text:
\section{Introduction}
		Active particles consume fuel locally to propel and generate persistent motions \cite{Ramaswamy,Bechinger et al}.  Examples of such self-propelled particles range from humans \cite{Silverberg et al} down to microorganisms \cite{Berg et al,Polin et al} and artificial swimmers \cite{Palacci et al,Paxton et al,Walsh et al}.  The propulsion and, most importantly, the persistence or correlation time of the direction of propulsion are responsible for out-of-equilibrium phenomena such as phase separation without attractive interactions \cite{Buttinoni et al} and preferential motion through funnel-shaped walls \cite{Galajda et al} or around gear-like objects \cite{Leonardo et al, Sokolov et al}.

		When an active particle collides with a solid boundary, it often surfs along the boundary until eventually turning around and propelling away \cite{Galajda et al,Volpe et al,Li Tang}.  This behavior is difficult to analyze mathematically because it often leads to singular behavior at a boundary.  One approach is to treat a system of active particles (without passive diffusion) in the presence of a wall as two coupled populations of particles: those stuck at the wall and those in the bulk \cite{Lee, Wagner et al,Ezhilan et al}.  This results in additional terms in the equations for density that capture the fluxes of particles from the bulk to the wall and similarly, from the wall back into the bulk.  This formulation is related to a class of models known as two-way diffusion equations in which one must specify how particles at the wall reenter the bulk \cite{Fisch Kruskal,Beals,Wagner Beals}.  An alternative approach is to represent boundaries as soft confining potentials, which has been useful for studying the pressure and distribution of active particles \cite{Solon et al pressure,Caprini Marconi,Marconi et al}.  However, compared to passive Brownian particles, for which we have the Boltzmann distribution, it is considerably more difficult to determine the distribution of active particles in arbitrary potentials.

		To gain insight into how activity affects the behavior of active particles near different types of boundaries, we consider the limit of weakly active particles, when passive Brownian diffusion cannot be neglected and activity can be treated pertubatively.  Passive Brownian diffusion due to thermal fluctuations is always present in any physical system.  As we will see from a mathematical perspective, including passive diffusion makes it easier for us to deal with boundaries by allowing us to define familiar Neumann, Dirichlet, or Robin type boundary conditions.  Physically, these may represent impenetrable or absorbing boundaries.  The limit of weak activity is also particularly useful because it allows us to apply perturbation theory to known results in the limit of passive particles or zero activity.

		The paper is structured as follows.  In Section \ref{sec:AOUP}, we introduce and summarize the active particle model of a Brownian particle driven by an exponentially correlated random force.  In Section \ref{sec:method outline}, we show how the Fokker-Planck equation describing the distribution of these active particles can be solved perturbatively by expanding the distribution in powers of the P\'{e}clet number and in terms of Hermite polynomials.  In Sections \ref{sec:cartesian}, \ref{sec:polar}, and \ref{sec:corrugated wall}, we use this approach to study several problems of noninteracting weakly active articles near impenetrable boundaries.  This includes simple confinement in 1D, confinement to a circular or wedge-shaped region, and confinement by a corrugated boundary.  Finally, in Section \ref{sec:spherical}, we consider an absorbing boundary problem of weakly active particles around a spherical absorber.  For each example geometry, we start the section with a brief description of the equations we solve and the boundary conditions we apply to obtain the distribution of active particles.

	\section{\label{sec:AOUP}Active Particles Driven By An Exponentially Correlated Random Force}
		We start by describing a Brownian particle driven by an exponentially correlated random force.  The equation of motion for its position is given by the overdamped Langevin equation
		\begin{equation}
			\gamma\dot{\boldsymbol{r}}=\boldsymbol{\eta}+\sqrt{2D_p\gamma^2}\boldsymbol{\xi}_r\label{eq:langevin r},
		\end{equation}
		where $\gamma$ is the friction coefficient, $D_p$ is the passive diffusivity, and $\boldsymbol{\xi}_r$ is a zero mean Gaussian white noise with $\langle\xi_{r,\alpha}(t)\xi_{r,\beta}(t')\rangle=\delta_{\alpha\beta}\delta(t-t')$.  Note that the passive diffusivity can be related to temperature through the Einstein relation $D_p\gamma=k_BT$.  The variable $\boldsymbol{\eta}$ is the active force that propels the particle.  We assume this force has first and second moments
		\begin{subequations}
			\begin{align}
				\langle\eta_{\alpha}(t)\rangle&=0\label{eq:1st moment eta},\\
				\langle\eta_{\alpha}(t)\eta_{\beta}(t')\rangle&=\delta_{\alpha\beta}\frac{\gamma^2v^2}{d}\exp\left(-\frac{1}{\tau}|t-t'|\right)\label{eq:2nd moment eta},
			\end{align}
		\end{subequations}
		where $\tau$ is the persistence time of the propulsion, $v$ is the swim speed, and $d$ is the spacial dimension.  In other words, this propulsion force on average has no preferred direction and is exponentially correlated in time.  The exponential correlation in time means that the active particle will have memory of its propulsion direction for roughly a time $\tau$ before orienting in a new direction.  Note that the magnitude of the correlations are chosen so that the characteristic propulsion force is $\sqrt{\langle\boldsymbol{\eta}^2\rangle}=\gamma v$, which is simply the force needed to move through a viscous environment at a speed $v$.

		One common way to generate an exponentially correlated random force is through an Ornstein-Uhlenbeck process given by
		\begin{equation}
			\tau\dot{\boldsymbol{\eta}}=-\boldsymbol{\eta}+\sqrt{\frac{2v^2\tau\gamma^2}{d}}\boldsymbol{\xi}_{\eta},
			\label{eq:langevin eta}
		\end{equation}
		where $\boldsymbol{\xi}_{\eta}$ is a zero mean Gaussian white noise independent of $\boldsymbol{\xi}_r$ with $\langle\xi_{\eta,\alpha}(t)\xi_{\eta,\beta}(t')\rangle=\delta_{\alpha\beta}\delta(t-t')$.  Eqs.\ (\ref{eq:langevin r}) and (\ref{eq:langevin eta}) thus describe the dynamics of an active particle driven by an exponentially correlated random force.  The mean-squared displacement of such a particle is given by
		\begin{equation}
			\langle\boldsymbol{r}^2(t)\rangle=2dD_pt+2v^2\tau^2\left[\frac{t}{\tau}-1+e^{-\frac{t}{\tau}}\right]
		\end{equation}
		There are two timescales: $\tau$ and $dD_p/v^2$.  The latter is the crossover between passive diffusion and ballistic motion.  On timescales longer than the persistence time $t\gg\tau$, the propulsion force becomes uncorrelated and the active particle effectively diffuses with $\langle\boldsymbol{r}^2\rangle\simeq2dD_{\textrm{eff}}t$, where the effective diffusivity $D_{\textrm{eff}}=D_p+v^2\tau/d$ is the sum of the passive and active diffusivities $D_p$ and $D_a=v^2\tau/d$.  For $t\ll dD_p/v^2$, the particle undergoes passive Brownian diffusion with $\langle\boldsymbol{r}^2\rangle\simeq2dD_pt$.  Finally for $dD_p/v^2\ll t\ll\tau$, the particle undergoes ballistic motion with $\langle\boldsymbol{r}^2\rangle\simeq v^2t^2$.  Note that this ballistic regime disappears when $\tau\ll dD_p/v^2$ or equivalently $v\tau\ll \sqrt{dD_p\tau}$, that is, when transport due to propulsion is much smaller than transport due to passive diffusion.  We refer to such particles as ``weakly active'', which will be our main focus here.

		The Langevin equations (Eqs.\ (\ref{eq:langevin r}) and (\ref{eq:langevin eta})) can be recast into a Fokker-Planck equation.  The distribution of a noninteracting system of these active particles satisfies the Fokker-Planck equation
		\begin{align}
			\begin{split}
				\frac{\partial\rho}{\partial t}&=-\frac{1}{\gamma}\boldsymbol{\eta}\cdot\boldsymbol{\nabla}_{r}\rho+D_p\nabla_r^2\rho+\frac{1}{\tau}\boldsymbol{\nabla}_{\eta}\cdot(\boldsymbol{\eta}\rho)+\frac{\gamma^2v^2}{d\tau}\nabla_{\eta}^2\rho\\
				&=-\boldsymbol{\nabla}_r\cdot\boldsymbol{J}_r-\boldsymbol{\nabla}_{\eta}\cdot\boldsymbol{J}_{\eta},
				\label{eq:fokker-planck}
			\end{split}
		\end{align}
		where $\rho=\rho(\boldsymbol{r},\boldsymbol{\eta},t)$ is the distribution of the active particles and
		\begin{subequations}
			\begin{align}
				\boldsymbol{J}_r&=\frac{1}{\gamma}\boldsymbol{\eta}\rho-D_p\boldsymbol{\nabla}_r\rho,\label{eq:current r}\\
				\boldsymbol{J}_{\eta}&=-\frac{1}{\tau}\boldsymbol{\eta}\rho-\frac{\gamma^2v^2}{d\tau}\boldsymbol{\nabla}_{\eta}\rho\label{eq:current eta}.
			\end{align}
		\end{subequations}
		are the currents for positions and propulsions.  Note that passive diffusion introduces the spacial gradient $\boldsymbol{\nabla}_r\rho$ in $\boldsymbol{J}_r$, which will be extremely useful for formulating boundary conditions.

		Our goal is to find a way to solve Eq.\ (\ref{eq:fokker-planck}) for the distribution of active particles.  Note that in the bulk far from any boundary, the spacial density of noninteracting active particles should be uniform and the propulsion force $\boldsymbol{\eta}$ for this model will be Gaussian distributed in steady-state.  The exact steady-state distribution in the bulk satisfying Eq.\ (\ref{eq:fokker-planck}) is
		\begin{equation}
			\rho(\boldsymbol{r},\boldsymbol{\eta})=\frac{\rho_{\textrm{bulk}}}{(2\pi\gamma^2v^2/d)^{d/2}}\exp\left(-\frac{\boldsymbol{\eta}^2}{2\gamma^2v^2/d}\right).
			\label{eq:exact bulk distribution}
		\end{equation}
		We wish to determine how this distribution changes near a boundary given certain conditions on the current or density at that boundary.

	\section{\label{sec:method outline}Perturbation Theory and Eigenfunction Expansion}
		To simplify the problem, let us define the dimensionless position and propulsion force to be
		\begin{subequations}
			\begin{align}
				\tilde{\boldsymbol{r}}&=\frac{\boldsymbol{r}}{\sqrt{2D_p\tau}}=\frac{\boldsymbol{r}}{\lambda},\\
				\tilde{\boldsymbol{\eta}}&=\frac{\boldsymbol{\eta}}{\sqrt{2\gamma^2v^2/d}}=\frac{\boldsymbol{\eta}}{\sigma},
			\end{align}
		\end{subequations}
		where the length scale $\lambda=\sqrt{2D_p\tau}$ is how far the particle passively diffuses in a persistence time and the force scale $\sigma=\sqrt{2\gamma^2v^2/d}$ is roughly the force needed to move through a viscous environment at speed $v$.  This length scale $\lambda$ is important and will show up again and again in subsequent sections.  In our context of boundaries, we can interpret it as the distance over which active particles will be persistent and still interact with a boundary through diffusion and propulsion.  In other words, within this distance, we will observe the influence of a boundary on, for example, the distribution of active particles.  Beyond this distance, however, the active particles may reorient many times without colliding with a boundary and thus behave as if they are in bulk.

		The resulting dimensionless Fokker-Planck equation in steady-state is
		\begin{equation}
			\nabla_r^2\tilde{\rho}+\nabla_{\eta}^2\tilde{\rho}+2\boldsymbol{\nabla}_{\eta}\cdot(\tilde{\boldsymbol{\eta}}\tilde{\rho})=2\epsilon\tilde{\boldsymbol{\eta}}\cdot\boldsymbol{\nabla}_r\tilde{\rho},
			\label{eq:dimensionless fokker-planck}
		\end{equation}
		where the dimensionless distribution is $\tilde{\rho}(\tilde{\boldsymbol{r}},\tilde{\boldsymbol{\eta}})=\rho(\lambda\tilde{\boldsymbol{r}},\sigma\tilde{\boldsymbol{\eta}})\lambda^d\sigma^d$.  Throughout the work, tildes will indicate a dimensionless quantity.  The only parameter that remains is
		\begin{equation}
			\epsilon=\sqrt{\frac{v^2\tau}{dD_p}}=\sqrt{\textrm{Pe}},
			\label{eq:epsilon}
		\end{equation}
		where Pe is the P\'{e}clet number which is the ratio of advective transport (swimming) to diffusive transport.  This parameter controls the level of activity of the particles; for example, $\epsilon=0$ corresponds to a passive particle while $\epsilon\gg1$, a strongly active particle.  As mentioned earlier, our main focus here will be on weakly active particles or $\epsilon\ll1$.

		To make further progress, we perform two expansions on the distribution: an expansion in powers of $\epsilon$ and an eigenfunction expansion in Hermite polynomials (Appendix \ref{subapp:hermite polynomials}).  To keep things simple here, we will only show the series solution for 1D, though it should be emphasized that the result can easily be extended to arbitrary dimensions (Appendix \ref{app:expansion arbitrary dimension}).  We start by writing the distribution in powers of $\epsilon$ as
		\begin{equation}
			\tilde{\rho}(\tilde{x},\tilde{\eta})=\sum_{n=0}^{\infty}\epsilon^n\tilde{\rho}^{(n)}(\tilde{x},\tilde{\eta}),
		\end{equation}
		which gives us for each order
		\begin{equation}
			\frac{\partial^2\tilde{\rho}^{(n)}}{\partial\tilde{x}^2}+\frac{\partial^2\tilde{\rho}^{(n)}}{\partial\tilde{\eta}^2}+2\frac{\partial}{\partial\tilde{\eta}}\Big(\tilde{\eta}\tilde{\rho}^{(n)}\Big)=2\tilde{\eta}\frac{\partial\tilde{\rho}^{(n-1)}}{\partial\tilde{x}}.
			\label{eq:dimensionless fokker-planck order 1D}
		\end{equation}
		The zeroth order solution $\tilde{\rho}^{(0)}$ is related to the density of passively diffusing particles, which is usually easy to find.  Thus, we can determine the effect of activity by iteratively computing higher-order terms starting from the solution for passive Brownian particles.

		There should be no active particles with an arbitrarily large propulsion force $\boldsymbol{\eta}$.  In other words, the distribution in $\boldsymbol{\eta}$ should decay sufficiently quickly as $|\boldsymbol{\eta}|\rightarrow\infty$.  We can therefore simplify the second and third terms on the left-hand side of Eq.\ (\ref{eq:dimensionless fokker-planck order 1D}) by expanding the distribution in terms of Hermite polynomials.  By writing each order of the distribution as
		\begin{equation}
			\tilde{\rho}^{(n)}(\tilde{x},\tilde{\eta})=\sum_{m=0}^{\infty}C_m^{(n)}(\tilde{x})e^{-\tilde{\eta}^2}H_m(\tilde{\eta}),
		\end{equation}
		we reduce the entire problem to solving for the coefficients $C_m^{(n)}(\tilde{x})$, which satisfy in 1D the simple ordinary differential equation
		\begin{equation}
			\frac{d^2C_m^{(n)}}{d\tilde{x}^2}-2mC_m^{(n)}=\frac{d}{d\tilde{x}}\left[C_{m-1}^{(n-1)}+2(m+1)C_{m+1}^{(n-1)}\right].
			\label{eq:1D coef equation}
		\end{equation}
		We will focus on two kinds of boundaries: impenetrable and absorbing.  At an impenetrable boundary, particles cannot pass through it and so the current normal to the boundary must be zero.  At an absorbing boundary, particles are removed from the system and so the density at the boundary is maintained to be zero.  By writing the current $\tilde{J}_x=2\epsilon\tilde{\eta}\tilde{\rho}-\frac{\partial\tilde{\rho}}{\partial\tilde{x}}$ or density $\tilde{\rho}$ in terms of the coefficients $C_m^{(n)}$, we can use the orthogonality of Hermite polynomials to determine the boundary conditions for the coefficients.  We now illustrate this approach with several examples of noninteracting weakly active particles in different geometries.

	\begin{figure*}
		\centering
		\includegraphics[scale=0.50]{various_geometries.png}
		\caption{Several of the geometries we consider: \textbf{a)} active particles near a single hard wall, \textbf{b)} active particles inside and outside a circular boundary, \textbf{c)} active particles confined to a wedge-shaped region, \textbf{d)} active particles near a corrugated boundary, and finally \textbf{e)} active particles near a spherical absorber.}
		\label{fig:various geometries}
	\end{figure*}
	\section{\label{sec:cartesian}Problems in Cartesian Coordinates}
		\subsection{\label{subsec:particles on line}Active particles on a line}
			Consider the simplest example of noninteracting weakly active particles freely propelling left or right on a line until they collide with an impenetrable wall.  We are interested in how the presence of such a wall modifies the bulk distribution of active particles (Eq.\ \ref{eq:exact bulk distribution}).  As outlined in Section \ref{sec:method outline}, the dimensionless distribution in 1D can be written as
			\begin{equation}
				\tilde{\rho}(\tilde{x},\tilde{\eta})=\sum_{n=0}^{\infty}\epsilon^n\sum_{m=0}^{\infty}C_m^{(n)}(\tilde{x})e^{-\tilde{\eta}^2}H_m(\tilde{\eta}),
			\end{equation}
			where the coefficients satisfy Eq.\ (\ref{eq:1D coef equation}).  We consider the cases of active particles confined by one solid wall and between two walls.

			\subsubsection{\label{subsec:semi-infinite}1D semi-infinite domain: one wall}
				We start with the case of an impenetrable wall at $x=0$ that confines the active particles to the region $x>0$.  The zero current condition at this wall $J_x(0,\eta)=0$ gives us the condition on the coefficients
				\begin{equation}
					\frac{dC_m^{(n)}(0)}{d\tilde{x}}=C_{m-1}^{(n-1)}(0)+2(m+1)C_{m+1}^{(n-1)}(0).
				\end{equation}
				Details of the solution can be found in Appendix \ref{subapp:semi-infinite}.  We are interested in how the presence of an impenetrable wall affects the distribution $\rho(x,\eta)$ and the currents $J_x(x,\eta),J_{\eta}(x,\eta)$ of the active particles.  In the bulk, the distribution is given by Eq.\ (\ref{eq:exact bulk distribution}) with $d=1$.  As we approach the wall, the distribution will no longer be independent of $x$.  Up to $\epsilon^2$, the distribution is
				\begin{widetext}
					\begin{equation}
						\rho(x,\eta)\simeq\frac{\rho_{\textrm{bulk}}}{\sigma\sqrt{\pi}}e^{-\frac{\eta^2}{\sigma^2}}\left\{1-\epsilon\sqrt{2}e^{-\frac{\sqrt{2}x}{\lambda}}\frac{\eta}{\sigma}+\epsilon^2\left[e^{-\frac{\sqrt{2}x}{\lambda}}+\left(\sqrt{2}e^{-\frac{2x}{\lambda}}-e^{-\frac{\sqrt{2}x}{\lambda}}\right)\left(\frac{2\eta^2}{\sigma^2}-1\right)\right]\right\},
						\label{eq:1d full solution}
					\end{equation}
				\end{widetext}
				where $\lambda=\sqrt{2D_p\tau}$ and $\sigma=\sqrt{2\gamma^2v^2}$.  Integrating out the active force $\eta$, we have for the spacial density
				\begin{align}
					\begin{split}
						\frac{\rho(x)}{\rho_{\textrm{bulk}}}\simeq&\,1+\epsilon^2e^{-\frac{\sqrt{2}x}{\lambda}}\\
						&+\epsilon^4\left[2\sqrt{2}e^{-\frac{2x}{\lambda}}+2\sqrt{2}\left(\frac{x}{4\lambda}-1\right)e^{-\frac{\sqrt{2}x}{\lambda}}\right].
					\end{split}
					\label{eq:1d density up to 4th}
				\end{align}
				\begin{figure}
					\centering
					\includegraphics[scale=0.4]{single_wall_epsilon_2_4.png}
					\caption{$\epsilon^2$ (blue) and $\epsilon^4$ (orange) contributions to the density of weakly active particles near a single impenetrable wall (Eq.\ \ref{eq:1d density up to 4th}).  The $\epsilon^2$ contribution gives an elevated density while the $\epsilon^4$ contribution slightly depletes the density near the wall.}
					\label{fig:single wall epsilon 2 4}
				\end{figure}%
				The $\epsilon^2$ and $\epsilon^4$ contributions to this density are shown in Figure \ref{fig:single wall epsilon 2 4}.  The density is elevated over the length scale $\lambda$, which, as we discussed earlier, is the distance over which the presence of the wall will be felt by the active particles.  Another way of thinking of this is that persistence of the active particles causes them to spend more time near the wall, thus elevating the density.  The excess number of particles near the wall is given by
				\begin{align}
					\begin{split}
						N_{\textrm{excess}}&=\int_{0}^{\infty}\left[\rho(x)-\rho_{\textrm{bulk}}\right]dx\\
						&\simeq\rho_{\textrm{bulk}}\sqrt{D_p\tau}\left[\epsilon^2-\frac{4\sqrt{2}-5}{2}\,\epsilon^4\right].
					\end{split}
					\label{eq:excess N}
				\end{align}
				It is interesting to note that the $\epsilon^4$ correction does not further increase the density near the wall and actually depletes it, as there is a decrease in the excess number of particles near the wall.  This suggests that at higher P\'{e}clet numbers, the accumulated density may become steeper.

				The currents in the position $x$ and active force $\eta$ are given by
				\begin{subequations}
					\begin{align}
						J_x(x,\eta)&\simeq\epsilon\frac{\lambda}{\tau}\left(\frac{\rho_{\textrm{bulk}}}{\sigma}\right)\frac{1}{\sqrt{\pi}}\left(1-e^{-\frac{\sqrt{2}x}{\lambda}}\right)\frac{\eta}{\sigma} e^{-\frac{\eta^2}{\sigma^2}},\label{eq:1D Jx current}\\
						J_{\eta}(x,\eta)&\simeq\epsilon\frac{\sigma}{\tau}\left(\frac{\rho_{\textrm{bulk}}}{\sigma}\right)\frac{1}{\sqrt{2\pi}}e^{-\frac{\sqrt{2}x}{\lambda}}e^{-\frac{\eta^2}{\sigma^2}}.\label{eq:1D Jeta current}
					\end{align}
				\end{subequations}
				These characterize how the positions and propulsions of our active particles change on average as they move near a wall.  There is circulation in the $x\eta$-plane (Figure \ref{fig:1D currents}), a signature of out-of-equilibrium systems \cite{Battle et al}.
				\begin{figure}
					\centering
					\includegraphics[scale = 0.55]{one_wall_currents.png}
					\caption{Currents $J_x(x,\eta),J_{\eta}(x,\eta)$ (Eqs.\ (\ref{eq:1D Jx current}) and (\ref{eq:1D Jeta current})) resulting from activity near a single impenetrable wall.  The circulation shows the simple behavior of active particles propelling towards the wall, spending time turning around, and then propelling away.}
					\label{fig:1D currents}
				\end{figure}
				In this case, the behavior is quite simple: active particles on average swim towards the wall, spend some time turning, and then swim away.  It is interesting to note that while we observe currents at order $\epsilon$, we do not observe any deviations from the passive density $\rho(x)=\rho_{\textrm{bulk}}$ until order $\epsilon^2$.  The currents result in an asymmetric distribution in $\eta$ at the wall.  Up to order $\epsilon$, the distribution at the wall is
				\begin{align}
					\rho(0,\eta)&\simeq\frac{\rho_{\textrm{bulk}}}{\sigma\sqrt{\pi}}e^{-\frac{\eta^2}{\sigma^2}}\left(1-\epsilon\frac{\sqrt{2}\eta}{\sigma}\right).
					\label{eq:wall distribution}
				\end{align}
				The correction shifts the mean to $\langle\eta\rangle_{x=0}\simeq-\epsilon\gamma v=-\gamma v\sqrt{\frac{v^2\tau}{D_p}}$.  Physically, this occurs because particles with $\eta<0$ swim towards the wall and have an increased density when they are slowed while particles with $\eta>0$ swim away and have a decreased density.  The result for $\langle\eta\rangle$ can also be obtained from a simple balancing of currents.  Recall that the density (Eq.\ (\ref{eq:1d density up to 4th})) is elevated by $\epsilon^2\rho_{\textrm{bulk}}$ over a length scale $\sqrt{D_p\tau}$, which gives a diffusive flux $J_{\textrm{diff}}\sim D_p\frac{\epsilon^2\rho_{\textrm{bulk}}}{\sqrt{D_p\tau}}$ away from the wall.  Setting this equal to the swim flux $J_{\textrm{swim}}\sim\frac{1}{\gamma}\langle\eta\rangle\rho_{\textrm{bulk}}$, we get an average propulsion of $\langle\eta\rangle\sim \gamma D_p\frac{\epsilon^2}{\sqrt{D_p\tau}}=\gamma v\sqrt{\frac{v^2\tau}{D_p}}$ towards the wall.  At higher P\'{e}clet numbers, we expect that the distribution at the wall will shift more towards $\eta<0$ and that the density $\rho(0,\eta>0)$ will become significantly depleted.  This occurs because the moment an active particle turns around, it immediately propels away from the wall and no longer contributes to $\rho(0,\eta>0)$.

			\subsubsection{\label{subsec:finite}1D finite domain: two walls}
				We now consider the case of active particles confined between two walls located at $x=\pm L$.  Unlike for a single wall, there is no bulk where the distribution of active particles is unaffected by the wall.  When there are two walls, each wall can have an effect on the distribution at the other.  The zero current boundary conditions $J_x(\pm L,\eta)=0$ give the following relation for the coefficients
				\begin{equation}
					\frac{dC_m^{(n)}(\pm\tilde{L})}{d\tilde{x}}=C_{m-1}^{(n-1)}(\pm\tilde{L})+2(m+1)C_{m+1}^{(n-1)}(\pm\tilde{L}).
				\end{equation}
				The steps for finding the solution (Appendix \ref{subapp:finite domain}) are similar to those of the single wall.  The main difference is that we now have a finite number of particles trapped between the two walls instead of an infinite bulk with constant density.  Taking $\rho(x,\eta)$ and integrating out $\eta$, we have for the density
				\begin{equation}
					\frac{\rho(x)}{N/2L}=1+\epsilon^2\left(\frac{\cosh\frac{\sqrt{2}x}{\lambda}}{\cosh\frac{\sqrt{2}L}{\lambda}}-\frac{\tanh\frac{\sqrt{2}L}{\lambda}}{\frac{\sqrt{2}L}{\lambda}}\right),
					\label{eq:density two walls}
				\end{equation}
				where $N$ is the number of particles between the walls.  Note that as a result of the accumulation at walls, there is a depletion of particles around the center of the confinement.  This is captured by the second term in the parenthesis, which vanishes in the limit of large separation of the walls.

				For large separations, each wall should not influence the other and we should obtain the result for a single wall in the previous section.  Indeed, taking $L\gg\lambda$, we find
				\begin{equation}
					\rho(x)\simeq\frac{N}{2L}\left[1+\epsilon^2\left(e^{\frac{\sqrt{2}(x-L)}{\lambda}}+e^{-\frac{\sqrt{2}(x+L)}{\lambda}}\right)\right],
					\label{eq:density 2 walls large L}
				\end{equation}
				which is simply the sum of the accumulations due to each wall if it were by itself.  We can also examine the opposite limit of two walls that are very close to each other ($L\ll\lambda$), for which we find the the parabolic profile
				\begin{equation}
					\rho(x)\simeq\frac{N}{2L}\left[1+\epsilon^2\left(\frac{x^2}{\lambda^2}-\frac{L^2}{3\lambda^2}\right)\right].
				\end{equation}

				We now look at the distribution of particles at each wall to see how one wall may influence the other.  The distribution at the walls to order $\epsilon$ is
				\begin{equation}
					\rho(\pm L,\eta)=\frac{N}{2L}\cdot\frac{e^{-\frac{\eta^2}{\sigma^2}}}{\sigma\sqrt{\pi}}\left(1\pm\epsilon\frac{\sqrt{2}\eta}{\sigma}\tanh\frac{\sqrt{2}L}{\lambda}\right),
				\end{equation}
				which gives for the average propulsion at each wall
				\begin{equation}
					\langle\eta\rangle(\pm L)\simeq\pm\epsilon\gamma v\tanh\frac{\sqrt{2}L}{\lambda}.
					\label{eq:avg eta two walls}
				\end{equation}
				Interestingly, we can interpret this average propulsion as a weighted average $\langle\eta\rangle(\pm L)\simeq\mp\epsilon\gamma vP_c\pm\epsilon\gamma vP_f$, where
				\begin{equation}
					P_c=\frac{1}{1+e^{-\frac{2\sqrt{2}L}{\lambda}}},\hspace{0.2in}P_f=\frac{e^{-\frac{2\sqrt{2}L}{\lambda}}}{1+e^{-\frac{2\sqrt{2}L}{\lambda}}},
				\end{equation}
				are the weights for the closest and farthest walls, respectively.  Note that these weights are proportional to how much the closest and farthest walls contribute to the accumulation.  When the two walls are far apart ($L\gg\lambda$), we get back the single wall result $\langle\eta\rangle(\pm L)\simeq\mp\epsilon\gamma v$ since the farthest wall contributes nothing ($P_f\ll P_c\approx1$).

			\subsubsection{Pressure on solid boundaries}
				One question of interest is how one relates the density of active particles to the pressure they exert on walls.  For an equilibrium system of noninteracting particles, the density is $\rho(x)=\rho_0$ everywhere and the pressure is simply the ideal gas pressure $P=\rho_0k_BT$.  To compute the pressures in our present case of noninteracting active particles, we start by replacing the impenetrable walls with soft confining potentials.  This approach of using soft potentials has been useful for computing the mechanical properties of active particles near boundaries \cite{Solon et al pressure,Caprini Marconi,Marconi et al,Zakine et al,Grosberg and Joanny,Duzgan Sellinger}.  The idea is that a confining potential, just like a wall, can prevent particles from moving a certain direction.  The pressure can then be computed by simply summing up the forces the potential exerts on the particles.  One can then take the limit as the potential becomes steep to obtain the pressure for an impenetrable wall.  For our purposes, we consider the ramp potentials (Appendix \ref{subapp:pressure})
				\begin{equation}
					U(x)=\begin{cases}
						-fx, & x < 0\\
						0, & x > 0
					\end{cases}
					\label{eq:U 1 ramp}
				\end{equation}
				for one wall at $x=0$ and
				\begin{equation}
					U(x)=\begin{cases}
						-fx-fL, & x<-L\\
						0, & -L<x<L\\
						fx-fL, & x>L
					\end{cases}
					\label{eq:U 2 ramp}
				\end{equation}
				for two walls at $x=\pm L$.  The mechanical pressure for both these cases is simply given by the integral
				\begin{equation}
					P=-\int U'(x)\rho(x)dx.
				\end{equation}
				over one of the wall regions where $U(x)\ne0$.  The distributions and the pressures obtained for these ramp potentials are left for Appendix \ref{subapp:pressure}.  We are interested in the limit of impenetrable walls or $f\rightarrow\infty$.  In this limit, we find the pressures
				\begin{equation}
					P=\rho_{\textrm{bulk}}D_p\gamma(1+\epsilon^2),
					\label{eq:P 1 wall}
				\end{equation}
				for one wall and
				\begin{equation}
					P\simeq\frac{ND_p\gamma}{2L}\left[1+\epsilon^2\left(1-\frac{\tanh\frac{\sqrt{2}L}{\lambda}}{\frac{\sqrt{2}L}{\lambda}}\right)\right],
					\label{eq:P 2 wall}
				\end{equation}
				for two.  We start by noting that the pressure for one wall (Eq.\ (\ref{eq:P 1 wall})) is actually exact even though it was obtained from perturbation theory.  A proof of this can be found in Appendix \ref{subapp:pressure}.  There are two interpretations of this pressure.  The first interpretation is $P=\rho_{\textrm{bulk}}[D_p(1+\epsilon^2)]\gamma=\rho_{\textrm{bulk}}D_{\textrm{eff}}\gamma$, which can be thought of as the ideal gas pressure of particles with an effective diffusivity $D_{\textrm{eff}}=D_p+v^2\tau$.  The second interpretation is $P=[\rho_{\textrm{bulk}}(1+\epsilon^2)]D_p\gamma\simeq\rho_{\textrm{wall}}D_p\gamma$, where $\rho_{\textrm{wall}}=\rho(0)$ is the density at the boundary.  This is just the the passive pressure due to the elevated density of particles (Eq.\ (\ref{eq:1d density up to 4th})) close to the wall.  For the case of confinement between two walls, the situation is different as there is no longer a bulk.  However, notice that we still have the relation $P\simeq\rho_{\textrm{wall}}D_p\gamma$, where $\rho_{\textrm{wall}}=\rho(\pm L)$.  Using Einstein's relation $D_p\gamma=k_BT$, we may write $P\simeq\rho_{\textrm{wall}}k_BT$ for noninteracting active particles.  While we have only shown that this relation for pressure holds up to order $\epsilon^2$, it is not inconceivable that it should hold in general.  In Appendix \ref{subapp:exactly solvable}, we show that for the exactly solvable model of noninteracting 1D run-and-tumble particles with passive diffusion, this relation for pressure holds without any approximations.  It is worth noting that a similar relation has been shown to hold for noninteracting active Brownian particles \cite{Duzgan Sellinger}, a different model of active particles.

		\subsection{\label{subsec:corner}Confinement in 2D: right-angled corner}
			In previous sections, we considered weakly active particles in 1D.  Those results can easily be extended to flat walls in higher dimensions such as an infinite line (2D) or infinite plate (3D).  In fact, the density profiles, particularly the exponential decays of density away from a flat wall, are exactly the same as those obtained in 1D.

			We are interested in going beyond these simple geometries and studying cases where the walls may be curved or are not parallel and meet at certain angles.  In this particular section, we will focus on the simpler case of two flat walls that meet at a right angle and confine active particles to a region $x>0,y>0$.  We write the dimensionless distribution as
			\begin{equation}
				\rho(\tilde{\boldsymbol{r}},\tilde{\boldsymbol{\eta}})=\sum_{n=0}^{\infty}\epsilon^n\sum_{\boldsymbol{m}}C_{\boldsymbol{m}}^{(n)}(\tilde{\boldsymbol{r}})e^{-\tilde{\boldsymbol{\eta}}^2}H_{m_x}(\tilde{\eta}_x)H_{m_y}(\tilde{\eta}_y),
			\end{equation}
			Notice that going to higher dimensions simply requires additional Hermite polynomials.  The coefficients satisfy
			\begin{align}
				\begin{split}
					\frac{\partial^2C_{\boldsymbol{m}}^{(n)}}{\partial\tilde{x}^2}+\frac{\partial^2C_{\boldsymbol{m}}^{(n)}}{\partial\tilde{y}^2}-2(m_x+m_y)C_{\boldsymbol{m}}^{(n)}=\frac{\partial w_x}{\partial\tilde{x}}+\frac{\partial w_y}{\partial\tilde{y}},
				\end{split}
			\end{align}
			where
			\begin{subequations}
				\begin{align}
					w_x&=C_{m_x-1,m_y}^{(n-1)}+2(m_x+1)C_{m_x+1,m_y}^{(n-1)},\\
					w_y&=C_{m_x,m_y-1}^{(n-1)}+2(m_y+1)C_{m_x,m_y+1}^{(n-1)}.
				\end{align}
			\end{subequations}
			The zero current boundary conditions $J_x(0,y,\boldsymbol{\eta})=J_y(x,0,\boldsymbol{\eta})=0$ give the following conditions on the coefficients
			\begin{subequations}
				\begin{align}
					\frac{\partial C_{\boldsymbol{m}}^{(n)}(0,\tilde{y})}{\partial\tilde{x}}=w_x(0,\tilde{y})
					\\
					\frac{\partial C_{\boldsymbol{m}}^{(n)}(\tilde{x},0)}{\partial\tilde{y}}=w_y(\tilde{x},0)
				\end{align}
			\end{subequations}
			Details of the solution can be found in Appendix \ref{subapp:2D corner}.  Integrating out $\boldsymbol{\eta}$ from $\rho(\boldsymbol{r},\boldsymbol{\eta})$, we find the density near the corner is
			\begin{align}
				\begin{split}
					\frac{\rho(\boldsymbol{r})}{\rho_{\textrm{bulk}}}\simeq&\ 1+\epsilon^2\left(e^{-\frac{\sqrt{2}x}{\lambda}}+e^{-\frac{\sqrt{2}y}{\lambda}}\right)\\
					&+\epsilon^4\left[2\sqrt{2}e^{-\frac{2x}{\lambda}}+2\sqrt{2}\left(\frac{x}{4\lambda}-1\right)e^{-\frac{\sqrt{2}x}{\lambda}}\right]\\
					&+\epsilon^4\left[2\sqrt{2}e^{-\frac{2y}{\lambda}}+2\sqrt{2}\left(\frac{y}{4\lambda}-1\right)e^{-\frac{\sqrt{2}y}{\lambda}}\right]\\
					&+\epsilon^4e^{-\frac{\sqrt{2}(x+y)}{\lambda}}\\
					=&\ \frac{\rho_{\textrm{1D}}(x)+\rho_{\textrm{1D}}(y)}{\rho_{\textrm{bulk}}}+\epsilon^4e^{-\frac{\sqrt{2}(x+y)}{\lambda}},
				\end{split}
				\label{eq:corner density up to 4th}
			\end{align}
			where $\rho_{\textrm{1D}}$ is the 1D density (Eq.\ (\ref{eq:1d density up to 4th})) if each wall were by itself.  The last term is new and enhances the accumulation near the corner.  This arises due to correlations between $\eta_x,\eta_y$ near the corner.  Indeed, if we study the full distribution, we find at order $\epsilon^2$ the new term
			\begin{align}
				\begin{split}
					\rho(\boldsymbol{r},\boldsymbol{\eta})\simeq&\ \textrm{single wall contributions}\\
					&+2\epsilon^2\frac{\rho_{\textrm{bulk}}}{\pi\sigma^2}e^{-\frac{\boldsymbol{\eta}^2}{\sigma^2}}\frac{\eta_x\eta_y}{\sigma^2}e^{-\frac{\sqrt{2}(x+y)}{\lambda}},
				\end{split}
			\end{align}
			which gives a nonzero correlation $\langle\eta_x\eta_y\rangle$ near the corner and is responsible for the extra accumulation at order $\epsilon^4$.  If we consider the average propulsion $\langle\boldsymbol{\eta}\rangle$ of the active particles near the corner, which is given by
			\begin{equation}
				\langle\boldsymbol{\eta}\rangle(x,y)=-\frac{\epsilon\gamma v}{\sqrt{2}}\left(e^{-\frac{\sqrt{2}x}{\lambda}},e^{-\frac{\sqrt{2}y}{\lambda}}\right),
				\label{eq:average eta corner}
			\end{equation}
			we also see that there is an increased tendency for active particles to orient and propel towards the corner (Figure \ref{fig:corner average propulsion}).
			\begin{figure}
				\centering
				\includegraphics[scale=0.55]{corner_average_eta.png}
				\caption{Average propulsion $\langle\boldsymbol{\eta}\rangle$ (Eq.\ (\ref{eq:average eta corner})) near a right-angled corner that confines active particles to $x>0,y>0$.  This orientation towards the corner leads to additional accumulation given by the last term of the density Eq.\ (\ref{eq:corner density up to 4th}).}
				\label{fig:corner average propulsion}
			\end{figure}

	\section{\label{sec:polar}Problems in Polar Coordinates}
		We have thus far considered simple examples of weakly active particles near flat walls or a right-angled corner.  We now want to study how the curvature of a surface or the sharpness of a corner affects the distribution of these particles.  We focus our attention on two cases: a circular boundary and a corner with an angle other than $\pi/2$.

		These examples are best studied using polar coordinates.  In polar coordinates, we write the dimensionless distribution as
		\begin{align}
			\tilde{\rho}(\tilde{r},\theta,\tilde{\boldsymbol{\eta}})=\sum_{n=0}^{\infty}\epsilon^n\sum_{\boldsymbol{m}}C_{\boldsymbol{m}}^{(n)}(\tilde{r},\theta)e^{-\tilde{\boldsymbol{\eta}}^2}H_{m_x}(\tilde{\eta}_x)H_{m_y}(\tilde{\eta}_y).
		\end{align}
		The coefficients satisfy
		\begin{align}
			\begin{split}
				&\frac{1}{\tilde{r}}\frac{\partial}{\partial\tilde{r}}\left(\tilde{r}\frac{\partial C_{\boldsymbol{m}}^{(n)}}{\partial\tilde{r}}\right)+\frac{1}{\tilde{r}^2}\frac{\partial^2C_{\boldsymbol{m}}^{(n)}}{\partial\theta^2}-2(m_x+m_y)C_{\boldsymbol{m}}^{(n)}\\
				&=\frac{1}{\tilde{r}}\frac{\partial}{\partial\tilde{r}}\left(\tilde{r}w_r\right)+\frac{1}{\tilde{r}}\frac{\partial w_{\theta}}{\partial\theta},
			\end{split}
			\label{eq:coef eq polar}
		\end{align}
		where the components of $\boldsymbol{w}$ in polar coordinates are
		\begin{subequations}
			\begin{align}
				\begin{split}
					w_r=&\ \left[C_{m_x-1,m_y}^{(n-1)}+2(m_x+1)C_{m_x+1,m_y}^{(n-1)}\right]\cos\theta\\
					&+\left[C_{m_x,m_y-1}^{(n-1)}+2(m_y+1)C_{m_x,m_y+1}^{(n-1)}\right]\sin\theta,
				\end{split}
				\\
				\begin{split}
					w_{\theta}=&\ -\left[C_{m_x-1,m_y}^{(n-1)}+2(m_x+1)C_{m_x+1,m_y}^{(n-1)}\right]\sin\theta\\
					&+\left[C_{m_x,m_y-1}^{(n-1)}+2(m_y+1)C_{m_x,m_y+1}^{(n-1)}\right]\cos\theta.
				\end{split}
			\end{align}
		\end{subequations}

		\subsection{\label{subsec:circular}Active particles around a circular boundary}
			Consider an impenetrable circular boundary of radius $R$ with active particles both inside and outside.  We will separately obtain the densities in both regions.  The radial current on both sides of the boundary must be zero, or $J_r(R,\theta,\boldsymbol{\eta})=0$.  We thus have the boundary condition for the coefficients
			\begin{equation}
				\frac{\partial C_{\boldsymbol{m}}^{(n)}(\tilde{R},\theta)}{\partial\tilde{r}}=w_x(\tilde{R},\theta)\cos\theta+w_y(\tilde{R},\theta)\sin\theta.
			\end{equation}
			Details of the solutions both inside and outside the region can be found in Appendix \ref{subapp:circular}.  The densities outside and inside the boundary to order $\epsilon^2$ are
			\begin{subequations}
				\begin{align}
					\rho_{\textrm{out}}(r,\theta)\simeq&\ \rho_{\textrm{bulk}}\left[1+2\epsilon^2\frac{K_0\left(\frac{\sqrt{2}r}{\lambda}\right)}{K_0\left(\frac{\sqrt{2}R}{\lambda}\right)+K_2\left(\frac{\sqrt{2}R}{\lambda}\right)}\right],\label{eq:circular density outside}\\
					\begin{split}
						\rho_{\textrm{in}}(r,\theta)\simeq&\ \frac{N}{\pi R^2}\left[1+2\epsilon^2\vphantom{\frac{I_0\left(\frac{\sqrt{2}r}{\lambda}\right)}{I_0\left(\frac{\sqrt{2}R}{\lambda}\right)}}\right.\\
						&\left.\times\frac{I_0\left(\frac{\sqrt{2}r}{\lambda}\right)-I_0\left(\frac{\sqrt{2}R}{\lambda}\right)+I_2\left(\frac{\sqrt{2}R}{\lambda}\right)}{I_0\left(\frac{\sqrt{2}R}{\lambda}\right)+I_2\left(\frac{\sqrt{2}R}{\lambda}\right)}\right],
					\end{split}
				\end{align}
			\end{subequations}
			where $I_{\mu},K_{\mu}$ are the modified Bessel functions of the first and second kinds and $N$ is the number of particles inside the circular region.

			Let us consider some limiting behaviors as $R\rightarrow\infty$ or $R\rightarrow0$.  Defining $\delta r=r-R$ as the radial distance from the circular boundary and taking $|\delta r|\ll R$, we obtain for the density near a large circular boundary ($R\rightarrow\infty$)
			\begin{subequations}
				\begin{align}
					\rho_{\textrm{out}}(R+\delta r,\theta)&\simeq\rho_{\textrm{bulk}}\left(1+\epsilon^2e^{-\frac{\sqrt{2}\delta r}{\lambda}}\right),\\
					\rho_{\textrm{in}}(R+\delta r,\theta)&\simeq\frac{N}{\pi R^2}\left(1+\epsilon^2e^{\frac{\sqrt{2}\delta r}{\lambda}}\right).
				\end{align}
			\end{subequations}
			Note that these are just the density profiles near a flat wall (Eqs.\ (\ref{eq:1d density up to 4th}) and (\ref{eq:density 2 walls large L})) since the curvature of the wall becomes negligible as $R\rightarrow\infty$.  In the opposite limit of a small circular boundary ($R\rightarrow0$), we have
			\begin{subequations}
				\begin{align}
					\rho_{\textrm{out}}(r\ll\lambda,\theta)&\simeq\rho_{\textrm{bulk}}\left[1+\frac{2\epsilon^2R^2}{\lambda^2}\left(\ln\frac{\sqrt{2}\lambda}{r}-\gamma_{em}\right)\right],\\
					\rho_{\textrm{in}}(r,\theta)&\simeq\frac{N}{\pi R^2}\left[1+\epsilon^2\left(\frac{r^2}{\lambda^2}-\frac{R^2}{2\lambda^2}\right)\right],
				\end{align}
			\end{subequations}
			where $\gamma_{em}\approx0.577$ is the Euler-Mascheroni constant.  The weak dependence of $\rho_{\textrm{out}}$ on $r$ is a result of the particles outside interacting with a small circular boundary, which should not affect the density much.  Finally, just as in the one dimensional case, the density inside the small region $\rho_{\textrm{in}}$ takes on a parabolic profile.

			We are interested in how the curvature of a boundary affects the accumulation of weakly active particles.  Let us consider the densities both inside and outside the circular boundary at $r=R$ and compare them with the density at a flat wall.  We focus on $R\gg\lambda$ when the curvature is small.  To start, recall that the density at a flat wall (Eq.\ (\ref{eq:1d density up to 4th})) is $\rho_{\textrm{flat}}\simeq\rho_{\textrm{bulk}}(1+\epsilon^2)$.  For outer part of the circular boundary, we have
			\begin{equation}
				\rho_{\textrm{out}}(R,\theta)-\rho_{\textrm{flat}}\simeq-\epsilon^2\rho_{\textrm{bulk}}\frac{\lambda}{\sqrt{2}R}.
				\label{eq:convex boundary density}
			\end{equation}
			For the inner part of the circular boundary, we have to be a bit more careful since there is not an infinite bulk.  We can mimic a bulk inside the region by maintaining the density at the center $r=0$ to be equal to the bulk density outside or $\rho_{\textrm{in}}(0,\theta)=\rho_{\textrm{bulk}}$.  The resulting density profile inside the circular boundary is
			\begin{equation}
				\rho_{\textrm{in}}(r,\theta)\simeq\rho_{\textrm{bulk}}\left[1+2\epsilon^2\frac{I_0\left(\frac{\sqrt{2}r}{\lambda}\right)-1}{I_0\left(\frac{\sqrt{2}R}{\lambda}\right)+I_2\left(\frac{\sqrt{2}R}{\lambda}\right)}\right].
				\label{eq:circular density inside}
			\end{equation}
			When compared to the flat wall, we have
			\begin{equation}
				\rho_{\textrm{in}}(R,\theta)-\rho_{\textrm{flat}}\simeq\epsilon^2\rho_{\textrm{bulk}}\frac{\lambda}{\sqrt{2}R}.
				\label{eq:concave boundary density}
			\end{equation}
			The $\epsilon^2$ contributions to the densities inside (Eq.\ (\ref{eq:circular density inside})) and outside (Eq.\ (\ref{eq:circular density outside})) the circular boundary are shown in Figure \ref{fig:circular density in out}.  On both sides of the circular boundary, the correction to density is proportional to $R^{-1}$, the curvature of the boundary.  In addition, the sign of the correction tells us that active particles accumulate more on concave surfaces than on convex ones.  This preference to accumulate on concave rather than convex surfaces is shown by a discontinuous drop in density going from inside to outside the circular boundary.

			The key observation here is that our approach, even though applied to weakly active particles, can recover some results beyond the weak limit such as the curvature dependence of the density of active particles near curved boundaries \cite{Duzgan Sellinger,Solon et al laplace,Wittmann et al,Nikola et al,Sandford et al,Fily et al}.

			\begin{figure}
				\centering
				\includegraphics[scale=0.45]{circular_boundary_in_out.png}
				\caption{$\epsilon^2$ contributions to the density inside (Eq.\ (\ref{eq:circular density inside})) and outside (Eq.\ (\ref{eq:circular density outside})) an impenetrable circular boundary where the density inside the boundary at $r=0$ is maintained to be $\rho_{\textrm{bulk}}$.  Here, $R=10\lambda$.  Active particles prefer to accumulate on concave surfaces such as the inner side of the circular boundary rather than convex surfaces such as the outer side.  This gives rise to a discontinuous drop in density between the inner and outer parts of a circular boundary that is proportional to the curvature $R^{-1}$.}
				\label{fig:circular density in out}
			\end{figure}

		\subsection{\label{subsec:wedge}Active particles inside a wedge-shaped region}
			In Section \ref{subsec:corner}, we considered active particles confined by two walls meeting at a right angle.  We here consider a more general and difficult problem of two walls meeting at an arbitrary angle $2\alpha$ (Figure \ref{fig:various geometries}c).  This problem is inspired by experiments and simulations \cite{Galajda et al, Kaiser et al} that showed that active particles could be trapped or directed by wedge-shaped obstacles.  For simplicity, we will focus on the particular case of weakly active particles trapped within a single wedge whose sides extend indefinitely.  The zero current boundary conditions $J_{\theta}(r,\pm\alpha,\boldsymbol{\eta})=0$ along the walls of the wedge give the following condition on the coefficients
			\begin{equation}
				\frac{1}{\tilde{r}}\frac{\partial C_{\boldsymbol{m}}^{(n)}(\tilde{r},\pm\alpha)}{\partial\theta}=w_{\theta}(\tilde{r},\pm\alpha).
			\end{equation}
			In order to make any progress on finding the coefficients in this geometry, we have to make use of the Kontorovich-Lebedev and Mellin transforms (Appendix \ref{subapp:KL M transforms}).  Details for computing the coefficients up to $\epsilon^2$ can be found in Appendix \ref{subapp:wedge}.  We will focus on two quantities: the average propulsion $\langle\boldsymbol{\eta}\rangle(r,\theta)$ and the density $\rho(r,\theta)$ within the wedge.  The general expressions for arbitrary wedge angle $2\alpha$ are quite cumbersome.  Note that $2\alpha=\pi$ and $2\alpha=\pi/2$ correspond to a single wall and two walls meeting at a right angle, respectively.  We have already considered these in Section \ref{sec:cartesian} and it is quite easy to check that the general solution Eq.\ (\ref{eq:wedge correction general}) reduces to those cases.  From here on, we will consider angles of the form $2\alpha=\pi/2^{l-1}$ with $l=3,4\dots$, for which we can make some analytical progress.

			\begin{figure*}
				\centering
				\includegraphics[scale=0.5]{wedge_tip_orientation.png}
				\caption{\textbf{a)} Average propulsion $\langle\boldsymbol{\eta}\rangle$ within a wedge-shaped region with angle $2\alpha=\pi/8$.  \textbf{b)} $x$-component of the average propulsion $\langle\eta_x\rangle$ along the center of the wedge $\theta=0$, also for $2\alpha=\pi/8$.  This propulsion towards the tip decays rapidly over a length scale $\frac{\lambda}{\sin\pi/2^l}$.}
				\label{fig:wedge tip orientation}
			\end{figure*}

			To start, the components of the average propulsion $\langle\boldsymbol{\eta}\rangle(r,\theta)=(\langle\eta_x\rangle(r,\theta),\langle\eta_y\rangle(r,\theta))$ within the wedge are given by
			\begin{widetext}
				\begin{subequations}
					\begin{align}
						\langle\eta_x\rangle(r,\theta)&\simeq-\frac{\epsilon\gamma v}{\sqrt{2}}\sin\frac{\pi}{2^l}\sum_{k=0}^{2^{l-2}-1}\left\{e^{-\frac{\sqrt{2}r}{\lambda}\sin\left[\frac{\left(2k+1\right)\pi}{2^l}-\theta\right]}+e^{-\frac{\sqrt{2}r}{\lambda}\sin\left[\frac{\left(2k+1\right)\pi}{2^l}+\theta\right]}\right\},\\
						\langle\eta_y\rangle(r,\theta)&\simeq\frac{\epsilon\gamma v}{\sqrt{2}}\cos\frac{\pi}{2^l}\sum_{k=0}^{2^{l-2}-1}(-1)^k\left\{e^{-\frac{\sqrt{2}r}{\lambda}\sin\left[\frac{\left(2k+1\right)\pi}{2^l}-\theta\right]}-e^{-\frac{\sqrt{2}r}{\lambda}\sin\left[\frac{\left(2k+1\right)\pi}{2^l}+\theta\right]}\right\}.
					\end{align}
				\end{subequations}
			\end{widetext}
			The average propulsion $\langle\boldsymbol{\eta}\rangle$ for $l=4$ or $2\alpha=\pi/8$ is shown in Figure \ref{fig:wedge tip orientation}a.  Note the interesting combination of exponentials in the expressions for $\langle\eta_x\rangle$ and $\langle\eta_y\rangle$.  For the simple case of a right-angled corner (Section \ref{subsec:corner}), we found that we could essentially treat each wall as independent up to $\epsilon^2$, that is, each wall contributed a single exponential decay $e^{-\frac{\sqrt{2}x}{\lambda}}$ or $e^{-\frac{\sqrt{2}y}{\lambda}}$ away from itself.  Here, for wedge angles smaller than $\pi/2$, the walls near the tip will influence each other and we find a multitude of exponentials with different length scales.  This has some consequences on the propulsion and accumulation of the active particles.  Let us focus on the propulsion along the center of the wedge $\theta=0$ on which we have $\langle\eta_y\rangle(r,0)=0$ and
			\begin{equation}
				\langle\eta_x\rangle(r,0)=-\frac{\epsilon\gamma v}{\sqrt{2}}\sin\frac{\pi}{2^l}\sum_{k=0}^{2^{l-2}-1}e^{-\frac{\sqrt{2}r}{\lambda}\sin\frac{(2k+1)\pi}{2^l}}.
				\label{eq:average eta_x along center}
			\end{equation}
			This is plotted in Figure \ref{fig:wedge tip orientation}b for $l=4$ or $2\alpha=\pi/8$.  The longest length scale is $\frac{\lambda}{\sin\pi/2^l}$, which grows with decreasing wedge angle.  Beyond this distance, the active particles along the center will be at least a distance $\lambda$ from the sides of the wedge and will effectively not interact with the boundaries.  Within this distance, however, the boundaries will on average orient the active particles towards the tip.  Thus, as the wedge angle decreases, active particles farther and farther from the tip will have some orientation towards it, which in turn should increase the density near the tip.

			To study the effect of wedge angle on the density near the tip, we write the density as $\rho(r,\theta)\simeq\rho_{\textrm{bulk}}[1+\epsilon^2\Delta(r,\theta)]$.  The correction to the density $\Delta(r,\theta)$ is given by
			\begin{widetext}
				\begin{align}
					\begin{split}
						\Delta(r,\theta)=&\int_{-\infty}^{\infty}a(s)\cosh s\theta\left(\frac{r}{\lambda}\right)^{-is}ds\\
						&+\sum_{k=0}^{2^{l-2}-1}(-1)^k\cos\left\{\frac{[2k+1-(-1)^k]\pi}{2^l}\right\}\left\{e^{-\frac{\sqrt{2}r}{\lambda}\sin\left[\frac{(2k+1)\pi}{2^l}-\theta\right]}+e^{-\frac{\sqrt{2}r}{\lambda}\sin\left[\frac{(2k+1)\pi}{2^l}+\theta\right]}\right\},
					\end{split}
					\label{eq:wedge correction}
				\end{align}
				where
				\begin{align}
					a(s)=\frac{\sqrt{2}^{-is}\Gamma(is+1)}{2\pi s\sinh\frac{s\pi}{2^l}}\left[1+2\sum_{k=1}^{2^{l-2}-1}\left(\sin^2\frac{\pi}{2^l}+(-1)^k\cos^2\frac{\pi}{2^l}\right)\left(\sin\frac{k\pi}{2^{l-1}}\right)^{-is+1}\right].
					\label{eq:wedge a}
				\end{align}
			\end{widetext}
			For angles $2\alpha=\pi/2^{l-1}$ with $l\ge3$, there is no known closed form solution and we have to numerically evaluate the integral in Eq.\ (\ref{eq:wedge correction}).  Let us focus on how the density at the tip of the wedge $\Delta_{\textrm{tip}}=\lim\limits_{r\rightarrow0}\Delta(r,\theta)$ depends on the angle of the wedge $2\alpha$.  This dependence is shown in Figure \ref{fig:wedge tip density}.
			\begin{figure}
				\centering
				\includegraphics[scale=0.5]{wedge_tip_density.png}
				\caption{\textbf{a)} Dependence of the correction to wedge tip density $\Delta_{\textrm{tip}}=\lim\limits_{r\rightarrow0}\Delta(r,\theta)$ on the wedge angle $2\alpha=\pi/2^{l-1}$ for $l=1,2,\dots,15$.  The black dotted line shows the linear dependence close to $2\alpha=\pi$, which was obtained using the approach developed in Section \ref{sec:corrugated wall}.  \textbf{b)} Weak scaling of $\Delta_{\textrm{tip}}$ with $2\alpha$ for small wedge angles.}
				\label{fig:wedge tip density}
			\end{figure}

			There are a couple of features to note.  The first is the increase in the density at the tip as we decrease the angle of the wedge, as shown in Figure \ref{fig:wedge tip density}a.  This is due to the effect of the sides of the wedge on the average propulsion $\langle\boldsymbol{\eta}\rangle$ discussed earlier (Eq.\ \ref{eq:average eta_x along center}).  As the wedge angle decreases, active particles farther and farther from the tip are on average directed towards it, thus increasing the density.  In addition to the increase in density for small wedge angles, we find that when the wedge is nearly a flat wall ($2\alpha\approx\pi$), the density has a linear dependence given by $\Delta_{\textrm{tip}}\approx1-0.44(2\alpha-\pi)$.  We will show how this is obtained using an approach developed in Section \ref{sec:corrugated wall}.

			The second feature is the scaling of the tip density $\Delta_{\textrm{tip}}$ with the wedge angle $2\alpha$.  As shown in Figure \ref{fig:wedge tip density}b, we find an interesting weak dependence of the form $\Delta_{\textrm{tip}}\sim\ln(2\alpha)^{-0.50}$.  This weak dependence is due to passive Brownian diffusion and can actually be obtained through a relatively simple scaling argument.  This argument goes as follows.  As we discussed earlier, there is a length scale $\lambda_l\sim\frac{\lambda}{\sin\pi/2^l}$ over which the active particles on average propel towards the tip.  This increases the density at the tip by $\rho_{\textrm{bulk}}\epsilon^2\Delta_{\textrm{tip}}$.  Thus, the diffusive flux over this length scale away from the tip goes as $J_{\textrm{diff}}\sim D_p\frac{\rho_{\textrm{bulk}}\epsilon^2\Delta_{\textrm{tip}}}{\lambda_l}$.  To estimate the advective flux, we compute a characteristic propulsion towards the tip by averaging $\langle\eta_x\rangle(r,0)$ (Eq.\ \ref{eq:average eta_x along center}) over the region $0<r\lesssim\lambda_l$.  Thus the advective flux towards the tip goes as $J_{\textrm{swim}}\sim\frac{1}{\gamma}\left|\frac{1}{\lambda_l}\int_{0}^{\lambda_l}\langle\eta_x\rangle(r,0)dr\right|\rho_{\textrm{bulk}}$.  To a good approximation, we can take the upper limit of the integral to $\infty$ since $\langle\eta_x\rangle(r,0)$ decays rapidly over $0<r\lesssim\lambda_l$ (Figure \ref{fig:wedge tip orientation}b).  Finally, the diffusive and advective fluxes in steady-state should balance ($J_{\textrm{diff}}\sim J_{\textrm{swim}}$) and so we obtain
			\begin{align}
				\begin{split}
					\Delta_{\textrm{tip}}&\sim\frac{1}{\gamma D_p\epsilon^2}\left|\int_{0}^{\infty}\langle\eta_x\rangle(r,0)dr\right|\\
					&=\sin\frac{\pi}{2^l}\sum_{k=0}^{2^{l-2}-1}\frac{1}{\sin\frac{(2k+1)\pi}{2^l}}\\
					&\approx\frac{1}{2}\int_{\frac{\pi}{2^l}}^{\frac{\pi}{2}}\frac{du}{\sin u}\approx-\frac{1}{2}\ln\frac{\pi}{2^{l+1}},
				\end{split}
			\end{align}
			where we assumed that $2^l\gg1$ for small wedge angles and approximated the sum as an integral.  Thus, up to a constant shift, we see that $\Delta_{\textrm{tip}}\sim\ln\left(\frac{\pi}{2^{l-1}}\right)^{-1/2}=\ln(2\alpha)^{-1/2}$, in close agreement with our numerical estimates in Figure \ref{fig:wedge tip density}b.

	\section{\label{sec:corrugated wall}Active particles near a corrugate wall}
		We now turn to one last example involving impenetrable walls.  We study how active particles behave near a corrugated wall (Figure \ref{fig:various geometries}d) and show how we can formulate the boundary condition for such a wall.  This example is inspired by experiments on asymmetric gears in bacterial baths showing that active particles can generate tangential forces on an asymmetric boundary \cite{Sokolov et al,Leonardo et al}, in addition to simulations showing that asymmetric boundaries can transport active particles \cite{Ghosh et al,Ai,Ai et al}.  Thus, we expect that a combination of asymmetry and activity should lead to directional motion of either the boundary or the particles.  However, one of the counterintuitive results we will find here is that there is actually no net transport of noninteracting active particles along an asymmetric corrugated wall if the amplitude of the corrugation is too small.  More precisely, the net tangential current of noninteracting active particles near an asymmetric corrugate wall does not decrease to zero linearly as the wall becomes flatter and flatter.

		To set up the problem, suppose we have a boundary with a shape $y=h(x)$ with period $2L$ and characteristic amplitude $\delta$ such that $|h(x)|\lesssim\delta$.  We can decompose the shape into Fourier modes as
		\begin{equation}
			h(x)=\delta\sum_{k=-\infty}^{\infty}h_ke^{\frac{i\pi kx}{L}},
		\end{equation}
		where $h_{k=0}=0$, that is, the shape of the boundary oscillates around $y=0$.  Assuming that the amplitude of the shape is small compared to the length scale of accumulation or $\delta\ll\lambda$, we can write the distribution as
		\begin{equation}
			\tilde{\rho}(\tilde{\boldsymbol{r}},\tilde{\boldsymbol{\eta}})=\sum_{n=0}^{\infty}\epsilon^n\sum_{\boldsymbol{m}}C_{\boldsymbol{m}}^{(n)}(\tilde{\boldsymbol{r}})e^{-\tilde{\boldsymbol{\eta}}^2}H_{m_x}(\tilde{\eta}_x)H_{m_y}(\tilde{\eta}_y),
		\end{equation}
		where we now expand the coefficients as
		\begin{equation}
			C_{\boldsymbol{m}}^{(n)}(\tilde{\boldsymbol{r}})\simeq a_{\boldsymbol{m}}^{(n)}(\tilde{y})+\tilde{\delta}\sum_{k=-\infty}^{\infty}b_{\boldsymbol{m};k}^{(n)}(\tilde{y})e^{\frac{i\pi k\tilde{x}}{\tilde{L}}}.
			\label{eq:corrugated c coefficient}
		\end{equation}
		The functions $a_{\boldsymbol{m}}^{(n)}(\tilde{y})$ are simply the solutions for a flat wall with no corrugation, which we have already computed in Section \ref{sec:cartesian}.  The resulting equation for the unknown coefficients $b_{\boldsymbol{m};k}^{(n)}(\tilde{y})$ is
		\begin{equation}
			\frac{d^2b_{\boldsymbol{m};k}^{(n)}}{d\tilde{y}^2}-\left[2(m_x+m_y)+\frac{\pi^2k^2}{\tilde{L}^2}\right]b_{\boldsymbol{m};k}^{(n)}=\frac{i\pi k}{\tilde{L}}w_x+\frac{dw_y}{d\tilde{y}},
			\label{eq:corrugated coefficient equation}
		\end{equation}
		where
		\begin{subequations}
			\begin{align}
				w_x&=b_{m_x-1,m_y;k}^{(n-1)}+2(m_x+1)b_{m_x+1,m_y;k}^{(n-1)},\\
				w_y&=b_{m_x,m_y-1;k}^{(n-1)}+2(m_y+1)b_{m_x,m_y+1;k}^{(n-1)}.
			\end{align}
		\end{subequations}
		For the boundary condition, we require the normal component of the current to be zero at the boundary or $\boldsymbol{J}(x,h(x))\cdot\hat{\boldsymbol{n}}=0$, where $\hat{\boldsymbol{n}}$ is the normal to the boundary.  Assuming that $\delta\ll\lambda$, we can linearize this boundary condition to get (see Appendix \ref{app:corrugated wall})
		\begin{widetext}
			\begin{align}
				\begin{split}
					&-\left[a_{m_x-1,m_y}^{(n-1)}(0)+2(m_x+1)a_{m_x+1,m_y}^{(n-1)}(0)\right]\frac{i\pi k}{\tilde{L}}h_k+\left[\frac{da_{m_x,m_y-1}^{(n-1)}(0)}{d\tilde{y}}+2(m_y+1)\frac{da_{m_x,m_y+1}^{(n-1)}(0)}{d\tilde{y}}-\frac{d^2a_{\boldsymbol{m}}^{(n)}(0)}{d\tilde{y}^2}\right]h_k\\
					&+b_{m_x,m_y-1;k}^{(n-1)}(0)+2(m_y+1)b_{m_x,m_y+1;k}^{(n-1)}(0)-\frac{db_{\boldsymbol{m};k}^{(n)}(0)}{d\tilde{y}}=0.
				\end{split}
				\label{eq:corrugated boundary condition}
			\end{align}
		\end{widetext}
		Determining the coefficients $b_{\boldsymbol{m};k}^{(n)}(\tilde{y})$ is quite straightforward and the expressions can be found in Appendix \ref{app:corrugated wall}.

		Let us start with the simplest case of $h(x)=\delta\cos\frac{\pi x}{L}$ or $h_{k=\pm1}=\frac{1}{2}$.  In particular, consider the density $\rho(x,h(x))$ along the boundary when the amplitude of the boundary is small $\delta\ll\lambda$ and the wavelength is large $L\gg\lambda$.  For this slow varying boundary, the density along the boundary to linear order in the amplitude is
		\begin{equation}
			\rho(x,h(x))\simeq\rho_{\textrm{bulk}}\left[1+\epsilon^2-\epsilon^2\delta\frac{\pi^2\lambda}{\sqrt{2}L^2}\cos\frac{\pi x}{L}\right].
		\end{equation}
		Notice that the last term, which captures the effect of corrugation, is proportional to $\frac{d^2h}{dx^2}$.  Thus, the change in density is related to the local curvature of the boundary with more active particles accumulating on the concave sections than on the convex sections.  The correction due to the corrugation can be written as $\epsilon^2\rho_{\textrm{bulk}}\frac{\lambda}{\sqrt{2}R}$, where $R=\left[1+\left(\frac{dh}{dx}\right)^2\right]^{\frac{3}{2}}\left|\frac{d^2h}{dx^2}\right|^{-1}\simeq\frac{L^2}{\pi^2\delta}$ is the radius of curvature at the maxima and minima of the corrugated boundary.  Note that this is in exact agreement with the result for the densities outside and inside a large circular boundary with radius $R\gg\lambda$ (Eqs.\ (\ref{eq:convex boundary density}) and (\ref{eq:concave boundary density})).

		Let us now consider a more complex example.  Suppose we have a sawtooth-shaped boundary (Figure \ref{fig:corrugated currents}, top) given by
		\begin{equation}
			h(x)=\begin{cases}
				-\delta+\frac{2\delta}{(1+\zeta)L}(x+L), & -L<x<\zeta L\\
				\delta-\frac{2\delta}{(1-\zeta)L}(x-\zeta L), & \zeta L<x<L
			\end{cases}.
			\label{eq:sawtooth shape}
		\end{equation}
		The asymmetry is controlled by $\zeta$, where $\zeta=0$ corresponds to a symmetric sawtooth.  Unlike the simple case of a cosine-shaped boundary, the sawtooth is not twice differentiable near the sharp tips, and so curvature is not well-defined.  In Section \ref{subsec:wedge}, we studied how the density at the tip of a wedge depended on the angle of the wedge.  Using a sawtooth-shaped boundary, we can obtain the dependence of the density on angles near $2\alpha=\pi$.  Taking the slow-varying symmetric sawtooth with $\delta\ll\lambda$ and $L\gg\lambda$, the active particles near the tip at $x=0$ effectively see a wedge with angle $2\alpha\simeq\pi+\frac{4\delta}{L}$, where $\delta>0$ and $\delta<0$ correspond to convex and concave, respectively.  Writing the density as $\rho(x,y)=\rho_{\textrm{bulk}}\left[1+\epsilon^2\Delta(x,y)\right]$, just as we did for the wedge, we find that the correction to density at the tip is (see Appendix \ref{app:corrugated wall})
		\begin{equation}
			\Delta_{\textrm{tip}}=\Delta(0,\delta)\approx1+\frac{4\delta}{L}S\simeq1-0.44(2\alpha-\pi).
		\end{equation}

		Before we conclude this section, we briefly discuss the currents of noninteracting weakly active particles in the presence of a corrugated wall.  The explicit forms of the currents can be found in Appendix \ref{app:corrugated wall}.  For a boundary with an asymmetric shape, one expects there to be a net flux of active particles along the boundary.  For example in suspensions of bacteria, it has been seen that swimming bacteria can be directed by funnels and can rotate gears with asymmetric teeth \cite{Galajda et al,Sokolov et al,Leonardo et al}.  However, for our case of an asymmetric sawtooth boundary and noninteracting weakly active particles, we find the surprising result that there is no net drift along the wall to linear order in the amplitude $\delta$ of the corrugation.  Mathematically, this is easily explained by noticing that the coefficient equation (Eq.\ (\ref{eq:corrugated coefficient equation})) and boundary condition (Eq.\ (\ref{eq:corrugated boundary condition})) are all independent for each mode $k$.  Since each mode is a symmetric sine or cosine wave, none of them contribute to a net drift.  In addition, there is also no net tangential force on the boundary by the same reasoning.

		In order to get a net tangential drift or force along the boundary, we need to couple modes with different $k$, which can be done by introducing nonlinearities.  There are two possible ways to do this.  The first way is going beyond the linearized boundary condition (Eq.\ (\ref{eq:corrugated boundary condition})) and considering higher orders in the amplitude $\delta$.  In fact, noting that the transformation $\delta\rightarrow-\delta$ should simply flip the direction of drift, the drift of noninteracting weakly active particles due to an asymmetric corrugated boundary should scale as $\delta^3$ for small amplitudes of corrugation.  This nonlinear scaling with amplitude of corrugation has been seen in simulations of noninteracting active particles \cite{Ghosh et al}.  The calculation for going beyond the linearized boundary condition is rather involved and will be reserved for a future work.  The second possible way is including interactions such as alignment between the active particles.  Interactions may make it easier for an asymmetric boundary to induce net fluxes.  In fact, it has been seen in simulations of aligning active particles in corrugated channels that the net currents along the channels can actually be linear in the amplitude of corrugation \cite{Ai et al}, contrasting our result for noninteracting active particles.

		Note that while there is no net tangential drift to linear order $\delta$, there is still a local circulation of active particles (Figure \ref{fig:corrugated currents}, bottom).  These local fluxes of active particles towards the concave parts and away from the convex parts of the boundary are responsible for the increases and decreases of the densities in those parts, respectively.

		\begin{figure*}
			\centering
			\includegraphics[scale=0.55]{corrugated_currents.png}
			\caption{\textbf{Top:} Sawtooth-shaped boundary defined by Eq.\ (\ref{eq:sawtooth shape}).  \textbf{Bottom:} Currents $\boldsymbol{J}(x,y)=(J_x(x,y),J_y(x,y))$ due to sawtooth-shaped boundaries with asymmetry $\zeta=0$ (\textbf{left}) and $\zeta=0.5$ (\textbf{right}) computed using the first 20 modes of $h(x)$.}
			\label{fig:corrugated currents}
		\end{figure*}

	\section{\label{sec:spherical}A problem in spherical coordinates: absorption of active particles around a sphere}
		For the final case, we will consider an absorbing boundary as opposed to a impenetrable boundary, which has been our focus thus far.  In particular, let us consider a uniform bath of weakly active particles in which we place an absorbing sphere (Figure \ref{fig:various geometries}e).  We are interested in determining the steady-state rate at which these active particles are absorbed and how that rate differs from that of passive Brownian particles.  For the absorption of passive Brownian particles, this is known as the Smoluchowski problem \cite{Smoluchowski}.

		In spherical coordinates, we write the dimensionless distribution as
		\begin{align}
			\begin{split}
				\tilde{\rho}(\tilde{r},\theta,\phi,\tilde{\boldsymbol{\eta}})=\sum_{n=0}^{\infty}\epsilon^n&\sum_{\boldsymbol{m}}C_{\boldsymbol{m}}^{(n)}(\tilde{r},\theta,\phi)\\
				&\times e^{-\tilde{\boldsymbol{\eta}}^2}H_{m_x}(\tilde{\eta}_x)H_{m_y}(\tilde{\eta}_y)H_{m_z}(\tilde{\eta}_z)
			\end{split}
		\end{align}
		The coefficients satisfy
		\begin{align}
			\begin{split}
				&\frac{1}{\tilde{r}^2}\frac{\partial}{\partial\tilde{r}}\left(\tilde{r}^2\frac{\partial C_{\boldsymbol{m}}^{(n)}}{\partial\tilde{r}}\right)-2(m_x+m_y+m_z)C_{\boldsymbol{m}}^{(n)}\\
				&+\frac{1}{\tilde{r}^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial C_{\boldsymbol{m}}^{(n)}}{\partial\theta}\right)+\frac{1}{\tilde{r}^2\sin^2\theta}\frac{\partial^2C_{\boldsymbol{m}}^{(n)}}{\partial\phi^2}\\
				&=\frac{1}{\tilde{r}^2}\frac{\partial}{\partial\tilde{r}}(\tilde{r}^2w_r)+\frac{1}{\tilde{r}\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta w_{\theta})+\frac{1}{\tilde{r}\sin\theta}\frac{\partial w_{\phi}}{\partial\phi},
			\end{split}
		\end{align}
		where the components of $\boldsymbol{w}$ in spherical coordinates are
		\begin{subequations}
			\begin{align}
				w_r&=w_x\sin\theta\cos\phi+w_y\sin\theta\sin\phi+w_z\cos\theta,\\
				w_{\theta}&=w_x\cos\theta\cos\phi+w_y\cos\theta\sin\phi-w_z\sin\theta,\\
				w_{\phi}&=-w_x\sin\phi+w_y\cos\phi,
			\end{align}
		\end{subequations}
		and
		\begin{subequations}
			\begin{align}
				w_x&=C_{m_x-1,m_y,m_z}^{(n-1)}+2(m_x+1)C_{m_x+1,m_y,m_z}^{(n-1)},\\
				w_y&=C_{m_x,m_y-1,m_z}^{(n-1)}+2(m_y+1)C_{m_x,m_y+1,m_z}^{(n-1)},\\
				w_z&=C_{m_x,m_y,m_z-1}^{(n-1)}+2(m_z+1)C_{m_x,m_y,m_z+1}^{(n-1)}.
			\end{align}
		\end{subequations}
		If the sphere has a radius $R$, then the absorbing boundary condition at $r=R$ gives us the condition on the coefficients  $C_{\boldsymbol{m}}^{(n)}(\tilde{R},\theta,\phi,\tilde{\boldsymbol{\eta}})=0$.  Details of the solution can be found in Appendix \ref{app:absorber}.  Taking $\rho(r,\theta,\phi,\boldsymbol{\eta})$ and integrating out $\boldsymbol{\eta}$, we obtain to order $\epsilon^2$ the density
		\begin{align}
			\frac{\rho(r,\theta,\phi)}{\rho_{\textrm{bulk}}}\simeq1-\frac{R}{r}+\epsilon^2\frac{\lambda}{\lambda+\sqrt{2}R}\left[1-e^{-\frac{\sqrt{2}(r-R)}{\lambda}}\right]\frac{R}{r},
		\end{align}
		where $\rho_{\textrm{bulk}}$ is the uniform density far from the sphere.  The first part is the familiar $r^{-1}$ solution for passive Brownian particles while the second part is the correction due to activity, which elevates the density near the absorbing sphere.  The radial current is
		\begin{equation}
			J_r(r,\theta,\phi)\simeq-\frac{\rho_{\textrm{bulk}}D_pR}{r^2}\left(1+\epsilon^2\frac{\sqrt{2}R}{\lambda+\sqrt{2}R}\right),
		\end{equation}
		from which we calculate the capture rate $\kappa=|\int J_r(R,\theta,\phi)R^2\sin\theta\,d\theta \,d\phi\,|$ as
		\begin{equation}
			\kappa\simeq\kappa_0\left(1+\epsilon^2\frac{\sqrt{2}R}{\lambda+\sqrt{2}R}\right),
		\end{equation}
		where $\kappa_0=4\pi\rho_{\textrm{bulk}}D_pR$ is the well-known capture rate for passive Brownian particles \cite{Smoluchowski}, which depends on the radius and not the surface area of the sphere.

		The correction due to activity is a new result.  There are two limits: a large target $R\gg\lambda$ and a small target $R\ll\lambda$.  For a large target, the time it takes the weakly active particles to passively diffuse over a distance comparable to the radius of the absorber is much longer than the persistence time ($\tau\ll R^2/D_p$).  On this time scale, the active particles appear effectively diffusive and we find the capture rate $\kappa\simeq4\pi\rho_{\textrm{bulk}}R(D_p+v^2\tau/3)$, which can be interpreted as just that of a diffusing particle with effective diffusivity $D_{\textrm{eff}}=D_p+v^2\tau/3$.  For a small target, however, the time it takes to diffuse over the radius of the absorber is much shorter than the persistence time ($\tau\gg R^2/D_p$).  On this time scale, the propulsions of the active particles appear persistent.  We find $\kappa\simeq4\pi\rho_{\textrm{bulk}}R(D_p+v^2\tau\sqrt{2}R/3\lambda)$.  Note that the correction to the rate due to activity scales as $R^2$, which is related to the surface area or cross-sectional area of the absorber.  This is reminiscent of the capture rate $\kappa\sim\rho_{\textrm{bulk}}vR^2$ for a spherical absorber in an ideal gas of ballistic particles \cite{Hubner Titulaer}, whose mean free paths are longer than the radius of the sphere.  One key difference of course is that while the propulsions of our weakly active particles appear persistent, their motions are still dominated by passive Brownian diffusion.  To summarize, in both cases of a small and large target, we find that activity increases the density and enhances capture rate of active particles near an absorbing boundary.

	\section{Discussion and Conclusion}
		We studied how noninteracting weakly active particles, for which activity can be treated perturbatively, behave near various types of boundaries in different geometries; for example, active particles moving on a line or in a wedge-shaped region, interacting with a corrugated wall, or absorbing around a sphere.  The key to making progress on this problem is to include passive Brownian diffusion, which allows us not only to cleanly formulate boundary conditions for the different types of boundaries but also to solve the problem systematically.  In other words, by treating activity perturbatively, we can take the solutions for passive Brownian particles, which are often known, and use them to iteratively compute the corrections due to the activity.  We formulated a relatively simple series solution for the distribution of active particles that consists of an expansion in powers of the P\'{e}clet number, which characterizes the strength of activity, and an expansion in terms of Hermite polynomials.  This series solution reduces the Fokker-Planck equation for the distribution of active particles to a simpler partial differential equation and in some cases, to an even simpler ordinary differential equation.  We summarize below some of our main results for the different geometries.

		By considering the simple cases of noninteracting weakly active particles confined by one or two impenetrable walls in 1D (Section \ref{subsec:particles on line}), we found that the active particles on average propel towards nearby walls.  This leads to accumulation and an increase in pressure exerted on the walls.  In particular, we found that the pressure follows the ideal gas law but instead of being proportional to the density in the bulk, it is proportional to the increased density at the walls.

		We also determined how the curvature of a boundary affects the accumulation of weakly active particles (Section \ref{sec:polar}).  For the case of a circular boundary, we found that the accumulation is proportional to the curvature of the boundary.  This last result has been shown to hold in the limit of strong activity \cite{Duzgan Sellinger,Solon et al laplace,Wittmann et al,Nikola et al,Sandford et al,Fily et al}.  Thus, our approach for studying the limit of weakly active particles can potentially gain us insight into the opposite limit.  For the case of a wedge-shaped region, we found that as the wedge angle decreases, active particles farther and farther from the tip gain on average some propulsion towards it.  Interestingly, while this propulsion does increase the density near the tip, we found that the accumulation has a rather weak dependence on the wedge angle.  This is due to passive Brownian diffusion which tends to smooth out variations in density.

		Finally, we have also obtained novel results for weakly active particles near a corrugated boundary and around an absorbing sphere.  Near a corrugated boundary (Section \ref{sec:corrugated wall}), particularly one shaped like an asymmetric sawtooth, we found that there is surprisingly no net transport of noninteracting weakly active particles along the boundary to linear order in the amplitude $\delta$ of the corrugation.  This is due to the linearity of the boundary condition and the Fourier modes that make up the shape of the wall.  We argued that in order to observe net currents we have to introduce nonlinearities such as going beyond the linearized boundary condition or including interactions between the active particles.  For our case of noninteracting weakly active particles, we expect to see net currents at order $\delta^3$.  A nonlinear dependence on $\delta$ has been seen in simulations of noninteracting active particles \cite{Ghosh et al}.  For interacting active particles, it is more difficult to determine the dependence.  However, it has been observed in simulations that for active particles with aligning interactions, the net current is linear in $\delta$ \cite{Ai et al}, suggesting that interactions may enhance the transport of active particles.  For an absorbing sphere placed in a bath of active particles (Section \ref{sec:spherical}), we computed the rate at which the weakly active particles are absorbed.  This is the active version of the Smoluchowski problem \cite{Smoluchowski} for passive Brownian particles.  We found that activity elevates the density near the sphere and enhances the absorption rate.  Thus, activity may be useful in enhancing the self-assembly of colloidal structures \cite{Mallory et al}.

		Before we end this story, there are some interesting future directions to consider.  The first direction is finding the exact solution for the distribution of noninteracting active particles near the simplest case of a flat wall.  By ``exact'', we mean a closed-form expression for the distribution or, at a minimum, for all the coefficients in our series solution.  With the current approach, one can systematically compute higher and higher orders.  However, the expressions, though straightforward, become increasingly cumbersome to write down.  The goal would be to find a pattern in the coefficients that one can exploit.  Finding a clean way to do this could aid us in finding more exact solutions in other interesting geometries.

		The second direction is going beyond the linearized boundary condition for a corrugated boundary.  As we found, there is no net transport of noninteracting active particles or net tangential force along an asymmetrically-shaped boundary to linear order in the amplitude of the corrugation.  In order to observe net tangential currents or forces, one will need to consider higher orders in the amplitude.  It would be interesting to perform this calculation and to analytically compute how fast active particles are transported by an asymmetric wall or how fast an asymmetric wall is pushed like a gear by active particles.

		Finally, it would be interesting to extend the approach developed here to more realistic models of active particles.  This includes studying other models of active particles such as active Brownian particles, which typically model many types of self-propelled colloids \cite{Palacci et al,Paxton et al}, and run-and-tumble particles, which typically model bacteria \cite{Berg et al}.  More generally, it may be interesting to study models where the correlations are not exponential or the persistence times have a broad distribution, as has been seen in some bacterial systems \cite{Figueroa-Morales et al}.  An important question is whether there are critical differences between the many models of active particles, for example, when interacting with boundaries.  In addition to studying different models of active particles, it would also be interesting to include interactions between particles in our approach.  It has been seen that a simple repulsive interaction can have significant effects on the density and pressure of active particles \cite{Buttinoni et al,Ginot et al}.  Similarly, as was discussed, interactions may affect the transport of active particles in corrugated channels \cite{Ghosh et al,Ai et al}.

	\section*{Acknowledgments}
		This work was supported primarily by the MRSEC Program of the National Science Foundation under Award Number DMR-1420073.  The author greatly thanks Alexander Grosberg for his invaluable comments and critical reading of the manuscript.

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Lee Hecht Harrison announces the appointment of its new Managing Director for Australia and New Zealand, James Mcilvena
Asia-Pacific Uncategorized
December 13, 2018 Presswire
Lee Hecht Harrison (LHH) today announces the appointment of its new Managing Director for Australia and New Zealand (ANZ), James Mcilvena. Joining on the 14th of January 2019, James will be leading the business for the ANZ region. James will serve as a member of the LHH Global Leadership Team and the Asia Pacific (APAC) Executive Committee.
James served as Head of Talent Acquisition – Wealth Management & Professional Services at Commonwealth Bank of Australia where he set strategy, led initiatives for hiring talent and drove continuous team improvement. Before Commonwealth Bank of Australia, he worked as Head of Accounts at PeopleScout where he was responsible for strategic direction, sales and overall account performance.
Commenting on James Mcilvena's appointment, Georg Hirschi, CEO of APAC for Lee Hecht Harrison says, "We are very pleased to have James on board. He brings a tremendous amount of knowledge and experience in the Human Resources industry that is key for our business as we continue our ambitious plans to grow. James will further strengthen our ability to deliver outstanding value to our customers, participate actively in the development of the APAC zone strategy and support the implementation of global initiatives."
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Profiles /
By Harry Thorne
In Profile: Sue Jones
The Director of Whitstable Biennale talks about her time in Kent and what can be expected from this year's programme
Harry Thorne You were director of Chisenhale Gallery in London from 1998 to 2000 before directing the Whitstable Biennale, a position you have held since 2005. How did you find that transition from the physical gallery space to the less-defined festival environment?
Sue Jones I first joined Chisenhale Gallery in 1990 as part of a small team assisting the then director Jonathan Watkins. It was when Rachel Whiteread's sculpture Ghost (1990) was showing, and the gallery was one of only four in east London (alongside Matt's Gallery, Maureen Paley and the Showroom). I grew up in east London, and it was interesting to watch the area become saturated with galleries during the 10 years that I was at Chisenhale. By the time Tate Modern opened in 2000 I wanted the space to do something different, to work with artists in a different way. The first project I curated outside London was Stephen Turner's Seafort (2005), where the artist spent 36 days in isolation on an abandoned sea fort defense from World War II in the Thames Estuary, just off the coast of Whitstable.
Having worked with Whitstable Biennale now for 10 years, I sometimes miss the level of formal control that comes with a venue, and the extended period of time that I had with works in a gallery – that's an amazing privilege for a curator. However, I like the fluidity of festivals, the way the shape can adapt from year to year. One of the things that hasn't changed with the transition from gallery to festival is that both are small organizations, where the whole team can work with the artists and are very closely involved with the process and production of new work. This is really important to me.
HT Was it easy to make the adjustment from ongoing programming to working towards an event held every two years?
SJ The process of commissioning new work is the same at the Biennale as it was at Chisenhale, but the obvious difference is that in Whitstable everything opens at the same time. The pace is different and there's something terrifying but also satisfying about seeing the results of two years work come together at the same time.
Lucy Pawlak, Academy Training Aids, video still, 2015. Courtesy: the artist
Whitstable's cultural infrastructure is developing rapidly but like many small towns there aren't the large venues and empty buildings that you find in cities, and so our venues are part of the fabric of the town: the odd halls and huts, the shops and bars, the alleyways and oyster beds, the working harbour and the steep shingle shoreline. I don't see the Biennale as a group exhibition, more a series of solo commissions that are able to speak to each other within the context of the town. Something that people often point out to me is the way that the Biennale brings artists together, not only in very direct contact with audiences, but also with each other. Critic John Slyce called it a 'sibling culture', which I think describes it really well.
'I like the way that things build over time. We have worked with many hundreds of artists over the last ten years, and it's satisfying that visitors are now revisiting their memories of previous works.'
HT How do you feel Whitstable Biennale has developed since you took over?
SJ There had been just two editions of the Biennale when I joined in 2005 and I've very much enjoyed the challenge of building the festival into an event which operates in various different contexts; one that has real value locally but also has relevance both nationally and internationally. I would like to keep the organization small, to retain the clear focus on working closely with artists, but as a small staff we are stretched, and we face that constant dilemma of judging how much growth is a good thing.
I like the way that things build over time. We have worked with many hundreds of artists over the last ten years, and it's satisfying that visitors are now revisiting their memories of previous works. This year when you go into the newsagents you won't find a work, but you may remember Katie Paterson's Dying Star (doorbell) from 2010. When you visit the Library lecture room to see Parlor Walls by young artists Webb-Ellis, you may remember Katrina Palmer's work from 2014, or Olivia Plender & Unnar Örn's work from 2010. When you see Evan Ifekoya's performances at the Sea Cadets' Hall, you might recall Ryan Gander's performance lecture that took place there in 2006.
Tessa Lynch, Stance, 2016, documentation of performance. Courtesy: the artist; photograph: Emma Sandström
HT Do you feel the Biennale has had an impact on the town itself?
SJ Unlike many contemporary art festivals we don't have a specific regeneration agenda. Our remit is to support experimental work and the development of artistic practice. Nevertheless, we do have an impact on the town: the local authorities calculate that we bring in over GBP£1 million per edition to Whitstable. With London recently named as the world's most expensive city in which to live and work, artists are increasingly moving out of the capital, and the seaside towns of east Kent are one of a few key places that provide a community where visual art is a strength.
HT The title for this year's edition of the Whitstable Biennale is 'The Faraway Nearby'. What motivated that choice of title?
SJ It comes from the title of Rebecca Solnit's wonderful book, The Faraway Nearby (2013), which in turn is a phrase Georgia O'Keefe used as her sign-off to letters sent from New Mexico back to New York. Solnit's book is about storytelling, a study of closeness and distance in a conceptual terrain that ranges from the Arctic landscape, scenes from Mary Shelley's Frankenstein (1813) to the fairytale-like mounds of apricots Solnit inherited when her mother's house was sold.
The Faraway Nearby has a local resonance in Kent, the nearest county in the UK to the European mainland, one which has come under increasing pressure from the refugee crisis. Without being heavy-handed, we wanted to say something about where we are and the humanitarian crisis that is happening around us. Boat People, a new film by Sarah Wood takes a statement made in 1946 in the aftermath of WWII by Martin Heidegger, that 'homelessness is coming to be the destiny of the world' and, 60 years on, considers whether displacement has become the norm. Boat People also questions the role that the moving image plays in the representation of human movement and the migration of ideas. Just as the invention of the telescopic lens brought near and far together for the very first time, Boat People is about the way that in the 21st century the near and far are mediated and transformed by the new 'perception accelerator': the digital image.
Mikhail Karikis, Ain't Got No Fear, 2016, video still. Courtesy: the artist
HT Can you talk about some more of this year's new commissions?
SJ Main new commissions for 2016 include works by Mikhail Karikis, Louisa Martin, Tessa Lynch, Evan Ifekoya, Marcia Farquhar, Lucy Pawlak, the artist duo Webb-Ellis, Richard Layzell, Alice Butler and Trish Scott. We also have a cinema designed by Stephen Beasley at Manalo & White architects, and a temporary café (open every day of the festival) which is a work by Matthew Herbert.
'We don't ask artists to respond to Whitstable; the starting point is always the idea, the work. We encourage artists to push an idea to its conclusion, and we do everything we can to make it a reality.'
The cinema will be showing a daily programme of films (including a number of premieres) and hosting talks, including Brian Dillon with Olivia Laing, and Rose Wylie with Ben Rivers and Skye Sherwin. There's also an artists' short film programme running every day, which includes works by Jessica Sarah Rinland and Xiaowen Zhu.
Offsite there is a symposium on ethics at University of the Creative Arts Canterbury, and an exhibition curated by Ben Judd at the Beaney Museum in Canterbury. We are asking people to sign up to read one chapter of The Faraway Nearby every day, and on Saturday 11 June there's an all day reading of David Seabrook's All the Devils Are Here.
HT In 2014, Laura Wilson's project made use of the local aggregates factory. How have the various artists responded to the landscape of Whitstable?
SJ The festival is a showcase for new and experimental work. We don't ask artists to respond to Whitstable; the starting point is always the idea, the work. We encourage artists to push an idea to its conclusion, and we do everything we can to make it a reality.
Evan Ifekoya, Let the Rhythm Pull you Towards Your Edges, 2015, video still. Courtesy: the artist
Some works are shown in blacked out rooms where everything around it disappears. Other works find their space through the logic of the project. For Clio Barnard's Plotlands (2008), the audience were passengers aboard passing trains through the Seasalter marshes, witnessing a film of a burning shack. For Bronwen Buckeridge's The Sorrowful and Immaculate Fall of One Hundred Grazing Sheep (2014), audience members had to find a nondescript garden gate, make their way through that garden to a room that overlooked the Isle of Sheppey across the water, and listen to a sound work about map dowsing. This year on 12 June, Matthew Herbert's temporary café for the festival will host a dinner that grows out of a chapter of a book he is writing about the profound musical revolution that has happened over the last 100 years between the invention of the microphone, the tape recorder and the sampler and computer – the changing structures and assumptions of music as we move from a form of impression to a form of documentary.
HT What can be expected from this year's festival fringe, the Whitstable Satellite?
SJ This year's edition of the Whitstable Satellite sees more than 100 artists contributing towards 45 events, performances, exhibitions and displays of work, co-ordinated by young artist Charley Vines. For the first time the Satellite is extending into Margate, with a cluster of projects happening across the two towns, collected together in a performance tour by Julia Riddiough. There are a number of walks, including Emily Whitebread's from Whitstable to Seasalter, which takes her research into John Betjeman's 'Shell Guides to British Counties' as a starting point to consider ideas around nationhood in the 21st century.
Webb-Ellis, Parlor Walls, 2016, video still. Courtesy: the artists
HT In 2014, Ellen Mara De Wachter noted how Whitstable's 'rural' status saved it from 'biennial fatigue'. Has the proliferation of globally-focused biennials led you to alter your approach?
SJ I think Whitstable Biennale exists at some distance from the globally-focused biennial trail. We are not a survey show. We commission predominantly UK based artists, and focus on emerging artists and experimental work. Our programme is all about depth.
Harry Thorne
Harry Thorne is a writer and editor based in London, UK.
Sue Jones is the Director of Whitstable Biennale. Originally from east London, she has lived and worked in the coastal towns of Ramsgate and Whitstable in east Kent since 2002. Jones studied Literature at London Metropolitan University (formerly North London Polytechnic) and in 1990 joined Chisenhale Gallery, working closely with Jonathan Watkins as part of a small team of three, before becoming Curator in 1995 and Director in 1998. She worked for five years as an independent curator (in London and east Kent) from 2000 to 2005, before joining Whitstable Biennale as Director in 2005.
Whitstable Biennale runs from 4 – 12 June 2016. For further details visit www.whitstablebiennale.com
Sue Jones
Whitstable Biennale
Chisenhale Gallery
Lucy Pawlak
Tessa Lynch
Evan Ifekoya
Webb-Ellis
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