# Gaps between prime divisors

Efthymios Sofos  Department of Mathematics
University of Glasgow
G12 8QQ United Kingdom<EMAIL_ADDRESS>
###### Abstract.

Erdős considered the second moment of the gap-counting function of prime
divisors in $1946$ and proved an upper bound that is not of the right order of
magnitude. We prove asymptotics for all moments.

###### 2010 Mathematics Subject Classification:

11K65; 11N37, 60F05, 60F15.

###### Contents

  1. 1 Introduction
  2. 2 The proof of Theorem 1.1

##  1\. Introduction

What are the typical gaps between the prime divisors of a randomly selected
integer? For $m\in\mathbb{N}$ we let $\omega(m)$ be the number of distinct
prime divisors of $m$ and $p_{i}(m)$ be the $i$-th smallest prime divisor of
$m$, so that

$\log\log p_{1}(m)<\ldots<\log\log p_{\omega(m)}(m)$

is a finite sequence that depends on $m$. It is not difficult to show that for
almost all $m$ and almost all $1\leqslant i\leqslant\omega(m)$, one has
$\log\log p_{i}(m)\sim i$, hence, $\log\log p_{i+1}(m)-\log\log p_{i}(m)$ is
typically bounded. A natural question is to count the number of gaps exceeding
a fixed constant $z\geqslant 0$, i.e. estimate

$\omega_{z}(m):=\sharp\left\\{1\leqslant i<\omega(m):\log\log
p_{i+1}(m)-\log\log p_{i}(m)>z\right\\}.$

Erdős [5, pg. 534] was the first to study this question. He showed that for
almost all $m$ the function $\omega_{z}(m)$ is well-approximated by
$\mathrm{e}^{-z}\omega(m)$ by proving an upper bound for the second moment:

$\frac{1}{n}\sum_{m\in\mathbb{N}\cap[1,n]}\left(\omega_{z}(m)-\mathrm{e}^{-z}\log\log
n\right)^{2}=o((\log\log n)^{3/2}),\text{ as }n\to+\infty.$

However, it turns out that this is not of the right order of magnitude. Here
we prove asymptotics not just for the second moment, but for all moments:

######  Theorem 1.1.

Fix any $z\geqslant 0$ and $r\geqslant 0$. Then

$\frac{1}{n}\sum_{m\in\mathbb{N}\cap[1,n]}\left(\omega_{z}(m)-\frac{\log\log
n}{\mathrm{e}^{z}}\right)^{r}=\mu_{r}((1-2z\mathrm{e}^{-z})\mathrm{e}^{-z}\log\log
n)^{r/2}(1+o(1)),\text{ as }n\to+\infty,$

where $\mu_{r}$ is the $r$-th moment of the standard normal distribution.

As a consequence, for all $\alpha<\beta\in\mathbb{R}$ one has

$\lim_{n\to+\infty}\frac{1}{n}\sharp\left\\{m\in\mathbb{N}\cap[1,n]:\frac{\omega_{z}(m)-\mathrm{e}^{-z}\log\log
m}{((1-2z\mathrm{e}^{-z})\mathrm{e}^{-z}\log\log
m)^{1/2}}\in(\alpha,\beta]\right\\}=\frac{1}{\sqrt{2\pi}}\int_{\alpha}^{\beta}\mathrm{e}^{-t^{2}/2}\mathrm{d}t.$

Setting $z=0$ we recover the much celebrated Erdős–Kac theorem [6]. Our method
is different from that of Erdős [5] in that it relies on Stein’s method on
normal approximations [12]. This allows us to deal with certain sums of
dependent random variables that arise when modelling $\omega_{z}(m)$. Stein’s
method has been rarely used in number theory, for example, by Harper [8].

There are many generalisations of the Erdős–Kac theorem to functions of the
form $\sum_{p\mid m}g(p)$ but they do not cover $\omega_{z}(m)$, as $g(p)$
would have to be a function of $m$ as well. Galambos [7, Theorem 2] studied
the values of a function that is somewhat related to our $\omega_{z}$, namely
the cardinality of $i<\omega(m)$ for which $\log\log p_{i+1}(m)-\log\log
p_{i}(m)>z+\log\log\log m$. His results and method are rather different as
they are suited to values of large gaps, while our result relates to small
gaps. A function similar to Galambos’ occurs in the recent work of
Chan–Koymans–Milovic–Pagano [2, §4] on the negative Pell equation.

######  Remark 1.2.

At the cost of a non-self sufficient argument, the number theoretic part of
the proof of Theorem 1.1, (namely, Lemma 2.9) can be alternatively verified
via the Kubilius model [4, §12]. The approximation of $\omega(m)$ by
$\mathrm{e}^{-z}\log\log m$ means that the gaps in the sequence $\\{\log\log
p_{i}(m)\\}_{i\geqslant 1}$ are Poissonian. It is worth mentioning that the
occurrence of Poisson distribution in other areas of Probabilistic Number
Theory is not uncommon, see the work of de Koninck–Galambos [3], Harper [8],
Granville [10] and Kowalski–Nikeghbali [11], for example.

######  Acknowledgements.

I wish to thank Daniel El-Baz for suggesting the use of Stein’s method. While
working on this paper I was supported by EPSRC New Horizons grant
EP/V048236/1.

##  2\. The proof of Theorem 1.1

###  2.1. Defining the model

The letter $z$ will denote a fixed non-negative real number throughout §2. As
usual, we denote $\exp(z):=\mathrm{e}^{z}$. For a prime $p$ and a positive
integer $m$ we define

$\delta_{p,z}(m):=\begin{cases}1,&\text{ if }p\mid m\text{ and }m\text{ is not
divisible by any prime in }(p,p^{\exp(z)}],\\\ 0,&\text{
otherwise.}\end{cases}$

In particular, $\omega_{z}(m)=\sum_{p}\delta_{p,z}(m)$, where the sum is over
all primes. Our plan, initially, is to follow the Kubilius model idea (see
Billingsley [1]) to define Bernoulli random variables $B_{p}$ that model the
behavior of $\delta_{p,z}$. For this we use the random variables $X_{p}$
defined by Kubilius, which in the notation of Billingsley’s work [1, Equation
(13)] are given as follows: for every prime $p$ the random variable $X_{p}$ is
defined so that

$P[X_{p}=1]=\frac{1}{p},\hskip 28.45274ptP[X_{p}=0]=1-\frac{1}{p}$

and such that $X_{p}$ are independent. In particular, the mean $E[X_{p}]$
equals $1/p$, thus, $X_{p}=1$ models the event that a random integer $m$ is
divisible by a fixed prime $p$. Let $[\cdot]$ denote the integer part. The
independence of $X_{p}$ is related to the Chinese Remainder Theorem.

To model $\delta_{p,z}$ we must also take into account the fact that each
prime $q$ in the range $(p,p^{\exp(z)}]$ must not divide $m$. Thus, we are
naturally led to define

${B_{p}:=X_{p}\prod_{\begin{subarray}{c}q\textrm{ prime }\\\ p<q\leqslant
p^{\exp(z)}\end{subarray}}(1-X_{q}).}$ (2.1)

We will later prove that $\sum_{p}B_{p}$ is a good model for
$\omega_{z}=\sum_{p}\delta_{p,z}$ in the sense that their moments agree
asymptotically.

######  Remark 2.1 (Independence break-down).

Definition (2.1) leads to a major difference between this paper and the proofs
of the Erdős–Kac theorem, namely, the variables $B_{p}$ are dependent. Indeed,
for all primes $p<q$ with $q\leqslant p^{\exp(z)}$, the quantity
$E[B_{p}B_{q}]$ vanishes while none of $E[B_{p}],E[B_{q}]$ does.

###  2.2. Distribution and moments of the model via Stein’s method

For any positive $N$ we define

$S_{N}=\sum_{p\leqslant N}B_{p}$

and denote its expectation and variance respectively by

$c_{N}:=E\left[S_{N}\right]\ \ \textrm{ and }\ \
s_{N}^{2}:=\textrm{Var}\left[S_{N}\right].$

Our goal in this section is to prove that $(S_{N}-c_{N})/s_{N}$ converges in
law to the standard normal distribution as $N\to\infty$ and that its moments
are asymptotically Gaussian. This will be done respectively in Propositions
2.5 and 2.7. We first need a few preparatory estimates.

######  Lemma 2.2.

We have

${E[B_{p}]=\frac{\mathrm{e}^{-z}}{p}+O\left(\frac{1}{p\log p}\right),}$ (2.2)
${c_{N}=\mathrm{e}^{-z}\log\log N+O(1)}$ (2.3)

and

${s_{N}^{2}=\left(1-\frac{2z}{\mathrm{e}^{z}}\right)\frac{\log\log
N}{\mathrm{e}^{z}}+O(1).}$ (2.4)

###### Proof.

Recall that Mertens’ theorem states that $\sum_{p\leqslant T}1/p=\log\log
T+c+O(1/\log T)$ for some constant $c$. The independence of $X_{p}$ yields

$E[B_{p}]=\frac{1}{p}\prod_{p<q\leqslant
p^{\exp(z)}}\left(1-\frac{1}{q}\right),$

which, by the approximation
$1-\varepsilon=\exp(-\varepsilon+O(\varepsilon^{2}))$ for
$|\varepsilon|\leqslant 1$ and Mertens’ theorem is

$\frac{1}{p}\exp\left(-\sum_{p<q\leqslant
p^{\exp(z)}}\frac{1}{p}+O\left(\sum_{p<q\leqslant
p^{\exp(z)}}\frac{1}{p^{2}}\right)\right)=\frac{\exp(-\log\log
p^{\exp(z)}+\log\log p+O(1/\log p))}{p}.$

Since $\exp\left(O(1/\log p)\right)=1+O(1/\log p)$, this is sufficient for
(2.2). The estimate (2.3) is directly deduced from it and the fact that
$\sum_{p}(p\log p)^{-1}$ converges. Next, denoting $h_{p}=E[B_{p}]$ we have

$s_{N}^{2}=\sum_{p\leqslant
N}E\left[(B_{p}-h_{p})^{2}\right]+2\sum_{p<q\leqslant
N}E\left[(B_{p}-h_{p})(B_{q}-h_{q})\right].$

First note that
$E\left[(B_{p}-h_{p})^{2}\right]=E\left[B_{p}\right]-h_{p}^{2}=h_{p}(1-h_{p})$.
Further, if $q>p^{\exp(z)}$ then $B_{p}$ and $B_{q}$ are independent, hence,
$E\left[(B_{p}-h_{p})(B_{q}-h_{q})\right]=0$. If $p<q\leqslant p^{\exp(z)}$
then $E[B_{p}B_{q}]$ vanishes, hence

$E\left[(B_{p}-h_{p})(B_{q}-h_{q})\right]=-E\left[B_{p}\right]h_{q}-h_{p}E\left[B_{q}\right]+h_{p}h_{q}=-h_{p}h_{q}.$

We obtain

$\displaystyle s_{N}^{2}$ $\displaystyle=\sum_{p\leqslant
N}h_{p}(1-h_{p})-2\sum_{\begin{subarray}{c}p<q\leqslant\min\\{N,p^{\exp(z)}\\}\end{subarray}}h_{p}h_{q}$
$\displaystyle=c_{N}-\sum_{p\leqslant
N}h_{p}^{2}-2\sum_{\begin{subarray}{c}p\leqslant N^{\exp(-z)}\\\ p<q\leqslant
p^{\exp(z)}\end{subarray}}h_{p}h_{q}-2\sum_{\begin{subarray}{c}N^{\exp(-z)}<p\leqslant
N\\\ p<q\leqslant N\end{subarray}}h_{p}h_{q}.$

By (2.2) we have $h_{p}\ll 1/p$, hence, $\sum_{p}h_{p}^{2}=O(1)$ and

$\sum_{\begin{subarray}{c}N^{\exp(-z)}<p\leqslant N\\\ p<q\leqslant
N\end{subarray}}h_{p}h_{q}\ll\left(\sum_{N^{\exp(-z)}<p\leqslant
N}\frac{1}{p}\right)^{2}=O(1).$

Hence, (2.3) gives

${s_{N}^{2}=\mathrm{e}^{-z}\log\log N-2\sum_{p\leqslant
N^{\exp(-z)}}h_{p}\sum_{p<q\leqslant p^{\exp(z)}}h_{q}+O(1).}$ (2.5)

Using (2.2) we see that

$\sum_{p\leqslant N^{\exp(-z)}}h_{p}\sum_{p<q\leqslant
p^{\exp(z)}}h_{q}=\sum_{p\leqslant N^{\exp(-z)}}h_{p}\sum_{p<q\leqslant
p^{\exp(z)}}\left(\frac{\mathrm{e}^{-z}}{q}+O\left(\frac{1}{q\log
q}\right)\right),$

which, by Mertens’ theorem and $\sum_{q>t}(q\log q)^{-1}\ll(\log t)^{-1}$,
equals

$\displaystyle\sum_{p\leqslant
N^{-\exp(z)}}h_{p}\left(\frac{z}{\mathrm{e}^{z}}+O\left(\frac{1}{\log
p}\right)\right)$ $\displaystyle=\sum_{p\leqslant
N^{-\exp(z)}}\left(\frac{\mathrm{e}^{-z}}{p}+O\left(\frac{1}{p\log
p}\right)\right)\left(\frac{z}{\mathrm{e}^{z}}+O\left(\frac{1}{\log
p}\right)\right)$
$\displaystyle=\frac{z}{\mathrm{e}^{2z}}\left(\sum_{p\leqslant
N^{-\exp(z)}}\frac{1}{p}\right)+O(1)=\frac{z}{\mathrm{e}^{2z}}(\log\log
N)+O(1).$

Injecting this into (2.5) concludes the proof.∎

######  Lemma 2.3.

For all $u\in\mathbb{N},\mathbf{r}\in\mathbb{N}^{u}$ and primes
$p_{1},\ldots,p_{u}$ we have

$E\left[\prod_{i=1}^{u}|B_{p_{i}}-E[B_{p_{i}}]|^{r_{i}}\right]=O_{\mathbf{r}}\left(\frac{1}{\mathrm{rad}(p_{1}\cdots
p_{u})}\right),$

where $\mathrm{rad}$ denotes the radical.

###### Proof.

First, we assume that $p_{1}<\ldots<p_{u}$ and prove the bound

${E\left[\prod_{i=1}^{u}(B_{p_{i}}-E[B_{p_{i}}])^{r_{i}}\right]=O_{\mathbf{r}}\left(\frac{1}{p_{1}\cdots
p_{u}}\right).}$ (2.6)

The binomial theorem shows that the expectation is

$\ll_{\mathbf{r}}\max_{\mathbf{t}\in[0,r_{1}]\times\cdots\times[0,r_{u}]}\\!E\left[\prod_{i=1}^{u}B_{p_{i}}^{t_{i}}\right]\prod_{i=1}^{u}E[B_{p_{i}}]^{r_{i}-t_{i}}\ll_{\mathbf{r}}\max_{\mathbf{t}}E\left[\prod_{\begin{subarray}{c}1\leqslant
i\leqslant u\\\
t_{i}>0\end{subarray}}X_{p_{i}}\right]\prod_{i=1}^{u}E[X_{p_{i}}]^{r_{i}-t_{i}},$

where we used $B_{p}\leqslant X_{p}$. By the independence of $X_{p}$ we get
the bound

$\ll_{\mathbf{r}}\max_{\mathbf{t}}\left(\prod_{\begin{subarray}{c}1\leqslant
i\leqslant u\\\
t_{i}>0\end{subarray}}E\left[X_{p_{i}}\right]\right)\prod_{i=1}^{u}E[X_{p_{i}}]^{r_{i}-t_{i}}=\max_{\mathbf{t}}\left(\prod_{\begin{subarray}{c}1\leqslant
i\leqslant u\\\
t_{i}>0\end{subarray}}\frac{1}{p_{i}}\right)\prod_{i=1}^{u}\frac{1}{p_{i}^{r_{i}-t_{i}}}.$

If $t_{i}>0$ then we get $1/p_{i}$ from the first product in the outmost right
hand side and if $t_{i}=0$ then we get a factor $1/p_{i}^{r_{i}}\leqslant
1/p_{i}$ from the second product. This concludes the proof of (2.6).

To verify our lemma it suffices to restrict to the cases with
$p_{1}<\ldots<p_{u}$. Observe that if $x\geqslant\max\\{0,y\\}$ then
$|y|\leqslant\max\\{y,x-y\\}$, therefore, for $y=B_{p}-E[B_{p}]$ and $x=B_{p}$
we obtain

$|B_{p}-E[B_{p}]|\leqslant\max\\{B_{p}-E[B_{p}],E[B_{p}]\\}\leqslant\max\\{B_{p}-E[B_{p}],1/p\\}.$

Combining this with (2.6) allows us to conclude the proof.∎

The following lemma is the main tool in the proof of Theorem 1.1. It is due to
Stein [12, Corollary $2$, page $110$].

######  Lemma 2.4 (Stein).

Let $T$ be a finite set and for each $t\in T$, let $Z_{t}$ be a real random
variable and $T_{t}$ a subset of $T$ such that $E[Z_{t}]=0$,
$E[Z_{t}^{4}]<\infty$ and $E[\sum_{t\in T}Z_{t}\sum_{s\in T_{t}}Z_{s}]=1$.
Then for all real $b$,

${\left|P\left[\sum_{t\in T}Z_{t}\leqslant
b\right]-\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{b}\mathrm{e}^{-t^{2}/2}\mathrm{d}t\right|\leqslant
4(\Psi_{1}+\Psi_{2}+\Psi_{3}),}$ (2.7)

where the terms $\Psi_{i}$ are defined through

$\Psi_{1}=E\left[\sum_{t\in T}\left|E\left[Z_{t}|Z_{s},s\notin
T_{t}\right]\right|\right],\Psi_{2}^{2}=E\left[\sum_{t\in
T}|Z_{t}|\left(\sum_{s\in T_{t}}Z_{s}\right)^{2}\right]$

and

$\Psi_{3}^{2}=E\left[\left\\{\sum_{t\in T}\sum_{s\in
T_{t}}(Z_{t}Z_{s}-E[Z_{s}Z_{t}])\right\\}^{2}\right].$

######  Proposition 2.5.

Fix $z\geqslant 0$ and $b\in\mathbb{R}$. For any $N\in\mathbb{N}$ we have

$\left|P\left[S_{N}\leqslant
c_{N}+bs_{N}\right]-\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{b}\mathrm{e}^{-t^{2}/2}\mathrm{d}t\right|\ll_{z}(\log\log
N)^{-1/2},$

where the implied constant depends at most on $z$. In particular,
$(S_{N}-c_{N})/s_{N}$ converges in law to the standard normal distribution as
$N\to\infty$.

###### Proof.

We will apply Lemma 2.4 with

  * •

$T$ being the set of primes in $[2,N]$,

  * •

$T_{p}$ being the set of primes in $[p^{\exp(-z)},p^{\exp(z)}]\cap[2,N]$,

  * •

$Z_{p}=(B_{p}-E[B_{p}])/s_{N}$ for $p\in T$.

Let $Y_{p}:=B_{p}-E[B_{p}]$. Note that if $q\notin T_{p}$ then $Z_{q}$ and
$Z_{p}$ are independent, hence, $E[Y_{p}Y_{q}]=0$. Therefore,

$s_{N}^{2}=\sum_{p,q\leqslant
N}E[Y_{p}Y_{q}]=\sum_{\begin{subarray}{c}p\leqslant N\\\ q\in
T_{p}\end{subarray}}E[Y_{p}Y_{q}],$

which verifies $E[\sum_{p\in T}Z_{p}\sum_{q\in T_{p}}Z_{q}]=1$. We next
observe that since for every $q\notin T_{p}$ the random variables $Z_{q}$ and
$Z_{p}$ are independent, one obtains $E\left[Z_{p}|Z_{q},q\notin
T_{t}\right]=E\left[Z_{p}\right]=0,$ therefore

${\Psi_{1}=0.}$ (2.8)

Next, we use Lemma 2.3 to obtain

$\displaystyle\Psi_{2}^{2}s_{N}^{3}=$
$\displaystyle\sum_{\begin{subarray}{c}p\leqslant N,q\in
T_{p}\end{subarray}}E\left[|Y_{p}|Y_{q}\right]+2\sum_{\begin{subarray}{c}p\leqslant
N,q_{1}<q_{2}\in T_{p}\end{subarray}}E\left[|Y_{p}|Y_{q_{1}}Y_{q_{2}}\right]$
$\displaystyle\ll$ $\displaystyle\sum_{\begin{subarray}{c}p\leqslant N,q\in
T_{p}\end{subarray}}\frac{1}{pq}+\sum_{\begin{subarray}{c}p\leqslant
N,q_{1},q_{2}\in T_{p}\end{subarray}}\frac{1}{pq_{1}q_{2}}.$

The sum $\sum_{q\in T_{p}}1/q$ is bounded only in terms of $z$ by Mertens’
theorem. It shows that

${\Psi_{2}^{2}\ll s_{N}^{-3}\sum_{p\leqslant N}\frac{1}{p}\ll(\log\log
N)^{-1/2},}$ (2.9)

owing to (2.4).

To bound $\Psi_{3}$ we write $\mathscr{C}_{p}:=\sum_{q\in
T_{p}}\left(Y_{p}Y_{q}-E[Y_{p}Y_{q}]\right)$ to obtain

${\Psi_{3}^{2}s_{N}^{4}=\sum_{p\leqslant
N}E\left[\mathscr{C}_{p}^{2}\right]+2\sum_{p_{1}<p_{2}\leqslant
N}E\left[\mathscr{C}_{p_{1}}\mathscr{C}_{p_{2}}\right].}$ (2.10)

Furthermore, $E\left[\mathscr{C}_{p}^{2}\right]$ can be written as

$\sum_{q\in
T_{p}}E\left[\left(Y_{p}Y_{q}-E[Y_{p}Y_{q}]\right)^{2}\right]+2\sum_{q_{1}<q_{2}\in
T_{p}}E\left[\left(Y_{p}Y_{q_{1}}-E[Y_{p}Y_{q_{1}}]\right)\left(Y_{p}Y_{q_{2}}-E[Y_{p}Y_{q_{2}}]\right)\right],$

which can be seen to be

$\ll\sum_{q\in T_{p}}\frac{1}{pq}+\sum_{q_{1}<q_{2}\in
T_{p}}\frac{1}{pq_{1}q_{2}}$

by Lemma 2.3. Alluding to $\sum_{q\in T_{p}}1/q\ll 1$ shows that

${\sum_{p\leqslant N}E\left[\mathscr{C}_{p}^{2}\right]\ll\sum_{p\leqslant
N}\frac{1}{p}\ll\log\log N.}$ (2.11)

Let us now observe that if $p_{2}>p_{1}^{2\exp(z)}$ then $T_{p_{1}}\cap
T_{p_{2}}=\varnothing$, therefore $\mathscr{C}_{p_{1}}$ and
$\mathscr{C}_{p_{2}}$ are independent. Since for every $p$ we have
$E[\mathscr{C}_{p}]=0$ by definition, we get
$E\left[\mathscr{C}_{p_{1}}\mathscr{C}_{p_{2}}\right]=\prod_{i=1}^{2}E\left[\mathscr{C}_{p_{i}}\right]=0$.
This means that

${\sum_{p_{1}<p_{2}\leqslant
N}E\left[\mathscr{C}_{p_{1}}\mathscr{C}_{p_{2}}\right]=\sum_{\begin{subarray}{c}p_{1}<p_{2}\leqslant
N\\\ p_{2}\leqslant
p_{1}^{2\exp(z)}\end{subarray}}\sum_{\begin{subarray}{c}q_{1}\in T_{p_{1}}\\\
q_{2}\in
T_{p_{2}}\end{subarray}}E\left[\left(Y_{p_{1}}Y_{q_{1}}-E[Y_{p_{1}}Y_{q_{1}}]\right)\left(Y_{p_{2}}Y_{q_{2}}-E[Y_{p_{2}}Y_{q_{2}}]\right)\right].}$
(2.12)

By Lemma 2.3 the contribution of the cases where
$p_{2}\notin\\{p_{2},q_{2}\\}$ is

$\ll\sum_{p_{1}\leqslant N}\frac{1}{p_{1}}\sum_{p_{1}<p_{2}\leqslant
p_{1}^{2\exp(z)}}\frac{1}{p_{2}}\prod_{i=1,2}\left(\sum_{q\in
T_{p_{i}}}\frac{1}{q}\right)\ll\sum_{p_{1}\leqslant
N}\frac{1}{p_{1}}\ll\log\log N,$

because the sums over $p_{2},q_{1},q_{2}$ are bounded in terms of $z$ by
Mertens’ theorem. The cases where $q_{1}\in\\{p_{2},q_{2}\\}$ are similarly
bounded by

$\ll\sum_{p_{1}\leqslant N}\frac{1}{p_{1}}\sum_{p_{1}<p_{2}\leqslant
p_{1}^{2\exp(z)}}\frac{1}{p_{2}}\sum_{q_{2}\in
T_{p_{2}}}\frac{1}{q_{2}}\ll\sum_{p_{1}\leqslant N}\frac{1}{p_{1}}\ll\log\log
N.$

This shows that the quantity in (2.12) is $\ll\log\log N$, which, when
combined with (2.11), can be fed into (2.10) to yield
$\Psi_{3}^{2}s_{N}^{4}\ll\log\log N$. Invoking (2.4) provides us with
$\Psi_{3}\ll 1/\sqrt{\log\log N}$. Together with (2.8)-(2.9) it implies that

$\left|P\left[S_{N}\leqslant
c_{N}+bs_{N}\right]-\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{b}\mathrm{e}^{-t^{2}/2}\mathrm{d}t\right|\ll_{z}(\log\log
N)^{-1/2}$

owing to Stein’s bound (2.7). Finally, letting $N\to\infty$ shows that
$(S_{N}-c_{N})/s_{N}$ converges in law to the standard normal distribution. ∎

######  Remark 2.6.

We next prove asymptotics for the moments of $(S_{N}-c_{N})/s_{N}$. This is
possibly the central proof in the present paper. The argument is a
modification of the one by Billingsley [1, Equations (28)-(29)], which relies
on a version of the dominated convergence theorem. However, the underlying
random variables are now dependent, thus, we need to introduce the notion of
linked indices.

######  Proposition 2.7.

Fix $z\geqslant 0$ and a positive integer $r$. Then we have

$\lim_{N\to\infty}E\left[\left(\frac{S_{N}-c_{N}}{s_{N}}\right)^{r}\right]=\mu_{r},$

where $\mu_{r}$ is the $r$-th moment of the standard normal distribution.

###### Proof.

Take $2k$ to be the least strictly positive integer with $r<2k$, so that
Proposition 2.5 with [13, Example 2.21] implies that it suffices to prove that

$\sup_{N\geqslant
1}\left|E\left[\left(\frac{S_{N}-c_{N}}{s_{N}}\right)^{2k}\right]\right|$

is bounded only in terms of $k$ and $z$. Equivalently, by (2.4) it suffices to
show

$E\left[\left(S_{N}-c_{N}\right)^{2k}\right]=E\left[\left(\sum_{p\leqslant
N}(B_{p}-E[B_{p}])\right)^{2k}\right]\ll_{k,z}(\log\log N)^{k}.$

The left side equals

$\sum_{u=1}^{2k}\sum_{\begin{subarray}{c}\mathbf{r}\in\mathbb{N}^{u}\\\
2k=\sum_{i=1}^{u}r_{i}\end{subarray}}\frac{(2k)!}{r_{1}!\cdots
r_{u}!}\sum_{p_{1}<\ldots<p_{u}\leqslant
N}E\left[\prod_{i=1}^{u}(B_{p}-E[B_{p}])^{r_{i}}\right].$

Using Lemma 2.3 we see that the contribution of the terms with $u\leqslant k$
is

$\ll_{k}\max_{1\leqslant u\leqslant k}\left(\sum_{p\leqslant
N}\frac{1}{p}\right)^{u}\ll_{k}(\log\log N)^{k}.$

Therefore,

${E\left[\left(S_{N}-c_{N}\right)^{2k}\right]\ll\max_{\begin{subarray}{c}u\in[k+1,2k]\\\
\mathbf{r}\in\mathbb{N}^{u}:=\sum_{i}r_{i}=2k\end{subarray}}\sum_{p_{1}<\ldots<p_{u}\leqslant
N}E\left[\prod_{i=1}^{u}(B_{p}-E[B_{p}])^{r_{i}}\right]+(\log\log N)^{k},}$
(2.13)

with an implied constant that is independent of $N$.

For given $u\in\mathbb{N}$, $z\geqslant 0$ and primes $p_{1}<\ldots<p_{u}$ we
say that two consecutive integers $i,i+1$ in $[1,u]$ are linked if and only if
$p_{i+1}\leqslant p_{i}^{\exp(z)}$. In particular, $p_{i+1}$ lies in a
relatively small interval, hence, its contribution will be small. Denote the
number of linked pairs $(i,i+1)$ by $\ell(\mathbf{p})$. By Lemma 2.3 we obtain

$\sum_{p_{1}<\ldots<p_{u}\leqslant
N}E\left[\prod_{i=1}^{u}(B_{p}-E[B_{p}])^{r_{i}}\right]\ll_{z}\left(\sum_{p\leqslant
N}\frac{1}{p}\right)^{u-\ell(\mathbf{p})}\ll(\log\log
N)^{u-\ell(\mathbf{p})},$

where we used the estimate
$\sum_{p_{i}<p_{i+1}<p_{i}^{\exp(z)}}1/p_{i}\ll_{z}1$ whenever $i$ and $i+1$
are linked. Hence, the contribution of all prime vectors
$(p_{1},\ldots,p_{u})$ with at least $\ell(\mathbf{p})\geqslant u-k$ linked
pairs is at most

$\ll(\log\log N)^{u-\ell(\mathbf{p})}\ll(\log\log N)^{k},$

which is acceptable. By (2.13) we obtain

${E\left[\left(S_{N}-c_{N}\right)^{2k}\right]\ll\max_{\begin{subarray}{c}u\in[k+1,2k]\\\
\mathbf{r}\in\mathbb{N}^{u}:=\sum_{i}r_{i}=2k\end{subarray}}\
\sum_{\begin{subarray}{c}p_{1}<\ldots<p_{u}\leqslant N\\\
\ell(\mathbf{p})<u-k\end{subarray}}E\left[\prod_{i=1}^{u}(B_{p}-E[B_{p}])^{r_{i}}\right]+(\log\log
N)^{k},}$ (2.14)

We will now show that every sum over $p_{i}$ in (2.14) vanishes. Denoting the
cardinality of $1\leqslant i\leqslant u$ with $r_{i}=1$ by $a$ we see that the
number of $i$ with $r_{i}\geqslant 2$ is $u-a$. Since $2k=\sum_{i=1}^{u}r_{i}$
we get $2k\geqslant a+2(u-a)$. Equivalently, $2(u-k)\leqslant a$, hence, by
$\ell(\mathbf{p})<u-k$ one gets

${2\ell(\mathbf{p})<\sharp\\{i\in[1,u]:r_{i}=1\\}.}$ (2.15)

We now partition the integers in $[1,u]$ into disjoint subsets
$\mathscr{A}_{1},\ldots,\mathscr{A}_{r}$ using the following rules:

  * •

if $i$ and $i+1$ are in $S_{j}$ then they are linked,

  * •

if $i\in S_{a}$ and $i+1\in S_{b}$ for some $a\neq b$ then $i$ and $i+1$ are
not linked.

The inequality $s\leqslant 2(-1+s)$ for $s\geqslant 2$ gives

$\sharp\\{i\in[1,u]:i\textrm{ linked to some
index}\\}=\sum_{\begin{subarray}{c}1\leqslant j\leqslant r\\\
2\leqslant\sharp\mathscr{A}_{j}\end{subarray}}\sharp\mathscr{A}_{j}\leqslant\sum_{\begin{subarray}{c}1\leqslant
j\leqslant r\\\
2\leqslant\sharp\mathscr{A}_{j}\end{subarray}}2(-1+\sharp\mathscr{A}_{j}).$

This equals $2\ell(\mathbf{p})$ since each $\mathscr{A}_{j}$ has
$-1+\sharp\mathscr{A}_{j}$ linked pairs and the total number of links is
$\ell(\mathbf{p})$. By (2.15) we infer that there exists an index $j$ for
which $r_{j}=1$ and that is not linked to any other index. This implies that
the following random variables are independent:

$\prod_{\begin{subarray}{c}1\leqslant i\leqslant u\\\ i\neq
j\end{subarray}}(B_{p_{i}}-E[B_{p_{i}}])^{r_{i}}\ \ \ \text{ and }\ \ \
(B_{p_{j}}-E[B_{p_{j}}])^{r_{j}}=B_{p_{j}}-E[B_{p_{j}}].$

Since $E\left[B_{p_{j}}-E[B_{p_{j}}]\right]=0$ we infer that every expectation
in the right hand side of (2.14) vanishes. This concludes the proof. ∎

###  2.3. Justifying the model

Let $n$ be a positive integer and denote by $\Omega_{n}$ the uniform
probability space $\mathbb{N}\cap[1,n]$. Our goal now becomes to show that, as
$n\to\infty$, the moments of $\omega_{z}(m)$ for $m$ in $\Omega_{n}$ are
asymptotically the same as the moments of $S_{N}$ for some parameter
$N=N(n)\to\infty$. Recall (2.1). For technical reasons we will first work with
a truncated version of $\omega_{z}$, namely,

${\omega_{z,N}(m)=\sum_{p\leqslant N}\delta_{p,z}(m),}$ (2.16)

where $N=N(n)$. The function $\delta_{p,z}$ imposes simultaneous coprimality
conditions of $m$ with several primes in large intervals and to deal with this
we shall need the Fundamental Lemma of Sieve Theory [9, Corollary 6.10].

######  Lemma 2.8 (Fundamental Lemma of Sieve Theory).

Let $\mathscr{P}$ be a set of primes. Given any sequence $a_{m}\geqslant 0$
for $m\in\mathbb{N}$ and any square-free $d\leqslant x$ that is only divisible
by primes in $\mathscr{P}$ we assume that

$\sum_{m\leqslant x\atop m\equiv 0\left(\textnormal{mod}\
d\right)}a_{m}=Xg(d)+r_{d}$

for some real numbers $X,r_{d}$ and a multiplicative function $g$. Assume that
$0\leqslant g(p)<1$ and that there exist constants $K>1,\kappa>0$ such that

$\prod_{w\leqslant p<y\atop p\in\mathscr{P}}(1-g(p))^{-1}\leqslant
K\left(\frac{\log y}{\log w}\right)^{\kappa}$

holds for all $2\leqslant w<y$. Then for all $D\geqslant y\geqslant 2$ we have

${\sum_{m\leqslant x\atop p\in\mathscr{P},p<y\Rightarrow p\nmid
m}a_{m}=X\left(\prod_{p<y\atop
p\in\mathscr{P}}(1-g(p))\right)\\{1+O(\mathrm{e}^{-s})\\}+O\left(\sum_{d<D\atop
p\mid d\Rightarrow p\in\mathscr{P}}\mu^{2}(d)|r_{d}|\right),}$ (2.17)

where $s=\log D/\log y$ and the implied constants depend at most on $\kappa$
and $K$.

######  Lemma 2.9.

Assume that there exists a function $N:[1,\infty)\to[1,\infty)$ satisfying

$\displaystyle\lim_{n\to\infty}N(n)=+\infty,$ (2.18)
$\displaystyle\limsup_{n\to\infty}\frac{(\log N(n))(\log\log\log N(n))}{\log
n}\neq+\infty.$ (2.19)

Fix $z\geqslant 0$ and $k\in\mathbb{N}$. Then we have

$\lim_{n\to\infty}E_{m\in\Omega_{n}}\left[\left(\frac{\omega_{z,N}-c_{N}}{s_{N}}\right)^{k}\right]=\mu_{k},$

where $\mu_{k}$ is the $k$-th moment of the standard normal distribution.

###### Proof.

By Proposition 2.7 and (2.18) it is sufficient to prove

${\lim_{n\to\infty}\left(E_{m\in\Omega_{n}}\left[\left(\frac{\omega_{z,N}(m)-c_{N}}{s_{N}}\right)^{k}\right]-E\left[\left(\frac{S_{N}-c_{N}}{s_{N}}\right)^{k}\right]\right)=0.}$
(2.20)

Let $r\in\mathbb{N}$. By (2.16), the fact that $\delta_{p,z}\in\\{0,1\\}$ and
the binomial theorem we obtain

${E_{m\in\Omega_{n}}\left[\omega_{z,N}(m)^{r}\right]=\sum_{u=1}^{r}\sum_{\begin{subarray}{c}r_{1},\ldots,r_{u}\in\mathbb{N}\\\
r_{1}+\ldots+r_{u}=r\end{subarray}}\frac{r!}{r_{1}!\cdots
r_{u}!}\sum_{p_{1}<\cdots<p_{u}\leqslant
N}E_{m\in\Omega_{n}}\left[\delta_{p_{1},z}(m)\cdots\delta_{p_{u},z}(m)\right].}$
(2.21)

Let $\mathscr{P}$ be the set of all primes in
$\bigcup_{i=1}^{u}(p_{i},p_{i}^{\exp(z)}]$ and let $a_{m}$ be the indicator
function of integers divisible by $p_{1}\cdots p_{u}$. In particular,

$E_{m\in\Omega_{n}}\left[\delta_{p_{1},z}(m)\cdots\delta_{p_{u},z}(m)\right]=\frac{1}{n}\sum_{\begin{subarray}{c}1\leqslant
m\leqslant n\\\ p\in\mathscr{P}\Rightarrow p\nmid m\end{subarray}}a_{m}.$

We assume that $p_{i+1}>p_{i}^{\exp(z)}$ for all $i=1,2,\ldots,u-1$ since
otherwise the sum clearly vanishes. We will now use Lemma 2.8 with
$X=n/(p_{1}\cdots p_{u}),g(d)=1/d,D=\sqrt{n},y=N^{2\exp(z)}.$ If $d$ is
divisible only by primes in $\mathscr{P}$ then it is coprime to $p_{1}\cdots
p_{u}$, hence,

$\sum_{m\leqslant n\atop m\equiv 0\left(\textnormal{mod}\
d\right)}a_{m}=\left[\frac{n}{p_{1}\cdots p_{u}d}\right],$

thus, $|r_{d}|\leqslant 1$ because $r_{d}$ is the fractional part of $X/d$.
Furthermore, we can take $K$ to be any large fixed positive constant and
$\kappa=1$, owing to

$\prod_{w\leqslant p<y\atop p\in\mathscr{P}}(1-g(p))^{-1}=\prod_{w\leqslant
p<y\atop p\in\mathscr{P}}(1-1/p)^{-1}\leqslant\prod_{w\leqslant
p<y}(1-1/p)^{-1}\ll\frac{\log y}{\log w}.$

The bound $|r_{d}|\leqslant 1$, means that $\sum_{d\leqslant
D}\mu^{2}(d)|r_{d}|\leqslant D=\sqrt{n}$. Since $p_{u}\leqslant N$, every
prime in $\mathscr{P}$ is strictly smaller than $y$, hence, (2.17) gives

${E_{m\in\Omega_{n}}\left[\delta_{p_{1}}\cdots\delta_{p_{u}}\right]=\left\\{\prod_{i=1}^{u}\frac{1}{p_{i}}\prod_{p_{i}<q\leqslant
p_{i}^{\exp(z)}}\left(1-1/p\right)\right\\}\left\\{1+O\left(\mathrm{e}^{-\frac{\log
n}{4\exp(z)\log N}}\right)\right\\}+O(n^{-1/2}),}$ (2.22)

where the implied constant depends at most on $r$ and $z$.

By the binomial theorem we get

$E\left[S_{N}^{r}\right]=E\left[\left(\sum_{p\leqslant
N}B_{p}\right)^{r}\right]=\sum_{u=1}^{r}\sum_{\begin{subarray}{c}r_{1},\ldots,r_{u}\in\mathbb{N}\\\
r_{1}+\ldots+r_{u}=r\end{subarray}}\frac{r!}{r_{1}!\cdots
r_{u}!}\sum_{p_{1}<\cdots<p_{u}\leqslant N}E\left[B_{p_{1}}\cdots
B_{p_{u}}\right]$

and we note that we can restrict the sum over $p_{i}$ to the terms with
$p_{i+1}>p_{i}^{\exp(z)}$ for all $i$, since otherwise $E\left[B_{p_{1}}\cdots
B_{p_{u}}\right]=0$. Under this restriction the random variables $B_{p_{i}}$
are independent, hence,

$\prod_{i=1}^{u}\frac{1}{p_{i}}\prod_{p_{i}<q\leqslant
p_{i}^{\exp(z)}}\left(1-1/p\right)=E\left[B_{p_{1}}\cdots B_{p_{u}}\right].$

We infer from (2.21) and (2.22) that

$\left|E_{m\in\Omega_{n}}\left[\omega_{z,N}(m)^{r}\right]-E\left[S_{N}^{r}\right]\right|\ll_{r}E\left[S_{N}^{r}\right]\mathrm{e}^{-\frac{\log
n}{4\exp(z)\log N}}+n^{-1/2}\sum_{u=1}^{r}\sum_{p_{1}<\ldots<p_{u}\leqslant
N}1.$

By (2.3) this is $\ll(\log\log N)^{r}\exp(-\frac{\log n}{4\exp(z)\log
N})+n^{-1/2}N^{r}$. Thus, the difference in (2.20) is

$\displaystyle\ll$ $\displaystyle s_{N}^{-k}\sum_{r=0}^{k}{k\choose
r}(-c_{N})^{k-r}\left(E_{\Omega_{n}}\left[\omega_{z,N}^{r}\right]-E\left[S_{N}^{r}\right]\right)$
$\displaystyle\ll$ $\displaystyle s_{N}^{-k}\left\\{\mathrm{e}^{-\frac{\log
n}{4\exp(z)\log N}}(c_{N}+\log\log N)^{k}+n^{-1/2}(N+c_{N})^{k}\right\\}.$

We need to show that this vanishes asymptotically and by (2.4) and (2.18) it
suffices to show

$(2\log\log N)^{k}\leqslant\exp\left(\frac{\log n}{4\exp(z)\log N}\right)\ \
\textrm{ and }\ \ (2N)^{k}\leqslant n^{1/2}.$

Both of these inequalities can be directly inferred from (2.18)-(2.19). ∎

######  Lemma 2.10.

Assume that there exists a function $N:[1,\infty)\to[1,\infty)$ satisfying

$\displaystyle\lim_{n\to\infty}\frac{\log\log N(n)}{\log\log n}=1,$ (2.23)
$\displaystyle\limsup_{n\to\infty}\frac{(\log N(n))\sqrt{\log\log n}}{\log
n}=+\infty.$ (2.24)

Fix $z\geqslant 0$. Then
${s_{N}}((1-2z\mathrm{e}^{-z}){\mathrm{e}}^{-z}\log\log n)^{-1/2}\to 1$ as
$n\to\infty$ and

$\lim_{n\to\infty}\frac{\max\left\\{\left|(\omega_{z}-\mathrm{e}^{-z}\log\log
n)-(\omega_{z,N}-c_{N})\right|:m\in\mathbb{N}\cap[1,n]\right\\}}{\sqrt{\log\log
n}}=0.$

###### Proof.

Combining (2.4) and (2.23) one immediately gets

$\lim_{n\to\infty}\frac{s_{N}}{((1-2z\mathrm{e}^{-z}){\mathrm{e}}^{-z}\log\log
n)^{1/2}}=1.$

For any $m\in[1,n]$ we have

$\left|(\omega_{z}-\mathrm{e}^{-z}\log\log
n)-(\omega_{z,N}-c_{N})\right|\leqslant\sum_{p>N}\delta_{p,z}(m)+|\mathrm{e}^{-z}(\log\log
n)-c_{N}|.$

Since $\delta_{p,z}$ takes only values in $\\{0,1\\}$ and $\delta_{p,z}(m)=1$
implies that $p$ divides $m$,we see that

$\sum_{p>N}\delta_{p,z}(m)\leqslant\sharp\\{p\mid m:p>N\\}\leqslant\frac{\log
m}{\log N}\leqslant\frac{\log n}{\log N}.$

Furthermore, (2.3) gives

$\mathrm{e}^{-z}(\log\log n)-c_{N}\ll 1+\log\frac{\log n}{\log N}\ll\frac{\log
n}{\log N}.$

The proof now concludes by using (2.24). ∎

###  2.4. Proof of Theorem 1.1

The function

$N(n):=n^{1/\log\log n}$

fulfills (2.18)- (2.19)-(2.23)-(2.24), hence, we can apply Lemmas 2.9-2.10.

For any $r\in\mathbb{N}$, $c\in\mathbb{C}$ any probability space $\Omega_{n}$
and any two sequences of random variables $X_{n},Y_{n}$ satisfying
$\lim_{n\to\infty}\sup_{m\in\Omega_{n}}|X_{n}(m)-Y_{n}(m)|=0$ and
$\lim_{n\to\infty}E_{m\in\Omega_{n}}[X_{n}(m)^{r}]=c$ it is easy to see by the
binomial theorem that $\lim_{n\to\infty}E_{m\in\Omega_{n}}[Y_{n}(m)^{r}]=c$.
Using this with $\Omega_{n}=\mathbb{N}\cap[1,n]$,

$X_{n}(m)=\frac{\omega_{z,N}(m)-c_{N}}{s_{N}}\ \ \textrm{ and }\ \
Y_{n}(m)=\frac{\omega_{z}(m)-\mathrm{e}^{-z}\log\log n}{s_{N}},$

in combination with Lemmas 2.9-2.10, shows that for every $k\in\mathbb{N}$ one
has

$\lim_{n\to\infty}E_{m\in\Omega_{n}}\left[\left(\frac{\omega_{z}(m)-\mathrm{e}^{-z}\log\log
n}{s_{N}}\right)^{k}\right]=\mu_{k}.$

Given any sequence $a_{n}\in\mathbb{R}$ having limit $1$ and any sequence of
random variables $X_{n}$ with $E[X_{n}]$ having limit $c$ it is clear that
$a_{n}E[X_{n}]$ has limit $c$. Using this with

$a_{n}=\frac{s_{N}}{((1-2z\mathrm{e}^{-z}){\mathrm{e}}^{-z}\log\log n)^{1/2}}\
\ \textrm{ and }\ \ X_{n}(m)=\frac{\omega_{z}(m)-\mathrm{e}^{-z}\log\log
n}{s_{N}}$

and invoking Lemma 2.10 shows that for every $k\in\mathbb{N}$ one has

${\lim_{n\to\infty}E_{m\in\Omega_{n}}\left[\left(\frac{\omega_{z}(m)-\mathrm{e}^{-z}\log\log
n}{((1-2z\mathrm{e}^{-z}){\mathrm{e}}^{-z}\log\log
n)^{1/2}}\right)^{k}\right]=\mu_{k}.}$ (2.25)

This proves Theorem 1.1 whenever $r$ is a positive integer and this is
sufficient. To see that, take any $r\in[0,\infty)$ and note that (2.25)
implies that

$T_{n}=\frac{\omega_{z}(m)-\mathrm{e}^{-z}\log\log
n}{((1-2z\mathrm{e}^{-z}){\mathrm{e}}^{-z}\log\log n)^{1/2}}$

converges in law to the standard normal distribution. Taking $p$ to be the
least even integer strictly exceeding $r$ in [13, Example 2.21] shows that the
$r$-th moment of $T_{n}$ converges to the $r$-th moment of the standard normal
distribution. ∎

## References

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&=  \sum_{\gamma \leq \alpha}{\alpha \choose \gamma} \eta^{(\gamma)}(z_n\varepsilon_j) \frac{(\varepsilon_j z_n)^{j-\alpha+\gamma}}{(j-\alpha+\gamma)!}\varepsilon_j^{\alpha-j}\del_{z'}^\beta f_j({z'})\,. \label{estimatel}
\end{align}

Note that the support properties of $\eta$ imply that $\abs{\eta^{(\gamma)}(z_n\varepsilon_j)(z_n\varepsilon_j)^{j-\alpha+\gamma}} < 2^{-(j-\alpha+\gamma)}$, and if we choose $\varepsilon_j$ (depending on $f_j$) in such a way that

\[
\abs{\sum_{\gamma\leq \alpha}{\alpha\choose\gamma} \varepsilon_j^{\alpha-j}\del_{z'}^\beta f_j({z'})} \leq 2^{-\alpha+\gamma}\,,
\]

then $\abs{\del_{z'}^\beta\del_{z_n}^\alpha \eta_j({z_n})f_j({z'})} \leq 2^{-j}$ for all $\alpha + \abs{\beta} < j$.

This means that if we define

\[
t(z',{z_n}) \coloneqq \sum_{j\geq 0}\eta_j({z_n})f_j({z'})\,,
\]

\[
\abs{\del_{z'}^\beta\del_{z_n}^\alpha l({z'},{z_n})} \leq \sum_{j\leq \alpha +\abs{\beta}}\abs{\del_{z'}^\beta\del_{z_n}^\alpha\eta_j({z_n})f_j({z'})} + \sum_{\alpha + \abs{\beta} < j}\abs{\del_{z'}^\beta\del_{z_n}^\alpha\eta_j({z_n})f_j({z'})}
\]

where the first is a finite sum and the second is bounded by $2$.

In conclusion this means that the series defining $t$ is absolutely convergent for every derivative.

Furthermore, from <ref> we can calculate to see that

\[
\del_{z_n}^j \eta_j(0) f_j({z'}) = f_j({z'}) \qquad \forall j\in\mathbb{N}\,.
\]

The conditions we desire for $t$ imply we take $f_0({z'}) = \tau_\Xi(\phi+i\psi)$, and $f_1({z'}) =-\tau_\Xi \del_{z_n}(\phi+i\psi)$.
Manually calculating, we have that

\begin{equation}\label{eq:nablatnablat}
\begin{aligned}
\nabla t \cdot \nabla t &= \sum_{w\geq 0} \sum_{j+k=w}\frac{1}{j!k!}\left[\eta({z_n}\varepsilon_j)\eta(\varepsilon_k)\nabla_{z'} f_j({z'})\nabla_{z'} f_k({z'})\right. \\
&+ \left.\left(\varepsilon_j\eta'({z_n}\varepsilon_j)f_j({z'})+\eta({z_n}\varepsilon_{j+1})f_{j+1}({z'})\right)\left(\varepsilon_k\eta'({z_n}\varepsilon_k)f_k({z'})+\eta({z_n}\varepsilon_{k+1})f_{k+1}({z'})\right)\right]
\end{aligned}
\end{equation}

which we want to be equal to $0$ approximately.

For $w = 0$ and $\abs{{z_n}} \leq (4\varepsilon_1)^{-1}$ the part of <ref> pertaining to $w=0$ is equal to

\[
\nabla_{z'} f_0 \nabla_{z'} f_0 + f_1({z'})f_1({z'}) = \nabla(\phi+i\psi)\cdot\nabla(\phi+i\psi) = 0\,.
\]

For $w\geq 1$ and $\abs{{z_n}} \leq (4\varepsilon_{w+1})^{-1}$ and $j+k = w$, we want to solve for

\[
0 = \sum_{j+k=w}\frac{1}{j!k!}\left(\nabla_{z'} f_j({z'})\nabla_{z'} f_k({z'}) + f_{j+1}({z'})f_{k+1}({z'})\right) = \frac{1}{w!}f_1({z'})f_w({z'}) + R_{j,k < w}
\]

where by $R_{j,k< w}$ we denote some term that only depends on $f_j$ with $j < w$. Thus if the $f_j$ have been defined recursively we must only solve for $f_w$, which we may do because the coefficient in front of it, being $f_1({z'})$, is bounded away from $0$.

This means that for any $w\in\mathbb{N}$ and $\abs{{z_n}} \leq (4\varepsilon_w)^{-1}$ we have that $t = \mathcal{O}(z_n^{w+1})$. Furthermore, <ref> is implied by <ref> which concludes the proof.

By direct computation we have that

\begin{align*}
e^{-t/h}h^2\Delta e^{\frac{t+ih\psi_h}{h}}b &= -e^{i\psi_h} \left(\nabla (t +ih\psi_h)\nabla(t+i\psi_h)b - h(\Delta (t+ih\psi_h))b+2 h\nabla(t+ih\psi_h)\nabla b - h^2\Delta b\right) \\
\shortintertext{where $\abs{\omega'_h} \leq Ch$ (see \cref{eq:defofomega}), and $\nabla t \cdot \nabla t = d(y,\Xi)^\infty$ by \cref{eq:constructlprops},  which means that}
e^{-t/h}h^2\Delta e^{\frac{t+ih\psi_h}{h}}b &= -e^{i\psi_h}\left(hb (\Delta t + 2i \xi\cdot \nabla t) + 2h\nabla b\cdot \nabla t + \mathcal{O}(h^2) + d(y,\Xi)^\infty\right)
\end{align*}

and so rephrasing <ref>, we want to approximately solve

\begin{equation}\label{eq:wantforb}
b(\Delta t + 2i\xi\cdot \nabla t) + 2\nabla b\cdot \nabla t = 0\,, \an \tau_\Xi b = 1\,.
\end{equation}

We will use the same recipe as in the above proof. If, after change of variables as above, we take $b({z'},{z_n}) = \sum_{j\geq 0}\eta_j({z_n})f_j({z'})$, we will want $f_0 = 1$ by the boundary condition we impose on $b$ and the first part of <ref> is rephrased as

\[
0 = \sum_{j\geq 0} \frac{z_n^j}{j!}\left[\eta({z_n}\varepsilon_j)(\Delta t+2i\xi\cdot\nabla t)f_j({z'}) + (\varepsilon_j\eta'({z_n}\varepsilon_j)f_j({z'})+\eta({z_n}\varepsilon_{j+1})f_{j+1}({z'}))\del_{z_n} t + \eta({z_n}\varepsilon_j)\nabla_{z'} f_j({z'})\nabla_{z'} t\right].
\]

For $\abs{{z_n}} \leq (4\varepsilon_1)^{-1}$ and $j=0$ we have the equation

\[
(\Delta t + 2i\xi \nabla t)f_0 + f_1({z'})\del_{z_n} t + \nabla_{z'} f_0 \nabla_{z'} t = 0\,,
\]

where $\del_{z_n} t({z'},0) = -\tau_\Xi\del_\nu(\phi+i\psi)$. Thus, for ${z_n}$ near $0$ this is bounded away from $0$ and so we may divide by it to get some $f_1 \in \mathcal{O}_{L^\infty}(1)$ with respect to $h$.

For every other $j \geq 1$, and having solved recursively for $f_1\dots,f_{j}$, we must solve an equation of the form

\[
(\Delta t + 2i\xi \nabla t)f_j + f_{j+1}({z'})\del_{z_n} t + \nabla_{z'} f_j \nabla_{z'} t = 0\,.
\]

This is possible because for small enough $z_n$, $\del_{z_n} t$ is non-zero. This recursive definition done, for $z_n$ small enough, $b$ is equal to $\mathcal{O}(z_n^{w+1})$, and to get the support properties of $b$, we cut it off with a smooth function.

§.§ Properties of The Partial Cauchy Data Set

The proof here is standard and follows <cit.>.

Fix $v\in H^{1/2}(\del\Omega)$ and assume $u \in C^\infty(\bar\Omega)$. By the standard extension theorem there is some $e_v \in H^1(\Omega)$ so that $e_v\vert_{\partial\Omega} = v$ and

\begin{equation}\label{eq:extensionbounded}
\norm{e_v}_{H^1(\Omega)} \leq C\norm{v}_{H^{1/2}(\partial\Omega)}\,.
\end{equation}

\[
w(v) \coloneqq \int_{\del\Omega} \bar v\del_\nu u\dd S\,,
\]

where $S$ is the surface measure and remark that by partial integration,

\[
w(v) = \int_\Omega \nabla u \nabla \bar e_v - \bar e_v \Delta u\dd x
\]

so that by Sobolev embedding $H^1(\Omega) \subset L^p(\Omega)$ and Hölder's inequality,

\begin{equation}\label{eq:boundedw}
\abs{w(v)} \leq C\norm{\nabla u}_{L^2}\norm{\nabla e_v}_{L^2} + \norm{e_v}_{L^p}\norm{\Delta u}_{L^{p'}} \leq C\norm{u}_{H^\sharp_\Delta}\norm{v}_{H^{1/2}(\del\Omega)}\,,
\end{equation}

where the second inequality in <ref> used Poincaré's inequality and <ref>. Note also that the definition of $w$ is independent of the choice of $e_v$.

Thus <ref> lets us conclude that $w \in H^{-1/2}(\del\Omega)$ and $C^\infty(\bar\Omega) \ni u \mapsto w \in H^{-1/2}(\del\Omega)$ is bounded in the $H^\sharp_\Delta(\Omega)$ norm. Using the density of $C^\infty(\bar \Omega)$ in $H^\sharp_\Delta(\Omega)$ concludes the proof.

§.§ Seminorm Bound on Symbols

Let $j \in S^{-\infty}_1(\RR^{n-1})$ be a symbol satisfying

\begin{equation}\label{eq:goodsymbol}
\abs{\del^{\alpha}_{x'}\del^\beta_{\xi'}j(x',\xi')} \leq C_{\alpha,\beta}\,,\an \supp j \subset \RR^{n-1}\times K\,,
\end{equation}

for some compact $K\subset \RR^{n-1}$ and some $C_{\alpha,\beta}\geq 0$.

Let $a \in S^{-\infty}_1(\RR^{n-1})$ be as in <ref>.
We have $a(x',hD) \colon L^r(\RR^{n-1}) \to W^{k,r}(\RR^{n-1})$ for all $k\in\mathbb{N}_0$, $1<r<\infty$ with the bound

\begin{equation*}
\norm{a(x',hD) u}_{W^{k,r}} \leq C_nC_K\max_{\abs{\beta}\leq 2n} C_{0,\beta} \norm{u}_{L^r}
\end{equation*}

where $C_{\alpha,\beta}$ are the seminorm bounds for the symbol $\langle \xi'\rangle^k a(x',\xi')$, $C_n$ depends only on the dimension $n$ and $C_K$ depends only on the measure of $K$.

Assume that $u\in \Ss(\RR^{n-1})$ and fix $1<r<\infty, k\in \mathbb{N}_0.$

Remark that by partial integration $a(x',hD')$ has the integral kernel

\begin{equation}\label{eq:kernel}
k_a(x',y') = (2\pi)^{-n+1}\int_{K} e^{i(x'-y')\cdot\xi'}(1+\Delta_{\xi'})^n a(x',h\xi')\dd\xi' (1+\abs{x'-y'}^2)^{-n}
\end{equation}

which satisfies the bounds

\begin{equation}\label{eq:boundonkernel}
\abs{k_a(x',y')} \leq \max_{\abs{\beta}\leq 2n} C_{0,\beta} C_n C_K(1+\abs{x'-y'}^2)^{-n}\,,
\end{equation}

where $C_K = \abs{K}$ is the measure of $K$, and $C_n = (2\pi)^{-n+1}$.

Using <ref>, we find that

\begin{equation}\label{eq:constskernel}
\int_{\RR^{n-1}} \abs{k_a(x,y)} \dd x' \leq C_n' C_K \max_{\abs{\beta}\leq 2n} C_{0,\beta}\,,\an \int_{\RR^{n-1}} \abs{k_a(x,y)} \dd y' \leq C_n' C_K \max_{\abs{\beta}\leq 2n} C_{0,\beta}
\end{equation}

where $C_n' = C_n \int_{\RR^{n-1}} (1+\abs{x'-y'}^2)^{-n}\dd x'$.

By <cit.> and <ref> we have

\begin{equation}
\norm{a(x',hD')u(x')}_{L^r} \leq C_n' C_K \max_{\abs{\beta}\leq 2n} C_{0,\beta} \norm{u}_{L^r}\,.
\end{equation}

Finally, if we replace the symbol $a$ by $\langle \xi'\rangle^{k} a(x',\xi')$, then the same arguments with different seminorm bounds $C_{\alpha,\beta}'$ will give

\begin{equation}
\norm{a(x',hD')u(x')}_{W^{k,r}} = \norm{\langle hD'\rangle^{k} a(x',hD')u(x')}_{L^r} \leq C_n' C_K \max_{\abs{\beta}\leq 2n} C_{0,\beta}' \norm{u}_{L^r}\,.
\end{equation}

where we used the known fact that Bessel potential spaces correspond to Sobolev spaces for integer regularity exponents. Using the density of $\Ss$ in $L^r$ concludes the proof.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

CRC Press

titleElectrical Impedance Tomography

journaltitleJ. Inverse Ill-Posed Probl.
titleRecovering a potential from Cauchy data in the two-dimensional case

journaltitleCommun. Partial Differ. Equations
titleRecovering a potential from partial Cauchy data

Soc. Brasil. Mat., Rio de Janeiro

booktitleSeminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980)
titleOn an inverse boundary value problem

titleA Partial Data Result for the Magnetic Schrodinger Inverse Problem

journaltitleJ. Fourier Anal. Appl.
titlePartial data for the Neumann-to-Dirichlet map

journaltitleTrans. Am. Math. Soc., Ser. B
titlePartial data inverse problem with \(L^{n/2}\) potentials

Springer, Cham

seriesGrundlehren Math. Wiss.
titleElliptic partial differential equations of second order

journaltitleDuke Math. J.
titleCalderón inverse problem with partial data on Riemann surfaces

Springer-Verlag, Berlin

noteDistribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]
seriesClassics in Mathematics
titleThe analysis of linear partial differential operators. I

titleInverse Boundary Value Problem by Partial data for the Neumann-to-Dirichlet-map in two dimensions

journaltitleJ. Am. Math. Soc.
titleThe Calderón problem with partial data in two dimensions

journaltitleAnn. Math. (2)
titleUnique continuation and absence of positive eigenvalues for Schrödinger operators. (With an appendix by E. M. Stein)

Providence, RI: American Mathematical Society (AMS)

booktitleInverse problems and applications. Proceedings of two conferences in honour of Gunther Uhlmann's 60th birthday, Irvine, CA, USA, June 18–22, 2012 and Hangzhou, China, September 17–21, 2012
isbn978-1-4704-1079-7; 978-1-4704-1659-1
titleRecent progress in the Calderón problem with partial data

journaltitleCommun. Pure Appl. Math.
titleDetermining conductivity by boundary measurements

journaltitleAnn. Math. (2)
titleReconstructions from boundary measurements

journaltitleCommun. Partial Differ. Equations
titleSemiclassical pseudodifferential calculus and the reconstruction of a magnetic field

journaltitleAnn. Math. (2)
titleA global uniqueness theorem for an inverse boundary value problem

New York, NY: Springer

edition2nd ed.
isbn978-1-4419-7051-0; 978-1-4419-7052-7
seriesAppl. Math. Sci.
titlePartial differential equations. II: Qualitative studies of linear equations

journaltitleInverse Probl.
noteId/No 123011
titleElectrical impedance tomography and Calderón's problem

American Mathematical Society, Providence, RI

seriesGraduate Studies in Mathematics
titleSemiclassical analysis
get arXiv to do 4 passes: Label(s) may have changed. Rerun


# Irreversiblity in Bacterial Turbulence: Insights from the Mean-Bacterial-
Velocity Model

Kolluru Venkata Kiran<EMAIL_ADDRESS>Centre for Condensed Matter
Theory, Department of Physics, Indian Institute of Science, Bangalore, 560012,
India.  Anupam Gupta<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology (IIT), Hyderabad, Kandi Sangareddy Telangana, 502285,
India  Akhilesh Kumar Verma<EMAIL_ADDRESS>Mathematics Institute,
Zeeman Building, University of Warwick, Coventry CV4 7AL, UK  Rahul Pandit
<EMAIL_ADDRESS>Centre for Condensed Matter Theory,Department of Physics,
Indian Institute of Science, Bangalore, 560012, India.

###### Abstract

We use the mean-bacterial-velocity model to investigate the irreversibility of
two-dimensional (2D) bacterial turbulence and to compare it with its 2D fluid-
turbulence counterpart. We carry out extensive direct numerical simulations of
Lagrangian tracer particles that are advected by the velocity field in this
model. Our work uncovers an important, qualitative way in which
irreversibility in bacterial turbulence is different from its fluid-turbulence
counterpart: For large positive (or large but negative) values of the friction
(or activity) parameter, the probability distribution functions of energy
increments, along tracer trajectories, or the power are positively skewed; so
irreversibility in bacterial turbulence can lead, on average, to particles
gaining energy faster than they lose it, which is the exact opposite of what
is observed for tracers in 2D fluid turbulence.

## I

Most fluid flows are turbulent; and they can attain a nonequilibrium, but
statistically steady, state (NESS), if the energy injection into the fluid,
say by an external force, is balanced by viscous dissipation. Far away from
boundaries, this NESS is statistically homogeneous and isotropic if we
consider length scales that are much smaller than the energy-injection scale
$l_{f}$[1, 2]. Two important characteristics of this NESS are (a) the
distribution of energy over a large range of length scales and (b) the
temporal irreversibility of turbulent flows. This irreversibility is not
easily apparent if we look at movies, played forward or backward in time, of
Lagrangian particles, or tracers, that are advected by turbulent flows;
however, the statistics of such tracers or inertial particles in turbulent
flows yields signatures of this irreversibility [3, 4, 5, 6, 7, 8, 9, 10]: if
we analyse (a) the increments

$W(t,\tau)\equiv E(t+\tau)-E(t)$ (1)

of the particle energy $E$ at time $t$ or (b) the power

$p_{L}(t)\equiv\frac{dE}{dt}=a_{L}v_{L},$ (2)

with $v_{L}$ the magnitude of the tracer velocity and $a_{L}$ the component of
its acceleration along its trajectory. It has been found that probability
distribution functions (PDFs) of $W$ and $p_{L}$, obtained by averaging over
$t$ and the trajectories of all tracers, are negatively skewed [7, 3, 9, 11,
12, 13, 14]; i.e., on average, such tracers lose energy faster than they gain
it. Is it possible to use these PDFs to characterise irreversibility in
bacterial turbulence [15, 16, 17, 18, 19, 20, 21, 22]? We show that this can,
indeed, be done. We illustrate this by carrying out an extensive study of the
irreversibility of bacterial turbulence in the mean-bacterial-velocity model
[15]. Our work uncovers an important, qualitative way in which irreversibility
in bacterial turbulence is different from its fluid-turbulence counterpart:
For large positive (or large but negative) values of the friction (or
activity) parameter $\alpha$ (see below), the PDFs of $W(\tau)$ or $p_{L}$ are
positively skewed; this implies that irreversibility in bacterial turbulence
can lead, on average, to particles gaining energy faster than they lose it,
for certain ranges of values of $\alpha$.

Dense bacterial suspensions, which are examples of active systems [15, 16, 20,
23, 24, 25], show spatiotemporal evolution that is reminiscent of flows in
turbulent fluids. Hydrodynamical models have been developed to describe
turbulence in dense, quasi-two-dimensional (2D) bacterial suspensions [17, 18,
19, 20, 21, 24, 26, 27]. We use the mean-bacterial-velocity model [15] or the
Toner-Tu-Swift-Hohenberg (TTSH) model [25, 28], for the incompressible
velocity field $\mbox{\boldmath$u$}(\mathbf{x},t)$; this model has been
employed to study turbulence in dense suspensions of B. subtilis:

$\displaystyle\frac{\partial\mbox{\boldmath$u$}}{\partial
t}+\lambda_{0}\mbox{\boldmath$u$}.\mathbf{\nabla}\mbox{\boldmath$u$}$
$\displaystyle=$ $\displaystyle-\nabla
P-(\alpha+\beta|u|^{2})\mbox{\boldmath$u$}$ $\displaystyle+$
$\displaystyle\Gamma_{0}\nabla^{2}\mbox{\boldmath$u$}-\Gamma_{2}\nabla^{4}\mbox{\boldmath$u$};$
$\displaystyle\nabla\cdot\mbox{\boldmath$u$}$ $\displaystyle=$ $\displaystyle
0.$ (3)

Here, $P(\mathbf{x},t)$ is the pressure at point $\mathbf{x}$ and time $t$;
the constant density $\rho$ is set to unity 111 Equation (I) is not Galilean
invariant; it reduces to the Navier-Stokes equation with friction for
$\Gamma_{0}>0,\alpha>0,\Gamma_{2}=0,\lambda_{0}=1$, and $\beta=0$.. We use
periodic boundary conditions in all directions because we concentrate on
statistically homogeneous and isotropic bacterial turbulence. We restrict
ourselves to two dimensions (2D) as most experiments in this field have been
conducted in quasi-2D systems.

The parameters $\Gamma_{0}<0$ and $\Gamma_{2}<0$; a spatial Fourier transform
of Eq. (3), followed by a linear-stability analysis about the spatially
uniform state, yields the wave vectors $\mathbf{k}$, with magnitude $k$, for
which there are linearly unstable modes. We define the following
characteristic length, velocity, and time scales, respectively:

$\Lambda=2\pi\sqrt{\frac{2\Gamma_{2}}{\Gamma_{0}}};\,v_{0}=\sqrt{\frac{|\Gamma_{0}|^{3}}{\Gamma_{2}}};\,\theta=\frac{\Lambda}{v_{0}}.$
(4)

These unstable modes provide a source of energy injection into the system
222This is similar to energy injection in the Kuramoto-Sivashinsky equation
see, e.g., Refs. [52, 53, 54].; this energy is dissipated by (a) the linearly
stable modes, (b) the cubic term with the coefficient $\beta>0$, and (c) the
linear term with the coefficient $\alpha$, if $\alpha>0$. Moreover, there is
energy injection, or activity, if $\alpha<0$; and $\Gamma_{0}<0$ and
$\lambda_{0}\neq 1$ also induce activity [28] [$\lambda_{0}>1$ for pusher
swimmers like B. subtilis (see, e.g., Refs. [15, 16, 31])]. The interplay
between these energy-injection and dissipation terms leads to a NESS with
self-sustained, turbulence-type patterns [22]. The effective viscosity

${}k^{2}\nu_{eff}(k)=\big{(}\alpha+2\beta
u^{2}_{rms}+\Gamma_{0}k^{2}+\Gamma_{2}k^{4}\big{)}$ (5)

can be used to rewrite Eq. (3) in a Navier-Stokes form (see the Supplemental
Material Supplemental Material and Ref. [22]). Clearly, the wave numbers $k$
at which energy is injected (dissipated) are those with
$\nu_{eff}(k)<0~{}(>0)$; the root-mean-square velocity $u_{rms}$ must be
obtained from a calculation (see below).

We solve Eq. (3) by a pseudospectral direct numerical simulation (DNS) [see,
e.g., Ref. [32]] with $N^{2}=1024^{2}$ collocation points and the parameters
in Table 1; we have checked in representative cases that our results are
unchanged if we use $N^{2}=2048^{2}$ collocation points. We hold
$\lambda_{0}$, $\beta$, and $\Gamma_{0}$ fixed, and we tune the activity
principally by varying $\alpha$.

Run | $\Gamma_{2}$ | $\alpha$ | $\delta t$ | $\Lambda$ | $\theta$ | $v_{0}$
---|---|---|---|---|---|---
A1-A12 | 9e-5 | $\ast$ | 2e-4 | 0.40 | 0.40 | 1.0
B | 9e-5 | -4 | 1e-5 | 0.40 | 0.40 | 1.0
C | 9e-5 | 1 | 5e-5 | 0.40 | 0.40 | 1.0
D | 3.6e-5 | 14 | 1e-4 | 0.25 | 0.16 | 1.60
Table 1:  Parameters for our DNSs:∗ Runs A1-A12 use
$\alpha=-4,\,-3.5,\,-2.5,\,-1.5,\,-1,\,-0.5,\,0,\,1,\,3,\,3.5,\,4,$ and $5$,
respectively. For all the Runs listed in the table, $\Gamma_{0}=-0.045$,
$\beta=0.5$, and $\lambda_{0}=3.5$.

(a)(b)(c)(d)(e)(f)(g)(h)(i)

Figure 1: (Color online) Plots for runs A1, A8, and D (Table 1): (a)-(c)
Filled contour plots of the vorticity
$\mbox{\boldmath$\omega$}(\mathbf{x},t))$, with some tracers (black points),
at a representative time in the NESS; log-log (base 10) plots versus
$k\Lambda$ of (d)-(f) the energy spectrum $\mathcal{E}(k)$ and (g)-(i) the
energy flux $\Pi(k)$ [Eq. (6)]; the gray-shaded areas indicate the ranges of
$k$ for which $\nu_{eff}(k)<0$ [Eq. (5)].

(a)(b)(c)(d)(e)(f)

Figure 2: (Color online) Plots versus $\alpha$ of (a) $u_{rms}/v_{0}$, (b)
$L_{I}/\Lambda$, (c) $T_{I}/\theta$, (d) $Re_{L_{I}}$, (e) the skewness
$\mathcal{P}_{Sk}$, and (f) $\mathcal{W}_{Sk}(\tau)$ for $\tau/\theta=0.025$;
in (e) and (f), blue and pink shading indicate, respectively, ranges of
$\alpha$ in which the skewnesses are positive and negative.

(a)(b)(c)

Figure 3: (Color online) Plots for run B: (a) Semi-log plot of the normalized
PDFs (a) $\mathcal{P}(p_{L})$ and (b) $\mathcal{P}(W(\tau))$, with
$\tau/\theta$ going from $0.025,0.08,0.13,0.25,0.38,$ to $0.50$, as we move
from the outermost to the innermost curve; in (a) negative values of $p_{L}$
(dashed) are reflected about the vertical axis to highlight the asymmetry of
$\mathcal{P}(p_{L})$. $(c)$ Log-Log (base 10) plot versus $\tau/\theta$ of the
skewness $\mathcal{W}_{Sk}(\tau)$. Inset: for the same range of $\tau/\theta$,
a log-log plot versus $\tau/\theta$ of $\langle W^{3}(\tau)\rangle/\langle
E\rangle^{3}$; the dashed black line is a fit to $\langle
W^{3}(\tau)\rangle/\langle E\rangle^{3}\sim(\tau/\theta)^{3}$.

(a)(b)(c)$p_{L}$

Figure 4:  (Color Online) Plot of the normalised energy $E(t)/\langle
E\rangle$ versus the normalized time $t/\theta$ along a representative
particle trajectory from Run A1; the colors along the trajectory indicate the
value of $p_{L}$.

In Figs. 1 (a), (b), and (c), we present filled contour plots of the vorticity
$\mbox{\boldmath$\omega$}(\mathbf{x},t)=\mathbf{\nabla}\times\mbox{\boldmath$u$}(\mathbf{x},t)$,
with some tracers shown via black points, for the representative Runs A1, A8,
and D, respectively (see Table 1); in Figs. 1 (d)-(i), we give log-log plots
versus $k\Lambda$ of the $k$-shell-averaged energy spectrum $\mathcal{E}(k)$
and energy flux $\Pi(k)$:

$\displaystyle\mathcal{E}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{k^{\prime}=k-1/2}^{k^{\prime}=k+1/2}\langle\widetilde{\mbox{\boldmath$u$}}(\mathbf{k}^{{}^{\prime}}).\widetilde{\mbox{\boldmath$u$}}(-\mathbf{k}^{{}^{\prime}})\rangle_{t};$
$\displaystyle\Pi(k)$ $\displaystyle=$
$\displaystyle-\lambda_{0}\sum_{k^{\prime}=0}^{k^{\prime}=k}\sum_{k^{\prime\prime}=k^{\prime}-1/2}^{k^{\prime\prime}=k^{\prime}+1/2}[\langle\widetilde{\mbox{\boldmath$u$}}(-\mathbf{k}^{{}^{\prime\prime}})\cdot\textbf{P}(\mathbf{k^{\prime\prime}})\cdot$
(6) $\displaystyle($
$\displaystyle\widetilde{\mbox{\boldmath$u$}.\nabla\mbox{\boldmath$u$})}({\mathbf{k^{\prime\prime}})}\rangle_{t}];$

here, tildes denote spatial Fourier transforms, $\langle\cdot\rangle_{t}$ is
the time average over the NESS, and the transverse projector
$\textbf{P}(\mathbf{k})$ has the components
$P_{ij}(\mathbf{k})=\delta_{ij}-\frac{k_{i}k_{j}}{k^{2}}$. The total fluid
energy, root-mean-square velocity, integral length scale, integral time scale,
and integral-scale Reynolds number are, respectively,

$\displaystyle\mathcal{E}_{T}$ $\displaystyle=$
$\displaystyle\sum_{k}\mathcal{E}(k);~{}u_{rms}=\sqrt{2\mathcal{E}_{T}};L_{I}=\frac{\sum_{k}[\mathcal{E}(k)/k]}{\sum_{k}\mathcal{E}(k)};$
$\displaystyle T_{I}$ $\displaystyle=$ $\displaystyle
L_{I}/u_{rms};\;Re_{L_{I}}\equiv u_{rms}L^{3}_{I}/\Gamma_{2}.$ (7)

The gray-shaded areas in Figs. 1 (d)-(i) indicate the ranges of $k$ for which
$\nu_{eff}(k)<0$. For the Runs in Table 1, there is no range of $k$ over which
$\Pi(k)$ remains constant, unlike its fluid-turbulence counterpart, so we
cannot identify inverse- or forward-cascade regimes in $\mathcal{E}(k)$;
however, $\mathcal{E}(k)$ is spread over a large range of $k$ and the temporal
evolution of $u$ is chaotic, so the bacterial-turbulence NESS for this model
[Eq. (3)] displays spatiotemporal chaos. In Figs. 2 (a)-(d) we present plots
versus $\alpha$ of $u_{rms}/v_{0},\,L_{I}/\Lambda,\,T_{I}/\theta,$ and
$Re_{L_{I}}$, respectively (Runs A1-A12); as $\alpha$ moves from the activity
regime ($\alpha<0$) to the frictional regime ($\alpha>0$),
$u_{rms}/v_{0},\,L_{I}/\Lambda,$ and $Re_{L_{I}}$ decrease, but $T_{I}/\theta$
first decreases and then increases because $u_{rms}$ decreases more rapidly
than $L_{I}$.

The velocity $\mathbf{v}_{L}(t)$ of a tracer at $\mathbf{x}_{L}(t)$ is

$\frac{d\mathbf{x}_{L}(t)}{dt}\equiv\mathbf{v}_{L}(t)=\mbox{\boldmath$u$}(\mathbf{x}_{L}(t),t).$
(8)

We track $N_{p}=10,000$ tracers, employ the second-order Runge-Kutta method
for time marching, and evaluate $\mbox{\boldmath$u$}(\mathbf{x}_{L}(t),t)$ at
off-grid points via bilinear interpolation [33, 34, 35]; to get good
statistics, we use very long runs ($3\times 10^{6}$ time steps per particle).
The acceleration of a tracer particle is

$\displaystyle\mathbf{a}(\mathbf{x}_{L},t)$ $\displaystyle\equiv$
$\displaystyle\frac{\partial\mbox{\boldmath$u$}}{\partial
t}+(\mbox{\boldmath$u$}.\nabla)\mbox{\boldmath$u$}\bigg{|}_{x_{L}(t)}$
$\displaystyle=$ $\displaystyle-\nabla
P_{eff}-(1-\lambda_{0})(\mbox{\boldmath$u$}\times\mbox{\boldmath$\omega$})$
$\displaystyle-(\alpha+\beta|u|^{2})\mbox{\boldmath$u$}$ $\displaystyle+$
$\displaystyle\Gamma_{0}\nabla^{2}\mbox{\boldmath$u$}-\Gamma_{2}\nabla^{4}\mbox{\boldmath$u$}\bigg{|}_{x_{L}(t)},$
(9)

where the effective pressure
$P_{eff}=P-\frac{1}{2}(1-\lambda_{0})\mbox{\boldmath$u$}.\mbox{\boldmath$u$}$;
the component of this acceleration along the tracer’s trajectory yields
$a_{L}$, whence we get $p_{L}$ [Eq. (2)] and its normalized PDF
$\mathcal{P}\big{(}\frac{p_{L}}{\langle p^{2}_{L}\rangle^{1/2}}\big{)}$. From
the time series of particle energies (Fig. 4) we obtain the energy-increment
PDFs $\mathcal{P}\big{(}\frac{W(\tau)}{\langle
W^{2}(\tau)\rangle^{1/2}}\big{)}$, for various values of $\tau<T_{I}$. Both
these PDFs have zero mean (Figs. 3 (a) and (b)), because we are considering a
statistically steady state in which the mean energy input is balanced by
dissipation, but they are asymmetrical; we characterize this asymmetry by
computing the skewnesses

$\mathcal{P}_{Sk}=\frac{\langle p^{3}_{L}\rangle}{\langle
p^{2}_{L}\rangle^{\frac{3}{2}}}\;{\rm{and}}~{}~{}\mathcal{W}_{Sk}(\tau)=\frac{\langle
W^{3}(\tau)\rangle}{\langle W^{2}(\tau)\rangle^{\frac{3}{2}}},$ (10)

which we plot versus $\alpha$ in Figs. 2(e) and (f), respectively. We observe
that $\mathcal{P}_{Sk}>0$ in the large-activity, $\alpha<-2$, and extreme-
friction, $\alpha>3.5$, regions (shaded blue). This is in stark contrast to 2D
fluid turbulence [3] where $\mathcal{P}_{Sk}<0$. The values of $\alpha$ for
which $\mathcal{P}_{Sk}<0$ lead to NESSs that are characterized by flight-
crash events in which, on average, $E(t)$ builds up slowly but decays
rapidly.[In 2D Faraday-wave experiments, $\mathcal{P}_{Sk}>0$ has been
attributed to the temporal coherence of these waves, and been removed by
filtering 333For conventional 2D fluid turbulence, Refs. [3, 51] discuss, for
both DNSs and experiments, the effects of different types of forcing on
$\mathcal{P}_{Sk}$; they report $\mathcal{P}_{Sk}<0$ in DNSs with white-noise
forcing; by contrast, in Faraday-wave experiments, they observe
$\mathcal{P}_{Sk}>0$, which they attribute to the temporal coherence of
Faraday waves. In the latter case they employ a filtering procedure that again
yields $\mathcal{P}_{Sk}<0$. In the Supplemental Material Supplemental
Material we investigate the effects of a similar filtering procedure for 2D
bacterial turbulence in Eq. (3)..] For runs B and D we also find
$\mathcal{P}_{Sk}>0$. In Fig. 4 (a) we plot the time series of $E/\langle
E\rangle$ of a typical particle. We also show magnified regions of this time
series to exhibit flight-crash events [Fig. 4 (c)], of the types that are
predominant in fluid turbulence, and the events in which $E(t)$ builds up
faster than it decays [Fig. 4 (b)]. In the large-activity and extreme-friction
regions mentioned above, the predominance of the events shown in Fig. 4 (b)
leads to $\mathcal{P}_{Sk}>0$ and, for small $\tau/\theta$,
$\mathcal{W}_{Sk}(\tau)=\frac{\langle W^{3}(\tau)\rangle}{\langle
W^{2}(\tau)\rangle^{\frac{3}{2}}}>0$, because $\lim_{\tau\to 0}W(\tau,t)\sim
p_{L}(t)$. Furthermore, for $\tau/\theta\ll 1$, we obtain the Taylor-expansion
result $\langle W^{3}(\tau)\rangle\sim\tau^{3}$, for which we give a
representative plot in the inset of Fig. 3 (c). As $\tau$ decreases, the tails
of $\mathcal{P}\big{(}\frac{W(\tau)}{\langle W^{2}(\tau)\rangle^{1/2}}\big{)}$
widen, as in fluid turbulence [7]444 This widening could be a signature of
intermittency effects, which we examine elsewhere [55]..

The sign of $\mathcal{P}_{Sk}$ (and, for small $\tau/\theta,$ the sign of
$\mathcal{W}_{Sk}(\tau)$) displays the following correlation with the scale-
by-scale energy budget in Fourier space, where we can identify the
$k$-dependence of the energy contributions from the terms with coefficients
$\alpha$, $\Gamma_{0}$, and $\lambda_{0}$ in Eq. (3). The contributions of the
first two terms dominate over those of the third term when
$\mathcal{P}_{Sk}>0$, as we show in detail in the Supplemental Material
Supplemental Material.

(a)(b)(c)

Figure 5: (Color online) (a) Semi-log plots of $\mathcal{P}(Q_{L})$ for runs
A1 (blue) and A13 (green). Inset gives the plot versus $\alpha$ of skewness,
$\mathcal{Q}_{Sk}$, for $\mathcal{P}(Q_{L})$. (b) Log-log plot of
$\mathcal{C}(Q^{+}_{L})$ for run A1; the shaded region shows a power-law and
the solid black line gives the fit
$\mathcal{C}(Q^{+}_{L})\sim[Q^{+}_{L}]^{-\vartheta}$, with $\vartheta=0.37\pm
0.04$. (c) Plots versus $\alpha$ of $\mathcal{P}_{Sk}$ for the conditioned
PDFs (see text) $\mathcal{P}(p_{L}|Q^{+}_{L})$ (violet) and
$\mathcal{P}(p_{L}|Q^{-}_{L})$ (maroon).

In 2D incompressible flows, the Okubo-Weiss parameter [38, 39, 40, 32]
distinguishes between vortical and strain-dominated regions. We define this,
along particle trajectories, as follows:

$Q_{L}(t)=\frac{\omega^{2}-\sigma^{2}}{4}\bigg{|}_{x_{L}(t)},$ (11)

where
$\omega^{2}=\frac{1}{2}\sum_{i,j}(\partial_{i}u_{j}-\partial_{j}u_{i})^{2}$
and
$\sigma^{2}=\frac{1}{2}\sum_{i,j}(\partial_{i}u_{j}+\partial_{j}u_{i})^{2}$,
with $i,j=1,2$. $Q_{L}>0~{}(\equiv Q^{+}_{L})$, in vortical regions, and
$Q_{L}<0~{}(\equiv Q^{-}_{L})$, in strain-dominated regions. The PDF
$\mathcal{P}\big{(}\frac{Q_{L}}{\langle Q^{2}_{L}\rangle^{1/2}}\big{)}$, is
positively skewed; its skewness $\mathcal{Q}_{sk}$ decreases with increasing
$\alpha$, but remains positive throughout the range of $\alpha$ for Runs
A1-A12 (inset of Fig. 5 (a)). For high activities, the cumulative PDF
$\mathcal{C}(Q^{+}_{L})$, shows a power-law tail for $Q^{+}_{L}$ (Fig. 5 (b)),
a unique feature of the bacterial turbulence we study 555Similar PDFs have
been obtained in Ref. [35], but the power-law form has not been noted.; in
contrast, for the high-friction regime ($\alpha>2$), the tail of
$\mathcal{P}(Q_{L})$ has a faster-than-exponential decay, for small and
positive values of $Q_{L}$, as in 2D fluid turbulence. Furthermore, in the
large-activity regime $\alpha\leq-2$, the positivity of $\mathcal{P}_{Sk}$
arises from vortical regions, whereas, in the high-friction regime $\alpha\geq
2$, this positive skewness comes from the strain-dominated regions, which we
surmise from Fig. 5(c), where we plot $\mathcal{P}_{Sk}$ for the conditioned
PDFs $\mathcal{P}(p_{L}|Q^{+}_{L})$ and $\mathcal{P}(p_{L}|Q^{-}_{L})$.
Quasi-2D experiments on dense suspension of aerobic bacteria, e.g., B.
subtilis, show that the average speed of bacterial flow increases with the
oxygen concentration [42, 43, 44]. We can increase the activity by making
$\alpha$ large and negative; in experiments, the activity can be increased by
enhancing the oxygen, because the polar-ordered velocity scale
$v_{p}=\sqrt{\frac{|\alpha|}{\beta}}$ is a measure of the swimming speed of
bacteria; $u_{rms}\propto\alpha$ (cf. [45]); and in the frictional or
$\alpha>0$ regime, the value of $\alpha$ can be tuned in experiments by
changing the bottom friction or the air-drag-induced friction (see the
Supplemental Material Supplemental Material for details). Therefore,
experiments on dense bacterial suspensions should be able to examine
irreversibility in bacterial turbulence as a function of the activity as we
have done in Fig. 2.

It is important to use the methods we describe here to explore irreversibility
of bacterial turbulence in other models [19, 24, 23] and also in models for
active fluids [46, 47] and active nematics [26, 27, 48]. We propose to carry
out such studies in the near future.

## Supplemental Material

In this Supplementary Material we provide details of the following:

  1. 1.

Our direct numerical simulations (DNSs).

  2. 2.

Different contributions to the energy spectrum and the role of the advective
term.

  3. 3.

The effects of filtering on the statistics of the power $p_{L}$.

  4. 4.

Additional figures and tables.

### Direct Numerical Simulation (DNS)

For two dimensional (2D) incompressible flows we rewrite Eq. 3, in the main
paper, in terms of the vorticity field
$\mbox{\boldmath$\omega$}(\textbf{x},t)$ as follows:

$\displaystyle\frac{\partial{\mbox{\boldmath$\omega$}}}{\partial
t}+\lambda_{0}\mbox{\boldmath$u$}\cdot\mathbf{\nabla}\mbox{\boldmath$\omega$}$
$\displaystyle=-\alpha{\mbox{\boldmath$\omega$}}-\beta\mathbf{\nabla}\times(|\mbox{\boldmath$u$}|^{2}\mbox{\boldmath$u$})$
$\displaystyle+\Gamma_{0}\nabla^{2}{\mbox{\boldmath$\omega$}}-\Gamma_{2}\nabla^{4}{\mbox{\boldmath$\omega$}};$
(12)

and we define the stream function $\Psi(\textbf{x},t)$ in terms of which

$\mbox{\boldmath$u$}=\mathbf{\nabla}\times\Psi\hat{\textbf{z}}\;\;\rm{and}\;\;\mbox{\boldmath$\omega$}=-\nabla^{2}\Psi,$
(13)

where $\hat{\textbf{z}}$ is the unit vector perpendicular to the plane
containing $u$. Our DNS of Eqs. (12) and (13) employs a pseudospectral method
[49, 32]; we use a square simulation domain, with sides of length $L=2\pi$,
periodic boundary conditions in all directions, and $N\times N$ collocation
points distributed uniformly over this domain. In most of our simulations we
use $N=1024$; we have checked in representative simulations that our results
are not altered significantly if we use $N=2048$. We employ the second-order
Runge-Kutta integrating-factor method IFRK2 for time marching [50]. We have
developed a CUDA Fortran code for our DNSs which are excuted on K20 and K80
GPUs. Once our DNS yields a turbulent, but statistically steady, state, we
introduce $N_{p}$ Lagrangian particles and follow their trajectories. The
position $\mathbf{x}_{L}(t)$ of such a particle evolves as follows:

$\frac{d\mathbf{x}_{L}(t)}{dt}\equiv\mathbf{v}_{L}(t)=\mbox{\boldmath$u$}(\mathbf{x}_{L}(t),t).$
(14)

We integrate Eq. (14) by using the second-order Runge-Kutta scheme and bi-
linear interpolation to obtain the particle velocity $\mathbf{v}_{L}(t)$ at
off-grid points. We evolve each particle trajectory for $3\times 10^{6}$ time
steps; and we store $\mathbf{x}_{L}$, $\mathbf{v}_{L}$, the particle
acceleration $\mathbf{a}_{L}$, and the power $p_{L}$ after every $50$
iterations.

We follow Refs. [45, 15] in restricting our model parameters to experimentally
realizable regimes. The average velocities observed in experiments on B.
subtilis are $\simeq 25~{}\mu m/s$, at normal oxygen concentrations; the
typical viewing area is $400\mu m\times 400\mu m$; we map these to the
constant velocity scale $v_{0}$ and the simulation box area $L\times L$,
respectively. This gives us the scale factors of $25/v_{0}$ and $4\times
10^{-2}/L$ for mapping velocities and lengths, respectively, in our DNS to
their experimental counterparts: specifically, $v_{0}$, $v_{p}$, and $u_{rms}$
are, respectively, $\simeq 25~{}\mu m/s$, $\simeq(25~{}\mu
m/s\;\rm{to}\;~{}70~{}\mu m/s)$ and $\simeq(4.5~{}\mu m/s\;\rm{to}\;65~{}\mu
m/s)$; similarly, $\Lambda$, which sets the linear scale for vortical regions,
$\simeq 25\mu m$.

### Energy budget

For the shell-averaged energy spectrum

$\mathcal{E}(k)=\frac{1}{2}\sum_{k^{\prime}=k-1/2}^{k^{\prime}=k+1/2}\langle\widetilde{\mbox{\boldmath$u$}}(\mathbf{k}^{{}^{\prime}}).\widetilde{\mbox{\boldmath$u$}}(-\mathbf{k}^{{}^{\prime}})\rangle_{t}$
(15)

we have [22]

$\displaystyle{}\partial_{t}\mathcal{E}(k)$ $\displaystyle=$ $\displaystyle
T^{a}(k)-T^{c}(k)-2\alpha\mathcal{E}(k)$ (16) $\displaystyle-$ $\displaystyle
2\Gamma_{0}k^{2}\mathcal{E}(k)-2\Gamma_{2}k^{4}\mathcal{E}(k),$

where $T^{a}(k)$ and $T^{c}(k)$, the $k$-shell averaged contributions from the
advective and cubic terms in Eq. (12), respectively, are

$\displaystyle{}T^{a}(k)$
$\displaystyle=-\lambda_{0}\sum_{k^{\prime}=k-1/2}^{k^{\prime}=k+1/2}\langle\widetilde{\mbox{\boldmath$u$}}(-\mathbf{k}^{{}^{\prime}}).\textbf{P}(\mathbf{k^{\prime}}).\widetilde{(\mbox{\boldmath$u$}.\mathbf{\nabla}\mbox{\boldmath$u$})}(\mathbf{k^{\prime}})\rangle_{t}$
$\displaystyle T^{c}(k)$
$\displaystyle=\beta\sum_{k^{\prime}=k-1/2}^{k^{\prime}=k+1/2}\langle\widetilde{\mbox{\boldmath$u$}}(-\mathbf{k}^{{}^{\prime}}).\textbf{P}(\mathbf{k^{\prime}})\widetilde{(|\mbox{\boldmath$u$}|^{2}\mbox{\boldmath$u$})}(\mathbf{k^{\prime}})\rangle_{t},$
(17)

with $P_{ij}(\mathbf{k})=\delta_{ij}-\frac{k_{i}k_{j}}{k^{2}}$ the transverse
projector and $\langle\cdot\rangle_{t}$ the average over time $t$. The flux of
energy arising from the advective term is

$\Pi(k)=-\sum_{k^{\prime}=0}^{k^{\prime}=k}T^{a}(k^{\prime}).$ (18)

The effective viscosity

$k^{2}\nu_{eff}(k)=\big{(}\alpha+2\beta
u^{2}_{rms}+\Gamma_{0}k^{2}+\Gamma_{2}k^{4}\big{)}$ (19)

can be used to rewrite Eq. (12), in a form that resembles the Navier-Stokes,
with the constant viscosity $\nu$ replaced by $\nu_{eff}(k)$. To obtain Eq.
(5), we use the approximation $T^{c}(k)\simeq-4\beta
u^{2}_{rms}\mathcal{E}(k)$ suggested in Ref. [22]; here, $u_{rms}$ must be
obtained from our calculation. Clearly, the wave numbers $k$ at which energy
is injected (dissipated) are those with $\nu_{eff}(k)<0~{}(>0)$.

(a)(b)(c)(d)

Figure 6: (Color online) Semi-log plots versus $k\Lambda$ of $T^{a}(k)$
(yellow), $-2\alpha\mathcal{E}(k)$ (blue) and
$-2\Gamma_{0}k^{2}\mathcal{E}(k)$ (orange); the gray-shaded areas indicate the
ranges of $k$ for which $\nu_{eff}(k)<0$ for (a) run A1, (b) run A7, (c) run
A8 and (d) run A11

The sign of $\mathcal{P}_{Sk}$ (and, for small $\tau/\theta,$ the sign of
$\mathcal{W}_{Sk}(\tau)$) displays the following correlation with the scale-
by-scale energy budget in Fourier space, where we can identify the
$k$-dependence of the energy contributions from the terms with coefficients
$\alpha$, $\Gamma_{0}$, and $T^{a}(k)$ from Eq. (16) and (Energy budget),
which we show in Fig. (3): the contribution to the energy budget (16) from the
active term, $-2\alpha\mathcal{E}(k)$, is significantly greater than
$T^{a}(k)$, for values of $\alpha<-2$. For values of $\alpha>2$, where
$\mathcal{P}_{Sk}>0$, the other active term, $-2\Gamma_{0}k^{2}\mathcal{E}(k)$
dominates over $T^{a}(k)$.

### Averaging and Filtering

For conventional 2D fluid turbulence, the effects of different types of
forcing, in both DNSs and experiments, on $\mathcal{P}_{Sk}$ have been
discussed in Refs. [3, 51], where it is noted that Faraday-wave experiments
yield $\mathcal{P}_{Sk}>0$; this sign is attributed to the temporal coherence
of Faraday waves. It is shown in Ref. [3] that, if $p_{L}$ is averaged over a
time that is comparable to this coherence time, then $\mathcal{P}_{Sk}<0$;
this averaging filters high-frequency components in $p_{L}(t)$. Specifically,
they use

$\overline{p}_{L}(t)=\frac{1}{\mathcal{T}}\int_{0}^{\mathcal{T}}p_{L}(t+t^{\prime})dt^{\prime};$
(20)

$\mathcal{T}$, the time over which $p_{L}(t)$ is averaged, is taken to be a
multiple (typically $0.5-5$) of the forcing-correlation time. In the mean-
bacterial-velocity model, which we consider, there is no external forcing; the
natural counterpart of $\mathcal{T}$ in Eq. (20) is the particle-acceleration
time $\tau_{a}$ that we can obtain from the first zero-crossing of the
normalised autocorrelation function

$C_{a}(\tau)=\frac{\langle a_{L}(t)a_{L}(t+\tau)\rangle}{\langle
a_{L}(t)a_{L}(t)\rangle},$ (21)

which we plot in the inset of Fig. 7 (a)for different values of $\alpha$. In
Fig. 7 (a) we show that $\tau_{a}$ increase monotonically with $\alpha$. We
define the smoothing parameter

$sm(\tau_{a})\equiv\frac{\mathcal{T}}{\tau_{a}},$ (22)

the multiple of $\tau_{a}$ over which we average $p_{L}(t)$. In Fig. 7 (b) we
plot $\mathcal{P}_{Sk}$ versus $\alpha$ for $sm(\tau_{a})=1$ (solid line) and
$sm(\tau_{a})=4$ (dashed line); for values of $\mathcal{P}_{Sk}>0$ (in both
high-activity and high-friction regimes), the averaging defined in Eq. 20
leads to a change in the sign of $\mathcal{P}_{Sk}$ for values of
$sm(\tau_{a})>1$. This averaging does not change the sign of
$\mathcal{P}_{Sk}$ if the unaveraged $p_{L}(t)$ leads $\mathcal{P}_{Sk}<0$.
This is also evident from the PDFs of the filtered $\overline{p}_{L}$; we
present representative plots for the PDFs for runs A1 and A8 in Figs. 8 (a)
and (b), respectively. From Figs. 7 and 8 we conclude that that the fast-gain-
slow-loss events, shown in Fig. 4 of the main text, are filtered out by the
averaging procedure, which we have described above, because these events occur
over time scales that are $\simeq\tau_{a}$.

(a)(b)

Figure 7: (Color online) (a) Plot versus $\alpha$ of the acceleration-
autocorrelation time $\tau_{a}$; the inset shows $C_{a}(\tau)$ as a function
of $\tau$, for $\alpha=-4$ (blue), $\alpha=1$ (red), and $\alpha=5$ (green).
(b) Plots versus $\alpha$ of $\mathcal{P}_{Sk}$ for $sm(\tau_{a})=1$ (solid
line) and $sm(\tau_{a})=4$ (dashed line).

(a)

(b)

Figure 8: Semi-log plots of the PDFs $\mathcal{P}(\overline{p}_{L})$ for
different values $sm(\tau_{a})$ for (a) Run A1 and (b) Run A8.

### Supplemental Tables and Figures

  * •

In Table 2 we list the parameters for Run D in the main paper.

  * •

In Fig. 9 we plot various moments of $p_{L}$ versus $\alpha$ for Runs A1-A12.

  * •

In Figs. 10 and 11 we present representative plots for Runs C and D.

  * •

In Fig. 12 we give a plot of the local slope in the power-law regime in
$\mathcal{C}(Q^{+}_{L})$ for Run A1.

Run | $u_{rms}$ | $L_{I}$ | $T_{I}$ | $\mathcal{P}_{Sk}$
---|---|---|---|---
D | 8.8e-2 | 4e-2 | 0.45 | 1.2e-2
Table 2:  DNS values of $u_{rms}$, $L_{I}$, $T_{I}$ and $\mathcal{P}_{Sk}$ for
Run D.

(a)

(b)

Figure 9:  For Runs A1-A12 (Table I in the main paper): (a) Absolute values of
the third moment $|\langle p^{3}_{L}\rangle|$ and the second moment $\langle
p^{2}_{L}\rangle$ plotted versus $\alpha$. (b) Plots of the normalized fifth
moment versus $\alpha$; these can also be used to quantify irreversibility and
to draw the same conclusions as we have in the main paper.

(a)

(b)

Figure 10: (Color online) Plots for Run C: $(a)$ Semi-log plot of the
normalized PDF $\mathcal{P}(p_{L})$; negative values of $p_{L}$ (dashed) are
reflected about the vertical axis to highlight the asymmetry of
$\mathcal{P}(p_{L})$. $(b)$ Semi-log plots of $\mathcal{P}(W(\tau))$, for
$\tau/\theta=0.025,0.08,0.13,0.25,0.38$ and $0.50$ (outermost to innermost
curve).

(a)

(b)

Figure 11: (Color online) Log-Log (base 10) plot versus $\tau/\theta$ of the
skewness $\mathcal{W}_{Sk}(\tau)$. Inset: for the same range of $\tau/\theta$,
a log-log plot versus $\tau/\theta$ of $\langle W^{3}(\tau)\rangle/\langle
E\rangle^{3}$; the dashed black line is a fit to $\langle
W^{3}(\tau)\rangle/\langle E\rangle^{3}\sim(\tau/\theta)^{3}$ for (a) Run C
and (b) Run D.

Figure 12: For Run A1, semi-log plot of the local slope $\chi$ of
$\mathcal{C}(Q^{+}_{L})$ in the shaded region that shows a power law.

### Acknowledgments

We thank J.K. Alageshan, V. Deshpande (NVIDIA), N.B. Padhan, S. Shukla, and
S.S.V. Kolluru for discussions, CSIR, the National Supercomputing Mission
(NSM), and SERB (India) for support, and SERC (IISc) for computational
resources.

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