem. In Fig. 4 we demonstrate how our
approximation of $u_{0}$ worsens as $\beta$ approaches 1, leading to the slow
convergence of the rate function and the propagator in such cases.

Figure 4: Approximation of $u_{0}$ (colored lines), and numerically computed
$u_{0}$ (corresponding colored symbols). While for $\beta=2$ our approximation
is exact, as $\beta\rightarrow 1$ the approximation works well only when $x\gg
t$.

##  4 Edgeworth Expansion

In the context of the summation of n IID random variables, the Edgeworth
expansion [64, 65, 66] provides corrections to the CLT. As a reminder, we
denote the PDF of finding the particle at $x$, conditioned on performing $n$
jumps, as $\phi(x|n)$. The leading order of the Edgeworth expansion when $n$
is large, is:

$\phi(x|n)\sim\frac{\exp(-x^{2}/2n)}{\sqrt{2\pi
n}}\left[1+\frac{\kappa_{4}\mathrm{He}_{4}(x/\sqrt{n})}{4!\ n}+\cdots\right]$
(20)

where $\kappa_{4}$ is the fourth cumulant of $f(\chi)$, namely
$\kappa_{4}=m_{4}-3\sigma^{2}$, and $m_{4}$ is the forth moment of $f(\chi)$.
In our example $\sigma=1$ while
$m_{4}=\Gamma(1/\beta)\Gamma(5/\beta)/\Gamma(3/\beta)^{2}$,
$\mathrm{He}_{4}(\xi)=\xi^{4}-6\xi^{2}+3$. In Eq. (20) the correction term is
proportional to the Hermite polynomial.

It is tempting to use the Edgeworth expansion in the summation presented in
Eq. (2). However, if $\kappa_{4}=0$, namely when $f(\chi)$ is Gaussian
corresponding to $\beta=2$, the correction term vanishes. We introduce a
modified Edgeworth expansion, where $t$ is the large parameter. We use the
small $k$ expansion $\widetilde{f}(k)=1-k^{2}/2+m_{4}k^{4}/4!+\cdots$ in Eq.
(4)

$\widetilde{P}(k,t)=\exp\left[-\frac{tk^{2}}{2}+\frac{tm_{4}k^{4}}{4!}+\cdots\right].$
(21)

Further expanding

$\widetilde{P}(k,t)=\exp\left(-\frac{tk^{2}}{2}\right)\left[1+k^{4}\frac{m_{4}t}{4!}+\cdots\right].$
(22)

Performing the inverse Fourier transform we obtain

$P(x,t)=\frac{\exp\left(-x^{2}/2t\right)}{\sqrt{2\pi
t}}\left[1+\frac{m_{4}}{4!\
t}\mathrm{He}_{4}\left(\frac{x}{\sqrt{t}}\right)+\cdots\right],$ (23)

Eq. (23) has the same structure as the original expansion in Eq. (20), where
we replaced large $n$ with large $t$. The difference is that now the
correction is proportional to the fourth moment of the distribution $f(\chi)$
and not to its cumulant. The former is always non-negative while the latter
can be negative zero or positive. Therefore, we see that the fluctuations in
the number of jumps have a significant effect.

The contribution of the Hermite polynomials within our model can be
demonstrated through a straightforward rearrangement of $P(x,t)$ from Eq.
(23), denoted as $\mathcal{R}(x,t)$. By defining the central limit theorem
component as $\mathrm{CLT}=\exp\left(-x^{2}/2t\right)/\sqrt{2\pi t}$, we
obtain the following form:

$\displaystyle\mathcal{R}(x,t)=\frac{4!t}{m_{4}}\frac{P(x,t)-CLT}{CLT},$ (24)

where, at sufficiently long times, this expression approaches the fourth
Hermite polynomial. This is shown in Fig. 5 for different values of $\beta$,
where $\mathcal{R}(x,t)$ is plotted versus the scaled variable $x/\sqrt{t}$.

The Edgeworth expansion uses the diffusive scaling $x/\sqrt{t}$ while the
large deviations theory relies on the scaled variable $x/t$. Both contribute
to the leading order of the central limit theorem when $x/t\ll 1$. Note that
when using the Edgeworth approach in the limit $x/\sqrt{t}\gg 1$, we have from
the Hermite function a correction term to the Gaussian $P(x,t)$ that can be
written as $tm_{4}(x^{4}/t^{4})$. This results in the same scaling $q=x/t$ as
found in the rate function $P(x,t)\propto\exp[-t{\cal{I}}(x/t)]$. Thus, as
expected, the Edgeworth expansion and large deviations theory are not
conflicting. They consider different limits of the problem. The large
deviations theory is valid for $\beta>1$ as it demands finite cumulants. The
convergence properties of the Edgeworth expansion and their $\beta$ dependence
are left for future work.

Figure 5: We showcase the leading order of the Edgeworth expansion by
plotting, for $t=10$ the rearranged result from the numerical CTRW as outlined
in Eq.(24) (colored lines). This is compared against the fourth Hermite
polynomial $\mathrm{He}_{4}(x/\sqrt{t})=(x/\sqrt{t})^{4}-6(x/\sqrt{t})^{2}+3$.
The excellent agreement is observed regardless of the value of $\beta$,
thereby confirmming the validity of this expansion for large time scales.

##  5 Big Jump Principle

###  5.1 Sum of IID Random Variables

The big jump principle (BJP) [67, 68, 69, 70, 71, 72, 73] describes how, in a
heavy-tailed process with a sub-exponential distribution of jump lengths, a
single “big” jump among a series of $n$ IID random variables can dominate the
asymptotic characteristics of the process. This principle connects the sum
$x=\sum_{i=1}^{n}\chi_{i}$ to the largest displacement
$\chi_{\text{max}}=\max{\chi_{1},\dots,\chi_{n}}$ [69]. Specifically, this
principle provides the asymptotic equality:

$\displaystyle\int_{\chi_{\text{large}}}^{\infty}\phi(x|n)dx$ $\displaystyle=$
$\displaystyle\text{Prob}(x>\chi_{\text{large}})=$ (25)
$\displaystyle\text{Prob}(\chi_{\text{max}}>\chi_{\text{large}}),$

in the limit of $\chi_{\text{large}}\rightarrow\infty$. To further elucidate,
we calculate:

Prob
$\displaystyle\left(\chi_{\text{max}}>\chi_{\text{large}}\right)=1-\text{Prob}\left(\chi_{i}<\chi_{\text{large}}\right)^{n}$
(26)
$\displaystyle\quad=1-\left[1-\int_{\chi_{\text{large}}}^{\infty}f(\chi)d\chi\right]^{n}.$

Taking the derivative with respect to $\chi_{\text{large}}$, we obtain the PDF
of $x$. Then, by approximating
$\int_{\chi_{\text{large}}}^{\infty}f(\chi)d\chi$ as approximately zero, which
is a leading-order approximation, we get the asymptotic result [74]:

$\displaystyle\phi(x|n)_{\beta<1}\approx nf(x).$ (27)

It is important to note that, in contrast with the CLT, which is applicable
primarily when $n$ is significantly large, Eq. (27) is valid for all values of
$n$, including instances when $n=2$. We will discuss the BJP for CTRW later,
but for now, our focus is on $\phi(x|n)$, specifically in the case where the
number of jumps, $n$, equals 2.

Figure 6: The graph of $g(y)=-|1-y|^{\beta}-|y|^{\beta}$, for different values
of $\beta$. It is evident that for $\beta>1$, the function is smooth with a
single global maximum at $y_{0}=1/2$. For $\beta=1$, the function is piecewise
linear, and for $\beta<1$, it exhibits two non-analytical cusps at $y=0$, and
$y=1$. This example serves as a foundational demonstration of the transition
from BJP $(\beta<1)$ to LDT $(\beta>1)$, while $\beta=1$ is a special
transition point [63].

###  5.2 Big Jump Principle for $n=2$

An understanding of the big jump principle is best demonstrated through the
simple yet comprehensive scenario of two jumps. Employing the convolution
theorem as seen earlier; $\phi(x|2)=f\ast
f=\widetilde{N}^{2}\int_{-\infty}^{\infty}\exp\left(-\alpha^{\beta}|\chi^{\prime}|^{\beta}-\alpha^{\beta}|x-\chi^{\prime}|^{\beta}\right)d\chi^{\prime}$,
we introduce the scaling variable $y=\frac{\chi^{\prime}}{x}$, and obtain the
following equation:

$\displaystyle\phi(x|2)=\widetilde{N}^{2}|x|\int_{-\infty}^{\infty}{\rm
e}^{\alpha^{\beta}|x|^{\beta}g(y)}dy,$ (28)

where we define $g(y)\equiv-|1-y|^{\beta}-|y|^{\beta}$. In Fig. 6 we
demonstrate the transition in the analytical properties of $g(y)$. For
$\beta>1$ where $f(\chi)$ is super-exponential, $g(y)$ is a smooth function
with a single global maximum at $y_{0}=1/2$, suitable for the Laplace method
of solving integrals. Conversely, as shown in Fig. 6, for $\beta<1$, $g(y)$
displays a pair of cusps, and two distinct maxima are observed. We proceed to
examine these cases. Beginning with the case of $\beta>1$, one can approximate
$g(y)\approx g(y_{0})+\frac{1}{2}g^{{}^{\prime\prime}}(y_{0})(y-y_{0})^{2}$,
facilitating a direct Gaussian integration:

$\displaystyle\phi(x|2)_{\beta>1}\approx\widetilde{N}^{2}|x|\sqrt{\frac{\pi
2^{\beta-2}}{\beta(\beta-1)\alpha^{\beta}|x|^{\beta}}}{\rm
e}^{-\frac{1}{2}^{\beta-1}\alpha^{\beta}|x|^{\beta}}.$

This result diverges as $\beta$ approaches 1. The critical case of $\beta=1$
marks a transition, where the cumulant generating function of $\phi(x|n)$,
exists only for $\beta>1$, and a piecewise linear function is displayed. At
$\beta=1$, we can exactly solve Eq. (28), yielding:
$\phi(x|2)=\frac{\sqrt{2}+2|x|}{4}{\rm
e}^{-\sqrt{2}|x|}=\left(\frac{1+\sqrt{2}|x|}{2}\right)f(x)$, which is not
described by either the BJP or the LDT.

However, for $\beta<1$, where $f(\chi)$ is sub-exponential, $g(y)$ exhibits
two non-analytical cusps, and a slightly modified version of the Laplace
method of solving integrals is needed. Due to the symmetry of $g(y)$, it is
easy to show that the main contributions from the two maxima to the
integration are identical. Opting for the left cusp, around $y=0$, we
approximate $g(y)\approx-1-|y|^{\beta}$, and insert it into Eq. (28), to
derive: $\phi(x|2)\approx\widetilde{N}^{2}|x|\int_{-\infty}^{\infty}{\rm
e}^{-\alpha^{\beta}|x|^{\beta}\left(1+|y|^{\beta}\right)}dy=2f(x)$, aligning
with the BJP.

Now, we calculate corrections to the BJP which are particularly important when
$\beta\to 1$. The improved approximation is given by
$g(y)\approx-1-|y|^{\beta}+\beta y+\frac{1}{2}\beta(1-\beta)y^{2}$, which we
can plug into Eq. (28) to obtain

$\displaystyle\phi(x|2)\approx 2\widetilde{N}^{2}|x|$
$\displaystyle\times\int_{-\infty}^{\infty}{\rm
e}^{\alpha^{\beta}|x|^{\beta}(-1-|y|^{\beta})}\big{(}1+{\rm d}_{1}(x)y+{\rm
d}_{2}(x)y^{2}\big{)}dy,$

where ${\rm d}_{1}(x)=\alpha^{\beta}|x|^{\beta}\beta$ vanishes due to
symmetry. The solution of Eq. (5.2) yields

$\phi(x|2)\approx 2\left(1+\frac{{\rm d}_{2}(x)}{x^{2}}\right)f(x),$ (31)

where

${\rm
d}_{2}(x)=\frac{1}{2}\alpha^{\beta}|x|^{\beta}\beta\left(1-\beta+\alpha^{\beta}|x|^{\beta}\beta\right).$
(32)

The first term of Eq. (31) is the BJP, as detailed in Eq. (27), while ${\rm
d}_{2}(x)/x^{2}$, serves as the correction term. The correction becomes
significant as $\beta$ approaches $1$ from below, since the magnitude of ${\rm
d}_{2}(x)/x^{2}$ is of the same order as the leading term of the BJP. In Fig.
7, we compare the numerically obtained $\phi(x|2)$, the leading term from Eq.
(27), and the correction term we have formulated in Eq. (31). This comparison
emphasizes the importance of additional correction terms as $\beta\to 1$ and
demonstrates their redundancy for $\beta\ll 1$.

Figure 7: The big jump principle for the pedagogical case of two jumps. The
figure shows the numerically calculated $\phi(x|2)$ (colored symbols), the
leading approximation $(2f(x))$ (colored lines), and the corrected form,
derived in Eq. (31) (colored dashed lines) for $\beta=0.2,0.7,0.9$, are
displayed on a semi-logarithmic scale. While the BJP is effective at the far
tails, it becomes apparent that convergence slows as $\beta$ nears the
critical value of 1.

###  5.3 BJP for CTRW

As we progress with the big jump principle, integrating Eq. (27) into Eq. (2)
leads us to a straightforward relation [63]:

$\displaystyle P(x,t)\approx\langle n_{t}\rangle f(x),$ (33)

where $\langle n_{t}\rangle$ represents the average number of jumps at time
$t$. This relationship holds true for every distribution of waiting times. In
this manuscript, our focus is specifically on exponential waiting times,
corresponding to Poisson statistics. Namely, we have $\langle n\rangle=t$,
hence $P(x,t)\approx tf(x)$. This signifies a pivotal behavior change: instead
of exhibiting exponential tails, the propagator will now display sub-
exponential characteristics. Thus, the universality observed in the super-
exponential case (for $\beta>1$) disappears for $\beta<1$. Similarly, if
$f(x)$ has a power law tail, Eq. (33) is still valid, and $P(x,t)$ will decay
as a power law for large x. In Fig. 8, we showcase this transition, observing
how the propagator, $P(x,t)$, shifts from sub-exponential features to
exponential, Laplace-like characteristics.

Figure 8: The propagator $P(x,t)$ is shown for different values of $\beta$
(shown in legend). The transition of $P(x,t)$ from exponential to Laplace-like
characteristics is associated with the large deviations theory. When
$\beta>1$, we observe a transition to sub-exponential behavior as per the big
jump principle when $\beta<1$. Circles denote the numerically obtained CTRW
data, while solid lines represent our approximations. For $\beta<1$, Eq. (33)
is employed, and for $\beta>1$, Eq. (15) is utilized.

##  6 Discussion

This research has centered on applying large deviations theory (LDT) to the
study of diffusive motion in the context of continuous time random walks with
exponential waiting times. A notable aspect of LDT is its traditional use in
probing rare events, specifically within the large $x$ limit of the
probability density function $P(x,t)$. While rate functions, a key component
in LDT, are often perceived as challenging to sample in empirical settings
[75, 76], our study suggests otherwise. By focusing on short time scales and
large $x$ values, we demonstrate that the nearly exponential decay of $P(x,t)$
\- albeit with logarithmic corrections - is more accessible for sampling than
previously assumed.

This accessibility is particularly pertinent in single tracer experiments,
where the microscopic time scale, characterized by the mean time between
jumps, is not significantly smaller than the observation period $t$. Under
these conditions, a universal exponential decay in the particle density is
observed. Additionally, our investigation delved into the realm of the big
jump principle, particularly its interplay and transition to LDT. The BJP is
critical for understanding heavy-tailed processes in which significant
fluctuations, or “big jumps”, dominate the characteristics of the random walk.
This principle becomes particularly relevant when the jump length distribution
is sub-exponential, leading to sub-exponential behavior in the propagator [74,
63, 77, 78, 79, 80]. The transition from the BJP regime $(\beta<1)$ to the LDT
regime $(\beta>1)$ highlights how different statistical frameworks govern the
behavior of diffusive processes under varying conditions. These findings are
crucial as they extend the utility of LDT and BJP beyond theoretical confines,
making them practical tools for capturing statistical behaviors of particle
dispersion that deviate from the normal distribution, inherent in various
physical systems.

While the CLT remains a cornerstone in statistical physics, our study
illustrates that deviations from CLT predictions, particularly in the tails,
are theoretically significant and experimentally observable. Furthermore, the
employment of the Edgeworth expansion has provided valuable corrections to the
CLT, leading to a more nuanced understanding of the propagator’s behavior in
different regimes. This approach has bridged the gap between the Gaussian
central, and the non-Gaussian tails of the distribution, capturing the essence
of diffusive dynamics more comprehensively.

The understanding of Laplace’s first law is not limited to the CTRW framework.
In Refs. [81, 29], a many-body scenario was studied, employing mainly
simulations. One example is the Hitchhiker model [81], a framework based on
experimental findings that the size of molecules may fluctuate due to
diffusion, aggregation, and fragmentation processes, resulting in exponential-
like tails for $P(x,t)$. Similarly, Ref. [29] argues that polymerization
processes lead to the demonstrations of Brownian yet non-Gaussian diffusion.

###  6.1 Open Questions

In the CTRW framework, a significant challenge is presented by the
intermediate regime, where neither the central limit theorem nor the large
deviations theory/big jump principle effectively approximates the results.
Rusciano et al.[16] introduced a scaling parameter, assuming exponential
statistics; $\exp{(-|x|/\lambda(t))}$, where $\lambda(t)\sim t^{1/3}$. An
opposing comment was quickly raised by Berthier et al.[82], yet the
phenomenon, particularly the non-trivial exponent $1/3$, was observed in
several other experiments [57, 23, 16, 24]. More specifically, in Ref. [57],
both the scaling parameter and the Lambert functions, as derived in this
manuscript, are demonstrated. Furthermore, in Ref. [14], a similar scaling
parameter, $\lambda(t)\sim t^{1/2}$, is proposed, adding further intrigue to
the subject. It remains unclear whether the experimental and numerical
evidence for $\lambda(t)\sim t^{b}$ represents merely a fitting issue due to
insufficient data, or if it describes some deep physics not fully understood
by the authors of this manuscript.

Chaudhuri et al. [22] proposed that the waiting time PDF is a combination of
two exponentials, with the first jump in the random walk being drawn from a
less frequent PDF compared to the subsequent jumps. Under certain conditions,
this results in more pronounced exponential tails. As mentioned in our model,
for the sake of mathematical simplicity and broad applicability, we employed a
Poisson process, while other works investigated a wider variety of processes
[47]. However, a practical question that remains largely unanswered is the
impact of non-exponential waiting times on the observations made in this
study. Similarly, the effect of changing the dimensionality of the problem on
our results is yet to be explored.

Another interesting approach to the tails of CTRW was recently pioneered by
Sokolov and Pacheco [55]. They use the rate function representation of the
temporal process (waiting times) and the rate function representation of the
spatial process (jumps) to study the properties of the rate function of
$P(x,t)$.

Finally, delving beyond sub- and super-exponential jump lengths, we still
expect Laplace-like tails, for example, the case of CTRW on a lattice [47].
One may find similar results to those presented in this manuscript, though the
Lambert function used extensively here, is not found to be generic. This
implies that the characterization of the rate function for more general jump
length distributions might yield further insights.

Acknowledgements: We thank Lucianno Defaveri for insightful discussions. We
also acknowledge the support of Israel Science Foundation’s grants 1614/21,
and 2796/20. †† *<EMAIL_ADDRESS>*<EMAIL_ADDRESS>**
<EMAIL_ADDRESS>
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