Numerical analysis of homogeneous and inhomogeneous intermittent search strategies

Random search processes for targets that are inhomogeneously distributed in a search domain require spatially inhomogeneous search strategies to find the target as fast as possible. Here, we compare systematically the efficiency of homogeneous and inhomogeneous strategies for intermittent search, which alternates stochastically between slow, diffusive motion in which the target can be detected and fast ballistic motion during which targets cannot be detected. We analyze the mean first-passage time of homogeneous and inhomogeneous strategies for three paradigmatic search problems: (1) the narrow escape problem, i.e., the searcher looks for a small area on the boundary of the search domain, (2) reaction kinetics, i.e., the detection of an immobile target in the interior of a search domain, and (3) the reaction-escape problem, i.e., the searcher first needs to find a mobile target before it can escape through a narrow area on the boundary. Using families of inhomogeneous strategies, partially motivated by the organization of the cytoskeleton in cells with a centrosome, we show that they are almost always more efficient than homogeneous strategies.

Figure 1

Sketch of boundary conditions in a search volume $V$ with escape area $A$ (dotted line). Gray wiggly lines represent diffusive motion, thick green arrows ballistic motion. (a) BB: a ballistically moving particle is reflected at the boundary and stays in the ballistic mode, i.e., $A$ is only detected if it is reached diffusively. (b) BD: a ballistically moving particle switches to diffusion at the boundary, i.e., $A$ will be detected if it reaches $A$ ballistically (trajectory, starting at ${\mathbf{r}}_0$) or diffusively (trajectory, starting at ${\mathbf{r}}_1$).

Figure 2

The probability density ${\rho }_x^{\alpha }$ [Eq. (17)] as a function of the variable $\alpha$ for different values of the spreading parameter $\sigma \left(x,\tilde{r}\right)$.

Figure 3

(a) The cytoskeleton transport network of a spherical cell with a centrosome: microtubules (long thick green lines), emanating from the MTOC of the cell close to the nucleus, are orientated predominantly radially to the cell membrane. Kinesin and dynein motor proteins transport cargo along them. The actin cortex (short, small, and crossed red lines) close to membrane (outer thick black line) is built by isotropically orientated actin filaments. Myosin motors transport cargo along. (b) Sketch of the stochastic process with the direction distribution ${\rho }_{p,\tilde{\mathrm{\Delta }}}^{\alpha }$. $\tilde{\mathrm{\Delta }}$ is the thickness of the outer region. ${\tilde{\mathbf{r}}}_{\mathbf{0}}$ is the starting point of the particle. Gray wiggly lines represent diffusive motion, thick green arrows ballistic radial motion (for $|\tilde{\mathbf{r}}|<1-\tilde{\mathrm{\Delta }}$, outward with probability $p$, inward with probability $1-p$), thin red arrows ballistic motion in random directions (for $1-\tilde{\mathrm{\Delta }}<\left|\tilde{\mathbf{r}}\right|<1$).

Figure 4

Illustration for the choices of protecting spheres and cones: the blue particle on the upper left is closer to the boundary of the simulation sphere than to any other particle, in consequence the radius of its protecting sphere is limited by the distance to the boundary. The green and the red particles near the origin are closer to each other than to the cell boundary. If they can react, their radius is limited by their distance. The gray particle on the upper right has reached the boundary. Hence, it is propagated in a cone, which is illustrated in the 2D projection.

Figure 5

Comparison of the Monte Carlo method and the solution of a FEM solver on the basis of the survival probability $S\left(t\right)$ for a purely diffusive narrow escape process with ${\vartheta }_{\mathrm{abso}}=\pi /12$ and $r_0=0$. The inset shows the very small relative deviation of these distributions.

Figure 6

${\tilde{T}}_{\mathrm{diff}}·\tilde{D}$ (red dots) and ${\tilde{T}}_{\mathrm{diff}}^{\text{appro}}·\tilde{D}$ (green line) as a function of ${\vartheta }_{\mathrm{abso}}$ for ${\tilde{\mathbf{r}}}_0=0$: each dot is the average of $2\times {10}^6–{10}^7$ Monte Carlo samples. It coincides very well with the analytic approximation of [33] (green line), given in Eq. (23). The inset shows the relative difference between the curves.

Figure 7

${\tilde{T}}_{\mathrm{v}}$ (red dots) and ${\tilde{T}}_{\mathrm{v}}^{\text{appro}}$ (green line) as a function of ${\vartheta }_{\mathrm{abso}}$ for ${\tilde{\mathbf{r}}}_0=0$: each dot is the average of ${10}^8$ Monte Carlo samples. It coincides very well with the analytic approximation ${\tilde{T}}_{\mathrm{v}}^{\text{appro}}$ (green line), given in Eq. (24). The inset shows the relative difference between the curves.

Figure 8

Narrow escape, BB, ${\rho }_{\mathrm{hom}}^{\alpha }$: the normalized MFPT $\mathbb{T}$ [Eq.$$(1)] is color coded as a function of the rates $\tilde{\gamma }$ and ${\tilde{\gamma }}^{\prime }$ for different values of $\tilde{D}$ and ${\vartheta }_{\mathrm{abso}}=\text{arcsin}(1/7)$ (interpolation from a grid of $41\times 41$ data points each). $2–5\times {10}^6$ samples have been done for each pair ($\tilde{\gamma }, {\tilde{\gamma }}^{\prime }$), which leads to relative stochastic fluctuations of $\mathbb{T}$, which are smaller than 0.2%. The black labeled isolines clarify the plotted data, especially in regions of a small gradient in the MFPT. The position of the minimum $({\tilde{\gamma }}_{\mathrm{opt}},{\tilde{\gamma }}_{\mathrm{opt}}^{\prime })$ is always shown by a (red) dot within the innermost isoline.

Figure 9

Narrow escape, BB, ${\rho }_{\mathrm{hom}}^{\alpha }$: the optimal rates ${\tilde{\gamma }}_{\mathrm{opt}}, {\tilde{\gamma }}_{\mathrm{opt}}^{\prime }$, and the resulting MFPT ${\mathbb{T}}_{\mathrm{opt}}$ (inset) as a function of $\tilde{D}$. For the diffusivities shown in Fig. 8, the data coincide with the coordinates of the minimum positions and its value of $\mathbb{T}$.

Figure 10

Narrow escape, BB, ${\rho }_x^{\alpha }$: $\mathbb{T}$ as a function of the spreading parameter $x$ for the transition rates $({\tilde{\gamma }}_{\mathrm{opt}},{\tilde{\gamma }}_{\mathrm{opt}}^{\prime })$ of the optimal solution of Fig. 9. The minimum position $x_{\mathrm{opt}}\approx 0.35$ (thick vertical gray bar) of all curves is almost identical. The (colored) numbers on the right show the ratio ${\mathbb{T}}_{\min }/{\mathbb{T}}_{\mathrm{opt}}$.

Figure 11

Narrow escape, BB: MFPT for purely diffusive search [Eq. (25), red dashed upper line]; optimal search with homogeneously distributed velocity direction (green crosses); optimal search with inhomogeneously distributed velocity direction for the fixed rates ${\tilde{\gamma }}_{\mathrm{opt}}, {\tilde{\gamma }}_{\mathrm{opt}}^{\prime }$ for the homogeneous scenario (blue squares); optimal search with inhomogeneously distributed velocity direction for rates ${\tilde{\mathrm{\Gamma }}}_{\mathrm{OPT}}, {\tilde{\mathrm{\Gamma }}}_{\mathrm{OPT}}^{\prime }$ (black circles).

Figure 12

Narrow escape, BD, ${\rho }_{\mathrm{hom}}^{\alpha }$: the normalized MFPT $\mathbb{T}$ [Eq.$$(1)] is color coded as a function of the parameters $\tilde{\gamma }$ and ${\tilde{\gamma }}^{\prime }$ for different values of $\tilde{D}$ and ${\vartheta }_{\mathrm{abso}}=\text{arcsin}(1/7)$ (interpolation from a grid of $41\times 41$ data points each). Between $2\times {10}^6$ and $5\times {10}^6$ samples have been done for each pair ($\tilde{\gamma }, {\tilde{\gamma }}^{\prime }$), which leads to relative stochastic fluctuations of $\mathbb{T}$ which are smaller than 0.2%. The black labeled isolines clarify the plotted data, especially in regions of a small gradient in the MFPT. The position of the minimum $({\tilde{\gamma }}_{\mathrm{opt}},{\tilde{\gamma }}_{\mathrm{opt}}^{\prime })$ is always shown by a (red) dot at the bottom of each plot.

Figure 13

Narrow escape, BD, ${\rho }_{\mathrm{hom}}^{\alpha }$: diagram for the choice of the best search strategy as a function of $\tilde{D}$ and ${\vartheta }_{\mathrm{abso}}$. In the red (a) and the green (b) domains, an intermittent search strategy is preferable, whereas in the white domain (c) pure diffusion is the best strategy. For the construction of the diagram, the behavior at the position of the dots was investigated.

Figure 14

Narrow escape, BD, ${\rho }_{\mathrm{hom}}^{\alpha }$: examples of the function $\mathbb{T}(\tilde{\gamma },{\tilde{\gamma }}_{\mathrm{opt}}^{\prime }=0$) for the three different areas (a,b,c) in Fig. 13.

Figure 15

Narrow escape, BD, ${\rho }_{\mathrm{hom}}^{\alpha }$: ${\tilde{T}}_{\mathrm{opt}}, {\mathbb{T}}_{\mathrm{opt}}$, and the corresponding ${\tilde{\gamma }}_{\mathrm{opt}}$ as a function of $\tilde{D}$ and $0.025<{\vartheta }_{\mathrm{abso}}<0.75$ (interpolation from the nonequidistant grid shown at the bottom of Fig. 13). The thick black line, called ${\tilde{D}}_{\text{be}}\left({\vartheta }_{\mathrm{abso}}\right)$, in each subfigure separates the area of intermittent search and purely diffusive search. It coincides with the boundary line between the areas a and c in Fig. 13. The black labeled isolines in each plot clarify the data, especially in regions of a small gradient. (a) ${\tilde{T}}_{\mathrm{opt}}$ in a log scale color plot, the dashed lines with transparent label show isolines of the purely diffusive search scenario [Eq. (23)] for the reason of comparison. (b) ${\mathbb{T}}_{\mathrm{opt}}$ is color coded plotted (c) ${\tilde{\gamma }}_{\mathrm{opt}}$ in a log scale color plot.

Figure 16

Narrow escape, BD, ${\rho }_{\mathrm{hom}}^{\alpha }$: (a) ${\mathbb{T}}_0(\tilde{\gamma },\tilde{D}=0.04)$ and $\mathbb{T}(\tilde{\gamma },\tilde{D}=0.04,{\vartheta }_{\mathrm{abso}})$ for different values of ${\vartheta }_{\mathrm{abso}}$. (b) ${\tilde{\gamma }}_{\mathrm{opt},0}\left(\tilde{D}\right)$ and ${\tilde{\gamma }}_{\mathrm{opt}}(\tilde{D},{\vartheta }_{\mathrm{abso}})$ for different values of ${\vartheta }_{\mathrm{abso}}$. (c) ${\mathbb{T}}_{\mathrm{opt},0}\left(\tilde{D}\right)$ and ${\mathbb{T}}_{\mathrm{opt}}(\tilde{D},{\vartheta }_{\mathrm{abso}})$ for different values of ${\vartheta }_{\mathrm{abso}}$.

Figure 17

Narrow escape, BD, ${\rho }_{\mathrm{hom}}^{\alpha }$: ${\tilde{\gamma }}_{\mathrm{opt}}$ and ${\mathbb{T}}_{\mathrm{opt}}$ (inset) as a function of $\tilde{D}$ for ${\vartheta }_{\mathrm{abso}}=\text{arcsin}(1/7)$. The corresponding curves $\mathbb{T}\left(\tilde{\gamma }\right)$ from which the minima are taken qualitatively all belong to case (a) in the diagram. For $2\times {10}^6$ samples for each investigated $\tilde{D}$ the position of the minimum ${\tilde{\gamma }}_{\mathrm{opt}}$ and its value ${\mathbb{T}}_{\mathrm{opt}}$ are shown.

Figure 18

Narrow escape, BD, ${\rho }_x^{\alpha }$: $\mathbb{T}$ as a function of the spreading parameter $x$ for the transition rate ${\tilde{\gamma }}_{\mathrm{opt}}$ of the optimal solution of Fig. 17. The (colored) numbers on the right show the ratio ${\mathbb{T}}_{\min }/{\mathbb{T}}_{\mathrm{opt}}$.

Figure 19

Narrow escape, BD: MFPT for purely diffusive search [Eq. (25), red dashed upper line]; search with homogeneously distributed velocity direction (green crosses); search with inhomogeneously distributed velocity direction for the fixed rates ${\tilde{\gamma }}_{\mathrm{opt}}, {\tilde{\gamma }}_{\mathrm{opt}}^{\prime }=0$ (blue squares); search with inhomogeneously distributed velocity direction for rates ${\tilde{\mathrm{\Gamma }}}_{\mathrm{OPT}}, {\tilde{\mathrm{\Gamma }}}_{\mathrm{OPT}}^{\prime }=0$ (black circles). The data at $\tilde{D}=0$ is based on random velocity simulations, which is the limit for $\tilde{D}\to 0$.

Figure 20

Narrow escape, BD, ${\rho }_{\mathrm{hom}}^{\alpha }$, and ${\rho }_{p=1,\tilde{D}}^{\alpha }$: (a) sketch of the process with the direction distribution ${\rho }_{p,\tilde{\mathrm{\Delta }}}^{\alpha }$ with a narrow escape region represented by the dotted segment on the surface of the spherical search volume (full black circle). Trajectories of the searcher are represented as in Fig. 3. (b) MFPT for the purely diffusive ${\tilde{T}}_{\mathrm{diff}}$ (red dashed upper line) and intermittent search ${\tilde{T}}_{\mathrm{opt}}$ (green full line), with ${\rho }_{\mathrm{hom}}^{\alpha }$ and optimal transition rates, as functions of the diffusion constant $\tilde{D}$ for ${\vartheta }_{\mathrm{abso}}=\text{arcsin}(1/7)$. The optimal rate ${\tilde{\gamma }}_{\mathrm{opt}}\left(\tilde{D}\right)$ is represented by the blue dotted lower line (and right $y$ axis), ${{\tilde{\gamma }}^{\prime }}_{\mathrm{opt}}$ vanishes for all $\tilde{D}$. The inset shows as an example for $\tilde{D}=0.05$ a color plot of the MFPT in dependence of the attachment and detachment rates $\tilde{\gamma }$ and ${\tilde{\gamma }}^{\prime }$, respectively. The black labeled isolines clarify the plotted data, especially in regions of a small gradient in the MFPT. The (red) dot at the bottom of the color plot indicates the optimal values (yielding the minimal MFPT) ${\tilde{\gamma }}_{\mathrm{opt}}(\tilde{D}=0.05)$ and ${\tilde{\gamma }}_{\mathrm{opt}}^{\prime }(\tilde{D}=0.05)$. Optimal attachment and detachment rates like this are used in the main plot for various diffusion constants. (c) MFPT for the inhomogeneous scenario ${\rho }_{1,\tilde{\mathrm{\Delta }}}^{\alpha }$ as a function of $\tilde{\mathrm{\Delta }}$ for different diffusion constants $\tilde{D}$ using the optimal rates for the homogeneous scenario. The values at $\tilde{\mathrm{\Delta }}=1$ coincide with the values of ${\tilde{T}}_{\mathrm{opt}}\left(\tilde{D}\right)$ in (b), as ${\rho }_{p,\tilde{\mathrm{\Delta }}=1}^{\alpha }={\rho }_{\mathrm{hom}}^{\alpha }$. (d) MFPT as a function of $\tilde{\gamma }$ and $\tilde{\mathrm{\Delta }}$ for $\tilde{D}=0.04, {\tilde{\gamma }}^{\prime }=0$. The (red) dot within the “114.5”-isoline indicates the global minimum ($\tilde{T}=113.7$) at $\tilde{\gamma }=42$ and $\tilde{\mathrm{\Delta }}=0.12$. The (green) dot (top left) indicates the minimum for the homogeneous case ($\tilde{\mathrm{\Delta }}=1, \tilde{T}=117.6$). (e) MFPT as a function of $\tilde{\mathrm{\Delta }}$ for fixed rates $\tilde{\gamma }={\tilde{\gamma }}^{\prime }=10$ for different $\tilde{D}$ and $p=1$.

Figure 21

Reaction kinetics, ${\tilde{\mathbf{r}}}_{\mathrm{tar}}=0, {\rho }_{\mathrm{hom}}^{\alpha }$: the upper line shows data for $\tilde{d}=0.2$ ($2\times {10}^6$ samples per data point), the lower line for $\tilde{d}=0.025$ ($5\times {10}^5$–$2\times {10}^6$ samples per data point). (a) $\mathbb{T}$ as a function of ${\tilde{\gamma }}^{\prime }$ for different $\tilde{D}$ with $\tilde{\gamma }={\tilde{\gamma }}_{\mathrm{opt}}^{\text{appro}}=1/\left(6\tilde{D}\right)$. The vertical dotted gray lines show the position of the suggested minimum $\tilde{\gamma }{{}^{\prime }}_{\mathrm{opt}}^{\text{appro}}$. (b) $\mathbb{T}$ is color coded as a function of $\tilde{\gamma }$ and ${\tilde{\gamma }}^{\prime }$ for the three smallest values of $\tilde{D}$ in (a). The black labeled isolines clarify the plotted data, especially in regions of a small gradient in the MFPT. The position of the global minimum $({\tilde{\gamma }}_{\mathrm{opt}},{\tilde{\gamma }}_{\mathrm{opt}}^{\prime })$ is always shown with a (red) dot within the innermost isoline. Each (colored) vertical dashed line corresponds to the curve of (a) with the same diffusivity. Each (green) dot on these vertical dashed lines shows the position of the proposed optimal rates ${\tilde{\gamma }}_{\mathrm{opt}}^{\text{appro}},\tilde{\gamma }{{}^{\prime }}_{\mathrm{opt}}^{\text{appro}}$ by [17, 18, 19].

Figure 22

${\tilde{T}}_{\mathrm{diff}}·\tilde{D}$ as a function of the reaction distance $\tilde{d}$. The dashed (red) line [Eq. (34)] belongs to a target position in the origin. The case of a homogeneously distributed target position is shown by the (green) dots ($2\times {10}^6$ Monte Carlo samples per dot). The inset shows the relative difference of both scenarios.

Figure 23

Reaction kinetics, ${\tilde{\mathbf{r}}}_{\mathrm{tar}}$ equally distributed, ${\rho }_{\mathrm{hom}}^{\alpha }$: $\mathbb{T}$ is color coded plotted as a function of $\tilde{\gamma }$ and ${\tilde{\gamma }}^{\prime }$ for the same $\tilde{D}$ and $\tilde{d}$ than in the middle column of Fig. 21. (Top: $\tilde{D}=1/420, \tilde{d}=0.025$, bottom: $\tilde{D}=1/52.5, \tilde{d}=0.2$). $2–4\times {10}^6$ samples have been done for each data point. The black labeled isolines clarify the plotted data, especially in regions of a small gradient in the MFPT. The position of the global minimum $({\tilde{\gamma }}_{\mathrm{opt}},{\tilde{\gamma }}_{\mathrm{opt}}^{\prime })$ is always shown with a (red) dot within the innermost isoline.

Figure 24

Reaction kinetics, ${\tilde{\mathbf{r}}}_{\mathrm{tar}}$ according to ${\rho }_w^{\text{init}}\left({\tilde{r}}_{\mathrm{tar}}\right), {\rho }_{\mathrm{hom}}^{\alpha }$ and ${\rho }_{p,\tilde{\mathrm{\Delta }}}^{\alpha }$: (a) sketch of the reaction process for ${\rho }_{p,\tilde{\mathrm{\Delta }}}^{\alpha }$ with an immobile target of diameter $\tilde{d}$ indicated by the gray circle. Trajectories of the mobile particle (searcher), starting at ${\tilde{\mathbf{r}}}_0$, are represented as in Fig. 3. (b) Homogeneous direction distribution: MFPTs as a function of $\tilde{\gamma }$ and ${\tilde{\gamma }}^{\prime }$ for a spatially homogeneous scenario ${\rho }_{\mathrm{hom}}^{\alpha }={\rho }_{p,1}^{\alpha }$, for (left) $\tilde{D}=0.01,\tilde{d}=0.1$, homogeneously distributed target position ${\tilde{\mathbf{r}}}_{\mathrm{tar}}$; (middle) $\tilde{D}=0.01,\tilde{d}=0.1$, target position ${\tilde{\mathbf{r}}}_{\mathrm{tar}}\le 0.5$ with probability $w=0.9$; (right) $\tilde{D}=1/300,\tilde{d}=0.025$, homogeneously distributed target position. The black labeled isolines clarify the plotted data, especially in regions of a small gradient in the MFPT. The position of the global minimum $({\tilde{\gamma }}_{\mathrm{opt}},{\tilde{\gamma }}_{\mathrm{opt}}^{\prime })$ is always shown with a (red) dot within the innermost isoline. (c) MFPT for the inhomogeneous distribution ${\rho }_{p,\tilde{\mathrm{\Delta }}}^{\alpha }$ with the optimal rates ${\tilde{\gamma }}_{\mathrm{opt}}(\tilde{D},\tilde{d}), {\tilde{\gamma }}_{\mathrm{opt}}^{\prime }(\tilde{D},\tilde{d})$ from the homogeneous case $\tilde{\mathrm{\Delta }}=1$ as function of $\tilde{\mathrm{\Delta }}$ for different values of the forward radial transport $p$ and different $w$. (d) MFPT as in (c) but now with fixed width $\tilde{\mathrm{\Delta }}=0.1$ as function of the forward probability $p$ for different fixed rates $\tilde{\gamma }$ and ${\tilde{\gamma }}^{\prime }$ and different values of $w$.

Figure 25

Reaction-escape, BD, ${\rho }_{\mathrm{hom}}^{\alpha }$, and ${\rho }_{p,\tilde{\mathrm{\Delta }}}^{\alpha }$: (a) sketch of the reaction-escape process, involving two particles, an intermittently searching particle (black, starting diffusively at random positions ${\tilde{\mathbf{r}}}_0$) and a diffusive target (blue, starting at random positions ${\tilde{\mathbf{r}}}_1$). Absorption at the narrow escape region is only possible for the searcher-target pair, and the two particles react when coming closer than a distance $\tilde{d}$ and form a pair (brown, at the bottom right), which will be absorbed at the escape region represented by the dotted segment on the boundary. (b) The MFPTs ${\tilde{T}}_{\mathrm{reac}},{\tilde{T}}_{\mathrm{esc}}$ and the resulting $\tilde{T}$ are color coded for the homogeneous direction distribution ${\rho }_{\mathrm{hom}}^{\alpha }$ for situation A and the parameters $\tilde{d}=0.1,\tilde{D}=0.01$. The black labeled isolines clarify the plotted data, especially in regions of a small gradient in the MFPT. (c) The MFPTs ${\tilde{T}}_{\mathrm{reac}},{\tilde{T}}_{\mathrm{esc}}$ and the resulting $\tilde{T}$ are color coded for the homogeneous direction distribution ${\rho }_{\mathrm{hom}}^{\alpha }$ for situation B and the parameters $\tilde{d}=0.1,\tilde{D}=0.01$. (d) The MFPTs ${\tilde{T}}_{\mathrm{reac}},{\tilde{T}}_{\mathrm{esc}}$ and the resulting $\tilde{T}$ as a function of $\tilde{\mathrm{\Delta }}$ in case of the inhomogeneous direction distributions ${\rho }_{1/2,\tilde{\mathrm{\Delta }}}^{\alpha }$ and ${\rho }_{1,\tilde{\mathrm{\Delta }}}^{\alpha }$ for the situations A and B with parameters $\tilde{d}=0.1,\tilde{D}=0.01$ and the optimal transition rates belonging to the corresponding homogeneous direction distribution scenario [position of dots in the right figures of (b) and (c)]. (e) The MFPTs ${\tilde{T}}_{\mathrm{reac}},{\tilde{T}}_{\mathrm{esc}}$ and the resulting $\tilde{T}$ as a function of $\tilde{\mathrm{\Delta }}$ in case of the inhomogeneous direction distributions ${\rho }_{1/2,\tilde{\mathrm{\Delta }}}^{\alpha }$ and ${\rho }_{1,\tilde{\mathrm{\Delta }}}^{\alpha }$ for the situations A and B with parameters $\tilde{d}=0.025,\tilde{D}=1/300$ and the optimal transition rates belonging to the corresponding homogeneous direction distribution scenario (data not shown). (f) The MFPTs ${\tilde{T}}_{\mathrm{reac}},{\tilde{T}}_{\mathrm{esc}}$ and the resulting $\tilde{T}$ as a function of $\tilde{\mathrm{\Delta }}$ in case of the inhomogeneous direction distributions ${\rho }_{1/2,\tilde{\mathrm{\Delta }}}^{\alpha }$ and ${\rho }_{1,\tilde{\mathrm{\Delta }}}^{\alpha }$ with parameters $\tilde{d}=0.025,\tilde{D}=1/300$ and the transition rates $\tilde{\gamma }=10, {\tilde{\gamma }}^{\prime }=2$ (A), respectively, $\tilde{\gamma }=5, {\tilde{\gamma }}^{\prime }=10$ (B).

Figure 26

Illustration for the construction of ${\rho }_h\left(\tau \right)$ for the cases ${\tau }_1=0.038$ ($\to N=25$) and ${\tau }_1=0.025$ ($\to N=332$). The case $N=25$ is created only to clarify the construction mechanism. The case $N=332$ is used in our implementation.

Figure 27

${\tilde{\rho }}_b\left(\tau \right)/\left[k{\rho }_h\left(\tau \right)\right]$: around $90\%$ of all random numbers ${\tau }_{\mathrm{cand}}$ will be produced in the interval $[0.05;0.354]$. The ratio ${\tilde{\rho }}_b/\left(k{\rho }_h\right)$ is bigger than 0.95 within this interval.

Figure 28

Illustration of the functions $g_{\mathrm{free}}\left(x\right|\tau )$ and $g\left(x\right|\tau )$ for different times $\tau$. Full lines represent $g_{\mathrm{free}}$, larger dotted lines $g$. For small $\tau$, the curves coincide almost perfectly.

Figure 29

Illustration of the empirically chosen preconditions: for more than $95\%$ of the sampled $x_c$, the sampled rejection random number $r_{\mathrm{rej}}$ will be within one of the three black dotted rectangles. There, $x_c$ can be accepted without a calculation of $p_a\left(x_{\mathrm{c}}\right|\tau )$.

Figure 30

Illustration of the construction of ${\rho }_9\left(x\right)$ for the choice $x_1^9=0.08$, which results in $N_9=194$ intervals $[x_i^9,x_{i+1}^9]$ and a scaling constant $k_9=1.02$.

Figure 31

(a) Transformation between the polar angles $\vartheta$ and ${\vartheta }^{\prime }$ of the two coordinate systems. The distance between the particle (shown as a dot) and the sphere's boundary in polar angle direction ${\vartheta }^{\prime }$ is called $\tilde{l}\left({\vartheta }^{\prime }\right)$. (b) Illustration of the polar angles ${\vartheta }_{\mathrm{tar}}$ and ${\vartheta }_{\mathrm{tar}}^{\prime }$. The center of the narrow escape region is located somewhere on the spherical ring (or cap for ${\vartheta }_{\mathrm{tar}}+{\vartheta }_{\mathrm{abso}}>\pi$), whose projection is shown by the thick (green) sector.

Figure 32

The angles ${\vartheta }_{+}({\vartheta }_{\mathrm{tar}},{\vartheta }_{\mathrm{abso}})$ and ${\vartheta }_{-}({\vartheta }_{\mathrm{tar}},{\vartheta }_{\mathrm{abso}})$ are shown. They define a spherical ring (or cap). As the narrow escape area is a subset of this ring, it is a necessary condition to reach the ring in order to reach the escape area.