One-dimensional run-and-tumble motions with generic boundary conditions

The study of solutions of the diffusion equation in the presence of different boundary conditions is a topic that has long been explored over the years [1–3]. Only recently has this type of analysis been extended to the case of persistent motions, characteristic of the realm of the so called active matter. This describes ensembles of self-propelled units which locally convert environmental energy into systematic motion; examples range from living organisms, such as motile bacteria, fish or birds, to nonliving objects, such as self-catalytic colloids or vibrating asymmetric particles [4–7]. The persistent character of the motion of active particles has dramatic consequences when considered in bounded domains. Contrary to the diffusion case, we have here the new phenomenon of accumulation at walls, affecting the overall behavior of the system and producing nontrivial and quite unexpected phenomena (see, for example, [4] and references therein). This peculiar property of active matter requires careful characterization of the boundary conditions. We focus our attention on one of the most widely used prototype models of active particles, the so called run-and-tumble (RT) model, inspired by the motion of Escherichia coli (E. coli) bacteria [8], describing a particle moving along straight lines at constant speed and reorienting its direction of motion at fixed rate [9–14]. Let us restrict attention to the one-dimensional (1D) case, where it is often possible to obtain analytical results that provide an in-depth understanding of the processes under consideration. Different boundary conditions have been analyzed in the past for 1D RT particles, in particular with respect to first-passage problems. Totally absorbing/reflecting boundaries are considered in [11, 15, 16]; the case of partial absorption has been investigated in [17, 18]; sticky boundaries, allowing particles accumulation, are analyzed in [19]. There are also many studies that deal with the boundary problem for RT particles in a variety of contexts, as, for example, considering resetting processes [20–24], diffusion terms in kinetic equations [25], space-dependent speed [26], confining potentials [27, 28], ratchet effects [29, 30], extremal statistics [31], fractional equations [32], encounter-based absorption [33, 34]. While previous work provides a fairly clear picture of the possible cases of boundary conditions, a thorough discussion of their role and a comprehensive treatment is still lacking.

Here we provide an in-depth discussion of the solutions of RT equations in the presence of generic boundary conditions at the two extremes of a bounded 1D domain. These include particle accumulation, absorption and reflection, regulated by appropriate rates. After formulating the problem in a general way (section 2), considering different boundary conditions in a unified way and discussing their various limits (section 3), we thoroughly analyze the exit processes (section 4). We obtain a universal expression of the exit time, valid for all boundary conditions, and we scrutinize several possible case studies, reporting the exit time expressions for each of them (section 5). Conclusions are given in section 6.

We consider a 1D RT motion, characterized by speed v and tumbling rate α, in a finite interval $[x_a,x_b]$ (we label with a and b the left and right boundaries). The equations of motion describing the time evolution of the probability density functions (PDFs) of right and left oriented RT particles, $P_{_R}(x,t)$ and $P_{_L}(x,t)$, are [9–14]

$$\begin{align} \frac{\partial P_{_R}}{\partial t}\left(x,t\right) & = - v \frac{\partial P_{_R} }{\partial x}\left(x,t\right) - \frac{\alpha}{2} \left[ P_{_R}\left(x,t\right) - P_{_L}\left(x,t\right) \right] , \end{align} \tag{ 1 }$$

$$\begin{align} \frac{\partial P_{_L}}{\partial t}\left(x,t\right) & = v \frac{\partial P_{_L} }{\partial x}\left(x,t\right) + \frac{\alpha}{2} \left[ P_{_R}\left(x,t\right) - P_{_L}\left(x,t\right) \right] . \end{align} \tag{ 2 }$$

In terms of total PDF $P = P_{_R} + P_{_L}$ and current $J = v (P_{_R} - P_{_L})$, the above equations correspond to

$$\begin{align} \frac{\partial P}{\partial t}\left(x,t\right) & = - \frac{\partial J }{\partial x}\left(x,t\right) , \end{align} \tag{ 3 }$$

$$\begin{align} \frac{\partial J}{\partial t}\left(x,t\right) & = - v^2 \frac{\partial P}{\partial x}\left(x,t\right) - \alpha J\left(x,t\right) , \end{align} \tag{ 4 }$$

which can be combined to get the so called telegrapher's equation [10, 35–37]:

$$\begin{equation} \frac{\partial^2 P}{\partial t^2}\left(x,t\right) + \alpha \frac{\partial P}{\partial t}\left(x,t\right) = v^2 \frac{\partial^2 P}{\partial x^2}\left(x,t\right) . \end{equation} \tag{ 5 }$$

We consider initial conditions

$$\begin{equation} P_{_R}\left(x,0\right) = P_{_L} \left(x,0\right) = \frac{\delta\left(x\right)}{2} , \end{equation} \tag{ 6 }$$

or, in terms of P and J,

$$\begin{align} P\left(x,0\right) & = \delta\left(x\right), \end{align} \tag{ 7 }$$

$$\begin{align} J\left(x,0\right) & = 0 , \end{align} \tag{ 8 }$$

corresponding to a particle that begins its motion at the origin x = 0 with equally distributed orientation (we are assuming, without loss of generality, that $x_a\lt0\lt x_b$).

We consider very general boundary conditions, which include accumulation, absorption and reflection [19, 34]. When the particle hits the wall it gets stuck at the boundary until it is absorbed with rate λ_w ($w = a,b$ labels the boundaries) or it turns back to the bulk after a tumble event which reverses its direction of motion (we assume a boundary dependent tumbling rate α_w , in general, different from that of the bulk α). Figure 1 shows a sketch of the problem we are analyzing. In order to describe accumulation at walls we introduce the probability $W_w(t)$ to find the particle stuck at the boundary point $w = a,b$. The time evolution of $W_{a,b}$ are described by the following equations

$$\begin{align} \frac{\partial W_a}{\partial t}\left(t\right) & = - J\left(x_a,t\right) - \lambda W_a\left(t\right) , \end{align} \tag{ 9 }$$

$$\begin{align} \frac{\partial W_b}{\partial t}\left(t\right) & = J\left(x_b,t\right) - \lambda W_b\left(t\right) , \end{align} \tag{ 10 }$$

with initial condition $W_a(0) = W_b(0) = 0$ (the particle starts its motion in the bulk). The above equations take into account the net incoming/outgoing flows of particles to/from the wall (±J) and the absorption rate at the boundary ($-\lambda W$). The boundary conditions reads

$$\begin{align} v P_{_R} \left(x_a,t\right) & = \frac{\alpha_a}{2} W_a\left(t\right) , \end{align} \tag{ 11 }$$

$$\begin{align} v P_{_L} \left(x_b,t\right) & = \frac{\alpha_b}{2} W_b\left(t\right) , \end{align} \tag{ 12 }$$

obtained imposing equality between outflow of particles from the boundary to the bulk (left-hand side in equations) and the rate of stuck particles which reverse their direction of motion leaving the wall (right-hand side).

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It is convenient to work in the Laplace domain

$$\begin{equation} {\tilde P}\left(x,s\right) = \int_0^{\infty} \mathrm{d}t \ e^{-st}\ P\left(x,t\right) , \end{equation} \tag{ 13 }$$

where the telegrapher's equation (5) can be written as

$$\begin{equation} v^2 \frac{\partial^2{\tilde P}}{\partial x^2} \left(x,s\right) - s \left(s+\alpha\right) {\tilde P} \left(x,s\right) = -\left(s+\alpha\right) \delta\left(x\right) . \end{equation} \tag{ 14 }$$

Boundary conditions now read

$$\begin{align} \left. v \frac{\partial {\tilde P}\left(x,s\right)}{\partial x}\right|_{x = x_a} & = g_a\left(s\right) {\tilde P}\left(x_a,s\right), \end{align} \tag{ 15 }$$

$$\begin{align} \left. v \frac{\partial {\tilde P}\left(x,s\right)}{\partial x}\right|_{x = x_b} & = - g_b\left(s\right) {\tilde P}\left(x_b,s\right) , \end{align} \tag{ 16 }$$

where

$$\begin{equation} g_w\left(s\right) = \frac{\left(s+\alpha\right)\left(s+\lambda_w\right)}{s+\alpha_w+ \lambda_w} , \quad w = a,b . \end{equation} \tag{ 17 }$$

Solutions of (14) are of the form

$$\begin{equation} {\tilde P}\left(x,s\right) = \left\{\begin{array}{lcl} A_+ e^{cx} + A_- e^{-cx} & \hspace{1cm} \mbox{for} ~ x > 0 ,\\ B_+ e^{cx} + B_- e^{-cx} & \hspace{1cm} \mbox{for} ~ x < 0, \end{array}\right. \end{equation} \tag{ 18 }$$

where

$$\begin{equation} v^2 c^2\left(s\right) = s \left(s+\alpha\right) . \end{equation} \tag{ 19 }$$

By imposing continuity of ${\tilde P}$ and discontinuity of $\partial_x {\tilde P}$ (integrating (14) from $0^-$ and $0^+$) at x = 0, and boundary conditions (15) and (16), we finally have

$$\begin{align} A_\pm & = \frac{c}{2s} \left(vc\mp g_b\right) \ e^{\mp cx_b} \ \frac{vc \cosh{\left(cx_a\right)} - g_a \sinh{\left(cx_a\right)}}{Q} , \end{align} \tag{ 20 }$$

$$\begin{align} B_\pm & = \frac{c}{2s} \left(vc\pm g_a\right) \ e^{\mp cx_a} \ \frac{vc \cosh{\left(cx_b\right)} + g_b \sinh{\left(cx_b\right)}}{Q} , \end{align} \tag{ 21 }$$

where c and $g_{a,b}$ are functions of s, see equations (17) and (19), and the quantity Q is

$$\begin{equation} Q = vc \left(g_a+g_b\right) \cosh{\left(cL\right)} + \left(v^2c^2+g_a g_b\right) \sinh{\left(cL\right)} \end{equation} \tag{ 22 }$$

with $L = x_b-x_a$ the box length. Substituting into (18) we obtain the solution of the bulk PDF P in the Laplace domain. The expression of the density of stuck particles at boundaries W_w can be obtained from (9)–(12) in the Laplace domain, getting

$$\begin{equation} {\tilde W}_w\left(s\right) = \frac{v}{s+\alpha_w+\lambda_w} {\tilde P}\left(x_w,s\right) , \quad w = a,b . \end{equation} \tag{ 23 }$$

The generic boundary conditions (15) and (16) are described by the function g_w (17), that we report again here for convenience

$$\begin{equation} g_w\left(s\right) = \frac{\left(s+\alpha\right)\left(s+\lambda_w\right)}{s+\alpha_w+ \lambda_w} . \end{equation} \tag{ 24 }$$

This very general expression encodes the boundary information of the problem and it is in agreement with that reported in [34], generalizing those derived in [18, 19]. Below, we show how it is possible to treat the various possible types of boundary by means of appropriate limits of the previous formula (a summary of results is shown in table 1).

Table 1. Different types of boundaries and the corresponding expressions of the g_w function, appearing in boundary conditions (15) and (16), obtained for particular choices of the parameters values λ_w and α_w in the general expression (24) given in the text. See figure 1 for the meaning of the different parameters.

Boundary type	λw	αw	$\displaystyle g_w = \frac{(s+\alpha)(s+\lambda_w)}{(s+\alpha_w+\lambda_w)}$
Absorbing	$\infty$	·	$s+\alpha$
Reflecting	·	$\infty$	0
Partially absorbing	$\infty$ a	$\infty$ a	$\epsilon(s+\alpha)$
Sticky	0	α	s
Sticky-reactive	0	αw	$\displaystyle\frac{s(s+\alpha)}{s+\alpha_w}$
Sticky-absorbing	λw	α	$\displaystyle\frac{(s+\alpha)(s+\lambda_w)}{s+\alpha+\lambda_w}$

^a With finite value of $\alpha_w/\lambda_w$. We define $\epsilon = (1+\alpha_w/\lambda_w)^{-1}$.

Absorbing boundary. The case of a fully absorbing boundary is obtained in the limit

$$\begin{equation} \lambda_w \to \infty , \end{equation} \tag{ 25 }$$

corresponding to instantaneous absorption when the particle hits the boundary In such a case the function g_w becomes

$$\begin{equation} g_w\left(s\right) = s+\alpha . \end{equation} \tag{ 26 }$$

Reflecting boundary. The reflecting boundary is obtained in the limit

$$\begin{equation} \alpha_w \to \infty , \end{equation} \tag{ 27 }$$

describing an instantaneous reflection of particle orientation when the particle arrive at the boundary point. The corresponding function g_w vanishes

$$\begin{equation} g_w\left(s\right) = 0. \end{equation} \tag{ 28 }$$

Partially absorbing/reflecting boundary. The situation of partial absorption/reflection at the boundary is achieved in the limits

$$\begin{equation} \left\{\begin{array}{l} \lambda_w \to \infty ,\\ \alpha_w \to \infty , \end{array}\right. \end{equation} \tag{ 29 }$$

with a finite value of their ratio

$$\begin{equation} \frac{\alpha_w}{\lambda_w} \to \text{finite} . \end{equation} \tag{ 30 }$$

In this limit the function g_w reads

$$\begin{equation} g_w\left(s\right) = \epsilon \left(s+\alpha\right) , \end{equation} \tag{ 31 }$$

where

$$\begin{equation} \epsilon = \left(1+\alpha_w/\lambda_w\right)^{-1} .\end{equation} \tag{ 32 }$$

This corresponds to the case treated in [18], by noting that $\alpha_w/\lambda_w = 2\gamma/\eta$, where γ is the reflection coefficient and $\eta = 1-\gamma$ the absorption coefficients. The quantity ε can then be expressed as

$$\begin{equation} \epsilon = \frac{1-\gamma}{1+\gamma} = \frac{\eta}{2-\eta} . \end{equation} \tag{ 33 }$$

Sticky boundary. The case of a normal sticky boundary is obtained considering the absence of absorption and a tumbling rate at wall equal to that of the bulk

$$\begin{equation} \left\{\begin{array}{l} \lambda_w = 0 ,\\ \alpha_w = \alpha . \end{array}\right. \end{equation} \tag{ 34 }$$

In this case we have

$$\begin{equation} g_w\left(s\right) = s , \end{equation} \tag{ 35 }$$

in agreement with [19].

Sticky-reactive boundary. The case of a sticky boundary which affects the tumbling rate property of the run-and-tumble particle but it is not absorbing is obtained for

$$\begin{equation} \lambda_w = 0 . \end{equation} \tag{ 36 }$$

The g_w function is now [19]

$$\begin{equation} g_w\left(s\right) = \frac{s\left(s+\alpha\right)}{s+\alpha_w} . \end{equation} \tag{ 37 }$$

Sticky-absorbing boundary. When the sticky boundary does not affect the tumbling rate but it is permeable, i.e. the particle can be absorbed at a given rate λ_w , we can put

$$\begin{equation} \alpha_w = \alpha \end{equation} \tag{ 38 }$$

The g_w is [19]

$$\begin{equation} g_w\left(s\right) = \frac{\left(s+\alpha\right)\left(s+\lambda_w\right)}{s+\alpha+\lambda_w} . \end{equation} \tag{ 39 }$$

We apply here the general formulation given above to analyze exit processes. The probability distribution of exit time $\varphi(t)$ can be obtained from the survival probability ${\mathbb{P}}(t)$, i.e. the probability that the particle has not yet left the domain until time t

$$\begin{equation} {\mathbb{P}}\left(t\right) = \int_{x_a}^{x_b} \mathrm{d}x \ P\left(x,t\right) + W_a\left(t\right) +W_b\left(t\right) . \end{equation} \tag{ 40 }$$

The survival probability can be written as a time integral of the probability distribution of exit times

$$\begin{equation} {\mathbb{P}}\left(t\right) = \int_t^\infty \mathrm{d}t^{^{\prime}} \varphi\left(t^{^{\prime}}\right)\ , \end{equation} \tag{ 41 }$$

leading to the relation

$$\begin{equation} \varphi\left(t\right) = -\frac{\partial {\mathbb{P}}}{\partial t}\left(t\right) . \end{equation} \tag{ 42 }$$

In the Laplace domain the above equation becomes

$$\begin{equation} {\tilde \varphi}\left(s\right) = 1 - s {\tilde {\mathbb{P}}}\left(s\right) , \end{equation} \tag{ 43 }$$

having used the initial condition ${\mathbb{P}}(0) = 1$. By using (43), (40) and (23) we obtain

$$\begin{equation} {\tilde \varphi}\left(s\right) = \frac{v \lambda_a}{s+\alpha_a+\lambda_a} {\tilde P}\left(x_a,s\right) +\frac{v \lambda_b}{s+\alpha_b+\lambda_b} {\tilde P}\left(x_b,s\right) , \end{equation} \tag{ 44 }$$

and, finally, thorough (18), after some algebra,

$$\begin{align} {\tilde \varphi}\left(s\right) & = \frac{\lambda_a g_a}{s+\lambda_a} \frac{vc \cosh{\left(cx_b\right)}+g_b \sinh{\left(cx_b\right)} }{Q} \nonumber\\ & \quad+ \frac{\lambda_b g_b}{s+\lambda_b} \frac{vc \cosh{\left(cx_a\right)}-g_a \sinh{\left(cx_a\right)} }{Q} , \end{align} \tag{ 45 }$$

with Q given in (22). The above expression is the very general form of the probability distribution of exit times with generic boundary conditions.

We now turn to calculate the mean exit time (MET) τ, defined as

$$\begin{equation} \tau = \int_0^\infty \mathrm{d}t \ t\ \varphi\left(t\right) = - \frac{\partial {\tilde \varphi}}{\partial s}\bigg|_{s = 0} .\end{equation} \tag{ 46 }$$

For the sake of simplicity, we specialize the calculation to the symmetric interval with respect to the origin $[-R,R]$, that is, we put in the previous formulas $x_b = -x_a = R$ (the generalization to the asymmetric case is reported in the appendix). After some algebra we finally arrive at the general expression of the mean exit time

$$\begin{equation} \tau = \frac{R}{v} + \frac{R^2}{2v^2} \alpha + \frac{h_b f_a + h_a f_b}{\lambda_b f_a+\lambda_a f_b} , \end{equation} \tag{ 47 }$$

where the functions h_w and f_w are defined by ($w = a,b$)

$$\begin{align} h_w & = 1+\alpha_w \frac{R}{v} , \end{align} \tag{ 48 }$$

$$\begin{align} f_w & = \alpha_w + \lambda_w \left(1+\alpha \frac{R}{v}\right) . \end{align} \tag{ 49 }$$

The first two terms of (47) are those typical of the first-passage problem of active particles, with the presence of a ballistic term and a diffusive-like one [16, 18]. The third term is the most interesting and non-trivial one, directly related to the different properties of the boundaries. The very general expression (47) is the main result of the present work and makes it possible to describe many situations with various kinds of boundaries at the two ends of a finite domain, from sticky to reflecting, absorbing and permeable cases, as summarized in section 3. In the following section we will provide expressions for the mean exit times in some case studies.

5.1. Equal boundaries

We first analyze the fully symmetric situation, in which the two boundaries, located symmetrically with respect to the origin, have the same properties, i.e.

$$\begin{equation} \left\{\begin{array}{l} \lambda_a = \lambda_b \equiv \lambda_w ,\\ \alpha_a = \alpha_b \equiv \alpha_w . \end{array}\right. \end{equation} \tag{ 50 }$$

In this case the MET (47) reduces to

$$\begin{equation} \tau = \frac{R}{v} + \frac12 \frac{R^2}{v^2} \alpha + \frac{1+\alpha_w R/v}{\lambda_w} , \end{equation} \tag{ 51 }$$

in agreement with the result reported in [34], generalizing the problem treated in [19], including wall dependent tumbling rate α_w . Some limiting cases are reported below.

5.1.1. Absorbing boundaries.

In the case of fully absorbing boundaries at the two ends of the interval, obtained in the limit $\lambda_w \to\infty$, we recover the standard first-passage problem and the expression of the mean first-passage time [16]

$$\begin{equation} \tau = \frac{R}{v} + \frac12 \frac{R^2}{v^2} \alpha . \end{equation} \tag{ 52 }$$

5.1.2. Partially absorbing boundaries.

In the case of two equal boundaries with partial absorption (see section 3) we obtain

$$\begin{equation} \tau = \frac{R}{\epsilon v} + \frac12 \frac{R^2}{v^2} \alpha \end{equation} \tag{ 53 }$$

with ε given in (32), in agreement with equation (58) of [18]. The factor $1/\epsilon$ in the first term takes into account the reduced absorption property of the boundary, leading to longer exit times. For ε = 1 (perfect absorption) we obtain the previous case (52), while for ε = 0 (two perfectly reflecting boundaries) the MET obliviously diverges. In figure 2 we show some typical behaviors of the reduced MET $\tau^*$ (in unit of α⁻¹) as a function of the adimensional quantity $x = (\ell/R)^2$, with $\ell = v/\alpha$ the persistent length of the RT motion. The three curves (red continuous lines), corresponding to different values of the coefficient ε, show the typical diffusive behavior at small x, $\tau^* \simeq x^{-1}$, and decrease as $x^{-1/2}$ for large values of x. In figure 3 the same quantity is reported as a function of ε for two values of x (red dashed curves), evidencing a different behavior in the two regimes of large and small x.

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5.1.3. Sticky-absorbing boundaries.

In the case of sticky-absorbing boundaries with a tumbling rate equal to that of the bulk $\alpha_w = \alpha$, we have

$$\begin{equation} \tau = \frac{R}{v} + \frac12 \frac{R^2}{v^2} \alpha + \frac{1+\alpha R/v}{\lambda_w}, \end{equation} \tag{ 54 }$$

in agreement with the expression reported in [19], equation (37). We note here the presence of a third term with respect to the previous expressions, which reflects the sticky property of the boundary and contributes to the increase in the time to exit the domain.

5.2. A fully absorbing boundary on one side

We now consider the case where there is a fully absorbing boundary on one side of the finite interval and a generic one on the other side. Let b the absorbing boundary, we can then use the previous general expression (47) in the limit

$$\begin{equation} \lambda_b \to \infty , \end{equation} \tag{ 55 }$$

corresponding to an instantaneous absorption when the particle hits the right boundary at x_b . The expression of the mean exit time becomes

$$\begin{equation} \tau = \frac{R}{v} + \frac{R^2}{2v^2} \alpha + \frac{\left(1+\alpha_a R/v\right)\left(1+\alpha R/v\right)}{\alpha_a + 2\lambda_a\left(1+\alpha R/v\right)} . \end{equation} \tag{ 56 }$$

We now discuss various boundary types at the left side of the domain.

5.2.1. Absorbing boundary.

The case where there is a completely absorbing boundary also at a is obtained in the limit $\lambda_a \to \infty$, and we recover the expression of the MET obtained previously (52)

$$\begin{equation} \tau = \frac{R}{v} + \frac12 \frac{R^2}{v^2} \alpha . \end{equation} \tag{ 57 }$$

5.2.2. Reflecting boundary.

For a reflecting boundary condition at a we have to put $\alpha_a \to \infty$. In this limit we have

$$\begin{equation} \tau = 2 \frac{R}{v} + \frac32 \frac{R^2}{v^2} \alpha . \end{equation} \tag{ 58 }$$

5.2.3. Partially absorbing boundary.

In the case of partially absorbing boundary in a we obtain

$$\begin{equation} \tau = \frac{R}{v} + \frac12 \frac{R^2}{v^2} \alpha + \frac{R}{v} \frac{\left(1-\epsilon\right)\left(1+\alpha R/v\right)}{1+\epsilon \left(1+2\alpha R/v\right)} \end{equation} \tag{ 59 }$$

with ε defined in (32). For ε = 1 (perfect absorption) and ε = 0 (perfect reflection) we obtain, respectively, the previous expressions (57) and (58). In figure 3 the reduced MET is shown as a function of ε for two values of x (black continuous lines). Contrary to the symmetric case of section 5.1.2, the MET has a finite value in the limit of reflecting boundary ε → 0, due to the presence of the opposite absorbing boundary that allows the particle to escape.

5.2.4. Sticky boundary.

The case of a normal sticky boundary is obtained considering $\lambda_a = 0$ and $\alpha_a = \alpha$. The MET is now given by

$$\begin{equation} \tau = \frac{1}{\alpha} + 3 \frac{R}{v} + \frac32 \frac{R^2}{v^2} \alpha . \end{equation} \tag{ 60 }$$

5.2.5. Sticky-reactive boundary.

The case of a (not-absorbing) sticky wall in a, that induces a boundary dependent tumbling rate α_a , is obtained considering $\lambda_a = 0$

$$\begin{equation} \tau = 2 \frac{R}{v} + \frac32 \frac{R^2}{v^2} \alpha + \frac{1+\alpha R/v}{\alpha_a} . \end{equation} \tag{ 61 }$$

We can treat different situations, by considering, for example, repelling ($\alpha_a\gt\alpha$) or attractive ($\alpha_a\lt\alpha$) walls, or, also, more general α-dependent tumbling rate at wall, e.g. $\alpha_a = c_1 \alpha+c_2$.

5.2.6. Sticky-absorbing boundary.

When the sticky boundary is also absorbing we can put $\alpha_a = \alpha$, and the MET reads:

$$\begin{equation} \tau = \frac{R}{v} + \frac12 \frac{R^2}{v^2} \alpha + \frac{\left(1+\alpha R/v\right)^2}{\alpha+2\lambda_a\left(1+\alpha R/v\right)} . \end{equation} \tag{ 62 }$$

Figure 2 show three typical curves for different values of boundary parameter λ_a (blue dotted-dashed lines). Again we observe a power law behavior at small and large values of x, respectively with power −1 and $-1/2$, but now there is a non-trivial intermediate behavior, with the presence of a shoulder-like shape for low enough values of λ_a . This corresponds to non-monotonic behavior of the mean exit time τ as a function of α for certain values of the boundary parameter λ_a (see discussion in the section 6).

5.3. A fully reflecting boundary on one side

We consider here the case of a fully reflecting boundary on one side together with a generic one on the other side. We obtain such a situation by putting (let b the reflecting boundary)

$$\begin{equation} \alpha_b \to \infty , \end{equation} \tag{ 63 }$$

leading to the MET expression

$$\begin{equation} \tau = 2 \frac{R}{v} + \frac32 \frac{R^2}{v^2} \alpha + \frac{1+2\alpha_a R/v}{\lambda_a}. \end{equation} \tag{ 64 }$$

Again, we report expressions for some choices of boundary type in a.

5.3.1. Absorbing boundary.

The absorbing case, $\lambda_a \to \infty$, leads to the same expression obtained above (58)

$$\begin{equation} \tau = 2 \frac{R}{v} + \frac32 \frac{R^2}{v^2} \alpha . \end{equation} \tag{ 65 }$$

5.3.2. Partially absorbing boundary.

By considering a partial absorption in a we obtain

$$\begin{equation} \tau = 2 \frac{R}{\epsilon v} + \frac32 \frac{R^2}{v^2} \alpha . \end{equation} \tag{ 66 }$$

with ε given by (32).

5.4. A sticky boundary on one side

The last case we report is that of a normal sticky boundary on one side, obtained for $\lambda_b = 0$ and $\alpha_b = \alpha$, giving rise to the MET

$$\begin{equation} \tau = \frac{1}{\alpha} + 3 \frac{R}{v} + \frac32 \frac{R^2}{v^2}\alpha + \frac{1+\alpha_a/\alpha+2\alpha_a R/v}{\lambda_a} .\end{equation} \tag{ 67 }$$

Some interesting boundary conditions at the other end of the domain are as follows.

5.4.1. Partially absorbing boundary.

For the partial absorption case the MET reads

$$\begin{equation} \tau = \frac{R}{v} + \frac32 \frac{R^2}{v^2}\alpha +\frac{1+2\alpha R/v}{\epsilon \alpha} . \end{equation} \tag{ 68 }$$

We note that for ε = 1 (fully absorption) we recover the expression (60). In figure 2 the reduced MET is reported as a function of $x = (\ell/R)^2$ for three different values of ε. Unlike the other cases reported in the figure, here we observe a different behavior at large x, where $\tau^* = \tau \alpha$ reaches a finite value $1/\epsilon$, corresponding to a divergence of τ at small α. Together with the usual divergence in the diffusive regime at small x (large α), this implies that there is a non-monotonic behavior of τ vs. α, with the presence of an optimal minimal value. In figure 3 we show the MET as a function of ε for two different values of x (blue dotted–dashed lines), obtaining curves similar to those of section 5.1.2, with upward shifted values.

5.4.2. Sticky-absorbing boundary.

In the case of a sticky-absorbing boundary in a, with $\alpha_a = \alpha$, the MET is

$$\begin{equation} \tau = \frac{1}{\alpha} + 3\frac{R}{v} + \frac32 \frac{R^2}{v^2}\alpha +2 \frac{1+\alpha R/v}{\lambda_a} . \end{equation} \tag{ 69 }$$

By using a general formulation of boundary conditions, we study the solutions of the RT equations in 1D finite domains. Boundary types include particles accumulation at walls, absorption (at rate λ_w ) and boundary-dependent tumbling rate α_w . By introducing the function $g_w(s)$ (in the Laplace domain)—see equation (17)—describing boundary conditions (15) and (16), we treat in a unified way many different types of boundaries, which can be obtained by suitable choices of the introduced parameters. The main result of the present work is the general expression of the mean exit time—see equation (47)—obtained for generic boundaries at the two ends of the finite domain. Various interesting case studies are analyzed, reporting the corresponding expressions of the mean exit times: from the case of equal boundaries to the situations in which the left and right boundaries have different characteristics. Typical behaviors of the mean exit times as a function of physical parameters are also shown for some of the cases analyzed. The reported analytical results allow us to infer some interesting features of confined active motions. Focusing on the mean exit time τ, it is easy to show that (47) is a growing function of R, being its first derivative strictly positive, $\partial \tau/\partial R\gt0$, regardless of the type of boundaries at either end. Monotonic (decreasing) behavior is present also as a function of v, being $\partial \tau/\partial v\lt0$, as deduced by the fact that τ is a function of $R/v$. These results are related to the trivial observation that larger domains and slower particles correspond to longer mean exit times. A less trivial behavior is observed, instead, as a function of the tumbling rate α. Now, the general expression (47) has no unique behavior, and different trends are possible, depending on the type of boundaries at either end of the domain. By considering the different expressions of τ obtained in this work, we can deduce that the MET is a monotonic increasing function of α in all the case in which the two boundaries are equal or one is reflecting, regardless of their properties (sections 5.1 and 5.3), or in the case in which they are different but not sticky-like (sections 5.2.2 and 5.2.3). Instead, non-monotonic behavior is definitely present whenever one of the boundaries is strictly sticky (section 5.4) and it is possible in the other sticky-like cases (sticky-reactive or sticky-absorbing), depending on the values of the parameters λ_w and α_w describing the boundary (sections 5.2.5 and 5.2.6). The sticky character of the boundary opposite to the absorbing one has then a crucial role in determine nontrivial trends of the MET. An in-depth treatment of this topic will be reported in a forthcoming paper [38], which will analyze RT motions in one- and two-dimensional bounded domains and discuss the general conditions for the existence of optimal escape rates in active matter.

I acknowledge financial support from the Italian Ministry of University and Research (MUR) under PRIN2020 Grant No. 2020PFCXPE.

All data that support the findings of this study are included within the article (and any supplementary files).

In this appendix we report the general expression of the MET when the boundaries are positioned asymmetrically with respect to the point where the particle begins its motion. We denote with $X_c = (x_b+x_a)/2$ the center coordinate of the domain of length $2R = x_b-x_a$. In the main text we derived the expression of the MET in the case of symmetrical position of the boundaries with respect to the origin (starting point of the particle motion), i.e. we considered $x_a = -x_b$, that is $X_c = 0$. In the general asymmetric case we can repeat the arguments of the section 4, and by using (46) and the general expression of the exit time PDF (45) we finally arrive at

$$\begin{equation} \tau = \frac{R}{v} + \frac{R^2-X_c^2}{2v^2} \alpha + \frac{h_b f_a + h_a f_b}{\lambda_b f_a+\lambda_a f_b} +\frac{\alpha X_c}{v} \frac{\lambda_b h_a - \lambda_a h_b}{\lambda_b f_a+\lambda_a f_b} , \end{equation} \tag{ A1 }$$

where h_w and f_w (with $w = a,b$) are defined in (48) and (49). The above expression generalizes (47) to the asymmetric case $X_c\neq 0$, showing the presence of two corrective terms, linear and quadratic in the variable X_c . For $X_c = 0$ the MET reduces to that reported in the main text (47). The first two terms in (A1) correspond to the first passage problem with asymmetric boundaries (perfectly absorbing boundaries placed asymmetrically with respect to the initial point). We note that in the case of partially absorbing boundaries the (A1) is in agreement—by recalling the definition of ε (32)—with the expression reported in [18], equation (36).

For comparison with the symmetric case we report here the explicit expressions of the MET for the case studies discussed in the main text. When the two boundaries are equal, $\lambda_a = \lambda_b = \lambda_w$ and $\alpha_a = \alpha_b = \alpha_w$ —see section 5.1, equation (51) in the symmetric case—the MET takes the simple form

$$\begin{equation} \tau = \frac{R}{v} + \frac{R^2-X_c^2}{2v^2}\alpha + \frac{1+\alpha_wR/v}{\lambda_w} , \end{equation} \tag{ A2 }$$

with the presence of only the quadratic term in X_c , due to the equal properties of the boundaries (starting the motion at a certain distance from one boundary is the same as starting it at the same distance from the other). In case of the presence of a fully absorbing boundary, $\lambda_b \to \infty$—see section 5.2, equation (56)— we have

$$\begin{equation} \tau = \frac{R}{v} + \frac{R^2-X_c^2}{2v^2} \alpha + \frac{\left(1+\alpha_a R/v\right)\left[1+\alpha \left(R+X_c\right)/v\right]}{\alpha_a + 2\lambda_a\left(1+\alpha R/v\right)} . \end{equation} \tag{ A3 }$$

Whit a fully reflecting boundary, $\alpha_b \to \infty$—see section 5.3, equation (64)—the MET is

$$\begin{equation} \tau = 2 \frac{R}{v} + \frac{3R^2-X_c^2}{2v^2} \alpha + \frac{1+2\alpha_a R/v}{\lambda_a} - \frac{X_c R \alpha}{v^2} . \end{equation} \tag{ A4 }$$

Finally, with a sticky boundary, $\lambda_b = 0$ and $\alpha_b = \alpha$—see section 5.4, equation (67)—we have

$$\begin{equation} \tau = \frac{1}{\alpha} + 3 \frac{R}{v} + \frac{3R^2-X_c^2}{2v^2}\alpha + \frac{1+\alpha_a/\alpha+2\alpha_a R/v}{\lambda_a} - \frac{X_c}{v} \left(1+\alpha R/v\right) . \end{equation} \tag{ A5 }$$