# The Dance of Odd-Diffusive Particles: A Fourier Approach

Amelie Langer  University of Augsburg, Institute of Physics, D-86159 Augsburg,
Germany  Abhinav Sharma  University of Augsburg, Institute of Physics, D-86159
Augsburg, Germany  Leibniz-Institute for Polymer Research, Institute Theory of
Polymers, D-01069 Dresden, Germany  Ralf Metzler  University of Potsdam,
Institute of Physics and Astronomy, D-14476 Potsdam, Germany  Asia Pacific
Centre for Theoretical Physics, KR-37673 Pohang, Republic of Korea  Erik Kalz
<EMAIL_ADDRESS>University of Potsdam, Institute of Physics and
Astronomy, D-14476 Potsdam, Germany

###### Abstract

Odd-diffusive systems are characterized by transverse responses and exhibit
unconventional behaviors in interacting systems. To address the dynamical
interparticle rearrangements in a minimal system, we here exactly solve the
problem of two hard disk-like interacting odd-diffusing particles. We
calculate the probability density function (PDF) of the interacting particles
in the Fourier-Laplace domain and find that oddness rotates all modes except
the zeroth, resembling a “mutual rolling” of interacting odd particles. We
show that only the first Fourier mode of the PDF, the polarization, enters the
calculation of the force autocorrelation function (FACF) for generic systems
with central-force interactions. An analysis of the polarization as a function
of time reveals that the relative rotation angle between interacting particles
overshoots before relaxation, thereby rationalizing the recently observed
oscillating FACF in odd-diffusive systems.

##  I Introduction

The description of dissipative systems with an inherent broken time-reversal
or parity symmetry has recently attracted considerable attention [1, 2, 3, 4,
5]. Relevant systems can be found in various domains of statistical physics,
and include Brownian particles under the effect of Lorentz force [6],
skyrmionic spin structures [7] and active chiral particles [1]. The transport
coefficients of these systems show a characteristic transverse response to
perturbations, which is encoded in antisymmetric off-diagonal elements in
transport tensors. These characteristic elements behave odd under the
transformation of the underlying (broken) symmetry, which serves as the
namesake for these odd systems. Interestingly, a transverse response does not
necessarily imply an anisotropic description of the system. In fact, in two
spatial dimensions, odd transport coefficients represent the most general
description of an isotropic physical system [8]. In this work, we specifically
consider odd-diffusive systems [1, 2], which in two spatial dimensions are
characterized by a diffusion tensor of the form

$\mathbf{D}=D_{0}(\mathbf{1}+\kappa\boldsymbol{\epsilon}),$ (1)

where $D_{0}$ is the bare diffusivity with physical diemensions
$[D_{0}]=\mathrm{m}^{2}/\mathrm{s}$, and $\kappa$ is the characteristic odd-
diffusion parameter with $[\kappa]=1$, encoding the transverse response.
$\mathbf{1}$ is the identity tensor and $\boldsymbol{\epsilon}$ the fully
antisymmetric Levi-Civita symbol in two dimensions
($\epsilon_{xy}=-\epsilon_{yx}=1$ and $\epsilon_{xx}=\epsilon_{yy}=0$).

While the oddness parameter enters explicitly in the diffusion tensor in Eq.
(1), the mean-squared displacement of a freely diffusing particle is
determined only by the symmetric part of the diffusion tensor. However, in a
system of interacting particles, oddness qualitatively alters the diffusive
behavior by affecting the self-diffusion [2, 9]. The self-diffusion measures
the dynamic of a tagged particle in a crowded system and explicitly accounts
for interactions of particles [10]. Independent of the microscopic details of
the inter-particle interactions, the self-diffusion is usually reduced [11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. In odd-diffusive systems, however, it
was recently shown that even purely repulsive interactions can enhance the
self-diffusion [2]. Via expressing the self-diffusion as a time-integral of
the (interaction) force autocorrelation function (FACF) [22], these findings
could further be related to the unusual microscopic particle rearrangements in
odd-diffusive systems [9].

For the overdamped equilibrium system under consideration here and consistent
with the reduction of the self-diffusion [22], autocorrelation functions in
general decay monotonically in time [23]. However, the enhancement of self-
diffusion for odd systems can occur only if the FACF switches sign, i.e.,
becomes non-monotonic in time. This apparent contradiction could be resolved
by recognizing that the time-evolution operator in an odd-diffusive system
becomes non-Hermitian when $\kappa\neq 0$ in Eq. (1), thereby breaking the
monotonicity requirements on the FACF [9]. Surprisingly, in this work, it was
further observed that the FACF even oscillates in time. While this is
consistent with the observed enhancement of the self-diffusion in odd systems,
a physical interpretation of this phenomenon remains elusive.

In the present work, we show that the non-monotonic FACF originates in the
unusual dynamics of interacting odd particles. When a pair of odd-diffusive
particles interacts, their motion resembles that of two mutually rotating
particles [2]. This is despite the fact that the interaction potential is
central-force, i.e., it acts along the vector that connects the centers of two
particles. Our approach is based on the exact analytical derivation of the
propagator of two interacting odd-diffusive particles. The joint probability
density function (PDF) for the two particles separates into a center-of-mass
PDF and a PDF of the relative coordinate, capturing the interplay of odd
diffusion and interactions. We can express the relative PDF in a Fourier
series to find that only certain modes enter the averages of observables, such
as in the FACF. In particular, it is the polarization mode of the relative
PDF, representing the positioning of the particles with respect to each other
in time, which determines the full tensorial force autocorrelation behavior.
By analyzing the polarization mode in time, we understand what originates the
oscillating FACF. Interacting odd-diffusive particles mutually rotate further
in time before they eventually relax. Figuratively they perform a dance,
reminiscent of the classical step of the Viennese waltz.

The remainder of this work is organized as follows: In Section II we set the
problem of two interacting odd-diffusive particles, which is then exactly
solved in Section II.1. The propagator can be put into the form of a Fourier
series, of which the numerical analysis of the modes is presented in Section
II.2. In Section II.3 we show that only certain Fourier modes enter in
averages of observabels, in particular into the FACF. Finally, in Section III
we summarize and give an extensive overview of systems, which can be subsumed
under the terminology of odd diffusion. In Appendix A we present the detailed
solution of the problem of interacting particles and Appendix B lists the
relevant integral relations.

##  II Interacting odd-diffusive particles

We study the dynamics of two odd-diffusive particles at positions
$\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ in two spatial dimensions. The particles
are assumed to interact with the potential energy $U$, which we assume to be
of the hard-disk type

$U(\mathbf{x}_{1},\mathbf{x}_{2})=U(r)=\begin{cases}\infty,&r\leq 1\\\
0,&r>1\end{cases},$ (2)

where $r=|\mathbf{x}_{1}-\mathbf{x}_{2}|/d$ is the rescaled relative distance
between the particles and $d$ is the particle diameter.

The conditional joint PDF, i.e., the propagator for the particles to be found
at positions $(\mathbf{x}_{1},\mathbf{x}_{2})$ at time $t$ given that they
were at positions $(\mathbf{x}_{1,0},\mathbf{x}_{2,0})$ at time $t_{0}$,
$P(t)=P(\mathbf{x}_{1},\mathbf{x}_{2},t|\mathbf{x}_{1,0},\mathbf{x}_{2,0},t_{0})$,
evolves according to the time-evolution equation

$\begin{split}\partialderivative{t}P(t)&=\nabla_{1}\cdot\,[\mathbf{D}\nabla_{1}+\boldsymbol{\mu}\nabla_{1}U(\mathbf{x}_{1},\mathbf{x}_{2})]\,P(t)\\\
&\quad+\nabla_{2}\cdot\,[\mathbf{D}\nabla_{2}+\boldsymbol{\mu}\nabla_{2}U(\mathbf{x}_{1},\mathbf{x}_{2})]\,P(t),\end{split}$
(3)

where the initial condition is given as
$P(t=t_{0})=\delta(\mathbf{x}_{1}-\mathbf{x}_{1,0})\,\delta(\mathbf{x}_{2}-\mathbf{x}_{2,0})$.
In Eq. (3), $\nabla_{1},\nabla_{2}$ are the partial differential operators for
the positions of the particles, and $\mathbf{D}$ is the odd-diffusion tensor
of Eq. (1). $\boldsymbol{\mu}$ is the mobility tensor, and we assume the
fluctuation-dissipation relation (FDR) to hold

$\mathbf{D}=k_{\mathrm{B}}T\boldsymbol{\mu},$ (4)

where $k_{\mathrm{B}}$ is the Boltzmann constant and $T$ the temperature of
the solvent. Note that even though Eq. (3) looks formally equivalent to a
Fokker-Planck equation for the joint PDF $P(t)$, strictly spoken it is not,
due to the antisymmetric (odd-diffusive) elements in the diffusion tensor
[24]. However, based on the assumption of the FDR Eq. (3) resembles
equilibrium dynamics with a unique steady-state solution [9, 25].

###  II.1 Analytical solution

We rescale space with the diameter of the particle $d$,
$\mathbf{x}_{i}\to\mathbf{x}_{i}/d$, and time by the natural time-scale of
diffusing the radial distance of a particle diameter
$\tau_{d}=d^{2}/(2D_{0})$, $t\to\tau=t/\tau_{d}$. Given the radial symmetry of
the interaction potential $U(r)$, the time-evolution equation (3) can be
written in terms of a center-of-mass coordinate
$\mathbf{x}_{\mathrm{c}}=(\mathbf{x}_{1}+\mathbf{x}_{2})/2$ and a relative
coordinate $\mathbf{x}=\mathbf{x}_{1}-\mathbf{x}_{2}$ as

$\begin{split}\partialderivative{\tau}P(\tau)&=\frac{1}{4}\nabla_{\mathbf{x}_{\mathrm{c}}}^{2}\,P(\tau)\\\
&\quad+\nabla_{\mathbf{x}}\cdot(\mathbf{1}+\kappa\boldsymbol{\epsilon})[\nabla_{\mathbf{x}}+\beta\,\nabla_{\mathbf{x}}U(r)]\,P(\tau),\end{split}$
(5)

where $\beta=1/k_{\mathrm{B}}T$ and
$\nabla_{\mathbf{x}_{\mathrm{c}}},\nabla_{\mathbf{x}}$ are the partial
differential operators corresponding to the center-of-mass and relative
coordinates. Note that
$(\mathbf{1}+\kappa\boldsymbol{\epsilon})=\mathbf{D}/D_{0}$ represents the
dimensionless odd-diffusion tensor of Eq. (1). As Eq. (5) decouples the
coordinates, the propagator can be written as
$P(\mathbf{x}_{1},\mathbf{x}_{2},\tau|\mathbf{x}_{2,0},\mathbf{x}_{1,0},\tau_{0})=p_{c}(\mathbf{x}_{\mathrm{c}},\tau|\mathbf{x}_{\mathrm{c},0},\tau_{0})\,p(\mathbf{x},\tau|\mathbf{x}_{0},\tau_{0})$
and the center-of-mass problem can be solved straight forwardly in the form

$p_{c}(\mathbf{x}_{\mathrm{c}},\tau|\mathbf{x}_{\mathrm{c},0})=\frac{1}{\pi\tau}\
\mathrm{exp}\left(-\frac{|\mathbf{x}_{\mathrm{c}}-\mathbf{x}_{\mathrm{c},0}|^{2}}{\tau}\right),$
(6)

where we have set $\tau_{0}=0$ as the underlying stochastic process is time-
translational invariant [26].

The relative PDF can be put into the form of a (Cartesian) multipole expansion
[27, 28, 29, 30, 31] which in polar coordinates $\mathbf{x}=(r,\varphi)$,
$\mathbf{x}_{0}=(r_{0},\varphi_{0})$, is given as

$\displaystyle
p(\mathbf{x},\tau|\mathbf{x}_{0})=\Theta(r-1)\bigg{[}\varrho(r,\tau|r_{0})+\boldsymbol{\sigma}(r,\tau|r_{0})\cdot\mathbf{e}(\Delta\varphi)$
$\displaystyle\quad+\boldsymbol{Q}(r,\tau|r_{0}):\left(\mathbf{e}(\Delta\varphi)\otimes\mathbf{e}(\Delta\varphi)-\frac{\mathbf{1}}{2}\right)+\ldots\bigg{]},$
(7)

where
$\mathbf{e}(\Delta\varphi)=(\cos(\Delta\varphi),\sin(\Delta\varphi))^{\mathrm{T}}$,
$\Delta\varphi=\varphi-\varphi_{0}$ is the angular difference.
$\mathbf{Q}\colon(\mathbf{e}\otimes\mathbf{e}-\mathbf{1}/2)=\sum_{\alpha,\beta=1}^{2}Q_{\alpha\beta}(e_{\beta}e_{\alpha}-\delta_{\beta\alpha}/2)$
denotes the full contraction, where $\mathbf{e}\otimes\mathbf{e}$ is the outer
product. We again set $\tau_{0}=0$ here. Note the multiplicative Heaviside
function $\Theta(r-1)$ in Eq. (II.1), which is defined as $\Theta(x)=1$, if
$x>1$ and $\Theta(x)=0$ otherwise. This ensures the no-overlap condition of
the hard-disk interaction potential in Eq. (2). The explicit Cartesian
multipole expansion for the relative PDF in Eq. (II.1) constitutes the first
terms of a Fourier expansion for $p(\mathbf{x},\tau|\mathbf{x}_{0})$ which
reads

$\displaystyle
p(\mathbf{x},\tau|\mathbf{x}_{0})=\frac{\Theta(r-1)}{2\pi}\bigg{[}$
$\displaystyle a_{0}(r,\tau|r_{0})$ (8)
$\displaystyle+2\sum_{n=1}^{\infty}\begin{pmatrix}a_{n}(r,\tau|r_{0})\\\
b_{n}(r,\tau|r_{0})\end{pmatrix}\cdot\mathbf{e}(n\,\Delta\varphi)\bigg{]}.$

Here $a_{n}$ and $b_{n}$ are the Fourier coefficients of order $n$,
$n\in\mathbb{N}_{0}$. We introduce the notation
$p=\Theta(r-1)\,[p_{0}+\sum_{n=1}^{\infty}p_{n}]$, where $p_{0}=a_{0}/2\pi$
and
$p_{n}=[a_{n}\,\cos\left(n\,\Delta\varphi\right)+b_{n}\,\sin\left(n\,\Delta\varphi\right)]/\pi$,
$n\geq 1$ for our subsequent shorthand notation of the Fourier modes. The
Cartesian modes in Eq. (II.1) are thus connected to the Fourier modes as

$\varrho(r,\tau|r_{0})=\frac{1}{2\mathrm{\pi}}\ a_{0}(r,\tau|r_{0}),$ (9)

which is the (scalar) mean positional PDF,

$\boldsymbol{\sigma}(r,\tau|r_{0})=\frac{1}{\mathrm{\pi}}\begin{pmatrix}a_{1}(r,\tau|r_{0})\\\
b_{1}(r,\tau|r_{0})\end{pmatrix},$ (10)

which is the (vectorial) polarization order PDF, and

$\boldsymbol{Q}(r,\tau|r_{0})=\frac{1}{\mathrm{\pi}}\begin{pmatrix}a_{2}(r,\tau|r_{0})&\phantom{-}b_{2}(r,\tau|r_{0})\\\
b_{2}(r,\tau|r_{0})&-a_{2}(r,\tau|r_{0})\end{pmatrix},$ (11)

which is the (tensorial) nematic order PDF.

In Appendix A we provide the full solution to the relative problem, thereby
following the original derivations in Refs. [13, 9], and show that the general
Fourier coefficients $a_{n},b_{n}$ are given by

$\displaystyle a_{n}(r,\tau|r_{0})$
$\displaystyle=\frac{\mathrm{e}^{-\frac{r_{0}^{2}+r^{2}}{4\tau}}}{2\tau}\
I_{n}\left(\frac{r_{0}r}{2\tau}\right)-\mathscr{L}^{-1}\Bigg{\\{}k_{n}(r,s|r_{0})\bigg{[}s\,K_{n}^{\prime}(\sqrt{s})\,I_{n}^{\prime}(\sqrt{s})+(n\kappa)^{2}K_{n}(\sqrt{s})\,I_{n}(\sqrt{s})-\delta(r_{0}-1)\,\frac{\sqrt{s}\,K_{n}^{\prime}(\sqrt{s})}{2K_{n}(\sqrt{s})}\bigg{]}\Bigg{\\}},$
(12) $\displaystyle b_{n}(r,\tau|r_{0})$
$\displaystyle=-n\kappa\,\left(1-\frac{\delta(r_{0}-1)}{2}\right)\,\mathscr{L}^{-1}\left\\{k_{n}(r,s|r_{0})\right\\},$
(13)

where we used the abbreviation

$k_{n}(r,s|r_{0})=\frac{K_{n}(r\sqrt{s})\,K_{n}(r_{0}\sqrt{s})}{(\sqrt{s}\,K^{\prime}_{n}(\sqrt{s}))^{2}+(n\kappa\,K_{n}(\sqrt{s}))^{2}}.$
(14)

Here $s$ denotes the (dimensionless) Laplace variable. The Laplace transform
for a function $f(\tau)$ is defined as
$\mathscr{L}\\{f\\}(s)=\int_{0}^{\infty}\mathrm{d}\tau\
\mathrm{exp}(-s\tau)f(\tau)$, using the rescaled time variable
$\tau=t/\tau_{d}$. The prime denotes the derivative
$g^{\prime}(a)=\mathrm{d}g(x)/\mathrm{d}x|_{x=a}$ and $I_{n}(x),\,K_{n}(x)$
are the modified Bessel functions of the first kind and second kind,
respectively [32]. As apparent from Eqs. (12) and (13), the inverse Laplace
transformations remain unfeasible at the moment, and we rely on established
numerical Laplace inversion methods.

We stress that the problem of solving for the full propagator of interacting
particles in Eq. (3) can be formulated in the language of scattering theory
[33], thus allowing to use well-developed tools from quantum field theory
[34]. Odd diffusion here adds the perspective of non-Hermicity with
potentially new insights [9].

###  II.2 Numerical results and Fourier modes

Figure 1: Plots of the first three Fourier modes $p_{0},p_{1}$ and $p_{2}$ of
the relative PDF of interacting (odd-diffusive) particles
$p(r,\tau|r_{0})=\Theta(r-1)\,\sum_{n=0}^{\infty}\ p_{n}(r,\tau|r_{0})$ at
fixed time $\tau=1$. (a) - (c) show the modes for normal particles
($\kappa=0$) and (d) - (f) for odd-diffusive particles ($\kappa=1$). The
initial positions are chosen as $(r_{0},\varphi_{0})=(1.01,0)$, i.e. the
particles are placed $1.01$ times their diameter along an arbitrarily defined
$x$-axis. The zeroth-order mode in (a) and (d) equals the mean positional PDF
for the relative coordinate $r$ (see Eq. (9)) which is unaffected by odd
diffusion. The first order mode in (b) and (e), which corresponds to the
(contracted) polarizational order PDF (see Eq. (10)), and the second order
mode in (c) and (f), which corresponds to the (contracted) nematic order PDF
(see Eq. (10)), as well as all higher order modes are affected by odd
diffusion such that the modes are rotated in time compared to the $\kappa=0$
case, see also Fig. 4. Note that the gray circle in the middle of each plot is
due to the excluded volume condition $\Theta(r-1)$ of the hard-disk
interaction of the particles. Figure 2: Significance $\delta_{n}(\tau)$ of a
mode $n$ as defined in Eq. (15), which measures the contribution of a Fourier
mode of order $n\geq 1$, $p_{n}(\tau)$, to the full relative PDF
$p(\tau)=\Theta(r-1)\,\sum_{n=0}^{\infty}\ p_{n}(\tau)$ as a function of time
$\tau$ for odd-diffusion parameter $\kappa=0,1$ and $5$ in (a), (b) and (c),
respectively. We see that higher order modes are consecutively ordered in
their contribution to $p$ in time and become of comparable importance for
$\tau\to 0$ as $p(\tau)\to p(0)\propto\delta(\mathbf{x}-\mathbf{x}_{0})$.
Comparing (a) with (c) we observe that $\kappa>0$ shifts the characteristic
decay to shorter times. Thus, for dynamics where one is not interested in the
$\tau\to 0$ limit, one can truncate the Taylor expansion at finite $n$.
Figure 3: PDF $p(r,\varphi,\tau|r_{0},\varphi_{0})$ for the relative
coordinate of two hard interacting particles. (a) to (c) show normal particles
($\kappa=0$) and (d) to (f) show odd-diffusive particles ($\kappa=1$) at times
$\tau=1,5$ and $50$. The initial positions are chosen as
$(r_{0},\varphi_{0})=(1.01,0)$, i.e. the particles are placed $1.01$ times
their diameter along an arbitrarily defined $x$-axis. The Fourier expansion of
$p(\tau)$ in Eq. (II.1) is truncated at $n=10$ in the numerical evaluation for
the plots, see also Fig. 2 for a rationale. (d) to (f) show the “mutual
rolling effect”, i.e. that odd-diffusive particles rotate around each other
after a collision [2], which is in contrast to a symmetric back-reflection for
normal particles, visible in (a) - (c). Note the grey circle in the middle of
each plot which is due to the excluded volume condition $\Theta(r-1)$ of the
hard-disk interaction of the particles.

In Fig. 1 we compare the positional mode $p_{0}=\Theta(r-1)\,\varrho$, the
polarization mode $p_{1}=\Theta(r-1)\,[\boldsymbol{\sigma}\cdot\mathbf{e}]$,
and the nematic mode
$p_{2}=\Theta(r-1)\,[\mathbf{Q}\colon(\mathbf{e}\otimes\mathbf{e}-\mathbf{1}/2)]$
for a normal ($\kappa=0$) and an odd-diffusive ($\kappa=1$) system of
interacting particles. The modes are evaluated at fixed time $\tau=1$ and for
$(r_{0},\varphi_{0})=(1.01,0)$, i.e. particles are initially placed at a
distance of $1.01$ times their diameter along the (arbitrarily) chosen
$x$-axis.

We observe that $\kappa\neq 0$ does not affect the interacting particles’ mean
positional distribution $p_{0}$, but rotates higher order modes in comparison
to $\kappa=0$. We can understand this by observing from Eq. (13) that
$b_{n}\propto n\kappa$, such that $p_{0}$ is not affected by $\kappa\neq
0$—but for higher order modes $b_{n}$ contributes for an odd-diffusive system.
It is of interest to analyse the contribution of the higher-order modes to the
time evolution of the relative PDF $p$. We define a measure of the
significance of a mode of order $n\geq 1$ as

$\delta_{n}(\tau)=\frac{\int\mathrm{d}\mathbf{x}\
|p_{n}(\mathbf{x},\tau|\mathbf{x}_{0})|}{\int\mathrm{d}\mathbf{x}\
p_{0}(\mathbf{x},\tau|\mathbf{x}_{0})}.$ (15)

We take the absolute value $|p_{n}|$ in the definition of $\delta_{n}(\tau)$,
as the full space integral for the polarization, nematic and all higher order
modes vanishes due to the orthogonality of the harmonic functions,
$\int\mathrm{d}\mathbf{x}\ p_{n}=0$, $n\geq 1$. In contrast,
$\int\mathrm{d}\mathbf{x}\ p_{0}=1$ (see Appendix B), which we only include in
Eq. (15) to avoid numerical discretization errors.

Fig. 2 shows $\delta_{n}(\tau)$ for $n=1,\ldots,5$ and for different values of
the odd-diffusion parameter $\kappa=0,1,5$. When considering $\delta_{n}$ as a
measure for the relevance of the $n$th order mode, we observe that the modes
are ordered consecutively in their contribution to the relative PDF $p$ and
that considering higher order modes becomes important for $\tau\to 0$ as
$p(\tau)\to p(0)\propto\delta(\mathbf{x}-\mathbf{x}_{0})$, but higher order
modes become less important for $\tau\gg 0$. As can be seen in Figs. 2(b),
(c), $\kappa>0$ shifts the decay of $\delta_{n}(\tau)$ to even shorter times.

In Fig. 3 we plot the full relative PDF $p(\tau)$ at times $\tau=1,5,50$ for a
normal ($\kappa=0$) and an odd-diffusive ($\kappa=1$) system. Again we choose
the initial condition to be $(r_{0},\varphi_{0})=(1.01,0)$ and, based on the
observation in Fig. 2, we truncated the Fourier series at $n=10$. Comparing
the normal and odd-diffusive relative PDF, we observe that the
$\kappa$-induced rotation of every but the zeroth order Fourier mode persists
into the full PDF. For $\kappa=0$, $b_{n}=0$ for all modes, and the Fourier
representation of the relative PDF therefore only contains cosine terms. The
relative PDF of a normal particle thus is symmetric around
$\Delta\varphi=0,\pi$ for all times. This symmetry implies that particles
encounter the space from both sides after a collision with equal likelihood.
However, this does not hold for odd-diffusive particles. $\kappa\neq 0$
introduces a handedness in the diffusive exploration of space. For interacting
odd-diffusive particles, the additional rotational probability flux introduced
via Eq. (1) results in a preferred direction after a collision depending on
the sign of the odd-diffusion parameter $\kappa$. This effect was recently
observed from Brownian dynamics simulations and termed as “mutual rolling”
[2]. The interaction-induced symmetry breaking has far-reaching consequences
for observable transport coefficients such as the self-diffusion coefficient.
In odd-diffusive systems, even though resembling equilibrium overdamped
dynamics, the self-diffusion can be enhanced by interactions instead of being
reduced as for a normal system [2]. Even though, seemingly contradicting
equilibrium statistical mechanics theorems [23, 35, 36], the interaction-
enhanced self-diffusion could recently be rationalized by observing that the
time-evolution in odd-diffusive systems (see Eq. (3)) becomes non-Hermitian
for finite $\kappa$ [9].

###  II.3 Relevance of the polarization mode for the force autocorrelation

The force autocorrelation tensor (FACT) $\mathbf{C}_{F}(\tau)$ encodes the
most detailed microscopic information in an interacting system. Via a Taylor-
Green-Kubo relation [37, 38, 39] it encodes the particle-particle interaction
effects in the self-diffusion. For a stationary system, $\mathbf{C}_{F}(\tau)$
is defined as

$\displaystyle\mathbf{C}_{F}(\tau)$
$\displaystyle=\langle\mathbf{F}(\vec{\mathbf{x}})\otimes\mathbf{F}(\vec{\mathbf{x}}_{0})\rangle$
(16)
$\displaystyle=\int\mathrm{d}\vec{\mathbf{x}}\int\mathrm{d}\vec{\mathbf{x}}_{0}\
\mathbf{F}(\vec{\mathbf{x}})\otimes\mathbf{F}(\vec{\mathbf{x}}_{0})\,P_{N}(\vec{\mathbf{x}},\tau,\vec{\mathbf{x}}_{0},\tau_{0}),$

where $\vec{\mathbf{x}}=\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{N}\\}$ and
similarly $\vec{\mathbf{x}}_{0}$ for a system of, in general, $N$ particles.
$P_{N}$ is the $N$-particle joint PDF, which can be rewritten as
$P_{N}(\vec{\mathbf{x}},\tau,\vec{\mathbf{x}}_{0},\tau_{0})=P_{N}(\vec{\mathbf{x}},\tau|\vec{\mathbf{x}}_{0})\,P_{\mathrm{eq}}(\vec{\mathbf{x}}_{0},\tau_{0})$,
assuming the particles where in equilibrium at time $\tau_{0}$, which we take
again to be $0$. The force on the tagged particle (particle one) is
$\mathbf{F}(\vec{\mathbf{x}})=-\nabla_{1}U_{N}(\vec{\mathbf{x}})$, and we
assume a pairwise additive and radially symmetric potential
$U_{N}(\vec{\mathbf{x}})=\sum_{i,j=1}^{N}U(r_{ij})/2$, where
$r_{ij}=|\mathbf{x}_{i}-\mathbf{x}_{j}|,i\neq j$. In the dilute limit, we can
safely assume that only two-body correlations are important and thus ignore
correlations between the untagged particles. The equilibrium PDF can be
approximated as
$P_{\mathrm{eq}}(\vec{\mathbf{x}}_{0})=1/(VZ_{N})\,\prod_{i=2}^{N}\mathrm{exp}(-\beta\,U(r_{1i,0})/2)$,
where $V\subset\mathbb{R}^{2}$ is the bounded space of diffusion and
$Z_{N}=\int\mathrm{d}\vec{\mathbf{x}}_{0}\
\prod_{i=2}^{N}\mathrm{exp}(-\beta\,U(r_{1i,0})/2)$ is the $N$-particle
partition function. The conditional PDF, i.e., the $N$-particle propagator can
be approximated as
$P_{N}(\vec{\mathbf{x}},\tau|\vec{\mathbf{x}}_{0})=(1/V)\,\prod_{i=2}^{N}p(\mathbf{x}_{1i},\tau|\mathbf{x}_{1i,0})$,
where $p(\mathbf{x}_{1i},\tau|\mathbf{x}_{1i,0})$ is the PDF of the relative
coordinate $\mathbf{x}_{1i}=\mathbf{x}_{1}-\mathbf{x}_{i}$, similar to Eq.
(II.1) but for a generic two-body interaction potential $U$. Following these
approximations, all but one particle (particle two) can be integrated out in
Eq. (16) and we denote as before, $\mathbf{x}=\mathbf{x}_{1}-\mathbf{x}_{2}$.
The central interaction force between the particles can be written as
$\mathbf{F}(\mathbf{x})=F(r)\,\mathbf{e}(\varphi)$, where
$\mathbf{e}(\varphi)=(\cos(\varphi),\sin(\varphi))^{\mathrm{T}}$, as before,
and coincides with the radial unit vector. In the dilute limit, the FACT of
Eq. (16) thus becomes

$\displaystyle\mathbf{C}_{F}(\tau)$
$\displaystyle=\frac{N-1}{V}\int\mathrm{d}\mathbf{x}\int\mathrm{d}\mathbf{x}_{0}\
F(r)\,F(r_{0})\,\frac{\mathrm{e}^{-\beta U(r_{0})}}{Z_{2}}$
$\displaystyle\quad\times
p(\mathbf{x},\tau|\mathbf{x}_{0})\,\mathbf{e}(\varphi)\otimes\mathbf{e}(\varphi_{0})$
(17) $\displaystyle=\frac{N-1}{V}\pi\int\mathrm{d}r\int\mathrm{d}r_{0}\
F(r)\,F(r_{0})\,\frac{\mathrm{e}^{-\beta U(r_{0})}}{Z_{2}}$
$\displaystyle\quad\times[a_{1}(r,\tau|r_{0})\,\mathbf{1}-b_{1}(r,\tau|r_{0})\,\boldsymbol{\epsilon}].$
(18)

Here we used the orthogonality of the Fourier modes in the relative PDF
$p(\tau)$ (analogously to Eq. (II.1)) from Eq. (II.3) to Eq. (II.3) to find
that only the polarization mode
$\boldsymbol{\sigma}(r,\tau|r_{0})=(a_{1}(r,\tau|r_{0}),b_{1}(r,\tau|r_{0}))^{\mathrm{T}}/\pi$
(analogously to Eq. (10)) contributes to the FACT. Note that relations
analogous to Eqs. (II.1) and (10) also hold in a system with generic radially
symmetric interaction potential $U(r)$, specifically also for forces with
transverse, odd components [40, 41, 7, 42].

We can specify Eq. (II.3) for the hard-disk interaction potential of Eq. (2)
by observing that $P_{\mathrm{eq}}(\mathbf{x}_{0})=\Theta(r_{0}-1)/V^{2}$ and
that we can rewrite the singular interaction force via the trick
$\beta\Theta(r-1)\,\mathbf{F}(r)=\delta(r-1)\,\mathbf{e}(\varphi)$ [43],
obeying the same generic form of a central force as before. We can perform the
radial integral of Eq. (II.3) and find that the FACT of a hard-disk system in
the dilute limit is given by
$\mathbf{C}_{F}(\tau)=\beta^{-2}\phi\,[a_{1}(1,\tau|1)\,\mathbf{1}-b_{1}(1,\tau|1)\,\boldsymbol{\epsilon}]$,
where $\phi=(\pi d^{2}/4)\,(N/V)$ is the area fraction in dimensional form [9,
22]. Thus we understand that the behavior of the polarization mode
$\boldsymbol{\sigma}(\tau)$ in time governs the behavior of the FACT for
interacting systems, in particular for a hard-disk interaction. Note
specifically here, that the off-diagonal correlation ($\propto
b_{1}\,\boldsymbol{\epsilon}$) is directly proportional to $\kappa$, and
therefore might serve as a characteristic to odd diffusion [3, 9].

To further analyze the polarization mode in the hard system, we define

$\gamma(\tau)=\arctan\left(\frac{b_{1}(1,\tau|1)}{a_{1}(1,\tau|1)}\right),$
(19)

as the mean angle between the polarization vector
$\boldsymbol{\sigma}(1,\tau|1)$ as a function of time and its initial
direction, which defines the $x$-axis in the system, see also inset in Fig.
4(a). We plot $\gamma(\tau)$ for different values of $\kappa$ as a function of
time in Fig. 4(a) and use the $\mathrm{arctan2}(\cdot)$-function for numerical
evaluation to obtain angles within the interval $(-\pi,\pi)$. We observe that
for $\kappa=0$ the initial direction of the polarization does not change with
time and stays constant. Recalling the polarization mode in Fig. 1(b), the
location of the extrema thus remain unchanged, meaning that the particles are
symmetrically back-reflected from the center of the collision. For $\kappa\neq
0$, however, $\gamma(\tau)$ changes in time (see again Fig. 1(e)). The mean-
direction of the polarization develops an extremum $\gamma_{\mathrm{ext}}$ and
relaxes to a constant, non-zero value $\gamma_{\infty}$ for long times. For
$\kappa>0.88$, $\gamma_{\mathrm{ext}}<-\pi/2$ and for $\kappa>1$ even
$\gamma_{\infty}<-\pi/2$. The interval $\kappa\in(0.88,1)$ thereby aligns with
the odd-diffusion interval, for which we reported an oscillating FACF earlier
[9]. These sign-switches in the FACF coincide with what we observe now for the
direction of $\boldsymbol{\sigma}(\tau)$, see also inset in Fig. 4(a):
$\gamma(\tau)$ crosses $-\pi/2$ for some time (negative FACF) and eventually
relaxes to angles larger than $-\pi/2$ again (positive FACF).

We interpret this behavior as interacting odd-diffusive particles, which for
$|\kappa|>0.88$ rotate more than $\pi/2$ but relax to a steady state with a
relative angle $|\gamma_{\infty}|<\pi/2$ as long as $|\kappa|<1$; see Fig.
4(b) for a sketch. This behavior also rationalizes the oscillating FACF in the
same interval, as $\langle\mathbf{F}(\tau)\cdot\mathbf{F}(0)\rangle=\langle
F(\tau)\,F(0)\,\cos(\pi/2)\rangle=0$ here and the particles oscillate around
that angle. However, a steady state angle $|\gamma_{\infty}|$ which is smaller
than the maximal angle $|\gamma_{\mathrm{ext}}|$ is a generic observation from
Fig. 4(a). On average particles always rotate further in time, as they finally
relax. Note that for $\kappa\to-\kappa$, we have that
$\gamma(\tau)\to-\gamma(\tau)$, as $a_{n}$ is even and $b_{n}$ is an odd
function of $\kappa$. Thus, $\kappa<0$ results in the same phenomenology for
the relaxation of $\gamma(\tau)$ and thus for the FACF. Further, if
$|\kappa|\to\infty$, we find that
$|\gamma_{\mathrm{ext}}|\to|\gamma_{\infty}|\to\pi$, which means, that at most
odd particles exchange positions but do not rotate any further.

##  III Discussion

Figure 4: (a) Angle $\gamma(\tau)$ between the polarization mode
$\boldsymbol{\sigma}(r=1,\tau|r_{0}=1)$, Eq. (10), and its initial direction
as function of time $\tau$ for different odd-diffusion parameters $\kappa$,
see Eq. (19) and the inset. The polarization mode constitutes the vectorial
mode of the relative PDF $p(\tau)$ for interacting particles and represents
relative particle rearrangements. In a normal system ($\kappa=0$),
$\gamma(\tau)=0$ and is constant in time, whereas it develops an extremum
$\gamma_{\mathrm{ext}}$ different from its asymptotic steady state value
$\gamma_{\infty}$ for $\kappa\neq 0$ in time. We observe that
$|\gamma_{\mathrm{ext}}|<|\gamma_{\infty}|$, indicating that particles on
average rotate further in time than they finally relax to. Specifically for
$\kappa\in(0.88,1.0)$, $\gamma_{\mathrm{ext}}<-\pi/2$, but
$\gamma_{\infty}>-\pi/2$ which quantitatively aligns with the oscillating
force autocorrelation function previously observed for this regime [9]. (b)
The sketch qualitatively illustrates the scenario, that odd-diffusive
particles with $0.88<\kappa<1$ collide at $\tau_{0}$ with each other along the
(arbitrary) $x$-axis and rotate for $\tau_{1}>\tau_{0}$ to
$\gamma_{\mathrm{ext}}<-\pi/2$, and eventually relax at $\tau_{2}>\tau_{1}$ to
$\gamma_{\infty}>-\pi/2$.

We here presented the exact analytical solution and numerical evaluation of
two hard disk-like interacting odd-diffusing particles in two spatial
dimensions. Odd diffusion thereby is characterized by antisymmetric elements
$\propto\kappa$ in the diffusion tensor
$\mathbf{D}=D_{0}(\mathbf{1}+\kappa\boldsymbol{\epsilon}$). Our analysis
showed that the two-particle problem separates into a center-of-mass and a
relative coordinate problem, of which the first can be solved
straightforwardly, and the latter incorporates the nontrivial effects of
interactions and odd diffusion. The relative PDF can be written as a Fourier
series, and we observe that oddness rotates all apart from the zeroth order
mode in time, which represents the unaffected positional distribution of the
relative coordinate. The modes are consecutively ordered in their contribution
to the relative PDF, and when summed up show a rotated form of the PDF. This
effect is a characteristic of odd diffusion and has been termed as “mutual
rolling” earlier [44] as the particles rotate around each other while
interacting, in contrast to normal diffusive ($\kappa=0$) particles, which are
symmetrically back-reflected when interacting.

The representation of the relative PDF in its Fourier modes becomes useful in
the analysis of microscopic correlation functions, specifically the FACF. For
any central interaction force, only the polarization mode of the relative PDF
determines the correlation function. We conjecture here that a similar
phenomenology holds for other (radially symmetric) observables of arbitrary
tensorial order in an interacting system; the corresponding mode of the
relative PDF might determine the expectation value and even the correlation
function. We used the analytical access to the polarization mode, to
understand the average configurations of the hard-disk-like interacting odd-
diffusive particles. The maximal relative rotation angle overshoots the final
relaxation on average, which quantitatively aligns with an oscillating FACF,
recently reported for these systems [9].

Our work shines light on the novel way of particle interactions in odd-
diffusive systems. Thereby it might serve as a reference case for interactions
in the various odd-diffusive systems such as in equilibrium, e.g., Brownian
particle under Lorentz force [45, 46, 47, 48, 49, 50, 6, 51, 52, 53, 54, 55,
56, 44], with a long-lasting history in statistical mechanics [57, 58, 59], or
skyrmionic spin structures [60, 61, 62]. Here numerical studies recently
reproduced the interaction-enhanced self-diffusion [7], which originates in
the mutual rolling effect [2]. But there exist also non-equilibrium odd-
diffusive systems, such as systems under shear [63, 64], or active chiral
particles [1, 65, 66, 67, 42, 68, 30, 69]. In general, for systems that break
time-reversal symmetry, the relevance of off-diagonal correlation functions
for transport properties has attracted considerable interest recently [70, 1,
3, 71, 67, 42, 5]. Odd diffusion further might serve as the unifying
terminology for systems showing transverse responses such as systems with
Magnus forces [72, 73, 74], Coriolis forces [75, 76], in complex (porous)
environments [77, 78, 79], or to describe non-conservative force fields which
are found in optical tweezer experiments [80, 81, 82, 83]. Systems with an
artificial transverse interaction component [41, 84] recently showed that odd
interactions enhance the sampling of configurations in dense systems,
originating again in the unique particle rearrangements in odd-diffusive
systems. Transverse forces further have been the subject of interest in so-
called linear-diffusive systems [85, 86, 87, 88, 89]. Odd diffusion is finally
also relevant for strongly rotating [90, 91] or magnetized plasmas, as it can
be found, e.g. in the realm of astrophysics, where antisymmetric transport is
relevant to describe the movement of energetic particles moving through
magnetized plasmas such as the interplanetary and interstellar medium [92, 93,
94].

###### Acknowledgements.

Acknowledgements. The authors acknowledge the help of Timo J. Doerries in the
creation of Fig. 4(b). A. S., R. M., and E. K. acknowledge support by the
Deutsche Forschungsgemeinschaft (grants No. SH 1275/5-1, ME 1535/16-1 and SPP
2332 - 492009952).

##  Appendix A Analytical solution of the relative problem

This Appendix closely follows Ref. [9] in its presentation of the analytical
solution to the relative problem, which itself adapted the work of Hanna, Hess
and Klein [13] to odd-diffusive systems.

The equation for the relative PDF $p(\tau)=p(\mathbf{x},\tau|\mathbf{x}_{0})$,
where $\mathbf{x}=\mathbf{x}_{1}-\mathbf{x}_{2}$, follows from Eq. (5) as

$\displaystyle\partialderivative{\tau}p(\tau)=\nabla_{\mathbf{x}}\cdot(\mathbf{1}+\kappa\boldsymbol{\epsilon})[\nabla_{\mathbf{x}}+\beta\,\nabla_{\mathbf{x}}U(r)]\,p(\tau),$
(20)

and reads in (rescaled) polar coordinates $\mathbf{x}=(r,\varphi)$ as

$\begin{split}\partialderivative{\tau}p(\tau)=\frac{1}{r^{2}}\left(r\partialderivative{r}r\partialderivative{r}+r\partialderivative{r}\,r\beta\,\partialderivative{U(r)}{r}\right.\\\
\left.\quad+\partialderivative[2]{\varphi}-\kappa\,r\beta\,\partialderivative{U(r)}{r}\partialderivative{\varphi}\right)\,p(\tau).\end{split}$
(21)

Note that space is rescaled with the diameter of the particle $d$,
$\mathbf{x}\to\mathbf{x}/d$, and time is rescaled by the natural time-scale of
diffusing the radial distance of a particle diameter
$\tau_{d}=d^{2}/(2D_{0})$, $t\to\tau=t/\tau_{d}$. The initial condition
$p(\tau=0)=\mathrm{\delta}(\mathbf{x}-\mathbf{x}_{0})\,\mathrm{\Theta}(r-1)$
in polar coordinates becomes

$p(\tau=0)=\frac{\mathrm{\delta}(r-r_{0})}{r_{0}}\,\mathrm{\delta}(\varphi-\varphi_{0})\,\mathrm{\Theta}(r-1).$
(22)

The angular part of the initial condition can be expanded into a Fourier
series
$\mathrm{\delta}(\varphi-\varphi_{0})=\sum_{n=-\infty}^{\infty}\mathrm{exp}(\mathrm{i}n(\varphi-\varphi_{0}))/2\pi$,
which, following Refs. [13, 9], we use as an ansatz to solve for $p(\tau)$ as

$p(\mathbf{x},\tau|\mathbf{x}_{0})=\frac{\mathrm{\Theta}(r-1)}{2\pi}\sum_{n=-\infty}^{\infty}R_{n}(r,\tau|r_{0})\,\mathrm{e}^{\mathrm{i}n(\varphi-\varphi_{0})}.\\\
$ (23)

Note that except for the radial functions $R_{n}(r,\tau|r_{0})$, all other
parts of the ansatz are time-independent.

Observing that for the hard-disk potential, see Eq. (2), we have that
$\mathrm{exp}(-\beta U(r))=\Theta(r-1)$, we can replace the otherwise singular
interaction force $\partial U(r)/\partial r$ in Eq. (21) via

$-\beta\mathrm{\Theta}(r-1)\,\partialderivative{U(r)}{r}=\partialderivative{r}\mathrm{e}^{-\beta
U(r)}=\mathrm{\delta}(r-1).$ (24)

Rewriting Eq. (21) for the radial functions $R_{n}(\tau)=R_{n}(r,\tau|r_{0})$,
we thus find

$\partialderivative{\tau}R_{n}(\tau)=\frac{1}{r^{2}}\left(r\partialderivative{r}r\partialderivative{r}-n^{2}\right)\,R_{n}(\tau)$
(25)

in the domain $r\geq 1$ and for each order $n\in\mathbb{Z}$, and

$\partialderivative{r}R_{n}(\tau)=-\frac{\mathrm{i}n\kappa}{r}R_{n}(\tau)$
(26)

to be satisfied additionally at $r=1$. Eq. (26) can be viewed as an extension
of an ordinary von Neumann no-flux boundary condition which is recovered for
$\kappa=0$. This generalized condition can be found in the literature under
the name of oblique boundary conditions, see for example [95]. Eq. (25) is
equipped with a second boundary condition, which can be found from the natural
boundary condition on $p(\tau)$, $\lim_{r\to\infty}p(\tau)=0$, as
$\lim_{r\to\infty}R_{n}(\tau)=0$ and the initial condition on $R_{n}(\tau)$
translates from Eq. (22) as $R_{n}(\tau=0)=\delta(r-r_{0})/r_{0}$. Note that
Eq.(25) together with the boundary condition (26) also forms the basis of
solving the problem by using the language of scattering theory [33], the only
difference being the introduction of the the generalized von-Neumann boundary
condition for interacting odd-diffusive particles.

For the particular solution of Eq. (25), $R_{n}^{\mathrm{part}}(\tau)$, we
make the ansatz

$R_{n}^{\mathrm{part}}(\tau)=\int\limits_{0}^{\infty}\differential{u}\,w_{n}(r,u|r_{0})\,\mathrm{e}^{-\tau
u^{2}},$ (27)

which when inserted into Eq. (25) shows that $w_{n}(r,u|r_{0})$ satisfy the
Bessel equation [32] with solutions $J_{n}(ur)$ and $Y_{n}(ur)$ as the Bessel
functions of first kind and second kind, respectively. Matching the particular
solution with the initial condition $R_{n}(\tau=0)=\delta(r-r_{0})/r_{0}$ and
noting that the delta-distribution $\delta(\cdot)$ can be expanded in Bessel
functions of the first kind (see Eq. (32)) we find that $Y_{n}(ur)$ does not
contribute to the particular solution.

After a Laplace transformation with (dimensionless) Laplace variable $s$, the
homogeneous part of Eq. (25) appears as the modified Bessel equation [32] with
solutions $I_{n}(r\sqrt{s})$ and $K_{n}(r\sqrt{s})$, as the modified Bessel
functions of the first and second kind, respectively. The Laplace
transformation for a function $f(\tau)$ thereby is defined as
$\mathscr{L}\\{f\\}(s)=\int_{0}^{\infty}\mathrm{d}\tau\
\mathrm{exp}(-s\tau)f(\tau)$, noting that $\tau=t/\tau_{d}$. As the
homogeneous solution has to satisfy the natural boundary condition on
$R_{n}(\tau)$, $I_{n}(x)$ is no suitable solution as it diverges for
$x\to\infty$ for every $n\in\mathbb{Z}$ [32, 9.7.1]. We conclude that the
homogeneous part is solved by
$\mathscr{L}\\{R_{n}^{\mathrm{hom}}\\}(s)=A_{n}(s|r_{0})\,K_{n}(r\sqrt{s})$
for some amplitude $A_{n}(s|r_{0})$ to be determined by matching with the
oblique boundary condition (26).

The relative solution $p(\tau)$ in the Laplace domain thus is given by

$\mathscr{L}\\{p\\}(\mathbf{x},s|\mathbf{x}_{0})=\mathrm{\Theta}(r-1)\sum_{n=-\infty}^{\infty}\frac{\mathrm{e}^{\mathrm{i}n(\varphi-\varphi_{0})}}{2\mathrm{\pi}}\int\limits_{0}^{\infty}\differential{u}\
\frac{u\,J_{n}(ur_{0})}{s+u^{2}}\left[J_{n}(ur)-K_{n}(r\sqrt{s})\,\frac{u\,J^{\prime}_{n}(u)+\mathrm{i}n\kappa\,J_{n}(u)}{\sqrt{s}\,K^{\prime}_{n}\left(\sqrt{s}\right)+\mathrm{i}n\kappa\,K_{n}(\sqrt{s})}\right].$
(28)

which was already found in Ref. [9]. There are two types of integrals
appearing in Eq. (28), which involve products of Bessel functions and which we
list in Appendix B. The integrals $\int_{0}^{\infty}\mathrm{d}u\
u\,J_{n}(ur_{0})J_{n}(ub)/(s+u^{2})$ can be evaluated using Eq. (33), where
$b\in\\{1,r\\}$. Depending on whether $b=r>r_{0}$, $r=r_{0}$ or $r<r_{0}$, Eq.
(33) gives different results but we only need to consider the latter two for
the cases $b=1=r_{0}$ and $1<r_{0}$. The second integral
$\int_{0}^{\infty}\mathrm{d}u\
u^{2}\,J_{n}(ur_{0})J^{\prime}_{n}(u)/(s+u^{2})$ involves the derivative of a
Bessel function and therefore is more involved. We list the integral in Eq.
(B) and again need to differentiate whether $r_{0}=1$ or $r_{0}>1$. Taking
into account these different cases, we find for the Laplace transformed
relative PDF

$\begin{split}\mathscr{L}\\{p\\}(\mathbf{x},s|\mathbf{x}_{0})=\frac{\Theta(r-1)}{2\pi}\sum_{n=-\infty}^{\infty}\mathrm{e}^{\mathrm{i}n(\varphi-\varphi_{0})}\bigg{[}\Theta(r_{0}-r)\,I_{n}(r\sqrt{s})\,K_{n}(r_{0}\sqrt{s})+\Theta(r-r_{0})\,I_{n}(r_{0}\sqrt{s})\,K_{n}(r\sqrt{s})\\\
-\frac{K_{n}(r\sqrt{s})\,K_{n}(r_{0}\sqrt{s})}{\sqrt{s}\,K_{n}^{\prime}(\sqrt{s})+\mathrm{i}n\kappa\,K_{n}(\sqrt{s})}\left(\sqrt{s}\,I_{n}^{\prime}(\sqrt{s})+\mathrm{i}n\kappa\,I_{n}(\sqrt{s})-\frac{\delta(r_{0}-1)}{2K_{n}(\sqrt{s})}\right)\bigg{]}.\end{split}$
(29)

We can Laplace invert some parts of Eq. (29) as

$\displaystyle\mathscr{L}^{-1}\\{I_{n}(r_{0}\sqrt{s})\,$ $\displaystyle
K_{n}(r\sqrt{s})\\}=\mathscr{L}^{-1}\left\\{I_{n}(r\sqrt{s})\,K_{n}(r_{0}\sqrt{s})\right\\}$
$\displaystyle=\frac{\exp\left(-\frac{r_{0}^{2}+r^{2}}{4\tau}\right)}{2\tau}I_{n}\left(\frac{r_{0}r}{2\tau}\right),$
(30)

[96, 13.96]. Together with decomposing
$\mathrm{e}^{\mathrm{i}x}=\cos{x}+\mathrm{i}\sin{x}$ in Eq. (29), this gives
the Fourier expanded relative PDF as

$\displaystyle
p(\mathbf{x},\tau|\mathbf{x}^{0})=\frac{\Theta(r-1)}{2\pi}\Bigg{[}a_{0}(r,\tau|r_{0})$
$\displaystyle+2\sum_{n=1}^{\infty}\begin{pmatrix}a_{n}(r,\tau|r_{0})\\\
b_{n}(r,\tau|r_{0})\end{pmatrix}\cdot\begin{pmatrix}\cos\left(n(\varphi-\varphi_{0})\right)\\\
\sin\left(n(\varphi-\varphi_{0})\right)\end{pmatrix}\Bigg{]},$ (31)

where $a_{n}$ and $b_{n}$ are given in Eqs. (12) and (13) in the main text.
Note that by using the symmetry in the order of the modified Bessel functions,
$I_{-n}(x)=I_{n}(x)$ and $K_{-n}(x)=K_{n}(x)$ [32, 9.6.6], we can restrict the
sum on positive modes only.

##  Appendix B Integral expressions

To evaluate the integrals in Eq. (28), we use the following tabulated
integrals, listed in this Appendix.

Following Arfken and Weber’s Mathematical Methods for Physicists [97], we find
that Bessel functions of (integer) order $n$, $J_{n}$, obey the integral
relation

$\int\limits_{0}^{\infty}\differential{u}\,u\,J_{n}(au)\,J_{n}(bu)=\frac{\mathrm{\delta}(a-b)}{a},$
(32)

valid for $n>-1/2$ and $a,b$ some real-valued constants [97, 11.59].

To evaluate the remaining integrals of Bessel-functions, we draw on Gradshteyn
and Ryzhik’s Table of Integrals, Series and Products [98] and Abramowitz and
Stegun’s Handbook of Mathematical Functions [32]. The first relevant integral
is

$\int\limits_{0}^{\infty}\differential{u}\ \frac{u\
J_{n}(au)J_{n}(bu)}{u^{2}+c^{2}}=\begin{cases}I_{n}(ac)\,K_{n}(bc),&0<a<b\\\
I_{n}(ac)\,K_{n}(ac),&0<a=b\\\ I_{n}(bc)\,K_{n}(ac),&0<b<a\end{cases},$ (33)

which is valid for $n>-1$ and $c>0$ [98, 6.541, 6.535]. Here $I_{n}$, $K_{n}$
are the modified Bessel functions of the first and second kind, and $a,b,c$
some (real-valued) constants. Together with the symmetry relations for the
order of Bessel functions, $J_{-n}(x)=(-1)^{n}J_{n}(x)$, $I_{-n}(x)=I_{n}(x)$
and $K_{-n}(x)=K_{n}(x)$ [32, 9.1.5, 9.6.6], Eq. (33) can be further extended
to the relevant cases of $n<0$.

The second relevant integral is $\int_{0}^{\infty}\differential{u}\
u^{2}\,J_{n}(au)\,J_{n}^{\prime}(bu)/(u^{2}+c^{2})$, for some real-valued,
positive constants $a,b,c$. Using
$\mathrm{d}J_{n}(x)/\mathrm{d}x=(J_{n-1}(x)-J_{n+1}(x))/2$ [98, 8.471], the
integral can be evaluated as

$\displaystyle\int\limits_{0}^{\infty}\differential{u}\ \frac{u^{2}\
J_{n_{1}}(au)J_{n_{2}}(bu)}{u^{2}+c^{2}}$
$\displaystyle=\begin{cases}(-1)^{\alpha_{+}}\,c\
I_{n_{1}}(ac)\,K_{n_{2}}(bc),&0<a<b\\\ (-1)^{\alpha_{-}}\,c\
I_{n_{2}}(bc)\,K_{n_{1}}(ac),&0<b<a\end{cases},$ (34)

which is valid for $n_{2}>-1$ [98, 6.577]. The exponents are
$\alpha_{\pm}=(1\pm(n_{1}-n_{2}))/2\in\mathbb{N}_{0}$. We use Eq. (B) for the
case of $n_{1}=n\pm 1$ and $n_{2}=n$. By relying on the symmetry relations for
the orders of the Bessel functions, we can extend Eq. (B) again to the
negative cases of $n_{1}=-(n\pm 1)$ and $n_{2}=-n$. Thus, together with [98,
8.468], we find

$\int\limits_{0}^{\infty}\differential{u}\ \frac{u^{2}\
J_{n}(au)J_{n}^{\prime}(bu)}{u^{2}+c^{2}}=\begin{cases}c\
K_{n}(bc)\,I_{n}^{\prime}(ac),&0<a<b\\\
\frac{n}{b}K_{n}(ac)\,I_{n}(bc),&0<b<a\end{cases},$ (35)

valid for all $n\in\mathbb{Z}$, but limited to $b\neq a$.

In order to generalize Eq. (35) to the case of $b=a>0$, which can not be found
in Ref. [98], we generalize the relation, which was already derived in Ref.
[9] for $n=1$ to an arbitrary $n\in\mathbb{Z}$. Therefore, by using the
indefinite integral $\int\differential{u}\
J_{n}(u)J_{n}^{\prime}(u)=J_{n}^{2}(u)/2+const$, we can partially integrate
and find an integral of the form of Eq. (33), which, using the Wronskian
$\mathcal{W}[I_{n}(x),K_{n}(x)]=I_{n}(x)\,K_{n+1}(x)+I_{n+1}(x)\,K_{n}(x)=1/x$
[32, 9.1.15], gives

$\displaystyle\int\limits_{0}^{\infty}\differential{u}\,\frac{u^{2}\
J_{n}(au)J_{n}^{\prime}(au)}{u^{2}+c^{2}}$ $\displaystyle=\begin{cases}c\
K_{n}(bc)\,I_{n}^{\prime}(ac),&0<a<b\\\ c\
K_{n}(ac)\,I_{n}^{\prime}(ac)-\frac{1}{2a},&0<a=b\\\
\frac{n}{b}K_{n}(ac)\,I_{n}(bc),&0<b<a\end{cases}.$ (36)

To prove the normalization of the zeroth order mode $\int\mathrm{d}\mathbf{x}\
p_{0}=1$, Eq. (9) in the main text, we transform the integrals over $I_{0}$
and $K_{0}$ into higher order Bessel functions evaluated at the boundaries
$r=1$ and $r\to\infty$ by using the recursion relations [98, 8.486]

$\left(\frac{1}{u}\ \derivative{u}\right)^{m}\left(u^{n}\
I_{n}(u)\right)=u^{n-m}\ I_{n-m}(u),$ (37)

and

$\left(\frac{1}{u}\ \derivative{u}\right)^{m}\left(u^{n}\
K_{n}(u)\right)=(-1)^{m}\ u^{n-m}\ K_{n-m}(u),$ (38)

which specifically imply for $m=1$ and $n=1$ to
$u\,I_{0}(u)=\mathrm{d}/\mathrm{d}u\,(u\,I_{1}(u))$ and
$u\,K_{0}(u)=-\mathrm{d}/\mathrm{d}u\,(u\,K_{1}(u))$. The integrals over
$I_{0}(u)$ and $K_{0}(u)$ thus turn into evaluating $u\,I_{1}(u)$ and
$u\,K_{1}(u)$ at the integration bounds and the upper bound vanishes by the
asymptotic expansion of
$\lim_{u\to\infty}uK_{n}(u)\sim\lim_{u\to\infty}u\,\exp(-u)=0$ [98, 8.451].
The remaining expressions from the lower bounds are evaluated by using the
Wronskian $\mathcal{W}[I_{n}(x),K_{n}(x)]$, to be exactly 1.

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