Emergent anomalous transport and non-Gaussianity in a simple mobile–immobile model: the role of advection

Fourier–Laplace transform $f(k,s)\equiv \int_{-\infty}^\infty \int_0^\infty f(x,t)\exp(-st+ikx)\mathrm{d}t\mathrm{d}x$ of the model equations (3) allows solving for the densities, as shown in appendix A. Subsequent Fourier inversion of equations (A.2) and (A.3) produces the expressions

$$\begin{align} n_\mathrm{m}(x,s)=\left(f\ _\mathrm{m}^0+f\ _\mathrm{im}^0\frac{1}{1+s \tau_\mathrm{im}}\right)\frac{\exp\left(\frac{vx}{2D}\right)}{\sqrt{v^2+ 4\phi(s)D}}\exp\left(-\sqrt{v^2+4\phi(s)D}\frac{|x|}{2D}\right) \end{align} \tag{ 4 }$$

$$\begin{align} n_\mathrm{im}(x,s)& =\frac{\tau_\mathrm{im}/\tau_\mathrm{m}}{1+ s\tau_\mathrm{im}}\left(f\ _\mathrm{m}^0+f\ _\mathrm{im}^0\frac{1}{1+ s\tau_\mathrm{im}}\right)\frac{\exp\left(\frac{vx}{2D}\right)}{\sqrt{v^2+ 4\phi(s)D}}\exp\left(-\sqrt{v^2+4\phi(s)D}\frac{|x|}{2D}\right)\nonumber\\ &\quad +f\ _\mathrm{im}^0\frac{\tau_\mathrm{im}}{1+s\tau_\mathrm{im}}\delta(x) \end{align} \tag{ 5 }$$

as well as (see equation (A.4))

$$\begin{align} n_\mathrm{tot}(x,s)& =\frac{f\ _\mathrm{ m}+f\ _\mathrm{im}^0\frac{1}{1+s\tau_\mathrm{im}}}{s} \frac{\phi(s)\exp\left(\frac{vx}{2D}\right)}{\sqrt{v^2+4\phi(s)D}}\exp\left( -\sqrt{v^2+4\phi(s)D}\frac{|x|}{2D}\right)\nonumber\\ &\quad +f\ _\mathrm{im}^0\frac{\tau_\mathrm{im}}{1+s\tau_\mathrm{im}} \end{align} \tag{ 6 }$$

with $\phi(s) = s[1+\tau_\mathrm{im}\tau_\mathrm{m}^{-1}/(1+s\tau_\mathrm{im})]$. The fraction $f_\mathrm{m}(t) = \int_{-\infty}^\infty n_\mathrm{m}(x,t)\mathrm{d}x$ of free and the fraction $f_\mathrm{im}(t) = \int_{-\infty}^\infty n_\mathrm{im}(x,t)\mathrm{d}x$ of immobile tracers are a function of time and the expressions are given in equation (A.8) in appendix A. From these relations we immediately deduce that the total density is normalised, $f_\mathrm{m}(t)+f_\mathrm{im}(t) = 1$. In the long-time limit, the equilibrium fraction $f\ _\mathrm{m}^\mathrm{eq} = \tau_\mathrm{m}/(\tau_\mathrm{m}+\tau_ \mathrm{im})$ of all tracers are mobile. We calculate the pth moment ($p\in\mathbb{N}$) using

$$\begin{align} \langle x^p(t)\rangle_j=\frac{1}{f_j(t)}\int_{-\infty}^\infty x^pn_j(x,t)\mathrm{d}x, \end{align} \tag{ 7 }$$

where j stands for $\mathrm{m}$, $\mathrm{im}$, or $\mathrm{tot}$ [54]. To shorten the notation, we use $\langle x^2(t)\rangle = \langle x^2(t)\rangle_\mathrm{tot}$ in the remainder of this work. The lengthy exact expressions for the MSD are given in appendix B. We will study their detailed behaviour below and in sections 3 and 4 for specific initial conditions. The first moment $\langle x(t)\rangle$ is related to the second moment $\langle x^2(t)\rangle_0$ in the advection-free setting with the second Einstein relation [58]

$$\begin{align} \langle x(t)\rangle = \frac{v}{2D}\langle x^2(t)\rangle_0. \end{align} \tag{ 8 }$$

This relation becomes obvious when we look at the moments in appendix B. Since $\langle x^2(t)\rangle_0$ was discussed in [54] in detail, we focus on the MSD here.

The concept of subordination was originally introduced by Bochner [59] and refers to a process $X[\tau(t)]$ with the operational time τ (in many random walk contexts the number of jumps [44]), which has random non-negative increments. For the operational time τ the propagator is known, in our case the Gaussian $G(x,\tau)$ (2). Then, the subordinator relates the stochastic increases of τ to the process (laboratory) time t measured in the real-world observation. In our model the stochasticity comes from the immobilisations, where t increases even while τ is stalling. Assume that we know of a single tracer when it is mobile and when it is immobile. Let i(t) be the index function that is unity if the tracer is mobile at time t and zero otherwise, as shown in figure 2. The index function follows a two-state continuous-time Markov process (telegraph process) with mean residence time $\tau_\mathrm{m}$ in state 1 and mean residence time $\tau_\mathrm{im}$ in state 0. As described in [60–62] (compare also [63]), this allows us to write a Langevin equation of the form

$$\begin{align} \frac{\mathrm{d}}{\mathrm{d}\tau} x(\tau)= v+ \sqrt{2D}\mathbf{\xi}(\tau) \end{align} \tag{ 9 }$$

$$\begin{align} \kern-33.4pt \frac{\mathrm{d}}{\mathrm{d}t}\tau(t)=i(t), \end{align} \tag{ 10 }$$

with the operational time τ, that is a random quantity, and zero-mean white Gaussian noise with correlation $\langle \xi(\tau)\xi(\tau+\Delta \tau)\rangle = \delta(\Delta \tau)$. Figure 2 shows two samples of $\tau(t)$ for $\tau_\mathrm{m} = 1$ and $\tau_\mathrm{im} = 2$ in the upper panel. The lower panel shows the corresponding trajectory x(t) for $D = 1/2$ and v = 2. The propagator with this new variable τ is given by $G(x,\tau)$ (2). In appendix D we show how to obtain the subordinator $P(\tau,t)$ and its moments using Laplace transforms. In figure 3 we show the numerical Laplace inversion of $P(\tau,s)$ (D.12) for three values of t and the parameters $\tau_\mathrm{m} = 1$ and $\tau_\mathrm{im} = 1/2$. In order to verify our approach, we compare the Laplace inversion to the solution $P(\tau,t)$ obtained in [23], that is presented in appendix D. We find good agreement between the two approaches, as shown in figure 3. In the long-time limit $P(\tau,t)$ converges to a Gaussian with mean $\mu = t\tau_\mathrm{m}/(\tau_\mathrm{m}+\tau_\mathrm{im})$ and variance $\sigma^2 = 2\tau_\mathrm{m}^2\tau_\mathrm{im}^2/(\tau_\mathrm{m} +\tau_\mathrm{im})^3t$, as shown by the dashed line following a Gaussian with that mean and variance (see the moments above). With $P(\tau,t)$ we can eliminate $\tau(t)$ and find using the well established method of subordination [60–65]

$$\begin{align} n_\mathrm{tot}(x,t)=\int_0^\infty P(\tau,t){G}(x,\tau)\mathrm{d}\tau, \end{align} \tag{ 11 }$$

where $G(x,t)$ denotes the propagator (18) in the mobile phase. Expression (11) can be written in the form (6) of $n_\mathrm{tot}(x,s)$ by inserting the subordinator $P(\tau,s)$ in Laplace space. In appendix D we show how to obtain $n_\mathrm{m}(x,t)$ and $n_\mathrm{im}(x,t)$ in a similar way.

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Our model (3) has the four parameters $\tau_\mathrm{m}$, $\tau_\mathrm{im},v$ and D. We now show that in a dimensionless form the model depends only on two free parameters, which significantly reduces the space of parameters we need to consider in order to obtain a full picture of the model. Moreover, this highlights the conceptual simplicity of the model. To this end, we define the new dimensionless variables $t^{^{\prime}} = t/\tau_\mathrm{m}$ and $x^{^{\prime}} = x/\sqrt{D\tau_\mathrm{m}}$. In these variables the model (3) turns into the set of equations

$$\begin{align} \frac{\partial}{\partial t^{^{\prime}}}n_\mathrm{m}(x^{^{\prime}},t^{^{\prime}})&=-n_\mathrm{m}(x^{^{\prime}},t^{^{\prime}}) +\frac{\tau_\mathrm{m}}{\tau_\mathrm{im}} n_\mathrm{im}(x^{^{\prime}},t^{^{\prime}})\nonumber\\ &\quad-\sqrt{2\mathrm{Pe}}\frac{\partial}{\partial x^{^{\prime}}}n_\mathrm{m}(x^{^{\prime}},t^{^{\prime}}) +\frac{\partial^2}{\partial {x^{^{\prime}}}^2}n_\mathrm{m}(x^{^{\prime}},t^{^{\prime}}) \end{align} \tag{ 12 }$$

$$\begin{align} \frac{\partial}{\partial t^{^{\prime}}}n_\mathrm{im}(x^{^{\prime}},t^{^{\prime}})= -\frac{\tau_\mathrm{m}}{\tau_\mathrm{im}} n_\mathrm{im}(x^{^{\prime}},t^{^{\prime}}) +n_\mathrm{m}(x^{^{\prime}},t^{^{\prime}}), \end{align} \tag{ 13 }$$

which only depends on the immobilisation ratio $\tau_\mathrm{im}/\tau_\mathrm{m}$ and the Péclet number $\mathrm{Pe} = v^2\tau_\mathrm{m}/(2D)$. The typical length scale for the latter is given by $v\tau_\mathrm{m}$. The factor $1/2$ in the Péclet number is introduced for convenience. To see this, we note that from the Péclet number we obtain the advection time scale $\tau_{v} = 2D/v^2$, which naturally arises from the solution to the advection-diffusion equation (2) as follows. The typical distance travelled due to advection and dispersion is given by $\Delta x_{v} = vt$ and $\Delta x_{D} = \sqrt{2Dt}$, respectively. Comparing these distances gives the time scale $\tau_{v} = 2D/v^2 = \tau_m/\mathrm{Pe}$, after which displacements due to advection dominate over diffusive displacements.

In the mobile phase tracers are being propagated with the Gaussian (2). As described above the displacements are diffusion dominated for $t\ll \tau_v$. This means that the tracers follow the same density as in the case without advection, which is described in detail in [54]. To emphasise the effects of advection we therefore choose $\tau_v\ll \tau_\mathrm{m},\tau_\mathrm{im}$. This defines the short-time limit $t\ll \tau_v$.

The densities of the total, mobile and immobile density are, up to a factor, the same in the long-time limit $t\gg \tau_\mathrm{m},\tau_\mathrm{im}$ [5]. In absence of advection, the effective long-time diffusivity is given by $D_\mathrm{eff} = D\tau_\mathrm{m}/(\tau_\mathrm{m}+\tau_\mathrm{im})$ for $t\gg\tau_\mathrm{m},\tau_\mathrm{im}$ [54]. Tracers disperse slower compared to the model of simple diffusion advection without immobilisation (1) with diffusivity D. We now obtain $D_\mathrm{eff}$ by analysing the asymptotic behaviour $t\gg \tau_\mathrm{m},\tau_\mathrm{im}$ of the expressions for the MSD. The exact expressions for the first and second moments are stated in appendix B. These results provide the long-time asymptote of the MSD for all initial conditions, namely,

$$\begin{align} \langle [x(t)-\langle x(t)\rangle]^2\rangle\sim 2 \left(D\frac{\tau_\mathrm{m}}{\tau_\mathrm{m}+\tau_\mathrm{im}} +\frac{v^2\tau_\mathrm{m}^2\tau_\mathrm{im}^2}{(\tau_\mathrm{m}+\tau_\mathrm{im})^3} \right)t,\mbox{~for }t\gg \tau_\mathrm{m},\tau_\mathrm{im}, \end{align} \tag{ 14 }$$

where ∼ denotes asymptotic equivalence with an additional spread $\propto v^2t$ compared to the case without advection. Remarkably, the asymptotic dispersion with the new effective diffusion coefficient

$$\begin{align} D_\mathrm{eff}=D\frac{\tau_\mathrm{m}}{\tau_\mathrm{m} +\tau_\mathrm{im}}+v^2\frac{\tau_\mathrm{m}^2\tau_\mathrm{im}^2}{(\tau_\mathrm{m}+\tau_\mathrm{im})^3} \end{align} \tag{ 15 }$$

can hence even be higher than the diffusivity D in the mobile state and overcompensate the slow-down of the spread due to immobile durations. In appendix H we explore the parameter regime that yields $D_\mathrm{eff}\gt D$. As shown in figure H1, the increase of $D_\mathrm{eff}$ is highest, when $\tau_\mathrm{m}$ and $\tau_\mathrm{im}$ are of the same order. A physical intuition for the additional spread due to advection arises from the special case D = 0, for which $x = v\tau$ with the mobile duration τ. Due to the stochastic switching between the mobile and immobile state, an ensemble of tracers will have a distribution of mobile durations τ (the density $P(\tau,t)$ in section 2.2), and hence the positions will have a finite spread. This additional spread was derived before [6, 13], albeit without a more detailed physical justification. The anomalous transport regime at intermediate times that necessarily has to exist to effect the crossover between the two normal-diffusive regimes, and the important physical consequences will not be explained ⁶ . We explore the physical mechanism behind the additional dispersion due to advection in the long-time asymptote (14) of the MSD. We start by discretising the advection-diffusion process into discrete steps $\Delta x_i$ that are normally distributed $\Delta x_i\stackrel{\mathrm{d}}{ = }\mathcal{N}(v\Delta t,2D\Delta t)$, where each step takes a small duration $\Delta t\ll t,\tau_\mathrm{m},\tau_\mathrm{im}$. After n steps the tracer position is given by $\sum_{i = 1}^n \Delta x_i = x$. In the long-time limit the number of steps n follows a Gaussian ${n\stackrel{\mathrm{d}}{ = }}\mathcal{N}(\mu/\Delta t, \sigma^2/(\Delta t)^2)$, as described in section 2.2 ⁷ . The number of steps n is independent of the steps $\Delta x_i$. With the expectation value $\mathbb{E}$ and the variance $\mathrm{var}$ we obtain the expression [66]

$$\begin{align} \mathrm{var}(x)=\mathbb{E}(n)\mathrm{var}(\Delta x_i)+(\mathbb{E}(\Delta x_i))^2\mathrm{var}(n) {\sim}2\left(D\frac{\tau_\mathrm{m}}{\tau_\mathrm{m}+\tau_\mathrm{im}} +\frac{v^2\tau_\mathrm{m}^2\tau_\mathrm{im}^2}{(\tau_\mathrm{m}+\tau_\mathrm{im})^3}\right)t, \end{align} \tag{ 16 }$$

for $t\gg\tau_{\mathrm{m}},\tau_{\mathrm{im}}$. Equation (16) demonstrates that in the case of diffusion without advection only the mean mobile duration affects the MSD. In contrast, advection couples to the mean in the first moment and to the variance of the duration that the tracers spend in the mobile state in the MSD (14).

All time scales in the model are finite, and therefore the density converges to the Gaussian

$$\begin{align} n_\mathrm{tot}(x,t)\sim\frac{1}{\sqrt{4\pi D_\mathrm{eff}t}} \exp\left(-\frac{(x-v_\mathrm{eff}t)^2}{4D_\mathrm{eff}t}\right) \quad\mathrm{for}\quad t\gg \tau_\mathrm{m},\tau_\mathrm{im} \end{align} \tag{ 17 }$$

in the long-time limit with the effective diffusion coefficient $D_\mathrm{eff}$ (equation (15)) and the effective advection velocity $v_\mathrm{eff} = v\tau_ \mathrm{m}/(\tau_\mathrm{m}+\tau_\mathrm{im})$ (see appendix B.1). In the next two sections we analyse the densities and MSDs for the specific choice of initially mobile and immobile tracers, respectively.

Initially mobile tracers that have not yet immobilised follow a Gaussian distribution corresponding to free Brownian motion without advection for $t\ll \tau_\mathrm{m}$. At short times $t\ll \tau_v$ advection is negligible in the Gaussian propagator and the density is the same Gaussian with mean $vt\ll \sqrt{2Dt}$ close to zero and variance 2 Dt. At intermediate times $\tau_v\ll t\ll \tau_\mathrm{m}$ advection becomes relevant, and the mobile density takes on the Gaussian form

$$\begin{align} n_\mathrm{m}(x,t)\sim\left(1-\frac{t}{\tau_\mathrm{m}}\right)\frac{\exp\left( -\frac{(x-vt)^2}{4Dt}\right)}{\sqrt{4\pi D t}} \end{align} \tag{ 18 }$$

with $\int_{-\infty}^\infty n_\mathrm{m}(x,t)\mathrm{d}x\sim 1-t/\tau_\mathrm{m}$ corresponding to a scaled solution of the simple diffusion advection equation (1) ⁸ . The Gaussian (18) is shown as a dashed red line in the first two panels of figure 4, in which a stacked position histogram from simulations is shown in which the colours denote the number of immobilisation events $N_\mathrm{im}$. The solid lines are obtained from Laplace inversion of $n_\mathrm{m}(x,s)$, $n_\mathrm{im}(x,s)$, and $n_\mathrm{tot}(x,s)$ from equations (5) and (6). We use the parameters $\tau_\mathrm{m} = 1$, $\tau_\mathrm{im} = 100$, $D = 1/2$ and v = 100. These parameters are specifically chosen to be able to resolve the multiple time regimes. In the short to intermediate time regime $t\ll\tau_\mathrm{m},\tau_\mathrm{im}$, where t can be shorter or longer than τ_v , the immobile density consists of tracers that immobilised at most once. In equation (F.2) in appendix F we arrive at the asymptotic expression

$$\begin{align} n_\mathrm{im}(x,t)&\sim \left[1+\mathrm{erf}\left(\frac{vt-|x|}{\sqrt{4Dt}}\right) +\exp\left(\frac{v|x|}{D}\right)\left(\mathrm{erf}\left(\frac{vt+|x|}{\sqrt{4Dt}}\right)-1\right)\right]\nonumber \\ &\quad\times \frac{\exp\left(\frac{vx-v|x|}{2D}\right)}{2v\tau_\mathrm{m}} \end{align} \tag{ 19 }$$

for $t\ll \tau_\mathrm{m},\tau_\mathrm{im}$. The mass corresponding to (19) is $\int_{-\infty}^\infty n_\mathrm{im}(x,t)\mathrm{d}x\sim t/\tau_\mathrm{m}$, corresponding to 10⁻³ in the left panel of figure 4. This value is very small and hence the immobile density is not visible in the first panel.

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The immobile density in figure 4 appears to be uniform at t = 0.1. Indeed, for $\tau_v\ll t$ the short-time density (19) approaches a uniform distribution. Using the properties of the error function we arrive at the asymptotic uniform distribution

$$\begin{align} n_\mathrm{im}(x,t)\sim \left\{\begin{array}{ll} \frac{1}{\tau_\mathrm{m}v} &\mathrm{for }\,\,0 < x < vt\\ 0 &\mathrm{otherwise} \end{array}\right., \end{align} \tag{ 20 }$$

valid for $\tau_v\ll t\ll\tau_\mathrm{m},\tau_\mathrm{im}$. This shape can indeed be seen in the second panel of figure 4, for which the immobile density remains almost constant for $0\lt x\lt10$. The approximation (19) is shown as the blue dashed line. In the same panel the Gaussian (18) is shown as the dashed red line. The total density is given by

$$\begin{align} n_\mathrm{tot}(x,t)\sim \left(1-\frac{t}{\tau_\mathrm{m}}\right)\frac{\exp\left(-\frac{(x-vt)^2}{4Dt}\right)}{\sqrt{4\pi D t}} +\left\{\begin{array}{ll} \frac{1}{\tau_\mathrm{m}v} &\mathrm{for }\,\,0 < x< vt \\ 0 &\mathrm{otherwise} \end{array}\right. \end{align} \tag{ 21 }$$

for $\tau_v\ll t\ll \tau_\mathrm{m},\tau_\mathrm{im}$ with $\int_{-\infty}^\infty n_\mathrm{tot}(x,t)\mathrm{d}x = 1$. The appearance of this regime is new, as compared to the case without advection [54]. A physical picture for the occurrence of the uniform density of immobile tracers is as follows. For $\tau_v\ll t\ll \tau_\mathrm{m}$ advective transport dominates over diffusion. Indeed, the typical distance a tracer moved due to advection is given by the mean position vt, while the typical distance travelled due to diffusion is given by the standard deviation $\sqrt{2Dt}$. The mobile tracers hence move with a narrow Gaussian distribution while a fraction of tracers immobilises with the constant rate $1/\tau_\mathrm{m}$, which generates the uniform part of the distribution (see figure 4 for t = 0.1).

Now we go to immobilisation dominated intermediate times $\tau_\mathrm{m}\ll t\ll\tau_\mathrm{im}$. This regime is easy to analyse when starting from the density in Laplace space. We have $\phi(s)\sim 1/\tau_\mathrm{m}$ for $s\tau_\mathrm{m}\ll 1\ll s \tau_\mathrm{im}$ and find the expression

$$\begin{align} n_\mathrm{tot}(x,s)\sim\frac{1}{s}\frac{\exp\left(\frac{vx}{2D}\right)}{\sqrt{v^2\tau_\mathrm{m}^2+4D\tau_\mathrm{m}}}\exp\left( -\sqrt{v^2+4D/\tau_\mathrm{m}}\frac{|x|}{2D}\right) \end{align} \tag{ 22 }$$

from $n_\mathrm{tot}(x,s)$ (6), which in time-domain corresponds to

$$\begin{align} n_\mathrm{tot}(x,t)\sim\frac{\exp\left(\frac{vx}{2D}\right)}{\sqrt{v^2\tau_\mathrm{m}^2+4D\tau_\mathrm{m}}}\exp\left( -\sqrt{v^2\tau_\mathrm{m}^2+4D\tau_\mathrm{m}}\frac{|x|}{2D\tau_\mathrm{m}}\right) {,} \end{align} \tag{ 23 }$$

for $\tau_\mathrm{m}\ll t\ll \tau_\mathrm{im}$. Expression (23) is a normalised distribution with time-independent parameters, that falls off exponentially in the positive and negative x direction with different coefficients. It is shown in the third panel in figure 4 for t = 10 as the grey dashed line. The density falls of quicker in the direction opposite to the advection velocity, as expected. Noteworthily, the Laplace distribution occurs for all values of the Péclet number, i.e. v can take on any value in the asymptote (23).

For long times $t\gg \tau_\mathrm{im}$, we show the Gaussian (17) in figure 4 as a red dashed line for $t = 10^4$. The Gaussian (17) contains the effective advection $v_\mathrm{eff} = v\tau_\mathrm{m}/(\tau_\mathrm{m}+\tau_\mathrm{im})$ and the effective diffusivity (15), that we obtained in section 2.5.

Now we turn to the case of short mean immobile residence times, $\tau_\mathrm{im}\ll \tau_\mathrm{m}$. For short times $t\ll \tau_v$ and intermediate times $\tau_v\ll t\ll \tau_\mathrm{im}\ll \tau_\mathrm{m}$ the mobile density follows the same Gaussian (18) as for long immobilisations. This can be seen in figure 5 in the first panel for t = 0.1. From simulations, we obtain position histograms that we colour-code according to the number of immobilisations. Most tracers have not immobilised at this time, as shown by the dominant black area that follows the Gaussian (18). The tracers that did immobilise follow the same asymptote (19) of $n_\mathrm{im}(x,t)$ as for long immobilisations, as depicted by the dashed blue line. Beyond $\tau_\mathrm{im}$ immobile tracers become mobile again and contribute to the mobile density. This can be seen in figure 5 in the second panel for t = 1 by the red area in the mobile density denoting tracers with a single immobilisation event. In figure 5, the total density appears to have an exponential tail in the opposite direction of the advection velocity for t = 10 and t = 20 in the third and fourth panel, respectively. We now investigate the origin of this phenomenon. For $t\ll \tau_\mathrm{m}$, the fraction $t/\tau_\mathrm{m}$ of mobile tracers is trapped for a short period τ drawn from $\gamma(\tau) = \exp(-\tau/\tau_\mathrm{im})/\tau_\mathrm{im}$. This is shown as the red area in figure 5 denoting a single immobilisation. Hence, these tracers were mobile for a total period of $t-\tau$. We convolute this with the propagator for advection diffusion. Together with tracers that have not immobilised this gives the total density for $\tau_\mathrm{m}\ll t\ll \tau_\mathrm{im}$,

$$\begin{align} n_\mathrm{tot}(x,t)\sim \left(1-\frac{t}{\tau_\mathrm{m}}\right)\frac{\exp\left(-\frac{(x-vt^{^{\prime}})^2}{4Dt^{^{\prime}}}\right)}{\tau_m\sqrt{4\pi D t^{^{\prime}}}} +\frac{t}{\tau_\mathrm{m}} \int_0^t \frac{\exp\left(-\frac{(x-vt^{^{\prime}})^2}{4Dt^{^{\prime}}}\right)}{\tau_\mathrm{im}\sqrt{4\pi D t^{^{\prime}}}}e^{-(t-t^{^{\prime}})/\tau_\mathrm{im}}\mathrm{d}t^{^{\prime}}. \end{align} \tag{ 24 }$$

The integral on the right-hand side can be solved and is given by

$$\begin{align} \int_0^t \frac{\exp\left(-\frac{(x-vt^{^{\prime}})^2}{4Dt^{^{\prime}}}\right)} {\tau_\mathrm{im}\sqrt{4\pi D t^{^{\prime}}}}e^{-(t-t^{^{\prime}})/\tau_\mathrm{im}}\mathrm{d}t^{^{\prime}} &\sim \frac{\exp\left(-\frac{t}{\tau_\mathrm{im}}\right)} {2v\tau_\mathrm{im}} \left[\exp\left(\frac{x}{v\tau_\mathrm{im}}\right)\mathrm{erfc}\left(\frac{x-tv} {2\sqrt{Dt}}\right)\right. \nonumber \\ &\quad\left.-\exp\left(\frac{vx}{D}\right)\mathrm{erfc}\left(\frac{x+vt} {2\sqrt{Dt}}\right)\right] , \end{align} \tag{ 25 }$$

for $v^2\tau_\mathrm{im}\gg D$ with the complimentary error function $\mathrm{erfc}(x) = 1-\mathrm{erf}(x)$. In appendix G we develop the full expression of the integral (25), that is also valid for $v^2\tau_\mathrm{im}\sim D$, i.e. intermediate Péclet numbers. Approximation (24) is shown in figure 5 as the dashed dark-yellow line and follows the density for x > 70 at t = 10 and matches almost the entire density at t = 20. For $x\ll vt$ and $v^2\tau_\mathrm{im}\gg D$ expression (24) simplifies to

$$\begin{align} n_\mathrm{tot}(x,t)\sim \frac{t}{\tau_\mathrm{m}} \frac{\exp\left(-\frac{t}{\tau_\mathrm{im}}+\frac{x}{v\tau_\mathrm{im}} \right)}{v\tau_\mathrm{im}}, \quad\mbox{for }\quad \tau_\mathrm{im}\ll t\ll \tau_\mathrm{m}\mbox{and }0 < x\ll vt. \end{align} \tag{ 26 }$$

This tail approximation is only valid close the origin. For this reason it is not normalised. It is shown in figure 5 as the solid grey line. The tail overlaps with expression (24) shown as the dashed dark-yellow line. The second exponent in expression (26) reveals that the slope of the tail does not depend on time. The slope decreases for larger values of $\tau_\mathrm{im}$. As described in detail in section 2.5 the long-time asymptote $t\gg \tau_\mathrm{m}$ of the density follows a Gaussian with an effective advection speed and an effective diffusivity.

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In this section we analyse the MSD of the total density. In appendices E.1.1 and E.1.2 we analyse the MSD of the immobile and mobile density, respectively. We now investigate how advection changes the MSD as compared to the results presented in [54]. Table C1 shows a series of the total, mobile and immobile MSDs for initially mobile and initially immobile tracers for $t\ll \tau_\mathrm{m},\tau_\mathrm{im}$. In all cases, the leading order term does not depend on v. Therefore, for short times the MSDs are equivalent to the MSD without advection.

In figure 6(a) the MSD for initially mobile tracers is shown for long immobilisations, $\tau_\mathrm{m}\ll \tau_\mathrm{im}$, where the solid black line corresponds to the total density's MSD. In panel (b) we show the MSDs for short immobilisations, $\tau_\mathrm{im}\ll \tau_\mathrm{m}$. For comparison, we show the MSD for the case without advection in grey. It can be seen that the corresponding MSDs with and without advection overlap for $t\ll \tau_v$. After a linear short-time behaviour for $t\ll \tau_\star$ the MSD crosses over to a cubic scaling for $\tau_\star\ll t\ll\tau_\mathrm{m},\tau_\mathrm{im}$. Our goal is to quantify the cubic scaling and determine the value of $\tau_\star$. From a series expansion of the exact expression for the MSD (exact expressions for the moments are given in appendix B), we obtain the asymptotic MSD

$$\begin{align} \langle [x(t)-\langle x(t)\rangle]^2\rangle \sim 2Dt+ \frac{v^2}{3\tau_\mathrm{m}}t^3, \mbox{for } t\ll \tau_\mathrm{m},\tau_\mathrm{im}~. \end{align} \tag{ 27 }$$

In figure E2 we compare equation (27) to the full expression of the MSD and find very nice agreement. A cubic scaling of the MSD (27) emerges at intermediate times when the cubic term dominates over the linear term in equation (27). This corresponds to the relation $\tau_\star\ll t\ll\tau_\mathrm{m}$ with $\tau_\star = \sqrt{3\tau_v\tau_\mathrm{m}}$. Indeed, the cubic scaling of the total MSD is shown in figure 6(a) in the domain $\tau_\star\ll t\ll \tau_\mathrm{m}$. Notably, it occurs also for $\tau_\mathrm{im}\ll \tau_\mathrm{m}$, as shown in figure 6(b). This cubic scaling is new compared to the case without advection. It lies within the domain where we found an advection dominated regime $\tau_v\ll t\ll \tau_\mathrm{m}$ in the previous section, in which the density (21) consists of a uniform and a Gaussian distribution with time-dependent weights. In figure E1 we show the MSD and choose such parameters that emphasise that the intermediate cubic scaling can be more than a bare crossover but corresponds to a distinct anomalous regime. We now show that the anomalous diffusion arises from this distribution. First we consider the uniform distribution ranging from zero to vt. It has the first moment $\langle x(t)\rangle_\mathrm{im} = \frac{vt}{2}$ and the second moment $\langle x(t)^2\rangle = \frac{v^2t^2}{3}$ with the MSD $\langle (x(t)-\langle x(t)\rangle)^2\rangle_\mathrm{im} = \frac{v^2t^2}{12}$. This is the dominating term of the immobile density's MSD for $\tau_v\ll t\ll \tau_\mathrm{m}, \tau_\mathrm{im}$, as shown in figures 6(a) and (b), where the quadratic scaling can be observed. Second we recall the first and second moments of the mobile Gaussian distribution $\langle x(t)\rangle_\mathrm{m} = vt$ and $\langle x(t)^2\rangle = v^2t^2+2Dt$. Now we consider the MSD of the total density by combining the moments with the normalisations $t/\tau_\mathrm{m}$ and $1-t/\tau_\mathrm{m}$ for the uniform and Gaussian distribution, respectively. This leads to the same asymptotic MSD (27), as obtained by the series expansion, as the following calculation shows:

$$\begin{align} &\langle x(t)\rangle=\langle x(t)\rangle_\mathrm{im}\frac{t}{\tau_\mathrm{m}} +\langle x(t)\rangle_\mathrm{m}\left(1-\frac{t}{\tau_\mathrm{m}}\right)=vt- \frac{vt^2}{2\tau_\mathrm{m}}\nonumber\\ &\langle x(t)^2\rangle=\langle x(t)^2\rangle_\mathrm{im}\frac{t}{\tau_\mathrm{m}} +\langle x(t)^2\rangle_\mathrm{m}\left(1-\frac{t}{\tau_\mathrm{m}}\right) =-\frac{2}{3}\frac{v^2t^3}{\tau_\mathrm{m}}+v^2t^2-2D\frac{t^2}{\tau_\mathrm{m}} +2Dt\nonumber\\ &\langle [x(t)-\langle x(t)\rangle]^2\rangle=2Dt-2D\frac{t^2}{\tau_\mathrm{m}} +\frac{v^2t^3}{3\tau_\mathrm{m}}-\frac{v^2t^4}{4\tau_\mathrm{m}^2}\nonumber\\ &\!\!\qquad\qquad\qquad\quad\sim2Dt+\frac{v^2}{3\tau_\mathrm{m}}t^3, \mbox{~for } t\ll \tau_\mathrm{m}\ll\tau_\mathrm{im}. \end{align} \tag{ 28 }$$

The asymptotic expression is identical to what we found from the exact expressions (27). This shows indeed that the uniform distribution and the Gaussian distribution can indeed explain the cubic scaling. The cubic scaling of the MSD emerges for $\tau_\star\ll t\ll \tau_\mathrm{m},\tau_\mathrm{im}$. This means that for long immobilisations $\tau_\mathrm{m}\ll\tau_\mathrm{im}$ the advection needs to be sufficiently large such that $\tau_v = 2D/v^2\ll \tau_\mathrm{m}$, as shown in figure 6(a). The cubic scaling emerges for short immobilisations $\tau_\mathrm{im}\ll \tau_\mathrm{m}$, as well. In that case the Péclet number needs to satisfy $\mathrm{Pe}\gg3\tau_\mathrm{m}^2/\tau_\mathrm{im}^2$. In appendix I we discuss the parameter regimes for which cubic scaling emerges in detail.

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Now we choose long immobilisations, $\tau_\mathrm{im}\gg \tau_\mathrm{m}$, and consider the intermediate immobilisation dominated regime $\tau_\mathrm{m}\ll t \ll\tau_\mathrm{im}$. In the absence of advection we found a plateau of the total MSD shown as the grey solid line in figure 6(a). Compared to [54] we here choose a high Péclet number, $v^2\tau_\mathrm{m}/2D\gg1$, and keep the time scale separation $\tau_\mathrm{m}\ll \tau_\mathrm{im}$ here. The plateau still exists at $\tau_\mathrm{m}\ll t\ll \tau_\mathrm{im}$, although at a higher value as shown by the black solid line in figure 6(a). The existence of the plateau comes as no surprise, because the physical mechanism of the plateau remains unchanged. All tracers are initially mobile and have immobilised for $\tau_\mathrm{m}\ll t\ll \tau_\mathrm{im}$, as shown in figure 4 for t = 10. When all tracers are immobile the density does not change and hence the MSD remains constant. Analytically, this can be seen most easily from the expressions of the moments in Laplace space

$$\begin{align} \langle x(s)\rangle&=\frac{v}{s\phi(s)} \end{align} \tag{ 29 }$$

$$\begin{align} \langle x^2(s)\rangle&=\frac{2v^2}{s\phi^2(s)}+\frac{2D}{s\phi(s)}, \end{align} \tag{ 30 }$$

that we obtain from expression (A.5). For $\tau_\mathrm{m}\ll t\ll \tau_\mathrm{im}$, we have $\phi(s)\sim 1/\tau_\mathrm{m}$, which is a constant. Since the Laplace inverse $\mathscr{L}^{-1}[1/s] = 1$ we obtain a constant MSD in this time-domain $\tau_\mathrm{m}\ll t\ll \tau_\mathrm{im}$, where we observe the asymmetric Laplace distribution (23). We emphasise that the plateau exists regardless of the values of v and D. At long times $t\gg \tau_\mathrm{m},\tau_\mathrm{im},\tau_v$ the MSD grows linearly with the effective diffusion coefficient $D_\mathrm{eff} = D\frac{\tau_\mathrm{m}}{\tau_\mathrm{m}+\tau_\mathrm{im}} +v^2\frac{\tau_\mathrm{m}^2\tau_\mathrm{im}^2} {(\tau_\mathrm{m}+\tau_\mathrm{im})^3}$, as described in section 2.5.

We assume long immobilisations, $\tau_\mathrm{m}\ll\tau_\mathrm{im}$, corresponding to the case studied in [54]. We choose a high Péclet number to emphasise the effect of advection, in contrast to the v = 0 case in [54]. At short times $t\ll\tau_v$ the propagator for mobile tracers is the same Gaussian as for the case v = 0. Therefore, we obtain the same short-time densities as in the v = 0 case, i.e. a δ-peak at the origin and an additional non-Gaussian distribution.

At short to intermediate times, $t\ll \tau_\mathrm{m},\tau_\mathrm{im}$, at which t can be shorter or longer than τ_v , most initially immobile tracers are concentrated at the origin and only gradually mobilise. This gives rise to the immobile density $n_\mathrm{im}(x,t)\sim (1-t/\tau_\mathrm{im})\delta(x)$. As described in detail in [54], immobile tracers that were initially mobile follow the same density as mobile tracers that were initially immobile, equation (19), up to a factor $\tau_\mathrm{m}/\tau_\mathrm{im}$. Therefore, we arrive at the total density for short to intermediate times

$$\begin{align} n_\mathrm{tot}(x,t)&\sim\left(1-\frac{t}{\tau_\mathrm{im}}\right)\delta(x) +\frac{\exp\left(\frac{vx-v|x|}{2D}\right)}{2v\tau_\mathrm{im}}\nonumber\\ &\quad\times\left[1+\mathrm{erf}\left(\frac{vt-|x|}{\sqrt{4Dt}}\right) +\exp\left(\frac{v|x|}{D}\right)\left(\mathrm{erf}\left(\frac{vt+ |x|}{\sqrt{4Dt}}\right)-1\right)\right] \end{align} \tag{ 31 }$$

for $t\ll \tau_\mathrm{m},\tau_\mathrm{im}$. The asymptote of the second summand corresponding to the mobile density in expression (31) is shown in figure 7 as the blue dashed line, which nicely matches the simulations and the Laplace inversions for $t = 10^{-3}$ and $t = 10^{-1}$. We note that $n_\mathrm{tot}(x,t)$ is always normalised, $\int_{-\infty}^\infty n_\mathrm{tot}(x,t)\mathrm{d}x = 1$, by construction. The same arguments as presented for initially mobile tracers explain the uniform density that appears at intermediate time scales for $\tau_v\ll t\ll \tau_\mathrm{m},\tau_\mathrm{im}$, corresponding to the second panel in figure 7. In the immobilisation dominated intermediate time domain $\tau_\mathrm{m}\ll t\ll\tau_\mathrm{im}$ the Laplace distribution with additional δ-peak

$$\begin{align} n_\mathrm{tot}(x,t)&=\frac{t}{\tau_\mathrm{im}}\frac{\exp\left(\frac{vx}{2D} \right)}{\sqrt{v^2\tau_\mathrm{m}^2+4D\tau_\mathrm{m}}}\exp\left( -\sqrt{v^2\tau_\mathrm{m}^2+4D\tau_\mathrm{m}}\frac{|x|}{2D\tau_\mathrm{m}} \right)\nonumber\\ &\quad +\left(1-\frac{t}{\tau_\mathrm{im}}\right)\delta(x) \end{align} \tag{ 32 }$$

emerges with the same scale parameter as for initially mobile tracers (23). The prefactor of the asymmetric Laplace distribution is now $t/\tau_\mathrm{im}$, and this asymmetric Laplace distribution is shown in figure 7 as the grey dashed line. The long-time limit does not depend on the initial conditions and follows the same density as the initially mobile tracers, equation (17), as shown by the red dashed line in figure 7 at $t = 10^4$.

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We now consider short immobilisations, $\tau_\mathrm{m}\ll \tau_\mathrm{im}$. At short to intermediate time scales $t\ll \tau_\mathrm{im}\ll\tau_\mathrm{m}$, at which t can be shorter or longer than τ_v , the same expression (31) holds as for long immobilisations. This can be seen in figure 8, where the asymptote (31) is shown as the blue dashed line. We find excellent agreement between the histogram based on the simulations and the density (31) for $t\ll \tau_\mathrm{im},\tau_\mathrm{m}$, see the first and second panels in figure 8 for $t = 2\times 10^{-6}$ and $t = 3\times 10^{-4}$, respectively. In contrast to the case v = 0 in [54], a new regime $\tau_\mathrm{im}\ll t\ll \tau_\mathrm{m}$ emerges, that we call advection induced subdiffusion. The total density follows an exponential distribution, as can be seen from the linear shape of the density in the semi-log plot for $t = 10^{-2}$. To explain this shape we solve the model equations (3) for the advection induced subdiffusion regime times $t\ll \tau_\mathrm{m}$, where t can be longer or shorter than τ_v . This produces the total density

$$\begin{align} n_\mathrm{tot}(x,t)\sim\delta(x)e^{-t/\tau_\mathrm{im}} +\int_0^t\frac{e^{-t^{^{\prime}}/\tau_\mathrm{im}}}{\tau_\mathrm{im}} \frac{\exp\left(-\frac{(x-vt^{^{\prime}})^2}{4Dt^{^{\prime}}}\right)}{\sqrt{4\pi Dt^{^{\prime}}}}\mathrm{d}t^{^{\prime}}, \mbox{for }t\ll\tau_\mathrm{m}, \end{align} \tag{ 33 }$$

where the integral is identical to expression (25) for immobile tracers that were initially mobile. We obtain the expression

$$\begin{align} n_\mathrm{tot}(x,t)\sim\delta(x)\exp\left(-\frac{t}{\tau_\mathrm{im}} \right)+\left\{\begin{array}{ll}\frac{1}{v\tau_\mathrm{im}}\exp\left( \frac{x}{v\tau_\mathrm{im}}-\frac{t}{\tau_\mathrm{im}}\right) & \mathrm{for }~0 < x\ll vt\\ 0 & \mathrm{otherwise} \end{array}\right., \end{align} \tag{ 34 }$$

for $v^2\tau_\mathrm{im}\gg D$ and $t\ll\tau_\mathrm{m}$. This comes as no surprise, as in both cases we have the same Gaussian propagator, in which the tracers have varying immobile durations. In the case of initially mobile tracers the immobile duration stems from an immobilisation at t > 0. In the present case of initially immobile tracers the immobile duration arises from the slow release at the origin at t = 0. In both cases the immobile duration is drawn from an exponential distribution with mean $\tau_\mathrm{im}$. The first term in (34) corresponds to initially immobile tracers that have not mobilised up to time t. The second term accounts for the slow release with rate $\tau_\mathrm{im}^{-1}\exp(-t/\tau_\mathrm{im})$ and motion in the mobile zone with advection only. The exponential distribution (34) is shown in figure 8 at $t = 2\times 10^{-2}$ as the grey dashed line, and we find good agreement with the Laplace inversion of $n_\mathrm{tot}(x,s)$ and simulations.

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We now analyse the MSD of the total density. In appendices E.2.1 and E.2.2 we analyse the MSD of the immobile and mobile density, respectively. We show the MSDs for long and short immobilisations in panels (a) and (b) of figure 9, respectively. From a series expansion of the MSD at t = 0 we obtain the asymptotic MSD at short to intermediate times,

$$\begin{align} \langle [x(t)-\langle x(t)\rangle]^2\rangle\sim \frac{Dt^2}{\tau_\mathrm{im}} +\frac{v^2}{3\tau_\mathrm{im}}t^3 ,\mbox{~for }t\ll\tau_\mathrm{m},\tau_\mathrm{im}, \end{align} \tag{ 35 }$$

in which t can be smaller or larger than τ_v . We compare the asymptote (35) to the full expression of the MSD in figure E2(b) and find very nice agreement. At short times $t\ll \tau_v$, the quadratic term dominates and we observe the same MSD as in the case without advection. This can be seen in figures 9(a) and (b). At intermediate times $\tau_v\ll t\ll \tau_\mathrm{m},\tau_\mathrm{im}$ the cubic term in the asymptotic MSD (35) dominates. As shown in figure 9 the cubic scaling emerges for short and long immobilisations.

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Now we go to the case of short immobilisation, $\tau_\mathrm{im}\ll \tau_\mathrm{m}$, for which advection induced subdiffusion emerges for $\tau_\mathrm{im}\ll t\ll \tau_\mathrm{m}$. As described in section 4.2, the total density follows an exponential distribution (34) and a δ-peak at the origin for $\tau_v\ll t\ll \tau_\mathrm{m}$. The MSD of that distribution is given by

$$\begin{align} \langle [x(t)-\langle x(t)\rangle]^2\rangle=v^2\tau_\mathrm{im} \left(\tau_\mathrm{im}-2t\exp\left(-\frac{t}{\tau_\mathrm{im}}\right) -\tau_\mathrm{im}\exp\left(-2\frac{t}{\tau_\mathrm{im}}\right)\right), \end{align} \tag{ 36 }$$

for $\tau_v\ll t \ll \tau_\mathrm{m}$ and $\tau_\mathrm{im}\ll \tau_\mathrm{m}$, which is shown in figure 9(b) as the blue line. From expression (36) we recover the intermediate time asymptote

$$\begin{align} \langle [x(t)-\langle x(t)\rangle]^2\rangle \sim \frac{v^2t^3}{3\tau_\mathrm{im}}, \mbox{~for } \tau_v \ll t\ll \tau_\mathrm{im}, \end{align} \tag{ 37 }$$

implying the same cubic scaling for $\tau_v\ll t\ll \tau_\mathrm{im}$ as we found from the series expansion of the full MSD (35) for the advection dominated regime. This can be seen in the MSD in figure 9(b). The cubic scaling of the MSD emerges in the domain $\tau_v\ll t \ll \tau_\mathrm{m},\tau_\mathrm{im}$, which limits the parameters to $\tau_v\ll \tau_\mathrm{m},\tau_\mathrm{im}$. In terms of the Péclet number this implies $\mathrm{Pe}\gg 1$ for long immobilisations $\tau_\mathrm{m}\ll \tau_\mathrm{im}$ and $\mathrm{Pe}\gg\tau_\mathrm{m}/\tau_\mathrm{im}$ for short immobilisations $\tau_\mathrm{im}\ll \tau_\mathrm{m}$. A detailed discussion of the parameter regimes and the coexistence of the plateau regime is presented in appendix J.

In the advection induced subdiffusion domain the MSD (36) reaches the plateau value

$$\begin{align} \langle [x(t)-\langle x(t)\rangle]^2\rangle\sim v^2\tau_\mathrm{im}^2, \mbox{for }\tau_\mathrm{im}\ll t \ll \tau_\mathrm{m}. \end{align} \tag{ 38 }$$

These two anomalous scaling regimes shown in figure 9(b), namely, the cubic scaling and the plateau behaviour can be explained as follows. For $\tau_v\ll t\ll \tau_\mathrm{im}$ the MSD grows due to the slow release and fast advection, where the spread due to diffusion is negligible. When all tracers mobilised at $t\gg \tau_\mathrm{im}$, this spread due to advection vanishes, and the distribution moves along the direction of advection without changing the shape significantly. This explains the plateau in the MSD for $\tau_\mathrm{im}\ll t\ll \tau_\mathrm{m}$. We highlight that this advection induced subdiffusion is a new behaviour compared to the case without advection. For comparison, we show the MSD for v = 0 in figure 9(b) as the grey solid line. It crosses over from the short-time scaling $\sim Dt^2/\tau_\mathrm{im}$ to the long-time asymptote $\sim 2D_\mathrm{eff}t$ without any intermediate regime.

For mobile initial conditions we find

$$\begin{align} \langle x(t)\rangle_{\mathrm{tot},f_{m}^0=1}=\frac{v}{1+\tau_\mathrm{im}/\tau_\mathrm{ m}}\left[t+\frac{\tau_\mathrm{im}^2/\tau_\mathrm{m}}{1+\tau_\mathrm{im}/ \tau_\mathrm{m}}\left(1-e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t} \right)\right], \end{align} \tag{ B.1 }$$

$$\begin{align} \langle x(t)\rangle_{\mathrm{m},f_m^0=1}&=\frac{v}{(1+\tau_\mathrm{im}/\tau_ \mathrm{m})\left(1+\tau_\mathrm{im}/\tau_\mathrm{m}e^{-(\tau_\mathrm{m}^{-1} +\tau_\mathrm{im}^{-1})t}\right)}\left[t\left(1+\frac{\tau_\mathrm{im}^2}{ \tau_\mathrm{m}^{2}}e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t} \right)\right.\nonumber\\ &\quad \left.+\frac{2\tau_\mathrm{im}^2/\tau_\mathrm{m}}{1+\tau_\mathrm{im}/\tau_ \mathrm{m}}\left(1-e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right)\right], \end{align} \tag{ B.2 }$$

and

$$\begin{align} \langle x(t)\rangle_{\mathrm{im},f_m^0=1}&=\frac{v}{1-e^{-(\tau_\mathrm{m}^{-1} +\tau_\mathrm{im}^{-1})t}}\left[ \frac{t}{1+\tau_\mathrm{im}/\tau_\mathrm{ m}}\left(1-\frac{\tau_\mathrm{im}}{\tau_\mathrm{m}}e^{-(\tau_\mathrm{m}^{-1} +\tau_\mathrm{im}^{-1})t}\right)\right.\nonumber\\ &\quad+\left.\frac{\tau_\mathrm{im}^2/\tau_\mathrm{m}-\tau_\mathrm{im}}{(1+ \tau_\mathrm{im}/\tau_\mathrm{m})^2}\left(1-e^{-(\tau_\mathrm{m}^{-1}+ \tau_\mathrm{im}^{-1})t}\right)\right]. \end{align} \tag{ B.3 }$$

For initially immobile tracers we find

$$\begin{align} \langle x(t)\rangle_{\mathrm{tot},f_\mathrm{im}^0=1}=\frac{v}{1+\tau_\mathrm{im}/\tau_\mathrm{ m}}\left[t-\frac{\tau_\mathrm{im}}{1+\tau_\mathrm{im}/\tau_\mathrm{m}}\left(1- e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right)\right], \end{align} \tag{ B.4 }$$

$$\begin{align} \langle x(t)\rangle_{\mathrm{im},f_\mathrm{im}^0=1}&=\frac{vt}{1+\tau_\mathrm{m}/\tau_\mathrm{ im}}\frac{1+e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}}{\tau_\mathrm{ im}/\tau_\mathrm{m}+e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}}\nonumber\\ &\quad -\frac{3v\tau_\mathrm{im}^2/\tau_\mathrm{m}}{(1+\tau_\mathrm{im}/\tau_\mathrm{ m})^2}\frac{1-e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}}{\tau_\mathrm{ im}/\tau_\mathrm{m}+e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}}, \end{align} \tag{ B.5 }$$

and

$$\begin{align} \langle x(t)\rangle_{\mathrm{m},f\ _\mathrm{im}^0=1}&=\frac{v}{1-e^{-(\tau_\mathrm{m}^{-1} +\tau_\mathrm{im}^{-1})t}}\left[ \frac{t}{1+\tau_\mathrm{im}/\tau_\mathrm{ m}}\left(1-\frac{\tau_\mathrm{im}}{\tau_\mathrm{m}}e^{-(\tau_\mathrm{m}^{-1} +\tau_\mathrm{im}^{-1})t}\right)\right.\nonumber\\ &\quad +\left.\frac{\tau_\mathrm{im}^2/\tau_\mathrm{m}-\tau_\mathrm{im}}{(1+ \tau_\mathrm{im}/\tau_\mathrm{m})^2}\left(1-e^{-(\tau_\mathrm{m}^{-1}+ \tau_\mathrm{im}^{-1})t}\right)\right]. \end{align} \tag{ B.6 }$$

Consider initially mobile tracers ($f\ _\mathrm{m}^0 = 1$ and $f\ _\mathrm{im}^0 = 0$). Then we obtain

$$\begin{align} \langle x(t)^2\rangle_{\mathrm{tot},f\ _\mathrm{m}^0=1}&= \frac{2D}{1+\tau_\mathrm{im}/\tau_\mathrm{m}}\left[t+\frac{\tau_\mathrm{im}^2 /\tau_\mathrm{m}}{1+\tau_\mathrm{im}\tau_\mathrm{m}}\left(1-e^{-(\tau_\mathrm{ m}^{-1}+\tau_\mathrm{im}^{-1})t}\right)\right]\nonumber\\ &\quad +\frac{v^2t^2\tau_\mathrm{m}^2}{(\tau_\mathrm{m}+\tau_\mathrm{im})^2} +\frac{\tau_\mathrm{im}^2\tau_\mathrm{m}^2v^2}{(\tau_\mathrm{im}+\tau_\mathrm{ m})^3}\left[2t\left(2-e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t} \frac{\tau_\mathrm{im}}{\tau_\mathrm{m}}\right)\right.\nonumber\\ &\quad \left.-2\frac{2\tau_\mathrm{im}\tau_\mathrm{m}-\tau_\mathrm{im}^2}{ \tau_\mathrm{m}+\tau_\mathrm{im}}+2e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{ -1})t}\frac{(2\tau_\mathrm{im}\tau_\mathrm{m})-\tau_\mathrm{im}^2}{\tau_\mathrm{ m}+\tau_\mathrm{im}}\right] \end{align} \tag{ B.7 }$$

and

$$\begin{align} f_\mathrm{m}(t)\langle x(t)^2\rangle_{\mathrm{m},f\ _\mathrm{m}^0=1}&= \frac{2D}{(1+\tau_\mathrm{im}/\tau_\mathrm{m})(1+\tau_\mathrm{im}/\tau_ \mathrm{m}e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t})}\nonumber\\ &\quad\times\left[t\left(1+\frac{\tau_\mathrm{im}^2}{\tau_\mathrm{m}^{2}}e^{- (\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right)+\frac{2\tau_\mathrm{im}^2 /\tau_\mathrm{m}}{1+\tau_\mathrm{im}/\tau_\mathrm{m}}(1-e^{-(\tau_\mathrm{m}^{ -1}+\tau_\mathrm{im}^{-1})t})\right]\nonumber\\ &\quad+\frac{v^2t^2}{(\tau_\mathrm{m}+\tau_\mathrm{im})^3}\left(\tau_\mathrm{m}^3 +\tau_\mathrm{im}^3e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right)\nonumber\\ &\quad+\frac{6v^2t\tau_\mathrm{im}^2\tau_\mathrm{m}^2}{(\tau_\mathrm{m}+\tau_\mathrm{im})^4} \left(\tau_\mathrm{m}-\tau_\mathrm{im}e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right)\nonumber\\ &\quad+\frac{6v^2\tau_\mathrm{im}^3(\tau_\mathrm{im}\tau_\mathrm{m}^3-\tau_\mathrm{m}^4)}{(\tau_\mathrm{m}+\tau_\mathrm{im})^5} \left(1-e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t})\right), \end{align} \tag{ B.8 }$$

along with

$$\begin{align} f_\mathrm{im}(t)\langle x(t)^2\rangle_{\mathrm{im},f\ _\mathrm{m}^0=1}&= \frac{2D}{1-e^{-(\tau_\mathrm{m}^{-1} +\tau_\mathrm{im}^{-1})t}}\left[ \frac{t}{1+\tau_\mathrm{im}/\tau_\mathrm{ m}}\left(1-\frac{\tau_\mathrm{im}}{\tau_\mathrm{m}}e^{-(\tau_\mathrm{m}^{-1} +\tau_\mathrm{im}^{-1})t}\right)\right.\nonumber\\ &\quad +\left.\frac{\tau_\mathrm{im}^2/\tau_\mathrm{m}-\tau_\mathrm{im}}{(1+ \tau_\mathrm{im}/\tau_\mathrm{m})^2}\left(1-e^{-(\tau_\mathrm{m}^{-1}+ \tau_\mathrm{im}^{-1})t}\right)\right.\nonumber\\ &\quad +\frac{2v^2\tau_\mathrm{im}}{\tau_\mathrm{m}}\left[ \frac{t^2\tau_\mathrm{m}\left(\tau_\mathrm{m}^2-e^{-(\tau_\mathrm{m}^{-1} +\tau_\mathrm{im}^{-1})t}\tau_\mathrm{im}^2\right)}{2(\tau_\mathrm{m}+\tau_\mathrm{im})^3}+\frac{t\tau_\mathrm{im}}{(\tau_\mathrm{ m}+\tau_\mathrm{im})^4}\right.\nonumber\\ &\quad \left.\times\left((2 \tau_\mathrm{im}\tau_\mathrm{m}^3-\tau_\mathrm{m}^4) -e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\tau_\mathrm{im}(\tau_\mathrm{im}\tau_\mathrm{m}^2-2\tau_\mathrm{m}^3)\right)\right.\nonumber\\ &\quad\left.+\frac{\tau_\mathrm{im}^4\tau_\mathrm{m}^3-4\tau_\mathrm{im}^3\tau_\mathrm{m}^4+\tau_\mathrm{im}^2\tau_\mathrm{m}^5}{(\tau_\mathrm{m}+\tau_\mathrm{im})^5} \left(1-e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right) \right]. \end{align} \tag{ B.9 }$$

For initially immobile tracers the moments read

$$\begin{align} \langle x(t)^2\rangle_{\mathrm{tot},f\ _\mathrm{im}^0=1}&= \frac{2D}{1+\tau_\mathrm{im}/\tau_\mathrm{ m}}\left[t-\frac{\tau_\mathrm{im}}{1+\tau_\mathrm{im}/\tau_\mathrm{m}}\left(1- e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right)\right]\nonumber\\ &\quad\frac{v^2t^2\tau_\mathrm{m}^2}{(\tau_\mathrm{im}+\tau_\mathrm{m})^2} +2\frac{tv^2\tau_\mathrm{im}^2\tau_\mathrm{m}^2}{(\tau_\mathrm{im}+ \tau_\mathrm{m})^3}\left(1-\frac{\tau_\mathrm{m}}{\tau_\mathrm{im}}+e^{ -(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right)\nonumber\\ &\quad -2\frac{v^2(2\tau_\mathrm{im}^3\tau_\mathrm{m}^3-\tau_\mathrm{im}^2 \tau_\mathrm{m}^4)}{(\tau_\mathrm{m}+\tau_\mathrm{im})^4}\left(1-e^{-( \tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right), \end{align} \tag{ B.10 }$$

$$\begin{align} f_\mathrm{m}(t)\langle x(t)^2\rangle_{\mathrm{m},f\ _\mathrm{im}^0=1}&= \frac{2D}{1-e^{-(\tau_\mathrm{m}^{-1} +\tau_\mathrm{im}^{-1})t}}\left[ \frac{t}{1+\tau_\mathrm{im}/\tau_\mathrm{ m}}\left(1-\frac{\tau_\mathrm{im}}{\tau_\mathrm{m}}e^{-(\tau_\mathrm{m}^{-1} +\tau_\mathrm{im}^{-1})t}\right)\right.\nonumber\\ &\quad +\left.\frac{\tau_\mathrm{im}^2/\tau_\mathrm{m}-\tau_\mathrm{im}}{(1+ \tau_\mathrm{im}/\tau_\mathrm{m})^2}\left(1-e^{-(\tau_\mathrm{m}^{-1}+ \tau_\mathrm{im}^{-1})t}\right)\right]\nonumber\\ &\quad +\frac{v^2t^2\tau_\mathrm{m}}{(\tau_\mathrm{m}+\tau_\mathrm{im})^3} \left(\tau_\mathrm{m}^2-\tau_\mathrm{im}^2e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right)\nonumber\\ &\quad +\frac{2v^2t\tau_\mathrm{im}\tau_\mathrm{m}^3}{(\tau_\mathrm{im}+\tau_\mathrm{m})^4} \left(2\tau_\mathrm{im}-\tau_\mathrm{m}-e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\tau_\mathrm{im}\left(\frac{\tau_\mathrm{im}}{\tau_\mathrm{m}}-2\right)\right)\nonumber\\ &\quad +\frac{2v^2\tau_\mathrm{im}^2\tau_\mathrm{m}^3((\tau_\mathrm{im}-\tau_\mathrm{m})^2-2\tau_\mathrm{im}\tau_\mathrm{m})}{(\tau_\mathrm{m}+\tau_\mathrm{im})^5} \left(1-e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right)\kern2.2pt \end{align} \tag{ B.11 }$$

and

$$\begin{align} f_\mathrm{im}(t)\langle x(t)^2\rangle_{\mathrm{im},f\ _\mathrm{im}^0=1}&= \frac{2Dt}{1+\tau_\mathrm{m}/\tau_\mathrm{im}}\frac{1+e^{-(\tau_\mathrm{m}^{ -1}+\tau_\mathrm{im}^{-1})t}}{\tau_\mathrm{im}/\tau_\mathrm{m}+e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}}\nonumber\\ &\quad -\frac{4D\tau_\mathrm{im}^2/\tau_\mathrm{m}}{(1+\tau_\mathrm{im}/\tau_\mathrm{ m})^2}\frac{1-e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}}{\tau_\mathrm{ im}/\tau_\mathrm{m}+e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}}\nonumber\\ &\quad +2v^2\frac{\tau_\mathrm{im}}{\tau_\mathrm{m}}\left[\frac{t^2\tau_\mathrm{m}^2}{2(\tau_\mathrm{m}+\tau_\mathrm{im})^3} \left(\tau_\mathrm{m}+\tau_\mathrm{im}e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right)\right.\nonumber\\ &\quad +\frac{t\tau_\mathrm{im}}{(\tau_\mathrm{m}+\tau_\mathrm{im})^4} \left((\tau_\mathrm{im}\tau_\mathrm{m}^3-2\tau_\mathrm{m}^4)+e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}(2\tau_\mathrm{im}\tau_\mathrm{m}^3-\tau_\mathrm{m}^4)\right)\nonumber\\ &\quad -\left. \frac{3\tau_\mathrm{im}^3(\tau_\mathrm{im}\tau_\mathrm{m}^4-\tau_\mathrm{m}^5)}{(\tau_\mathrm{m}+\tau_\mathrm{im})^5} \left(1-e^{-(\tau_\mathrm{m}^{-1}+\tau_\mathrm{im}^{-1})t}\right)\right]. \end{align} \tag{ B.12 }$$

We assume initially mobile tracers and obtain asymptotic expressions in sections appendices E.1.1 and E.1.2.

E.1.1. MSD of the immobile density

Now we turn to the MSD of the immobile density. A series expansion of the exact MSD produces

$$\begin{align} \langle [x(t)-\langle x(t)\rangle]^2\rangle_\mathrm{im} \sim Dt+ \frac{v^2}{12}t^2 ,\mbox{for } t\ll \tau_\mathrm{m}\ll\tau_\mathrm{im}. \end{align} \tag{ E.1 }$$

Expression (E.1) implies a ballistic scaling of the immobile MSD for $6\tau_v\ll t\ll \tau_\mathrm{m},\tau_\mathrm{im}$. This matches the MSD of the uniform distribution (20) of $n_\mathrm{im}(x,t)$ shown in figure 4 where the right border moves at a constant speed. The immobile MSD with the ballistic scaling is shown in figure 6(a) as the dotted line. Panel (c) depicts the anomalous diffusion exponent α, which is close to two for $6\tau_v \ll t\ll \tau_\mathrm{m}$. Figure 6(a) shows the MSD of the mobile density as a dashed black line. For almost all times $t\ll\tau_\mathrm{m}$ it coincides with 2 Dt corresponding to the case without advection and Brownian diffusion. The reason for this is that at $t\lt\tau_\mathrm{m}\ll \tau_\mathrm{im}$ almost all mobile tracers have never immobilised, as shown in figure 4.

E.1.2. MSD of the mobile density

The MSD of the mobile density has the series expansion

$$\begin{align} \langle [x(t)-\langle x(t)\rangle]^2\rangle = 2Dt+\frac{v^2}{12\tau_\mathrm{m}\tau_\mathrm{im}}t^4,\mbox{for }t\ll\tau_\mathrm{m},\tau_\mathrm{im}. \end{align} \tag{ E.2 }$$

The short-time behaviour 2 Dt dominates up to $(2D\tau_\mathrm{m}\tau_\mathrm{im}/v^2)^{1/3}$, as shown in figures 6(a) and (b). The time scale $(2D\tau_\mathrm{m}\tau_\mathrm{im}/v^2)^{1/3}$ is very close to $\tau_\mathrm{m}$ in panel (a) and close to $\tau_\mathrm{im}$ in panel (b), therefore the t⁴ scaling is not observed in figure 6. In the appendix in figure E1(a), we choose a higher Péclet number, for which the t⁴ scaling is distinct. After $\tau_\mathrm{m}$ the MSD has a peak and decreases afterwards for some time. This can be explained similarly to the advection free case in [54]. Up to $\tau_\mathrm{m}$, most of the mobile density consists of mobile tracers that have never immobilised. This can be seen in the histogram for t = 0.1 in figure 4, where most of the mobile density follows a Gaussian which is coloured black corresponding to zero immobilisation events $N_\mathrm{im} = 0$. When the mobile density mostly consists of tracers that were immobile once the MSD decreases because the leading Gaussian peak is missing.

We assume initially immobile tracers and obtain asymptotic expressions in sections appendices E.2.1 and E.2.2.

E.2.1. MSD of the immobile density

From a series expansion of the immobile MSD we obtain the short to intermediate time asymptote of the immobile population,

$$\begin{align} \langle [x(t)-\langle x(t)\rangle]^2\rangle_\mathrm{im}\sim \frac{Dt^3}{3\tau_\mathrm{im}\tau_\mathrm{m}} +\frac{v^2}{12\tau_\mathrm{im}\tau_\mathrm{m}}t^4 ,\mbox{for }t\ll\tau_\mathrm{m},\tau_\mathrm{im}, \end{align} \tag{ E.3 }$$

where t can be shorter or longer than τ_v . In figure E2(b) we verify this asymptote and it shows good agreement with the full analytical solution. For a high Péclet number $\mathrm{Pe} = \frac{v^2\tau_\mathrm{m}}{2D}\gg 1$ the quartic term in the asymptotic MSD (E.3) dominates for $\tau_v\ll t\ll \tau_\mathrm{m},\tau_\mathrm{im}$, as shown in figures 9(a) and (b).

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E.2.2. MSD of the mobile density

Here we analyse for which parameter regimes the cubic scaling (27) of the total MSD appears for initially mobile tracers. First, long immobilisations are assumed in section appendix I.2 $\tau_\mathrm{m}\ll\tau_\mathrm{im}$, followed by short immobilisations in section appendix I.3.

A necessary condition for the cubic regime to emerge is that the lower bound $\tau_\star$ of the time regime is smaller than the upper bound $\tau_\mathrm{m}$. We define t₃ as the duration between the upper and lower bound. This restricts the parameter space in which the cubic regime appears to positive values of the time difference t₃. As shown in section 2.3, the whole parameter space can be spanned by the Péclet number $\mathrm{Pe} = v^2\tau_\mathrm{m}/2D$ and the ratio $\tau_\mathrm{im}/\tau_\mathrm{m}$. We use the decadic logarithm of these quantities as axis for a map in figure I1, similar to figure H1. The border of the region with $t_3\gt0$ in the top right of figure H1 is shown as a black dotted line. We notice that $\mathrm{Pe}\gt1$ is a necessary condition for $t_3\gt0$. We analyse the parameter space for $\tau_\mathrm{im}\gt \tau_\mathrm{m}$ first and for $\tau_\mathrm{im}\lt\tau_\mathrm{m}$ afterwards. Rearranging the condition $\tau_\star\lt\tau_\mathrm{m}$ for the Péclet number reveals the lower bound $27/16\lt\mathrm{Pe}$ independent of $\tau_\mathrm{im}/\tau_\mathrm{m}$. Indeed, the black dotted line for $t_3 = 0$ is vertical in the map (I1) for $\tau_\mathrm{im}\gg \tau_\mathrm{m}$, indicating no dependence on $\tau_\mathrm{im}/\tau_\mathrm{m}$ in that regime.

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So far, we looked at the necessary condition that the lower bound for the interval of the cubic regime is lower than the upper bound. Another approach is to analyse the instantaneous anomalous diffusion exponent $\alpha(t)$. It can be obtained by calculating the slope in a double-logarithmic plot. This is shown in figure E1(c), where α remains close to unity for $t\lt\tau_\star$ and reaches the value three for $\tau_\star\ll t \ll\tau_\mathrm{m},\tau_\mathrm{im}$. The alternative criterion for the cubic regime to exist is then simply that the maximum value of $\alpha_\mathrm{max}$ is close to three. With this definition of $\alpha(t)$, superdiffusive parameter regimes are shown with the red colour-coding ranging from one to three in figure I1. This means that the more intense the red in an area is, the closer the parameters are to the asymptotic limits in which the cubic regime emerges. We notice that the red area lies entirely in the region with $t_3\gt0$ and its border follows the same asymptote $\mathrm{Pe}^{-1/2}$ for $\tau_\mathrm{m}\gg \tau_\mathrm{im}$. The cubic regime appears both for $D_\mathrm{eff}\gt D$ and for $D_\mathrm{eff}\lesssim D$ for long immobilisations $\tau_\mathrm{im}\gg \tau_\mathrm{m}$ and for $\mathrm{Pe}\gg 1$, as shown in the top right corner of the map in figure I1. For reference, a contour plot of $D_\mathrm{eff}/D$ is shown in figure H1. The reason for the appearance of the cubic regime regardless of the ratio $D_\mathrm{eff}/D$ is the existence of the plateau for $\tau_\mathrm{im}\gg \tau_\mathrm{m}$, which we explain now in detail. We know that regardless of the ratio $\tau_\mathrm{im}/\tau_\mathrm{m}$ the MSD of initially mobile tracers has the asymptote 2 Dt for short times $t\ll \tau_\mathrm{m},\tau_\mathrm{im},\tau_v$. In the long-time limit it has the asymptote $2D_\mathrm{eff}t$. The MSD of the total density is a strictly monotonic function. This means that if $D_\mathrm{eff}\lt D$, any growth faster than linear must be compensated by a regime with a sublinear growth. This slower than linear growth arises for long immobilisations only and constitutes the plateau for a sufficiently large ratio $\tau_\mathrm{im}/\tau_\mathrm{m}$.

Finally, we analyse the case of short immobilisations, $\tau_\mathrm{im}\ll \tau_\mathrm{m}$. The same series expansion of the total MSD holds as in the case with long immobilisations (27). The difference between the upper and lower bound of the cubic scaling is therefore $t_3 = \tau_\mathrm{im}-\tau_\star$. Notably, the duration of the cubic regime is shorter as compared to the case with long immobilisations in figure 6(a), because $\tau_\star = \sqrt{6D\tau_\mathrm{m}/v^2}$ is close to the lower bound $\tau_\mathrm{im}$ for $\tau_\mathrm{m}\gg \tau_\mathrm{im}$. We emphasise that the cubic regime appears for short immobilisations, whereas for the case without advection the MSD is close to normal, as shown by the grey line in figure 6(b). For $\tau_\mathrm{im}\ll \tau_\mathrm{m}$, i.e. for negative ordinates in figure H1, the condition $\tau_\star\ll\tau_\mathrm{im}$ simplifies to $\sqrt{\frac{D}{v^2\tau_\mathrm{m}}} = (2\mathrm{Pe})^{-1/2}\ll \sqrt{2/3} \frac{\tau_\mathrm{im}}{\tau_\mathrm{m}}$. This means that for each ratio $\tau_\mathrm{im}/\tau_\mathrm{m}$ there exists a lower bound for the Péclet number for the cubic regime to exist. Indeed, the black dotted line on the map in figure I1 for $t_3\gt0$ has the asymptotic scaling $\mathrm{Pe}^{-1/2}$ for $\tau_\mathrm{m}\gg \tau_\mathrm{im}$. The cubic regime appears for $t_3\gt0$ only, i.e. only in the top right part from the black dashed line in figure I1. In figure I1 the red colour denotes the maximum anomalous diffusion coefficient $\alpha_\mathrm{max}$. The condition $D_\mathrm{eff}\gt100D$ has the same asymptotic regime $\mathrm{Pe}^{-1/2}$ for $\tau_\mathrm{m}\gg \tau_\mathrm{im}$, as shown by the black dashed line in figure I1. For $\tau_\mathrm{m} \gg \tau_\mathrm{im}$ the dark red region with $\alpha_\mathrm{max}$ close to three is almost entirely contained within this region. This means that for $\tau_\mathrm{m}\gg \tau_\mathrm{im}$ the cubic regime always coexists with $D_\mathrm{eff}\gg D$. In contrast, we found that the cubic regime appears regardless of the ratio $D_\mathrm{eff}/D$ for long immobilisations.

In addition to the maximum value of the instantaneous anomalous diffusion exponent, we consider the minimum value $\alpha_\mathrm{min}$ in figure I1, in which values close to zero correspond to very slow intermediate growth, i.e. a plateau. This allows us to analyse how the plateau at $\tau_\mathrm{m}\ll t\ll \tau_\mathrm{im}$ is influenced by the presence of advection and the cubic regime. The coexistence of the cubic regime and the plateau can be seen in the MSD in figure 6(a). The top left blue corner in figure I1 corresponds to the case considered in [54] with long immobilisations and (almost) no advection. The blue subdiffusive area does not significantly change for growing Péclet numbers and it overlaps with the red superdiffusive region in the top right corner of the map in figure I1. In the lower left part of figure I1(a) white area dominates, in which diffusion is close to normal, with both $\alpha_\mathrm{min}$ and $\alpha_\mathrm{max}$ close to unity. This can be seen from the solid grey line in figure 6(b).

In section 4.3 we found an anomalous scaling of the MSD, and now we analyse for which values of the Péclet number and the characteristic time scales the cubic scaling of the MSD (35) emerges at intermediate times. The cubic term dominates for $\tau_v \ll t\ll \tau_\mathrm{m},\tau_\mathrm{im}$. This implies the time scale separation $\tau_v \ll \tau_\mathrm{m},\tau_\mathrm{im}$. For long immobilisations $\tau_\mathrm{im}\gg \tau_\mathrm{m}$ the parameter space for the cubic scaling to occur is therefore restricted to $\mathrm{Pe}\gg1$. This can be seen in figure J1 in the top right corner, where we show the decadic logarithm of the Péclet number as the abscissa and $\mathrm{log}_{10}(\tau_\mathrm{im}/\tau_\mathrm{m})$ as the ordinate. The red colour coding denotes the maximal anomalous diffusion coefficient, which is close to three in that top right region. For clarity, the contour line for $\alpha_\mathrm{max} = 2.5$ is shown. For short immobilisations, $\tau_\mathrm{im}\ll \tau_\mathrm{m}$, the cubic term in the asymptote (35) dominates for $\tau_v\ll t\ll \tau_\mathrm{im}$, which limits the parameter space for the cubic scaling to occur to $\frac{\tau_\mathrm{im}}{\tau_\mathrm{m}} \gg \mathrm{Pe}^{-1}$. This can be seen in the map in figure J1, where the relation $\frac{3}{2}\mathrm{Pe}^{-1}\ll \frac{\tau_\mathrm{im}}{\tau_\mathrm{m}}$ is shown as a dashed black line. Below this line no cubic scaling is possible, meaning that for a given Péclet number the fraction $\tau_\mathrm{m}/\tau_\mathrm{im}$ needs to be sufficiently small.

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After the cubic scaling an advection induced subdiffusion regime may emerge. The time-domain of the subdiffusion is given by $\tau_v\ll \tau_\mathrm{im}\ll t \ll \tau_\mathrm{m}$. Therefore, the subdiffusive regime appears only for short immobilisations, $\tau_\mathrm{im}\ll \tau_\mathrm{m}$. Furthermore, the condition $\tau_v\ll \tau_\mathrm{im}$ translates to $\mathrm{Pe}^{-1}\ll \tau_\mathrm{im}/\tau_\mathrm{m}$, shown as the dotted line in figure J1. This restricts the parameter space for the plateau. It is bound by $\tau_\mathrm{im}\ll\tau_\mathrm{m}$, corresponding to the horizontal boundary in figure J1. In the map in figure J1 we show $\alpha_\mathrm{min}$ for the total MSD as a blue colour map and observe a region with $\alpha_\mathrm{min}\lt1/2$ in the bottom right corner for a high Péclet number and short immobilisations. As a guide to the eye we show a contour line for $\alpha_\mathrm{min} = 1/2$. A corresponding MSD is shown in figure 9(b), where a plateau occurs for $\tau_\mathrm{im}\ll t \ll \tau_\mathrm{m}$. This is a new behaviour that does not occur in the case without advection, as is shown by the grey line in figure J1 for a small Péclet number.

We here compare the results obtained from the MIM (3) to a CTRW with advection and exponentially distributed sojourn times. The relation between MIM and CTRW has been discussed in detail, for example, in [13, 48, 70]. Let us define the sojourn time density $\psi(t) = \exp(-t/\tau)/\tau$ and the Gaussian displacement density $\lambda(x)$ with mean µ and variance σ². This corresponds to the model considered in [46]. In [71] the solution of the density function $p(x,t)$ in this CTRW framework is given as

$$\begin{align} p(x,t)= \sum_{j=1}^\infty \frac{(t/\tau)^j\exp\left(-\frac{t}{\tau}\right)}{j!} \frac{\exp\left(-\frac{x^2}{2j\sigma^2}\right)}{\sqrt{2\pi j\sigma^2}}, \end{align} \tag{ L.1 }$$

for µ = 0. We incorporate the non-zero mean µ and obtain the solution

$$\begin{align} p(x,t)= \sum_{j=1}^\infty \frac{(t/\tau)^j\exp\left(-\frac{t}{\tau}\right)}{j!} \frac{\exp\left(-\frac{(x-j\mu)^2}{2j\sigma^2}\right)}{\sqrt{2\pi j\sigma^2}}. \end{align} \tag{ L.2 }$$

Let us turn to the moments of $p(x,t)$ (L.2). The fist moment is given by

$$\begin{align} \langle x(t)\rangle &=\sum_{j=1}^\infty \frac{(t/\tau)^j\exp\left(-\frac{t}{\tau}\right)}{j!}j\mu\nonumber\\ & =\mu \frac{t}{\tau} \exp\left(-\frac{t}{\tau}\right) \sum_{j=0}^\infty \frac{(t/\tau)^{j}}{j!}\nonumber\\ & =\mu \frac{t}{\tau}. \end{align} \tag{ L.3 }$$

In the same way we obtain the second moment

$$\begin{align} \langle x^2(t)\rangle=\sigma^2\frac{t}{\tau}+\mu^2\frac{t^2}{\tau^2}+\mu^2\frac{t}{\tau}, \end{align} \tag{ L.4 }$$

resulting in the MSD

$$\begin{align} \langle [x(t)-\langle x(t)\rangle]^2\rangle=\frac{\sigma^2}{\tau}t+\frac{\mu^2}{\tau}t, \end{align} \tag{ L.5 }$$

which is linear at all times, in contrast the results obtained from our model.

In this section we compare our model to the two-state CTRW introduced in [51] in which two transition densities

$$\begin{align} f_i(x,t)=\frac{1}{\sqrt{4\pi D_i t}}\exp\left(-\frac{(x-v_it)^2}{4D_i t}\right)\psi_i(t),\mbox{}i=1,2 \end{align} \tag{ L.6 }$$

are present with average speeds v_i and diffusion constants D_i . In an alternating way jumps are drawn from $f_1(x,t)$ and $f_2(x,t)$. The long-time diffusion coefficient presented in [51] is given by

$$\begin{align} D_\mathrm{eff}&=\frac{D_1\tau_1+D_2\tau_2}{\tau_1+\tau_2} +\frac{1}{2(\tau_1+\tau_2)}\nonumber\\ &\quad\times\left[\sigma_1^2 \left(v_1-\frac{v_1\tau_1+v_2\tau_2}{\tau_1+\tau_2}\right) +\sigma_2^2 \left(v_2-\frac{v_1\tau_1+v_2\tau_2}{\tau_1+\tau_2}\right)\right] \end{align} \tag{ L.7 }$$

with $\tau_i = \int_0^\infty t \psi_i(t)\mathrm{d}t$ and $\sigma_i^2 = \int_0^\infty t^2\psi_i(t)\mathrm{d}t-\tau_i^2$, $i = 1,2$. The long-time effective diffusion coefficient (L.7) matches our result (15), if we formally choose $D_1 = D$, $v_1 = v$, $D_2 = 0$, $v_2 = 0$, $\psi_1(t) = \exp(-t/\tau_\mathrm{m})/\tau_\mathrm{m}$ and $\psi_2(t) = \exp(-t/\tau_\mathrm{im})/\tau_\mathrm{im}$.

In this section we compare our model to the CTRW analysed in [52], in which the waiting time distribution function is given by the weighted sum

$$\begin{align} \psi(t)=\frac{p}{\tau_\mathrm{D}}\exp\left(-\frac{t}{\tau_\mathrm{D}}\right) +\frac{(1-p)}{\tau_\mathrm{B}}\exp\left(-\frac{t}{\tau_\mathrm{B}}\right) \end{align} \tag{ L.8 }$$

of two exponentials, where $\tau_\mathrm{B}\leqslant \tau_\mathrm{D}$ and $p\in [0,1]$. The jump length distribution we consider is given by the Gaussian

$$\begin{align} \lambda(x)=\frac{1}{\sqrt{2\pi \sigma^2}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right), \end{align} \tag{ L.9 }$$

with mean µ and variance σ². Note that in [52] $\lambda(x)$ is restricted to the case µ = 0. Formally, we choose the parameters $\tau_\mathrm{B} = \tau_\mathrm{im}$, $\tau_\mathrm{D}\ll \tau_\mathrm{B},\tau_\mathrm{im}$, and $p = 1-\tau_\mathrm{D}/\tau_\mathrm{m}$. We stress that this means that $\tau_\mathrm{D}\neq \tau_\mathrm{m}$ and $\tau_\mathrm{D}$ does not have the meaning of the mean residence time in a mobile state. By choosing $\tau_\mathrm{D}\ll \tau_\mathrm{m}$ and $p = 1-\tau_\mathrm{D}/\tau_\mathrm{m}$ close to unity there is a high probability to draw from the exponential distribution with short mean duration. This effectively models the Brownian motion of a tracer in the mobile state. After each step, the tracer immobilises with probability $1-p$. If there was no waiting time drawn from the long exponential, on average, there are $t/\tau_\mathrm{D}$ steps at time t. This gives an effective immobilisation rate of $1/\tau_\mathrm{m}$, which is the same as in MIM. In figure L1 we compare the MSD of MIM for $\tau_\mathrm{m} = 1$, $\tau_\mathrm{im} = 100$, v = 0 or v = 10 and D = 1 with simulations of CTRW with $\tau_\mathrm{D} = 2\times 10^{-2}$, $\tau_\mathrm{B} = \tau_\mathrm{im}$, $\mu = v\tau_\mathrm{D}$ and $\sigma = \sqrt{2D\tau_\mathrm{D}}$. We find good agreement at all times for the advection-free case, while the CTRW yields higher values than MIM for $t\lt\tau_\mathrm{D}$ in the case with advection.

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The reason why the MSDs are identical in the advection-free case and not in the case with advection can be understood by considering the MSD in terms of step numbers n [66]

$$\begin{align} \mathrm{var}(x)=\mathbb{E}(n)\mathrm{var}(\Delta x_i)+(\mathbb{E}(\Delta x_i))^2\mathrm{var}(n), \end{align} \tag{ L.10 }$$

where $x = \sum_{i = 1}^n \Delta x_i$, as introduced in section 2.5. We emphasise that expression (L.10) is exact. From expression (L.10) we see that in the advection-free case $\mathbb{E}(\Delta x_i) = 0$ only the mean number of steps is relevant for the MSD and not the variance of step numbers. Indeed, in figure L2 we show the mean number of steps $\mathbb{E}(n)$ (calculated by setting D = 0), where the dots for CTRW and the line for MIM overlap at all times. Therefore, the MSDs in the advection-free case of CTRW and MIM in figure L1 are equivalent. In contrast, in the case of advection, the variance of step numbers $\mathrm{var}(n)$ couples to the MSD, as can be seen in expression (L.10). The variance of step numbers differs for $t\lt\tau_\mathrm{D}$. We explain this as follows. If a tracer is in the mobile state of MIM, it will have a fixed number of steps $n = t/\Delta t$ at time t. The stochasticity of step numbers arises solely from immobilisations. By setting D = 0 we obtain a cubic short-time growth of the variance from the short-time asymptote (27). In contrast, in the CTRW case, the number of steps is random even in the case of a simple CTRW with only one exponential waiting time distribution, as described in appendix L.1. It can be seen in the variance of CTRW (L.5) for σ = 0 and µ = 1, where the variance of step numbers grows linearly. This is due to the fact that after each step a random waiting time is drawn from the exponential distribution. Therefore, the variance of step numbers is higher for small times and hence the MSDs of MIM and CTRW do not overlap for $t\lt\tau_\mathrm{D}$, as shown in figure L1. We point out that the disagreement between the MSD of MIM and CTRW with advection lasts until $\approx 3\times 10^{-1}$, which is notably larger than the chosen $\tau_\mathrm{D} = 2\times 10^{-2}$.

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