††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Anisotropic Brownian particles under resetting

Subhasish Chaki  Kristian Stølevik Olsen  Hartmut Löwen  Institut für
Theoretische Physik II - Weiche Materie, Heinrich-Heine-Universität
Düsseldorf, D-40225 Düsseldorf, Germany

###### Abstract

We study analytically the dynamics of an anisotropic particle subjected to
different stochastic resetting schemes in two dimensions. The Brownian motion
of shape-asymmetric particles in two dimensions results in anisotropic
diffusion at short times, while the late-time transport is isotropic due to
rotational diffusion. We show that the presence of orientational resetting
promotes the anisotropy to late times. When the spatial and orientational
degrees of freedom are reset, we find that a non-trivial spatial probability
distribution emerges in the steady state that is determined by the initial
orientation, particle asymmetry and the resetting rate. When only spatial
degrees of freedom are reset while the orientational degree of freedom is
allowed to evolve freely, the steady state is independent of the particle
asymmetry. When only particle orientation is reset, the late-time probability
density is given by a Gaussian with an effective diffusion tensor, including
off-diagonal terms, determined by the resetting rate. Generally, the coupling
between the translational and rotational degrees of freedom, when combined
with stochastic resetting, gives rise to unique behaviour at late times not
present in the case of symmetric particles. Considering recent developments in
experimental implementations of resetting, our results can be useful for the
control of asymmetric colloids, for example in self-assembly processes.

###### pacs:

Valid pACS appear here

††preprint: ApS/123-QED

##  I Introduction

A wide variety of processes in physics, chemistry, and biology are comprised
of diffusing particles that are highly anisotropic in shape. Prominent
examples include rod-like bacteria or viruses [1, 2, 3, 4], nematic
macromolecules [5, 6], and shape-asymmetric colloids [7, 8, 9, 10, 11, 12].
There are several recent research avenues in soft condensed matter, where
asymmetric colloidal micro- and nano-particles play a prominent role. In
colloidal self-assembly, control of shape-asymmetric particles is crucial for
the assembly of materials with novel functions and properties [13, 14, 15,
16]. In nanomedicine, how to externally control the dynamics of an asymmetric
colloid moving inside the human body is a central question [17].

The diffusive motion of shape-asymmetric objects is a topic with a long
history, dating back to the work of Perrin almost a century ago [18, 19, 20].
The translational motion of these anisotropic particles can be quite different
from those of spherical particles. When left to diffusive, the anisotropic
particles show a crossover from short-time anisotropic diffusion, due to the
coupling of rotational and translational motion, to effective isotropic
diffusion at late times [21]. The crossover between these regimes depends on
the timescale set by the rotational diffusion coefficient, which determines
the directional memory along the particle’s long axis. Furthermore, the
transient behaviour of an anisotropic particle is longer-lived in two
dimensions when compared to the three-dimensional case. This suggests that the
transient behaviour may modulate the reaction rate of diffusion-limited
reactions in intracellular membranes, where reaction time is frequently of the
order of microseconds and the environment is dimensionally restricted [22,
23]. Experimental methods are also available for synthesizing and
characterizing the dynamics of ellipsoidal colloids [21]. Experimentally, it
has been shown that the transition from anisotropic to isotropic diffusion in
two dimensions occurs over a period of a few seconds for a free micrometer-
sized ellipsoidal particle [24].

The Brownian motion of an anisotropic particle has been studied in many
situations over the years, including dynamics in confinement and external
potentials [24, 25], first passage analysis [26, 27], active matter [28, 29,
30] and stochastic thermodynamics [31]. So far, to the best of our knowledge,
there has been little or no work on anisotropic particles subjected to
_stochastic resetting_. Stochastic resetting is a process whereby a system’s
state is brought back to its initial condition at a constant rate [32, 33,
34]. A wide range of intriguing phenomena results from this simple rule of
intermittent interruption, for example, anomalous relaxation properties [35,
36] and non-trivial steady states resulting from the confining effect of
resetting [37, 38]. Over recent years, a myriad of aspects has been
investigated, including target search processes under resets [39, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55], stochastic thermodynamics
[56, 57, 58, 59, 60, 61, 62, 63, 64, 65], and active matter [66, 67, 68, 69,
70]. Recently, the effects of resetting in systems with multiple coupled
degrees of freedom have been studied. In particular, the case where an
observed degree of freedom experiences indirect effects of resets due to a
coupling to a resetting variable has been investigated [71, 72].

In the majority of past studies, resetting is applied to symmetric or point-
like particles. Here, we extend past studies by studying the effect of various
resetting schemes applied to a shape-asymmetric Brownian particle. In
particular, we consider the case of a Brownian anisotropic particle, where a
rotational-translational coupling is present which separates the dynamics from
the symmetric counterpart. We study the dynamics of a Brownian anisotropic
particle under various resetting schemes, all of which may be implemented
experimentally. Resetting of the translational degrees of freedom can be
achieved by optical tweezers [73, 74, 75, 65], while resetting of the
particle’s orientation can be performed using techniques developed for
magnetic anisotropic colloid assembly [76]. We seek to understand the
dynamical consequences of the combined effect of particle asymmetry and
resetting in two dimensions, which can be characterised by moments and
marginal distributions, available, for example, in colloidal experiments using
particle tracking.

This paper is structured as follows. Section II introduced the necessary
background theory, including the Brownian motion of anisotropic particles in
two dimensions, as well as the renewal approach to stochastic resetting when
multiple degrees of freedom is present. We then proceed to investigate the
dynamical and stationary properties under various resetting schemes, namely
complete resetting of all degrees of freedom, translational resetting, and
orientational resetting in sections III, IV and V respectively. Section VI
offer a concluding discussion.

##  II General theory

###  II.1 Diffusion of an anisotropic particle

Here we consider the dynamics of an anisotropic free Brownian particle in two
dimensions with mobilities $\Gamma_{||}$ and $\Gamma_{\perp}$ along the longer
and the shorter axes of the particle, respectively. Additionally, the motion
of the particle is subjected to rotational diffusion, characterized by the
mobility $\Gamma_{\theta}$. The translational and rotational diffusivities are
defined through Einstein-Smoluchowski relations $D_{||}=k_{B}T\Gamma_{||}$ and
$D_{\perp}=k_{B}T\Gamma_{\perp}$, and $D_{\theta}=k_{B}T\Gamma_{\theta}$,
where $T$ is the temperature of the bath. The particle at any given time $t$
can be described by the position vector $\vec{r}(t)$ of its centre of mass,
which can be decomposed as $(\delta\tilde{x},\delta\tilde{y})$ in the body
frame and $(\delta x,\delta y)$ in the laboratory frame. The angle between the
lab and body frames is denoted as $\theta(t)$, which evolves diffusively as

$\begin{split}\frac{\partial\theta}{\partial
t}=\sqrt{2k_{B}T\Gamma_{\theta}}\tilde{\eta}_{\theta}(t).\end{split}$ (1)

The translational motions of the anisotropic particle are decoupled in the
body frame, and are described by the Langevin equations,

$\begin{split}\frac{\partial\tilde{x}}{\partial
t}=\sqrt{2k_{B}T\Gamma_{\parallel}}\tilde{\eta}_{x}(t);\,\,\,\frac{\partial\tilde{y}}{\partial
t}=\sqrt{2k_{B}T\Gamma_{\perp}}\tilde{\eta}_{y}(t).\end{split}$ (2)

The translational and rotational noise terms $\tilde{\eta}_{x}(t)$,
$\tilde{\eta}_{y}(t)$ and $\tilde{\eta}_{\theta}(t)$ in equation (2) are
assumed to be Gaussian and characterized by the following mean and variances

$\begin{split}&\langle\tilde{\eta}_{i}(t)\rangle=0,\,\,\,\mathrm{and}\,\,\,\langle\tilde{\eta}_{i}(t)\tilde{\eta}_{j}\left(t^{\prime}\right)\rangle=\delta_{ij}\delta\left(t-t^{\prime}\right)\end{split}$
(3)

The displacements in the lab and the body frames are related by the following
equations,

$\begin{split}\delta x=\cos\theta\delta\tilde{x}-\sin\theta\delta\tilde{y},\\\
\delta y=\cos\theta\delta\tilde{y}+\sin\theta\delta\tilde{x}.\\\ \end{split}$
(4)

Dividing the equation (LABEL:eq:coordinate_transformation) by $\delta t$,
taking the limit $\delta t\rightarrow 0$, and substituting the translational
velocities in the body frame from equation (2), we get the equations of motion
in the lab frame as follows

$\begin{split}\frac{\partial{x}}{\partial
t}={\xi}_{x}(t);\,\,\,\frac{\partial{y}}{\partial t}={\xi}_{y}(t)\end{split}$
(5)

The random variables $\xi_{i}$ have zero mean and the correlations at fixed
$\theta(t)$ are given by

Figure 1: a) Anisotropic particle with three mobilities; parallel to body axis
$\Gamma_{||}$, perpendicular to the body axis $\Gamma_{\perp}$, and a
rotational motility $\Gamma_{R}$. b) Under complete resetting, all degrees of
freedom are reset to their initial conditions. c) For translational resets,
only the particle’s center-of-mass position is reset. d) Under orientational
resetting, the particle orientation angle is reset, while the center-of-mass
positions remain unchanged.

$\begin{split}\langle{\xi}_{i}(t){\xi}_{j}\left(t^{\prime}\right)\rangle=2k_{B}T\Gamma_{ij}\left[\theta(t)\right]\delta\left(t-t^{\prime}\right),\end{split}$
(6)

where
$\Gamma_{ij}\left[\theta(t)\right]=\frac{\bar{\Gamma}}{2}+\frac{\Delta\Gamma}{2}\left[\begin{array}[]{cc}\cos
2\theta(t)&\sin 2\theta(t)\\\ \sin 2\theta(t)&-\cos
2\theta(t)\end{array}\right]$ with
$\bar{\Gamma}=\frac{\Gamma_{||}+\Gamma_{\perp}}{2}$ and
$\Delta{\Gamma}=\frac{\Gamma_{||}-\Gamma_{\perp}}{2}$.

Integrating equation (5) with respect to time for $x$ direction, one obtains

$\begin{split}x(t)=\int_{0}^{t}dt^{\prime}\xi_{x}\left(t^{\prime}\right)+x_{0}\end{split}$
(7)

The rotational degree of freedom evolves on the unit circle following an
ordinary Brownian that is independent of the translational degrees of freedom.
Hence, the second moment of the particle, conditional on an initial
orientation angle $\theta_{0}$, can be explicitly calculated from equation (7)
as

$\begin{split}\langle
x^{2}(t)|\theta_{0}\rangle&=\int_{0}^{t}dt^{\prime}\int_{0}^{t}dt^{\prime\prime}\langle\xi_{i}\left(t^{\prime}\right)\xi_{i}\left(t^{\prime\prime}\right)\rangle\\\
&=2k_{B}T\int_{0}^{t}dt^{\prime}\int_{0}^{t}dt^{\prime\prime}\langle\Gamma_{ij}\left[\theta(t^{\prime})\right]\rangle_{\eta_{\theta}}\delta\left(t^{\prime}-t^{\prime\prime}\right)\\\
&=2k_{B}T\int_{0}^{t}dt^{\prime}\left[\bar{\Gamma}+\frac{\Delta{\Gamma}}{2}\langle\cos\theta\left(t^{\prime}\right)|\theta_{0}\rangle\right]\\\
&=2k_{B}T\left[\bar{\Gamma}t+\frac{\Delta{\Gamma}}{2}\cos
2\theta_{0}\left(\frac{1-e^{-4D_{\theta}t}}{4D_{\theta}}\right)\right]\end{split}$
(8)

Here, and throughout, we assume the particle is initially at the origin
$x_{0}=y_{0}=0$. Above, we used the following identity

$\begin{split}\langle\cos 2\theta\left(t\right)|\theta_{0}\rangle=Re\langle
e^{i2\theta}|\theta_{0}\rangle=Re\left[\int d\theta
e^{i2\theta}\frac{e^{-\frac{\left(\theta-\theta_{0}\right)^{2}}{4D_{\theta}t}}}{\sqrt{4\pi
D_{\theta}t}}\right]=\cos(2\theta_{0})\>e^{-4D_{\theta}t}\end{split}$ (9)

Similarly, one can explicitly calculate the second moment for $y(t)$ direction

$\begin{split}\langle
y^{2}(t)|\theta_{0}\rangle=2k_{B}T\left[\bar{\Gamma}t-\frac{\Delta{\Gamma}}{2}\cos
2\theta_{0}\left(\frac{1-e^{-4D_{\theta}t}}{4D_{\theta}}\right)\right]\end{split}$
(10)

The first term in equations (8) and (10) is attributed to the diffusion of the
center-of-mass and the second term results from the displacement of the
particle due to the orientational fluctuations. At late times much larger than
the rotational timescale $t\gg D_{\theta}^{-1}$, the diffusion is isotropic
with $\langle x^{2}(t)\rangle=\langle y^{2}(t)\rangle=2k_{B}T\bar{\Gamma}t$.
At short timescales, however, we have the first-order expansions

$\displaystyle\langle x^{2}(t)|\theta_{0}\rangle$
$\displaystyle=2k_{B}T\left(\bar{\Gamma}+\frac{\Delta\Gamma}{2}\cos(2\theta_{0})\right)t+\mathcal{O}(t^{2})$
(11) $\displaystyle\langle y^{2}(t)|\theta_{0}\rangle$
$\displaystyle=2k_{B}T\left(\bar{\Gamma}-\frac{\Delta\Gamma}{2}\cos(2\theta_{0})\right)t+\mathcal{O}(t^{2})$
(12)

where we see that the shape asymmetry gives rise to anisotropic diffusion at
early times, depending on the particle’s initial orientation $\theta_{0}$. It
is interesting to compare the mean squared displacements along the $x$ and $y$
directions. The ratio of the variances is given by $K=\frac{{\langle
x^{2}(t)|\theta_{0}\rangle}}{{\langle
y^{2}(t)|\theta_{0}\rangle}}=\frac{\left(\bar{\Gamma}+\frac{\Delta\Gamma}{2}\cos(2\theta_{0})\right)}{\left(\bar{\Gamma}-\frac{\Delta\Gamma}{2}\cos(2\theta_{0})\right)}$.

The anisotropy in diffusion also generates non-zero cross-correlation between
$x$ and $y$ through their mutual coupling to the orientation,

$\begin{split}\langle
x(t)y(t)|\theta_{0}\rangle=2k_{B}T\left[\frac{\Delta{\Gamma}}{2}\sin
2\theta_{0}\left(\frac{1-e^{-4D_{\theta}t}}{4D_{\theta}}\right)\right]\end{split}$
(13)

It is evident that the correlation between $x$ and $y$ in equation (13)
becomes constant at $t\rightarrow\infty$ due to the decorrelation between
$\theta_{0}$ and $\theta(t)$ after a timescale proportional to
$D_{\theta}^{-1}$.

###  II.2 Renewal approach to resetting

The anisotropic particle in two dimensions can be described by its center-of-
mass positions, $(x,y)$ as well as an orientation angle $\theta$. We collect
these three degrees of freedom into a single phase space variable
$X\equiv(x,y,\theta)$. Upon stochastic resetting, we must prescribe a rule for
how all phase space variables are reset. In general, we assume that upon a
reset event the phase space point $X(t)$ resets to

$X(t)\longrightarrow X_{R}(X_{0},X(t))$ (14)

where the resetting location $X_{R}(X_{0},X(t))$ can be a mix of the
coordinates $X(t)$ before the reset and the initial coordinates $X_{0}$
depending on the type of resetting considered. For Poissonian resets, i.e. the
waiting time between resets follows an exponential distribution with rate $r$,
the full phase space propagator $P_{r}(X,t|X_{0})$ can be expressed in terms
of a renewal equation

$\displaystyle P_{r}(X,t|X0)$
$\displaystyle=e^{-rt}P_{0}(X,t|X_{0})+r\int_{0}^{t}d\tau e^{-r\tau}\int
dX^{\prime}P_{r}(X^{\prime},t-\tau|X_{0})P_{0}(X,\tau|X_{R}(X_{0},X^{\prime}))$
(15)

Here, $P_{0}(X,t|X_{0})$ is the propagator of the underlying system in the
absence of resetting. The first term in equation (15) corresponds to
trajectories where no resets have occurred up to time $t$, which happens with
probability $e^{-rt}$. The second term takes into account trajectories where
resetting takes place, and decomposed the path at the time of the last reset.
By changing the functional form of the resetting point in phase space
$X_{R}(X_{0},X(t))$, we can study different resetting schemes and how they
affect the dynamics. We consider the following resetting schemes:

  * i)

Complete resetting: all phase space variables are reset to their initial
values $X_{R}(X_{0},X^{\prime}(t))=X_{0}$.

  * ii)

Translational resetting: Only the spatial coordinates are reset, i.e.
$X_{R}(X_{0},X^{\prime}(t))=(x_{0},y_{0},\theta^{\prime}(t))$.

  * iii)

Orientational resetting: Only the orientation is reset
$X_{R}(X_{0},X^{\prime}(t))=(x^{\prime}(t),y^{\prime}(t),\theta_{0})$.

In the following, we consider these cases separately. In particular, we are
interested in the reduced dynamics of one of the spatial coordinates, for
which reduced renewal equations are derived in each case.

##  III Complete resetting

For complete resetting, the propagator $P_{r}(X,t|X_{0})$ in the presence of
Poissonian resetting can be written through the last renewal equation

$\begin{split}P_{r}(X,t|X_{0})=e^{-rt}P_{0}(X,t|X_{0})+r\int_{0}^{t}d\tau
e^{-r\tau}\int dX_{0}P(X_{0})P_{0}(X,\tau|X_{0})\end{split}$ (16)

Here we have assumed that the initial condition $X_{0}$ is drawn from a
distribution $P(X_{0})$. The last renewal equation can predict the steady
state as

$\begin{split}P_{\textrm{st}}(X,t|X_{0})=r\int_{0}^{\infty}d\tau
e^{-r\tau}\int dX_{0}P(X_{0})P(X,\tau|X_{0})=r\int
dX_{0}P(X_{0})\tilde{P_{0}}(X,r|X_{0})\end{split}$ (17)

Here $\tilde{P}(X,r|X_{0})$ is the Laplace transform of $P(X,\tau|X_{0})$.
From the last renewal equation, Eq.(16), we know that the moments also satisfy
a renewal equation

$\begin{split}{\langle X^{n}(t)\rangle}_{r}=e^{-rt}{\langle
X^{n}(t)\rangle}_{0}+r\int_{0}^{t}d\tau e^{-r\tau}\int dX_{0}P(X_{0}){\langle
X^{n}(\tau)\rangle}_{0}\end{split}$ (18)

For our case, $X$ represents three dynamical variables
$\left(x,y,\theta\right)$. For a fixed initial state
$X_{0}=\left(0,0,\theta_{0}\right)$, we have

$\begin{split}P_{r}(x,y,\theta,t|0,0,\theta_{0})=e^{-rt}P_{0}(x,y,\theta,t|0,0,\theta_{0})+r\int_{0}^{t}d\tau
e^{-r\tau}P_{0}(x,y,\theta,\tau|0,0,\theta_{0})\end{split}$ (19)

The effect of resetting on the variable $x$ can be obtained by integrating
over $y$ and $\theta$ in the above renewal equation:

$\begin{split}P_{r}(x,t|0,\theta_{0})=e^{-rt}P_{0}(x,t|0,\theta_{0})+r\int_{0}^{t}d\tau
e^{-r\tau}P_{0}(x,\tau|0,\theta_{0})\end{split}$ (20)

The corresponding moment for $x$ in that case would be

$\begin{split}{\langle x^{n}(t)|\theta_{0}\rangle}_{r}=e^{-rt}{\langle
x^{n}(t)|\theta_{0}\rangle}_{0}+r\int_{0}^{t}d\tau e^{-r\tau}{\langle
x^{n}(\tau)\rangle}_{\theta_{0}}\end{split}$ (21)

Next, we calculate the second moment of $x$ using equation (8) and $n=2$ in
the equation (21),

$\begin{split}{\langle
x^{2}(t)|\theta_{0}\rangle}_{r}=\frac{2k_{B}T\bar{\Gamma}}{r}\left(1-e^{-rt}\right)+\frac{k_{B}T\Delta\Gamma\cos
2\theta_{0}}{\left(r+4D_{\theta}\right)}\left(1-e^{-\left(r+4D_{\theta}\right)t}\right)\end{split}$
(22)

Similarly, one can also calculate ${\langle y^{2}(t)|\theta_{0}\rangle}_{r}$
and the cross-correlation ${\langle x(t)y(t)|\theta_{0}\rangle}_{r}$ in the
presence of resetting. We find

$\begin{split}{\langle
y^{2}(t)|\theta_{0}\rangle}_{r}=\frac{2k_{B}T\bar{\Gamma}}{r}\left(1-e^{-rt}\right)-\frac{k_{B}T\Delta\Gamma\cos
2\theta_{0}}{\left(r+4D_{\theta}\right)}\left(1-e^{-\left(r+4D_{\theta}\right)t}\right)\end{split}$
(23)

$\begin{split}{\langle
x(t)y(t)|\theta_{0}\rangle}_{r}=\frac{k_{B}T\Delta\Gamma\sin
2\theta_{0}}{\left(r+4D_{\theta}\right)}\left(1-e^{-\left(r+4D_{\theta}\right)t}\right)\end{split}$
(24)

Figure 2: Dynamical exponents $\zeta_{x}(t)$ and $\zeta_{y}(t)$ for the mean
squared displacement in the $x$ and $y$ direction for various values of
resetting rate. The black line shows the exponents without resetting.
Parameters are set to $\bar{\Gamma}=1.0$, $\Delta\Gamma=0.001$, with initial
orientation $\theta_{0}=0$.

To gain further insights into the dynamics, we consider the dynamical
exponents $\zeta_{x}(t)=\frac{d\left[\log\langle
x^{2}(t)\rangle\right]}{d\left[\log t\right]}$ and
$\zeta_{y}(t)=\frac{d\left[\log\langle y^{2}(t)\rangle\right]}{d\left[\log
t\right]}$ for both $x$ and $y$ directions respectively. These are plotted in
Fig. (2). In Fig. (2), we chose the particle initially oriented along the $x$
direction. Without resetting, the exponent $\zeta(t)$ clearly exhibits three
distinct regimes – short time diffusive regime $\left(\zeta_{x,y}(t)\approx
1\right)$ for both the $x$ and $y$ directions, intermediate superdiffusion
$\left(\zeta_{y}(t)>1\right)$ for $y$ direction and subdiffusion
$\left(\zeta_{x}(t)<1\right)$ for $x$ direction and long time diffusive motion
$\left(\zeta_{x,y}(t)\approx 1\right)$ for both the $x$ and $y$ directions. A
transition to confined motion $\left(\zeta_{x,y}(t)\approx 0\right)$ occurs in
the presence of resetting. At a very short times,
$t\ll\text{min}(r^{-1},D_{\theta}^{-1})$, the particle moves diffusively in
each direction with different mobilities $\Gamma_{||}$ and $\Gamma_{\perp}$
for $\theta_{0}=0$ as shown in equations (12), and the motion is not yet
affected by resetting. After the characteristic timescale set by rotational
diffusion, the particle performs diffusive motion with an effective diffusion
constant $\bar{D}$ due to the decorrelation in the orientational degrees of
freedom. Since the mobility in the $x$ and $y$ directions are larger and
smaller than $\bar{D}$ respectively, the motion in these directions must be
decelerated or accelerated to reach the isotropic diffusivity $\bar{D}$ in the
long time limit. In the presence of resetting, a transition to confinement
($\zeta_{x,y}\approx 0$) occurs at a typical time $r^{-1}$, potentially
interrupting the other dynamical regimes.

Since resetting bring the particle orientation back to its initial angle, the
anisotropy found in the short-time expansions Eq. (12) is partially promoted
to late times. To see this, we consider as a measure for the late-time
anisotropy the ratio of the mean squared displacements in the $x$ and $y$
directions. In the absence of resetting, the steady-state second moment in the
presence of resetting for a spherical Brownian particle with diffusivity
$\bar{D}$ is $\frac{2\bar{D}}{r}$ along both the $x$ and $y$ directions.
However, the steady-state second moments in the presence of resetting for the
anisotropic particle are modified and become larger or smaller than
$\frac{2\bar{D}}{r}$ in the $x$ and $y$ directions. Indeed, at late times
$\langle
x^{2}|\theta_{0}\rangle=\frac{2\bar{D}}{r}+\frac{k_{B}T\Delta\Gamma\cos
2\theta_{0}}{\left(r+4D_{\theta}\right)}$ and $\langle
y^{2}|\theta_{0}\rangle=\frac{2\bar{D}}{r}-\frac{k_{B}T\Delta\Gamma\cos
2\theta_{0}}{\left(r+4D_{\theta}\right)}$. The ratio of the variances is given
by

$K_{r}=\frac{{\langle x^{2}(t)|\theta_{0}\rangle}_{r}}{{\langle
y^{2}(t)|\theta_{0}\rangle}_{r}}=\frac{\frac{2\bar{D}}{r}+\frac{k_{B}T\Delta\Gamma\cos
2\theta_{0}}{\left(r+4D_{\theta}\right)}}{\frac{2\bar{D}}{r}-\frac{k_{B}T\Delta\Gamma\cos
2\theta_{0}}{\left(r+4D_{\theta}\right)}}$ (25)

In the case of a spherical particle $(\Delta\Gamma=0)$, one obtains $K_{r}=1$
independent of the stochastic resetting. The motion is not isotropic for
$\Delta\Gamma\neq 0$ under the resetting of all the variables, and the degree
of anisotropy increases with $r$ and eventually saturates, as shown in Fig.
(3). The timescale of the cross-correlation between $x$ and $y$ directions is
renormalized in the presence of resetting, but the form remains invariant (see
Eq.(13) and Eq. (24)).

Figure 3: Ratio of variances $K_{r}$ as a function of resetting rate. The
anisotropy grows with resetting rate, and saturates in the $r\to\infty$ limit.

###  III.1 Perturbative steady state for near-symmetric particles

While we can derive exact expressions for the lower-order moments as discussed
above, we would also like to gain insights into the overall shape of the non-
equilibrium steady state under resetting. To achieve this, we consider for
simplicity a perturbative approach where we calculate the propagator in the
absence of resetting, and then use the renewal approach to resetting in order
to find the steady state.

In the absence of resetting, the dynamics can be described by the
Smoluchowski-Perrin equation

$\partial_{t}P_{0}(X,t|X_{0})=D_{\theta}\partial_{\theta}^{2}P_{0}(X,t|X_{0})+\nabla\cdot\left[\mathbb{D}\cdot\nabla\right]P_{0}(X,t|X_{0}),$
(26)

where $\mathbb{D}$ is the diffusion tensor
$\mathbb{D}_{ij}=k_{B}T\Gamma_{ij}=D_{||}n_{i}n_{j}+D_{\perp}(\delta_{ij}-n_{i}n_{j})$with
$\bm{n}=(\cos\theta,\sin\theta)$ the unit vector pointing along the particle’s
major axis. Performing a Fourier transform,

$\hat{P}_{0}(\bm{k},\theta,t|\theta_{0})\equiv\int
d\bm{x}e^{-i\bm{k}\cdot\bm{x}}P_{0}(\bm{x},\theta,t|\theta_{0}),$ (27)

we find the transformed equation

$\partial_{t}\hat{P}_{0}(\bm{k},\theta,t|\theta_{0})=D_{\theta}\partial_{\theta}^{2}\hat{P}_{0}(\bm{k},\theta,t|\theta_{0})+\bm{k}\cdot\left[\mathbb{D}\cdot\bm{k}\right]\hat{P}_{0}(\bm{k},\theta,t|\theta_{0}).$
(28)

For simplicity, we here consider the steady state in the $x$-direction. Hence,
we consider a wave vector $\bm{k}=k\bm{e}_{x}$, with $\bm{e}_{x}$ the unit
vector in the $x$-direction. The marginal distribution in the $x$ direction
satisfies [77]

$\partial_{t}P_{0}(k,\theta,t|\theta_{0})=\left[D_{r}\partial_{\theta}^{2}-k^{2}\left(\bar{D}+\frac{\Delta
D}{2}\cos(2\theta)\right)\right]P_{0}(k,\theta,t|\theta_{0})$ (29)

A natural interpretation of this equation is that the effectively experienced
diffusivity in the $x$-direction $\bar{D}+\frac{\Delta D}{2}\cos(2\theta)$
depends on the orientation of the particle. To find an analytical solution
that provides some insight into the shape of the steady state under resetting,
we consider a perturbative approach, where we expand in power of the
dimensionless asymmetry parameter $\varepsilon\equiv\Delta D/\bar{D}$. We
write the marginal Smoluchowski-Perrin equation as

$\partial_{t}P_{0}(k,\theta,t|\theta_{0})=D_{r}\partial_{\theta}^{2}\hat{P}(k,\theta,t|\theta_{0})-k^{2}\bar{D}\hat{P}_{0}-\varepsilon\>k^{2}\bar{D}\frac{\cos(2\theta)}{2}\hat{P}_{0}(k,\theta,t|\theta_{0})$
(30)

We expand the solution in powers of the asymmetry as

$P_{0}(k,\theta,t|\theta_{0})=\sum_{n=0}^{\infty}p_{n}(k,\theta,t|\theta_{0})\varepsilon^{n}.$
(31)

where $p_{n}(k,\theta,t|\theta_{0})$ is the coefficient function of the power
series. Inserting this ansatz into the Smoluchowski-Perrin equation gives the
coupled equations

$\displaystyle\partial_{t}p_{0}(k,\theta,t|\theta_{0})$
$\displaystyle=D_{r}\partial_{\theta}^{2}p_{0}(k,\theta,t)-k^{2}\bar{D}p_{0}(k,\theta,t|\theta_{0}),$
(32) $\displaystyle\partial_{t}p_{n}(k,\theta,t|\theta_{0})$
$\displaystyle=-[D_{r}\partial_{\theta}^{2}+k^{2}\bar{D}]p_{n}(k,\theta,t|\theta_{0})-\frac{k^{2}\bar{D}\cos(2\theta)}{2}p_{n-1}(k,\theta,t|\theta_{0})$
(33)

First, we note that $p_{0}(k,\theta,t)$ is nothing but the Gaussian solution
of a symmetric colloid. As we are interested in the spatial steady state to
first order, we integrate over the angular degree of freedom for $n=0,1$
above, resulting in

$\displaystyle\partial_{t}p_{0}(k,t)$ $\displaystyle=-k^{2}\bar{D}p_{0}(k,t),$
(34) $\displaystyle\partial_{t}p_{1}(k,t|\theta_{0})$
$\displaystyle=-k^{2}\bar{D}p_{1}(k,t|\theta_{0})-k^{2}\bar{D}p_{0}(k,t)\int
d\theta\frac{\cos(2\theta)}{2}\frac{\exp\left(-\frac{(\theta-\theta_{0})^{2}}{4D_{\theta}t}\right)}{\sqrt{4\pi
D_{\theta}t}}$ (35)

where we used the fact that the zeroth order solution is Gaussian. Here the
zeroth order solution reads

$p_{0}(k,t)=e^{-\bar{D}k^{2}t}$ (36)

The integration in the equation for the first order correction can be carried
out exactly, yielding

$\partial_{t}p_{1}(k,t|\theta_{0})=-k^{2}\bar{D}p_{1}(k,t|\theta_{0})-k^{2}\bar{D}\cos(2\theta_{0})p_{0}(k,t)\frac{e^{-4D_{\theta}t}}{2}$
(37)

Using Eq. (36) and performing a Laplace transform results in

$\hat{\tilde{p}}_{1}(k,s|\theta_{0})=-\frac{k^{2}\bar{D}\cos(2\theta_{0})}{2(s+4D_{\theta}+\bar{D}k^{2})(s+k^{2}\bar{D})}$
(38)

Inverting the Fourier transform, we finally arrive at the first order
correction to the propagator as

$\tilde{p}_{1}(x,s|\theta_{0})=\cos(2\theta_{0})\frac{\alpha(s)}{16D_{\theta}}e^{-\alpha(s)|x|}-\cos(2\theta_{0})\frac{\alpha(s+4D_{\theta})}{16D_{\theta}}e^{-\alpha(s+4D_{\theta})|x|}$
(39)

where we introduced the inverse lengthscale $\alpha(s)\equiv\sqrt{s/\bar{D}}$.
Since $p_{0}$ takes care of normalization, $p_{1}$ integrates to zero as
required.

Figure 4: Steady state marginal density under complete resetting. a) Zeroth
order solution corresponding to the standard Laplacian solution for a normal
Brownian particle with diffusivity $\bar{D}$. b) First order correction to the
steady state for a Brownian anisotropic particle for various $\theta_{0}$. The
lengthscale is set by $\ell^{2}=2k_{B}T\bar{\Gamma}/D_{\theta}$, and the
parameters are set to $D_{\theta}=\bar{D}=r=1$.  Figure 5: Steady state
marginal density under complete resetting. a) Zeroth order solution at
different resetting rates. b) First order correction to the steady state for a
Brownian anisotropic particle for various resetting rates.The lengthscale is
set by $\ell^{2}=2k_{B}T\bar{\Gamma}/D_{\theta}$, and the parameters are set
to $D_{\theta}=\bar{D}=r=1$, $\theta_{0}=0$.

We can now easily obtain the steady state under the effect of resetting by
using the renewal equation Eq. (20), and letting $t\to\infty$. This
immediately yields

$P_{r}(x|\theta_{0})=\frac{\alpha(r)}{2}e^{-\alpha(r)|x|}+\varepsilon\cos(2\theta_{0})\left[\frac{r\alpha(r)}{16D_{\theta}}e^{-\alpha(r)|x|}-\frac{r\alpha(r+4D_{\theta})}{16D_{\theta}}e^{-\alpha(r+4D_{\theta})|x|}\right].$
(40)

Several things are worth noting. The zeroth order solution of the steady state
probability distribution is determined by a single length scale $\alpha(r)$
set by resetting. It shows a peak at $x=0$ in Fig. (4 a) due to the confining
effect of the resetting towards the origin. This peak value increases with
increasing the resetting rate $r$, as is expected. Once particle asymmetry is
introduced, the solution also depends on the length scale set by
$\alpha(r+4D_{\theta})$, and in addition depends on the initial orientation
$\theta_{0}$. Here, the rotational dynamics enter into the spatial steady
state due to the coupling between the translational and rotational degrees of
freedom, which persists to late times as discussed above. Fig. (4 b) shows
that the first order correction to the distribution is negative at $x=0$ for
$\theta_{0}=0$, which is consistent with the enhanced motion along the $x$
direction; hence the steady state widens in this case. When
$\theta_{0}=\pi/2$, the long axis of the anisotropic particle will be along
the $y$ axis, resulting in the suppression of the mobility along the $x$
direction, and consequently, the first order correction becomes positive at
$x=0$. The peak value of the first-order correction also increases with the
resetting rate $r$, as seen in Fig. (5). This is consistent with the
observation that the degree of anisotropy $K_{r}$ increases with increasing
resetting rate $r$.

##  IV Translational resetting

Next, we consider the case where $x$ and $y$ undergo resets while the
orientational degree of freedom $\theta$ variable does not reset. Following
the discussion in section II.2, we have the renewal equation

$\begin{split}P_{r}(x,y,\theta,t|0,0,\theta_{0})&=e^{-rt}P_{0}(x,y,\theta,t|0,0,\theta_{0})\\\
&+r\int_{0}^{t}d\tau e^{-r\tau}\int dx^{\prime}\int dy^{\prime}\int
d\theta^{\prime}P_{r}(x^{\prime},y^{\prime},\theta^{\prime},t-\tau|0,0,\theta_{0})P_{0}(x,y,\theta,\tau|0,0,\theta^{\prime})\end{split}$
(41)

The propagator for $x$ for this case can be obtained by integrating the above
equation over $y$, $\theta$, $x^{\prime}$ and $y^{\prime}$,

$\begin{split}P_{r}(x,t|0,\theta_{0})&=e^{-rt}P_{0}(x,t|0,\theta_{0})\\\
&+r\int_{0}^{t}d\tau e^{-r\tau}\int
d\theta^{\prime}P_{0}(\theta^{\prime},t-\tau|\theta_{0})P_{0}(x,\tau|0,\theta^{\prime})\end{split}$
(42)

The corresponding moment for $x$ in that case would be

$\begin{split}{\langle x^{n}(t)|\theta_{0}\rangle}_{r}=e^{-rt}{\langle
x^{n}(t)|\theta_{0}\rangle}_{0}+r\int_{0}^{t}d\tau e^{-r\tau}\int
d\theta^{\prime}P_{0}(\theta^{\prime},t-\tau|\theta_{0}){\langle
x^{n}(\tau)|\theta^{\prime}\rangle}\end{split}$ (43)

Next, we calculate the second moment of $x$ for the anisotropic particle using
equation (8) and $n=2$ in equation (43), resulting in

$\begin{split}{\langle
x^{2}(t)|\theta_{0}\rangle}_{r}&=e^{-rt}2k_{B}T\left[\bar{\Gamma}t+\frac{\Delta{\Gamma}}{2}\cos
2\theta_{0}\left(\frac{1-e^{-4D_{\theta}t}}{4D_{\theta}}\right)\right]\\\
&+r\int_{0}^{t}d\tau e^{-r\tau}\int
d\theta^{\prime}\frac{e^{-\frac{\left(\theta^{\prime}-\theta_{0}\right)^{2}}{4D_{\theta}(t-\tau)}}}{\sqrt{4\pi
D_{\theta}(t-\tau)}}2k_{B}T\left[\bar{\Gamma}\tau+\frac{\Delta{\Gamma}}{2}\cos
2\theta^{\prime}\left(\frac{1-e^{-4D_{\theta}\tau}}{4D_{\theta}}\right)\right]\\\
&=\frac{2k_{B}T\bar{\Gamma}}{r}-\left(\frac{2k_{B}T\bar{\Gamma}}{r}-\frac{k_{B}T\Delta\Gamma\cos
2\theta_{0}}{4D_{\theta}-r}\right)e^{-rt}-\frac{k_{B}T\Delta\Gamma\cos
2\theta_{0}}{4D_{\theta}-r}e^{-4D_{\theta}t}\end{split}$ (44)

Similarly, one can also calculate ${\langle y^{2}(t)|\theta_{0}\rangle}_{r}$
and the cross-correlation ${\langle x(t)y(t)|\theta_{0}\rangle}_{r}$ in the
presence of resetting. We find

$\begin{split}{\langle
y^{2}(t)|\theta_{0}\rangle}_{r}=\frac{2k_{B}T\bar{\Gamma}}{r}-\left(\frac{2k_{B}T\bar{\Gamma}}{r}+\frac{k_{B}T\Delta\Gamma\cos
2\theta_{0}}{4D_{\theta}-r}\right)e^{-rt}+\frac{k_{B}T\Delta\Gamma\cos
2\theta_{0}}{4D_{\theta}-r}e^{-4D_{\theta}t}\end{split}$ (45)

$\begin{split}{\langle
x(t)y(t)|\theta_{0}\rangle}_{r}&=\frac{2k_{B}T\Delta\Gamma\sin
2\theta_{0}}{2(4D_{\theta}-r)}\left(e^{-rt}-e^{-4D_{\theta}t}\right)\end{split}$
(46)

Note that in the limit $r\to 4D_{\theta}$, the above equations remain well-
defined if the limit is taken carefully. In the long time limit, both
${\langle x^{2}(t)|\theta_{0}\rangle}_{r}$ and ${\langle
y^{2}(t)|\theta_{0}\rangle}_{r}$ approach $\frac{2k_{B}T\bar{\Gamma}}{r}$, and
the steady state ratio of the variances, $K_{r}$ becomes $1$. The cross-
correlation ${\langle x(t)y(t)|\theta_{0}\rangle}_{r}$ will be zero in the
long time limit, but at intermediate times, ${\langle
x(t)y(t)|\theta_{0}\rangle}_{r}$ shows a non-monotonic behaviour against time.
The initial growth is similar to what is seen in the absence of resetting,
while resets exponentially decrease the correlations at later times.
Therefore, as the state at late time has no cross-correlations and no
anisotropy, the steady-state distribution for $x$ will become
$P_{r}(x)=\frac{\alpha(r)}{2}\exp\left(-\alpha(r)|x|\right)$, with
$\alpha(r)=\sqrt{r/\bar{D}}$, which is independent of $D_{\theta}$. Due to
isotropy at late time, the distribution in the $y$ direction will be
identical.

##  V Orientational resetting

Finally, we consider the case of orientational resetting, where $\theta$
undergoes resets while the $x$ and $y$ variables do not undergo resets. Again,
we follow the general discussion in section II.2, leading to the last renewal
equation

$\begin{split}P_{r}(x,y,\theta,t|0,0,\theta_{0})&=e^{-rt}P_{0}(x,y,\theta,t|0,0,\theta_{0})\\\
&+r\int_{0}^{t}d\tau e^{-r\tau}\int dx^{\prime}\int dy^{\prime}\int
d\theta^{\prime}P_{r}(x^{\prime},y^{\prime},\theta^{\prime},t-\tau|0,0,\theta_{0})P_{0}(x,y,\theta,\tau|x^{\prime},y^{\prime},\theta_{0})\end{split}$
(47)

At the time of the last resetting $t-\tau$, the system is in some arbitrary
state $\left(x^{\prime},y^{\prime},\theta^{\prime}\right)$. As only $\theta$
is reset to $\theta_{0}$, the system proceeds to evolve towards
$\left(x,y,\theta\right)$ from the new initial condition
$\left(x^{\prime},y^{\prime},\theta_{0}\right)$. The propagator for $x$ for
this case can be obtained by integrating the above equation over $y$,
$y^{\prime}$ $\theta$, and $\theta^{\prime}$,

$\begin{split}P_{r}(x,t|0,\theta_{0})&=e^{-rt}P_{0}(x,t|0,\theta_{0})\\\
&+r\int_{0}^{t}d\tau e^{-r\tau}\int
dx^{\prime}P_{r}(x^{\prime},t-\tau|0,\theta_{0})P_{0}(x,\tau|x^{\prime},\theta_{0})\end{split}$
(48)

In the case of complete and translational resetting, the renewal equations for
the spatial moments could be expressed explicitly in terms of the
corresponding densities without resetting. However, in the present case the
probability density $P_{r}(x,t|0,\theta_{0})$ is not given explicitly, as it
appears both on the left-hand-side and inside the integral on the right-hand-
side of the renewal equation.

In order to calculate the late-time moments, one can decouple the equation
through the use of Fourier transforms in space and a Laplace transform in time
under the assumption of spatial homogeneity. Recently, Ref. [72] studied the
late-time behaviour of the moments for this class of problems, where it was
shown that the effective diffusion coefficients take the form

$\displaystyle D_{\text{eff}}^{(x)}$
$\displaystyle=\lim_{t\to\infty}\frac{{\langle
x^{2}(t)\rangle}_{r}}{2t}=\frac{r^{2}}{2}\mathcal{L}_{r}[\langle
x^{2}(t)|\theta_{0}\rangle_{0}]$ (49) $\displaystyle D_{\text{eff}}^{(y)}$
$\displaystyle=\lim_{t\to\infty}\frac{{\langle
y^{2}(t)\rangle}_{r}}{2t}=\frac{r^{2}}{2}\mathcal{L}_{r}[\langle
y^{2}(t)|\theta_{0}\rangle_{0}]$ (50)

where $\mathcal{L}_{r}$ denotes a Laplace transform evaluated at $r$. Using
the mean squared displacement for the problem without resetting, we
immediately arrive at

$\displaystyle D_{\text{eff}}^{(x)}$
$\displaystyle=k_{B}T\left(\bar{\Gamma}+\frac{r\Delta\Gamma\cos
2\theta_{0}}{2(r+4D_{\theta})}\right)$ (51) $\displaystyle
D_{\text{eff}}^{(y)}$
$\displaystyle=k_{B}T\left(\bar{\Gamma}-\frac{r\Delta\Gamma\cos
2\theta_{0}}{2(r+4D_{\theta})}\right)$ (52)

In order to obtain a full time-dependent solution of the moments, we make use
of the fact that the positions are themselves not coupled, but determined
solely by the orientation $\theta$. Hence, the mean squared displacement takes
the same mathematical form as in the case of no resetting, Eq. (8), but now
with the orientational dynamics subject to resets:

$\begin{split}{\langle
x^{2}(t)|\theta_{0}\rangle}_{r}=2k_{B}T\int_{0}^{t}dt^{\prime}\left[\bar{\Gamma}+\frac{\Delta{\Gamma}}{2}{\langle\cos\theta\left(t^{\prime}\right)|\theta_{0}\rangle}_{r}\right]\end{split}$
(53)

To calculate
${\langle\cos\theta\left(t^{\prime}\right)|\theta_{0}\rangle}_{r}$ in the
presence of resetting, we use the last renewal equation only for $\theta$,
which takes the simple form

$\begin{split}P_{r}(\theta,t|\theta_{0})=e^{-rt}P_{0}(\theta,t|\theta_{0})+r\int_{0}^{t}d\tau
e^{-r\tau}P_{0}(\theta,\tau|\theta_{0})\end{split}$ (54)

Multiplying with $\cos 2\theta$ and integrating results in

$\begin{split}{\langle\cos
2\theta(t)|\theta_{0}\rangle}_{r}&=e^{-rt}{\langle\cos
2\theta(t)|\theta_{0}\rangle}_{0}+r\int_{0}^{t}d\tau e^{-r\tau}{\langle\cos
2\theta(\tau)|\theta_{0}\rangle}_{0}\\\
&=\cos(2\theta_{0})e^{-\left(r+4D_{\theta}\right)t}+\frac{r\cos
2\theta_{0}}{r+4D_{\theta}}\left(1-e^{-\left(r+4D_{\theta}\right)t}\right)\end{split}$
(55)

Using Eq. (53), we find

$\begin{split}{\langle
x^{2}(t)|\theta_{0}\rangle}_{r}&=2k_{B}T\int_{0}^{t}dt^{\prime}\left[\bar{\Gamma}+\frac{\Delta{\Gamma}}{2}{\langle\cos\theta\left(t^{\prime}\right)|\theta_{0}\rangle}_{r}\right]\\\
&=2k_{B}T\left[\left(\bar{\Gamma}+\frac{r\Delta\Gamma\cos
2\theta_{0}}{2(r+4D_{\theta})}\right)t+\frac{2D_{\theta}\Delta\Gamma\cos
2\theta_{0}}{\left(r+4D_{\theta}\right)^{2}}\left(1-e^{-\left(r+4D_{\theta}\right)t}\right)\right]\end{split}$
(56)

Figure 6: Dynamical exponents $\zeta_{x}(t)$ and $\zeta_{y}(t)$ for the mean
squared displacement in the $x$ and $y$ direction for various values of
resetting rate, in the case of orientational resets. The black line shows the
exponents without resetting. Parameters are set to $\bar{\Gamma}=1.0$,
$\Delta\Gamma=0.001$, with initial orientation $\theta_{0}=0$.

Similarly, one can also calculate ${\langle y^{2}(t)|\theta_{0}\rangle}_{r}$
and the cross-correlation ${\langle x(t)y(t)|\theta_{0}\rangle}_{r}$ in the
presence of resetting, resulting in

$\begin{split}{\langle
y^{2}(t)|\theta_{0}\rangle}_{r}=2k_{B}T\left[\left(\bar{\Gamma}-\frac{r\Delta\Gamma\cos
2\theta_{0}}{2(r+4D_{\theta})}\right)t-\frac{2D_{\theta}\Delta\Gamma\cos
2\theta_{0}}{\left(r+4D_{\theta}\right)^{2}}\left(1-e^{-\left(r+4D_{\theta}\right)t}\right)\right]\end{split}$
(57)

$\begin{split}{\langle
x(t)y(t)|\theta_{0}\rangle}_{r}&=2k_{B}T\int_{0}^{t}dt^{\prime}\left[\frac{\Delta{\Gamma}}{2}{\langle\sin\theta\left(t^{\prime}\right)|\theta_{0}\rangle}_{r}\right]\\\
&=2k_{B}T\left[\frac{r\Delta\Gamma\sin
2\theta_{0}}{2\left(r+4D_{\theta}\right)}t+\frac{2D_{\theta}\Delta\Gamma\sin
2\theta_{0}}{\left(r+4D_{\theta}\right)^{2}}\left(1-e^{-\left(r+4D_{\theta}\right)t}\right)\right]\end{split}$
(58)

In the long time limit, since all expressions grow linearly in time, we can
neglect the constant offset in the above equations, and the variances of the
particle’s position in the $xy$ plane and the cross-correlations between $x$
and $y$ can be written as

$\begin{split}{\langle x^{2}(t)|\theta_{0}\rangle}_{r}\approx
2k_{B}T\left(\bar{\Gamma}+\frac{r\Delta\Gamma\cos
2\theta_{0}}{2(r+4D_{\theta})}\right)t\end{split}$ (59)

$\begin{split}{\langle y^{2}(t)|\theta_{0}\rangle}_{r}\approx
2k_{B}T\left(\bar{\Gamma}-\frac{r\Delta\Gamma\cos
2\theta_{0}}{2(r+4D_{\theta})}\right)t\end{split}$ (60)

which agrees with the predictions made in Eqs. (51) and (52). The transient
behaviour can once again be understood by considering the dynamical exponents
of the motion in the $x$ and $y$ directions. Fig (6) shows these exponents for
the case $\theta_{0}=0$. In this case, the motion in the $x$ direction
undergoes a subdiffusive regime before re-entering a diffusing regime at late
times. The dynamics in the $y$ direction conversely goes through
superdiffusive regime, before also re-entering a diffusive regime.

In contrast to the other resetting schemes considered so far, we now also have
a cross correlation that at late times grows as

$\begin{split}{\langle x(t)y(t)|\theta_{0}\rangle}_{r}\approx
2k_{B}T\frac{r\Delta\Gamma\sin
2\theta_{0}}{2\left(r+4D_{\theta}\right)}t\end{split}$ (61)

Linear growth of the cross correlations has also been seen, for example, in
multithermostat Brownian systems under the effect of a Lorentz force [78].

The steady state ratio of the variances, $K_{r}$, in the presence of resetting
is given as

$\begin{split}K_{r}=\frac{{\langle x^{2}(t)|\theta_{0}\rangle}_{r}}{{\langle
y^{2}(t)|\theta_{0}\rangle}_{r}}=\frac{\left(\bar{\Gamma}+\frac{r\Delta\Gamma\cos
2\theta_{0}}{2(r+4D_{\theta})}\right)}{\left(\bar{\Gamma}-\frac{r\Delta\Gamma\cos
2\theta_{0}}{2(r+4D_{\theta})}\right)}\end{split}$ (62)

Figure 7: Fixed-time ($tD_{\theta}=4$) two-dimensional positional densities
under orientational resetting at different rates: $r=0$ (panel a),
$r=2D_{\theta}$ (panel b) and $r=5D_{\theta}$ (panel c). Lengthscale is set by
$\ell^{2}=2k_{B}T\bar{\Gamma}/D_{\theta}$. The dashed line shows initial angle
$\theta_{0}$. Parameters are set to
$\theta_{0}=\pi/8,D_{\theta}=1.\Gamma_{||}=1$, $\Gamma_{\perp}=0.001$.

Using $\bar{D}=k_{B}T\bar{\Gamma}$, we see that this is exactly the degree of
anisotropy as in the case of complete resetting. While the degree of
anisotropy is the same, the dynamics is different. In particular, we obtain a
non-zero cross-correlation between $x$ and $y$ which also grows linearly with
time, while for complete resetting the same cross correlations saturates.

In the long time limit, we can expect that the two-dimensional probability
distribution in the Cartesian components will be Gaussian due to the central
limit theorem, given as

$\begin{split}P(x,y,t)=\frac{1}{4\pi
t\sqrt{\textrm{Det}(\mathbb{D}_{\text{eff}})}}\exp\left[-\frac{1}{4t}\left(x,y\right)^{\textrm{T}}\mathbb{D}_{\text{eff}}^{-1}\left(x,y\right)\right],\end{split}$
(63)

where the effective diffusion tensor is given as

$\mathbb{D}_{\text{eff}}=2k_{B}T\left[\begin{array}[]{cc}\bar{\Gamma}+\frac{r\Delta\Gamma\cos
2\theta_{0}}{2(r+4D_{\theta})}&\frac{r\Delta\Gamma\sin
2\theta_{0}}{2\left(r+4D_{\theta}\right)}\\\ \frac{r\Delta\Gamma\sin
2\theta_{0}}{2\left(r+4D_{\theta}\right)}&\bar{\Gamma}-\frac{r\Delta\Gamma\cos
2\theta_{0}}{2(r+4D_{\theta})}\end{array}\right].$ (64)

The marginal probability distribution for $x$ can be obtained by integrating
the $y$ variable, given as

$\begin{split}P(x,t)=\int dyP(x,y,t)=\frac{1}{\sqrt{4\pi
D_{\text{eff}}^{(x)}t}}\exp\left[-\frac{x^{2}}{4D_{\text{eff}}^{(x)}t}\right]\end{split}$
(65)

The same argument holds for the marginal probability distribution for $y$.
However, the long-time probability distribution in equation (63) cannot be
separated into a product of two independent distributions in the $x$ and $y$
directions. Fig. (7) shows the density plots of the steady state probability
distribution of the position of the particle. The steady state probability
distribution is isotropic without resetting, and the degree of anisotropy
grows along the direction of initial angle $\theta_{0}$ (dashed line in Fig.
(7) ).

It is worth considering a somewhat more intuitive approach to understand the
unboundedness of the cross-correlation between $x$ and $y$ with time. We
consider the correlation coefficient defined as $C(t)=\frac{\langle
x(t)y(t)|\theta_{0}\rangle}{\sqrt{\langle
x^{2}(t)|\theta_{0}\rangle}\sqrt{\langle y^{2}(t)|\theta_{0}\rangle}}$. In the
presence of only orientational resetting, $C(t)$ will become constant in the
long-time limit, as can be readily verified by using the above results. This
is similar to the case of the resetting all the variable, where $C(t)$ will
also be a constant, which is an immediate consequence of the fact that a
steady state is reached in this case. This non-zero correlation between the
two spatial directions results from the presence of orientational resetting,
while for translational resetting, $C(t)$ will be zero in the long-time limit.

##  VI Discussion

In this article, we have explored the effects of shape asymmetry on the two-
dimensional diffusive motion of an anisotropic particle in the presence of
stochastic resetting. In particular, we studied the short- and long-time
behaviour of moments, cross-correlations and steady-state distributions under
various resetting protocols. It is well known that anisotropic diffusion
persists only for very short times, and that isotropic diffusion is recovered
at long-times. Hence, for practical purposes the short-time behaviour can
often be neglected and the movement of an asymmetric particle can be described
by the Langevin equations for a point particle with an isotropic translational
diffusion coefficient given by the average of the diffusion coefficients along
the major and minor axes of the particle. This has been shown to be the case
for example for a free particle and in a presence of a harmonic confinement in
two dimensions [25].

We have shown that these assumptions are generally not valid for resetting
schemes that include orientational resets, and can be restored only if we
apply resetting of the translational degrees of freedom. In this case, the
steady-state distribution, moments and cross-correlations are exactly equal to
the case of a spherical particle in the presence of stochastic resetting. On
the other hand, when orientational resetting takes place, the short-time
anisotropy of the motion is promoted to late times. Both the case of complete
resetting and orientational resetting display this, with the degree of
anisotropy $K_{r}$ being the same in the two cases. For complete resetting, we
calculated both time-dependent moments and a perturbative steady state. For
orientational resetting, we calculated the effective (anisotropic) diffusion
tensor, which determines the effective Gaussian behaviour at late times.

In future works, it would be interesting to study the first passage problem
for an anisotropic diffusive particle in a two-dimensions in the presence of
resetting. It would also be interesting to study resetting of other asymmetric
particles, or to study the dynamics of resetting colloids in the presence of
interactions. Intriguing effects of resetting could also be explored in three
dimensions, both for rod-like particle with one orientational degree of
freedom and for full anisotropic particles [79].

###### Acknowledgements.

SC acknowledges support from the Alexander Von Humboldt foundation. KSO and HL
acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) within the
project LO 418/29-1.

## References

  * Bawden  _et al._ [1936] F. Bawden, N. Pirie, J. Bernal, and I. Fankuchen, Liquid crystalline substances from virus-infected plants, Nature 138, 1051 (1936).
  * Wen  _et al._ [1989] X. Wen, R. B. Meyer, and D. Caspar, Observation of smectic-a ordering in a solution of rigid-rod-like particles, Physical Review Letters 63, 2760 (1989).
  * Graf and Löwen [1999] H. Graf and H. Löwen, Phase diagram of tobacco mosaic virus solutions, Physical Review E 59, 1932 (1999).
  * Doostmohammadi  _et al._ [2016] A. Doostmohammadi, S. P. Thampi, and J. M. Yeomans, Defect-mediated morphologies in growing cell colonies, Physical Review Letters 117, 048102 (2016).
  * Gray  _et al._ [1984] C. G. Gray, K. E. Gubbins, and C. G. Joslin,  _Theory of Molecular Fluids: Volume 2: Applications_ , Vol. 2 (International Monographs on Ch, 1984).
  * Narayan  _et al._ [2007] V. Narayan, S. Ramaswamy, and N. Menon, Long-lived giant number fluctuations in a swarming granular nematic, Science 317, 105 (2007).
  * Ivlev  _et al._ [2012] A. Ivlev, G. Morfill, H. Lowen, and C. P. Royall,  _Complex plasmas and colloidal dispersions: particle-resolved studies of classical liquids and solids_ , Vol. 5 (World Scientific Publishing Company, 2012).
  * Löwen [1994] H. Löwen, Brownian dynamics of hard spherocylinders, Physical Review E 50, 1232 (1994).
  * Bolhuis and Frenkel [1997] P. Bolhuis and D. Frenkel, Tracing the phase boundaries of hard spherocylinders, The Journal of Chemical Physics 106, 666 (1997).
  * Dijkstra [2014] M. Dijkstra, Entropy-driven phase transitions in colloids: From spheres to anisotropic particles, Advances in Chemical Physics 156, 35 (2014).
  * Löwen [1999] H. Löwen, Anisotropic self-diffusion in colloidal nematic phases, Physical Review E 59, 1989 (1999).
  * Roller  _et al._ [2021] J. Roller, A. Laganapan, J.-M. Meijer, M. Fuchs, and A. Zumbusch, Observation of liquid glass in suspensions of ellipsoidal colloids, Proceedings of the National Academy of Sciences 118, e2018072118 (2021).
  * Wu  _et al._ [2018] N. Wu, D. Lee, and A. Striolo,  _Anisotropic Particle Assemblies: Synthesis, Assembly, Modeling, and Applications_ (Elsevier, 2018).
  * Glotzer and Solomon [2007] S. C. Glotzer and M. J. Solomon, Anisotropy of building blocks and their assembly into complex structures, Nature Materials 6, 557 (2007).
  * Sacanna and Pine [2011] S. Sacanna and D. J. Pine, Shape-anisotropic colloids: Building blocks for complex assemblies, Current Opinion in Colloid & Interface Science 16, 96 (2011).
  * Thorkelsson  _et al._ [2015] K. Thorkelsson, P. Bai, and T. Xu, Self-assembly and applications of anisotropic nanomaterials: A review, Nano Today 10, 48 (2015).
  * Simeonidis  _et al._ [2016] K. Simeonidis, M. P. Morales, M. Marciello, M. Angelakeris, P. de La Presa, A. Lazaro-Carrillo, A. Tabero, A. Villanueva, O. Chubykalo-Fesenko, and D. Serantes, In-situ particles reorientation during magnetic hyperthermia application: Shape matters twice, Scientific Reports 6, 38382 (2016).
  * Perrin [1934] F. Perrin, Mouvement brownien d’un ellipsoide-i. dispersion diélectrique pour des molécules ellipsoidales, Journal de Physique et le Radium 5, 497 (1934).
  * Perrin [1936] F. Perrin, Mouvement brownien d’un ellipsoide (ii). rotation libre et dépolarisation des fluorescences. translation et diffusion de molécules ellipsoidales, Journal de Physique et le Radium 7, 1 (1936).
  * Koenig [1975] S. H. Koenig, Brownian motion of an ellipsoid. a correction to perrin’s results, Biopolymers: Original Research on Biomolecules 14, 2421 (1975).
  * Han  _et al._ [2006] Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, Brownian motion of an ellipsoid, Science 314, 626 (2006).
  * Bicout and Field [1996] D. J. Bicout and M. J. Field, Stochastic dynamics simulations of macromolecular diffusion in a model of the cytoplasm of escherichia coli, The Journal of Physical Chemistry 100, 2489 (1996).
  * Schnell and Turner [2004] S. Schnell and T. Turner, Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws, Progress in Biophysics and Molecular Biology 85, 235 (2004).
  * Han  _et al._ [2009] Y. Han, A. Alsayed, M. Nobili, and A. G. Yodh, Quasi-two-dimensional diffusion of single ellipsoids: Aspect ratio and confinement effects, Physical Review E 80, 011403 (2009).
  * Grima and Yaliraki [2007] R. Grima and S. Yaliraki, Brownian motion of an asymmetrical particle in a potential field, The Journal of Chemical Physics 127 (2007).
  * Levernier  _et al._ [2016] N. Levernier, O. Bénichou, and R. Voituriez, Mean first-passage time of an anisotropic diffusive searcher, Journal of Physics A: Mathematical and Theoretical 50, 024001 (2016).
  * Ghosh  _et al._ [2022] A. Ghosh, S. Mandal, and D. Chakraborty, Persistence of an active asymmetric rigid Brownian particle in two dimensions, The Journal of Chemical Physics 157 (2022).
  * Ghosh and Chakraborty [2020] A. Ghosh and D. Chakraborty, Persistence in Brownian motion of an ellipsoidal particle in two dimensions, The Journal of Chemical Physics 152 (2020).
  * Anchutkin  _et al._ [2024] G. Anchutkin, V. Holubec, and F. Cichos, Run-and-tumble motion of ellipsoidal swimmers, arXiv preprint arXiv:2402.04697 (2024).
  * Shemi and Solomon [2018] O. Shemi and M. J. Solomon, Self-propulsion and active motion of Janus ellipsoids, The Journal of Physical Chemistry B 122, 10247 (2018).
  * Marino  _et al._ [2016] R. Marino, R. Eichhorn, and E. Aurell, Entropy production of a Brownian ellipsoid in the overdamped limit, Physical Review E 93, 012132 (2016).
  * Evans and Majumdar [2011a] M. R. Evans and S. N. Majumdar, Diffusion with stochastic resetting, Physical Review Letters 106, 160601 (2011a).
  * Evans and Majumdar [2011b] M. R. Evans and S. N. Majumdar, Diffusion with optimal resetting, Journal of Physics A: Mathematical and Theoretical 44, 435001 (2011b).
  * Evans  _et al._ [2020] M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic resetting and applications, Journal of Physics A: Mathematical and Theoretical 53, 193001 (2020).
  * Majumdar  _et al._ [2015] S. N. Majumdar, S. Sabhapandit, and G. Schehr, Dynamical transition in the temporal relaxation of stochastic processes under resetting, Physical Review E 91, 052131 (2015).
  * Gupta [2019] D. Gupta, Stochastic resetting in underdamped Brownian motion, Journal of Statistical Mechanics: Theory and Experiment 2019, 033212 (2019).
  * Pal [2015] A. Pal, Diffusion in a potential landscape with stochastic resetting, Physical Review E 91, 012113 (2015).
  * Nagar and Gupta [2016] A. Nagar and S. Gupta, Diffusion with stochastic resetting at power-law times, Physical Review E 93, 060102 (2016).
  * Reuveni [2016] S. Reuveni, Optimal stochastic restart renders fluctuations in first passage times universal, Physical Review Letters 116, 170601 (2016).
  * Besga  _et al._ [2020a] B. Besga, A. Bovon, A. Petrosyan, S. N. Majumdar, and S. Ciliberto, Optimal mean first-passage time for a Brownian searcher subjected to resetting: experimental and theoretical results, Physical Review Research 2, 032029 (2020a).
  * Faisant  _et al._ [2021] F. Faisant, B. Besga, A. Petrosyan, S. Ciliberto, and S. N. Majumdar, Optimal mean first-passage time of a Brownian searcher with resetting in one and two dimensions: experiments, theory and numerical tests, Journal of Statistical Mechanics: Theory and Experiment 2021, 113203 (2021).
  * Pal  _et al._ [2020] A. Pal, Ł. Kuśmierz, and S. Reuveni, Search with home returns provides advantage under high uncertainty, Physical Review Research 2, 043174 (2020).
  * Durang  _et al._ [2019] X. Durang, S. Lee, L. Lizana, and J.-H. Jeon, First-passage statistics under stochastic resetting in bounded domains, Journal of Physics A: Mathematical and Theoretical 52, 224001 (2019).
  * Tucci  _et al._ [2022] G. Tucci, A. Gambassi, S. N. Majumdar, and G. Schehr, First-passage time of run-and-tumble particles with noninstantaneous resetting, Physical Review E 106, 044127 (2022).
  * Tal-Friedman  _et al._ [2020a] O. Tal-Friedman, A. Pal, A. Sekhon, S. Reuveni, and Y. Roichman, Experimental realization of diffusion with stochastic resetting, The Journal of Physical Chemistry Letters, 11, 7350 (2020a).
  * Toledo-Marin and Boyer [2022] J. Q. Toledo-Marin and D. Boyer, First passage time and information of a one-dimensional Brownian particle with stochastic resetting to random positions, arXiv preprint arXiv:2206.14387 (2022).
  * Ahmad  _et al._ [2022] S. Ahmad, K. Rijal, and D. Das, First passage in the presence of stochastic resetting and a potential barrier, arXiv preprint arXiv:2202.03766 (2022).
  * Ahmad  _et al._ [2019] S. Ahmad, I. Nayak, A. Bansal, A. Nandi, and D. Das, First passage of a particle in a potential under stochastic resetting: A vanishing transition of optimal resetting rate, Physical Review E 99, 022130 (2019).
  * Xu  _et al._ [2022] P. Xu, T. Zhou, R. Metzler, and W. Deng, Stochastic harmonic trapping of a lévy walk: transport and first-passage dynamics under soft resetting strategies, New Journal of Physics 24, 033003 (2022).
  * Ray [2020] S. Ray, Space-dependent diffusion with stochastic resetting: A first-passage study, The Journal of Chemical Physics 153, 234904 (2020).
  * Pal and Prasad [2019] A. Pal and V. Prasad, First passage under stochastic resetting in an interval, Physical Review E 99, 032123 (2019).
  * Pal  _et al._ [2022] A. Pal, S. Kostinski, and S. Reuveni, The inspection paradox in stochastic resetting, Journal of Physics A: Mathematical and Theoretical 55, 021001 (2022).
  * Mercado-Vásquez  _et al._ [2022] G. Mercado-Vásquez, D. Boyer, and S. N. Majumdar, Reducing mean first passage times with intermittent confining potentials: a realization of resetting processes, Journal of Statistical Mechanics: Theory and Experiment 2022, 093202 (2022).
  * Bressloff [2020a] P. C. Bressloff, Directed intermittent search with stochastic resetting, Journal of Physics A: Mathematical and Theoretica 53, 105001 (2020a).
  * Bressloff [2020b] P. C. Bressloff, Modeling active cellular transport as a directed search process with stochastic resetting and delays, Journal of Physics A: Mathematical and Theoretica 53, 355001 (2020b).
  * Fuchs  _et al._ [2016] J. Fuchs, S. Goldt, and U. Seifert, Stochastic thermodynamics of resetting, EPL (Europhysics Letters) 113, 60009 (2016).
  * Pal and Rahav [2017] A. Pal and S. Rahav, Integral fluctuation theorems for stochastic resetting systems, Physical Review E 96, 062135 (2017).
  * Pal  _et al._ [2021] A. Pal, S. Reuveni, and S. Rahav, Thermodynamic uncertainty relation for systems with unidirectional transitions, Physical Review Research 3, 013273 (2021).
  * Gupta  _et al._ [2020] D. Gupta, C. A. Plata, and A. Pal, Work fluctuations and Jarzynski equality in stochastic resetting, Physical Review Letters 124, 110608 (2020).
  * Gupta and Plata [2022] D. Gupta and C. A. Plata, Work fluctuations for diffusion dynamics submitted to stochastic return, New Journal of Physics 24, 113034 (2022).
  * Mori  _et al._ [2023] F. Mori, K. S. Olsen, and S. Krishnamurthy, Entropy production of resetting processes, Physical Review Research 5, 023103 (2023).
  * Olsen  _et al._ [2023] K. S. Olsen, D. Gupta, F. Mori, and S. Krishnamurthy, Thermodynamic cost of finite-time stochastic resetting (2023), arXiv:2310.11267 [cond-mat.stat-mech] .
  * Pal  _et al._ [2023] P. S. Pal, A. Pal, H. Park, and J. S. Lee, Thermodynamic trade-off relation for first passage time in resetting process, arXiv preprint arXiv:2305.04562 (2023).
  * Olsen and Gupta [2024] K. S. Olsen and D. Gupta, Thermodynamic work of partial resetting, Journal of Physics A: Mathematical and Theoretical 57, 245001 (2024).
  * Goerlich  _et al._ [2023] R. Goerlich, M. Li, L. B. Pires, P.-A. Hervieux, G. Manfredi, and C. Genet, Experimental test of landauer’s principle for stochastic resetting, arXiv preprint arXiv:2306.09503 (2023).
  * Kumar  _et al._ [2020] V. Kumar, O. Sadekar, and U. Basu, Active Brownian motion in two dimensions under stochastic resetting, Physical Review E 102, 052129 (2020).
  * Evans and Majumdar [2018] M. R. Evans and S. N. Majumdar, Run and tumble particle under resetting: a renewal approach, Journal of Physics A: Mathematical and Theoretical 51, 475003 (2018).
  * Santra  _et al._ [2020] I. Santra, U. Basu, and S. Sabhapandit, Run-and-tumble particles in two dimensions under stochastic resetting conditions, Journal of Statistical Mechanics: Theory and Experiment 2020, 113206 (2020).
  * Baouche  _et al._ [2024] Y. Baouche, T. Franosch, M. Meiners, and C. Kurzthaler, Active Brownian particle under stochastic orientational resetting (2024), arXiv:2405.06769 [cond-mat.soft] .
  * [70] K. Goswami and R. Chakrabarti, Stochastic resetting and first arrival subjected to Gaussian noise and Poisson white noise Physical Review E 104, 034113 (2021).
  * Abdoli and Sharma [2021] I. Abdoli and A. Sharma, Stochastic resetting of active Brownian particles with lorentz force, Soft Matter 17, 1307 (2021).
  * Olsen and Löwen [2024] K. S. Olsen and H. Löwen, Dynamics of inertial particles under velocity resetting, Journal of Statistical Mechanics: Theory and Experiment 2024, 033210 (2024).
  * Tal-Friedman  _et al._ [2020b] O. Tal-Friedman, A. Pal, A. Sekhon, S. Reuveni, and Y. Roichman, Experimental realization of diffusion with stochastic resetting, The Journal of Physical Chemistry letters 11, 7350 (2020b).
  * Besga  _et al._ [2020b] B. Besga, A. Bovon, A. Petrosyan, S. N. Majumdar, and S. Ciliberto, Optimal mean first-passage time for a Brownian searcher subjected to resetting: experimental and theoretical results, Physical Review Research 2, 032029 (2020b).
  * Besga  _et al._ [2021] B. Besga, F. Faisant, A. Petrosyan, S. Ciliberto, and S. N. Majumdar, Dynamical phase transition in the first-passage probability of a Brownian motion, Physical Review E 104, L012102 (2021).
  * Tierno [2014] P. Tierno, Recent advances in anisotropic magnetic colloids: realization, assembly and applications, Physical Chemistry Chemical Physics 16, 23515 (2014).
  * Mayer  _et al._ [2021] D. B. Mayer, E. Sarmiento-Gómez, M. A. Escobedo-Sánchez, J. P. Segovia-Gutiérrez, C. Kurzthaler, S. U. Egelhaaf, and T. Franosch, Two-dimensional Brownian motion of anisotropic dimers, Physical Review E 104, 014605 (2021).
  * Abdoli  _et al._ [2020] I. Abdoli, E. Kalz, H. D. Vuijk, R. Wittmann, J.-U. Sommer, J. M. Brader, and A. Sharma, Correlations in multithermostat Brownian systems with lorentz force, New Journal of Physics 22, 093057 (2020).
  * Wittkowski and Löwen [2012] R. Wittkowski and H. Löwen, Self-propelled Brownian spinning top: dynamics of a biaxial swimmer at low reynolds numbers, Physical Review E 85, 021406 (2012).