In this section, we derive a macroscopic PDE-based model for the SPPs and the obstacles. The IBM model in (1) serves as the starting point. The derivation is a two-step process: First we formally derive a kinetic description for both the SPPs and the obstacles by taking a mean-field limit. In the second step, we use a hydrodynamic scaling for the SPPs and derive equations for the SPP density and orientation. For this step, we use previous work (Degond and Motsch 2008; Degond et al. 2015). For the obstacles, we focus on a particular parameter regime and assume to have low obstacle noise and strong obstacle spring stiffness. The main technical difficulty and new derivation strategy lie in this last step. Figure 4 summarises the different derivation steps. Throughout the document, the domains of integrations are understood to mean the whole domain, unless specified otherwise.
The Mean-Field Limit
We start by defining \(g( x, \alpha ,t)\), the distribution of the SPPs at position \(x \in {\mathbb {R}}^n\), time \(t \ge 0\) with direction of the self-propelled velocity \(\alpha \in {\mathbb {S}}^{n-1}\) and let f(x, y, t) be the distribution of obstacles with position \(x \in {\mathbb {R}}^n\), tethered at \(y \in {\mathbb {R}}^n\) at time \(t \ge 0\).
We consider the empirical distribution associated with the dynamics of the SPPs and tethered obstacles given by system (1).
$$\begin{aligned} g^M ( x, \alpha , t)&= \frac{ 1}{ M} \sum _{k=1}^M \delta _{Z_k(t)} ( x) \otimes \delta _{ \alpha _k(t)} ( \alpha ) \, , \nonumber \\ f^N ( x, y, t)&= \frac{ 1}{ N} \sum _{i=1}^N \delta _{X_i(t)} ( x) \otimes \delta _{ Y_i(t)} ( y), \end{aligned}$$
(3)
where \(\delta _{A}\) denotes the Dirac delta in \({\mathbb {R}}^n\) (for \(A=X_i, Y_i, Z_k\)) or in \({\mathbb {S}}^{n-1}\) (for \(A=\alpha _k\)) concentrated at A.
Lemma 1
(Kinetic Model) Formally, as \(N, M \rightarrow \infty \), \(f^N \rightarrow f\) and \(g^M \rightarrow g\), where the distributions f(x, y, t) and \(g(x,\alpha ,t)\) fulfil the following Kolmogorov–Fokker–Planck equations
$$\begin{aligned}&\partial _tf + \nabla _x \cdot \big ( {\mathcal {W}} f \big ) = d_o \varDelta _x f, \end{aligned}$$
(4a)
$$\begin{aligned}&\partial _tg + \nabla _x \cdot \big ( {\mathcal {U}} g \big ) + \nabla _\alpha \cdot \bigg ( P_{\alpha ^\perp } \left[ \nu {\bar{\alpha }}_g \right] g \bigg ) = d_s \varDelta _\alpha g, \end{aligned}$$
(4b)
where
$$\begin{aligned}&{\bar{\alpha }}_g(x,t) = \frac{ J_g ( x, t)}{| J_g ( x, t)|} \qquad \text { with} \qquad J_g ( x, t) = \int _{|x-z|\le r_A}\!\!\! \alpha \, g ( z, \alpha , t) \,\mathrm{d}z \,\mathrm{d}\alpha . \end{aligned}$$
(5)
For the (space and time dependent) velocities, we have
$$\begin{aligned} {\mathcal {W}}&= - \frac{\kappa }{\eta } ( x - y) - \frac{1}{\eta } \nabla _x {\bar{\rho }}_g(x,t),\nonumber \\ {\mathcal {U}}&= \alpha - \frac{ 1}{ \zeta } \nabla _x {\bar{\rho }}_f(x,t)- \frac{ 1}{\zeta } \nabla _x \breve{\rho }_g(x,t), \end{aligned}$$
(6)
where we have introduced the densities of obstacles and SPPs
$$\begin{aligned} \rho _g(x,t)=\int g ( x, \alpha , t) \,\mathrm{d}\alpha ,\qquad \rho _f(x,t)=\int f( x, y, t) \,\mathrm{d}y, \end{aligned}$$
(7)
as well as an abbreviation for densities convoluted with kernels
$$\begin{aligned} {{\bar{\rho }}}(x,t):=(\phi * \rho )(x,t), \qquad \breve{\rho }(x,t):=(\psi * \rho )(x,t). \end{aligned}$$
Further f fulfils
$$\begin{aligned} \int f(x,y,t) \,\mathrm{d}x = \rho _A(y), \end{aligned}$$
(8)
where \(\rho _A (y)\) is a given, time-independent function of obstacle anchor positions.
Proof
The limit is purely formal and uses standard techniques. We observe that \(f^N\) and \(g^M\) fulfil the equations for all N and M and then pass to the limit. \(\square \)
Remark 2
Note that since f and g are probabilities, they also fulfil
$$\begin{aligned} \int f(x,y,t) \,\mathrm{d}x \,\mathrm{d}y = \int g(x, \alpha , t) \,\mathrm{d}x \,\mathrm{d}\alpha \equiv 1, \end{aligned}$$
and consequently
$$\begin{aligned} \int \rho _A (y) \,\mathrm{d}y = 1. \end{aligned}$$
Interpretation At this point, we have a system of coupled kinetic equations for the obstacle distribution f(x, y, t) and the SPP distribution \(g(x,\alpha ,t)\). The interactions between the obstacles and the SPPs lead to the terms of the form \(\nabla _x{{\bar{\rho }}}\) in the speeds \({\mathcal {W}}\) and \({\mathcal {U}}\) in (6). An easy way to understand these terms is by assuming that the interaction force is of repulsive nature and purely local, in which case \(\nabla _x{{\bar{\rho }}}=\nabla _x \rho \). We then see that the interaction force moves obstacles and SPPs in the opposite direction of the gradient of each other. The convolution with \(\phi \) accounts for the potential non-locality of this interaction, which will be crucial later on. The remaining terms in \({\mathcal {W}}\) and \({\mathcal {U}}\) show the influence of the tethers and the self-propulsion for obstacles and the SPPs, respectively. In \({\mathcal {U}}\), we also see the influence of SPP repulsion. The term involving \(\nabla _\alpha \) in (4b) reflects the effect of SPP alignment. The terms on the right-hand side of (4) are results of the stochasticity in the obstacle position (for f) and in the SPP orientation (for g).
Scaling Assumptions
To derive the macroscopic equations for the SPP–obstacle interactions, we make a number of scaling assumptions for both the SPPs and the obstacles.
Scaling Assumptions for the SPPs Following previous work Degond and Motsch (2008), Degond et al. (2015), we introduce a small parameter \(\varepsilon \) and specify the relative order of the various terms. We mostly follow Degond et al. (2015), with a few small differences: Firstly we assume the effect of alignment to be purely local, i.e. \(r_A={\mathcal {O}}(\varepsilon )\), as has been done e.g. in Degond and Motsch (2008). Alternatively one could choose a weakly non-local scaling \(r_A={\mathcal {O}}(\sqrt{\varepsilon })\), which would lead to an additional viscous term in the SPP orientation equation (13b) below. As in Degond et al. (2015), we also assume the SPP self-repulsion to be purely local, i.e. \(r_R={\mathcal {O}}(\varepsilon )\) and assume that
$$\begin{aligned} \int \psi (x)\,\mathrm{d}x=:\mu <\infty . \end{aligned}$$
However, we do not make any smallness assumption with regard to the SPP–obstacle interaction scale \(r_I\). This is because we are interested in studying the effect of the non-locality of this interaction. Otherwise we proceed as in Degond et al. (2015), i.e. assuming the alignment frequency \(\nu \) and orientational diffusion \(d_s\) to be of order \(1/\varepsilon \), and their ratio to be of order one.
Scaling Assumptions for the Obstacles From (6), we see that it is only the macroscopic obstacle density \(\rho _f(x,t)\) that enters the SPP equation. Unfortunately, we cannot obtain a closed system for the macroscopic obstacle density \(\rho _f(x,t)\) of f by integrating (4a). Instead we make assumptions about the time scales of the obstacle dynamics. From now on, we also assume to have a constant anchor density, i.e. \(\rho _A \equiv 1\) is constant in space and time. We note that the results can be generalised to non-uniform \(\rho _A\). We introduce the following quantities
$$\begin{aligned} \gamma =\eta /\kappa ,\qquad \delta =d_o \gamma . \end{aligned}$$
For the derivation, we will assume both \(\gamma \) and \(\delta \) to be small. For \(\gamma \), this means that the obstacle spring relaxation time scale is small compared to the SPP domain crossing time. We will sometimes refer to this assumption as ‘stiff obstacles’, since it can be realised with a large spring constant \(\kappa \). For \(\delta \), smallness means that the obstacle spring relaxation time scale is small compared to the obstacle diffusion time scale, which we refer to as ‘low obstacle noise’. Next we rewrite (4a) as
$$\begin{aligned} \partial _tf + \nabla _x \cdot \left( {{\tilde{v}}}(x,t) f\right) =\frac{1}{\gamma } {\mathcal {A}}_y(f), \end{aligned}$$
(9)
where we have defined the ‘external’ velocity as
$$\begin{aligned} {{\tilde{v}}}(x,t)=-\frac{1}{\eta }\nabla _x{\bar{\rho }}_g(x,t) \end{aligned}$$
(10)
and the operator \({\mathcal {A}}_y\) by
$$\begin{aligned} {\mathcal {A}}_y(f):=\nabla _x \cdot \left[ (x-y)f + \delta \nabla _x f \right] . \end{aligned}$$
(11)
We can rewrite the operator as
$$\begin{aligned} {\mathcal {A}}_y(f)=\delta \nabla _x \cdot \left[ M_\delta (x-y)\nabla _x\left( \frac{f}{M_\delta (x-y)}\right) \right] , \end{aligned}$$
where \(M_{\delta }(z)\) is a Gaussian with variance \(\delta \) centred around 0, whose mass is normalised to one, i.e.
$$\begin{aligned} M_{\delta }(z)=\frac{1}{Z_\delta }e^{-\frac{|z|^2}{2\delta }}, \qquad Z_\delta =(2\pi \delta )^{n/2}. \end{aligned}$$
(12)
The above also shows that \(M_\delta (x-y)\) is in the kernel of \({\mathcal {A}}_y\).
Remark 3
Note that the rescaling of the diffusion term \(\delta =d_o\gamma \) ensures the operator \({\mathcal {A}}_y\) is a Fokker–Planck-type operator. Without it, we would obtain \({\mathcal {A}}_y(f)=\nabla _x \cdot \left[ (x-y)f\right] \), whose kernel contains Dirac deltas, making the analysis much more tedious. Eventually, however, we are interested in the small noise limit. This, of course raises several questions, which are beyond the scope of this work, e.g. does the order of the limits \(\gamma \rightarrow 0\) and \(\delta \rightarrow 0\) matter?
The Macroscopic SPP–Obstacle Equation
Using the scaling and notation above, we now state the main result of this section, which we prove in Sect. 3.4.
Theorem 1
(SPP–Obstacle Macromodel) Let \(\rho _A\equiv 1\) be constant and f(x, y, t) fulfill (9) with \(\gamma \ll 1\) and \(\delta \ll 1\). Further let \(g^\varepsilon (x,\alpha ,t)\) be the solution of (4b) using the scaling involving \(\varepsilon \) described above and let \(g^0(x,\alpha ,t)\) be its (formal) limit as \(\varepsilon \rightarrow 0\). Then it holds that
$$\begin{aligned} g^0(x,\alpha ,t)=\rho _g(x,t)N_{\Omega _g(x,t)}(\alpha ), \end{aligned}$$
where \(N_{\Omega }\) is the von Mises–Fisher distribution defined by
$$\begin{aligned} N_{\Omega }(\alpha )=\frac{1}{K_{d}}e^{\frac{\Omega \cdot \alpha }{d}},\qquad K_{d}=\int e^{\frac{\Omega \cdot \alpha }{d}} \,\mathrm{d}\alpha , \qquad d=\frac{d_s}{\nu }, \qquad \text {for}\quad \Omega \in {\mathbb {S}}^{n-1}. \end{aligned}$$
Note that \(K_d\) is a normalisation constant and is independent of \(\Omega \). Further the macroscopic SPP density \(\rho _g(x,t)\) and the macroscopic SPP orientation \(\Omega _g(x,t)\) fulfil
$$\begin{aligned}&\partial _t\rho _g + \nabla _x \cdot \left( U \rho _g \right) = 0, \end{aligned}$$
(13a)
$$\begin{aligned}&\rho _g \partial _t\Omega _g + \rho _g \left( V \cdot \nabla _x\right) \Omega _g + d P_{\Omega _g^\perp } \nabla _x \rho _g = 0, \end{aligned}$$
(13b)
$$\begin{aligned}&U=c_1\Omega _g-\frac{1}{\zeta }\nabla _x {\bar{\rho }}_f-\frac{\mu }{\zeta }\nabla _x \rho _g, \quad V=c_2\Omega _g-\frac{1}{\zeta }\nabla _x {\bar{\rho }}_f-\frac{\mu }{\zeta }\nabla _x \rho _g, \end{aligned}$$
(13c)
The constants \(c_1>0\) and \(c_2>0\) depend only on \(d=d_s/\nu \) and are defined as in Degond et al. (2015). The macroscopic obstacle density \(\rho _f(x,t)\) is given by
$$\begin{aligned} \rho _f(x,t)=1&- \frac{\gamma }{\delta \eta } \bigg [ {\bar{\rho }}_g(x)-\big [M_{2\delta }*{\bar{\rho }}_g\big ](x) \bigg ] \nonumber \\&-\frac{\gamma ^2}{\eta }\partial _t \varDelta _x {{\bar{\rho }}}_g +\frac{\gamma ^2}{\eta ^2}{\mathcal {N}}({{\bar{\rho }}}_g)+{\mathcal {O}}(\gamma ^2\delta )++{\mathcal {O}}(\gamma ^3), \end{aligned}$$
(14)
where the nonlinear term \({\mathcal {N}}\) is defined by
$$\begin{aligned} {\mathcal {N}}({{\bar{\rho }}}_g)=\frac{1}{2}\left[ (\varDelta _x \bar{\rho }_g)^2-{\mathbb {H}}({{\bar{\rho }}}_g):{\mathbb {H}}({{\bar{\rho }}}_g)\right] \, , \end{aligned}$$
where \({\mathbb {H}}({{\bar{\rho }}}_g)\) denotes the Hessian of the function \({{\bar{\rho }}}_g\), i.e. \(\{ {\mathbb {H}}({{\bar{\rho }}}_g)\}_{i,j} = \partial _i \partial _j {{\bar{\rho }}}_g\), and given two n by n matrices \({\mathbb {A}}\) and \({\mathbb {B}}\), their scalar product is defined as \({\mathbb {A}}:{\mathbb {B}} = \sum _{i,j=1}^n A_{i,j} B_{i,j}\).
Equations (13a) and (13b) give the evolution for the particle density \(\rho _g\) and mean orientation \(\Omega _g\), respectively. Without the term \(\nabla _x{{\bar{\rho }}}_f\) appearing in U and V in Eq. (13c), these equations correspond to the so-called Self-Organised Hydrodynamics with Repulsion (SOHR) and their derivation can be found in Degond et al. (2015). The additional terms in Eq. (13c) account for the influence of the obstacles density \(\rho _f\).
The equation for the obstacle density, expanded in the small variables \(\delta \) and \(\gamma \), is given in (14). It is important to note that the obstacle density given in (14) can in principle become negative, which is not physically meaningful. This is a consequence of the assumption that \(\gamma \) is small and indicates that the validity of the model will be limited to certain parameter regimes. We see that for infinitely strong springs, i.e. \(\gamma \rightarrow 0\), \(\rho _f(x,t)\equiv \rho _A\equiv 1\), i.e. obstacles remain exactly at their anchor points and since those are assumed to be uniformly distributed, the obstacles have no effect on the SPPs (\(\nabla _x{{\bar{\rho }}}_f\equiv 0\)). For small, but finite \(\gamma \) the feedback from the SPPs leads to non-uniform obstacles.
Influence of Obstacle Noise The influence of the obstacle noise \(\delta \) is contained in the order \(\gamma \) term in (14). We note that
$$\begin{aligned} - \frac{1}{\delta \eta } \bigg [ {\bar{\rho }}_g(x)-\left( M_{2\delta }*{\bar{\rho }}_g\right) (x) \bigg ]\rightarrow \frac{1}{\eta }\varDelta _x {\bar{\rho }}_g(x)\qquad \text {as}\quad \delta \rightarrow 0. \end{aligned}$$
We see that the noise adds an additional form of non-locality. Whether the obstacle density is reduced or increased depends on whether \({\bar{\rho }}_g\), the convoluted SPP density at x is smaller or larger than the ‘blurred’, convoluted SPP density \({\bar{\rho }}_g\), where the amount of blurring depends on the obstacle noise. In the absence of obstacle noise, (14) simplifies to
$$\begin{aligned} \rho _f(x,t)=1 +\frac{\gamma }{\eta }\varDelta _x {\bar{\rho }}_g(x)-\frac{\gamma ^2}{\eta }\partial _t \varDelta _x {{\bar{\rho }}}_g +\frac{\gamma ^2}{\eta ^2}{\mathcal {N}}({{\bar{\rho }}}_g)+{\mathcal {O}}\big (\gamma ^3\big ). \end{aligned}$$
(15)
SPP Dynamics Deform Obstacle Volume Elements In the absence of obstacle noise, we can rewrite (15) as
$$\begin{aligned} \rho _{f}(x)&=\det {J_Y}-\frac{\gamma ^2}{\eta }\partial _t \varDelta _x \bar{\rho }_g+{\mathcal {O}}\big (\gamma ^3\big ), \end{aligned}$$
(16)
where \(J_Y\) is the Jacobian of the map
$$\begin{aligned}&Y(x,t)=x+\frac{\gamma }{\eta }\nabla _x{\bar{\rho }}_g(x,t). \end{aligned}$$
The map Y can be interpreted as an estimate of the anchor position of an obstacle at position x moved under the influence of the SPP density. Then the determinant of the Jacobian reflects the deformation of a volume element of obstacles due to the SPPs. Note that for \(n=3\) \(\det {J_Y}\) contains also order \(\gamma ^3\) terms, for \(n=2\) only order \(\gamma ^2\) terms and lower.
Higher-Order Terms Account for SPP Movement Finally, we comment on the time derivative appearing in (14). The time derivative leads to a form of delay, i.e. the obstacles retain a memory of where SPPs were. This can be seen by Taylor expanding the SPP density in time using the time scale of obstacle relaxation \(\gamma \). Then the linear terms in (15) can be written as
$$\begin{aligned} \frac{\gamma }{\eta }\varDelta _x\left( {{\bar{\rho }}}_g-\gamma \partial _t{{\bar{\rho }}}_g\right) =\frac{\gamma }{\eta }\varDelta _x {{\bar{\rho }}}_g(x,t-\gamma )+{\mathcal {O}}\big (\gamma ^2\big ). \end{aligned}$$
Finally in preparation for the analytical and numerical investigation of Sects. 4 and 5, we state the following:
Corollary 1
(1D equations.) Let the assumptions of Theorem. 1 hold. Then for \(n=1\), the equations for the SPP density \(\rho _g(x,t)\) and the obstacle density \(\rho _f(x,t)\) with \(x\in {\mathbb {R}}\) and \(t\ge 0\) are given by
$$\begin{aligned}&\partial _t\rho _g+c_1\partial _x \rho _g=\frac{1}{\zeta }\partial _x\left( \mu \rho _g\partial _x\rho _g+\rho _g\partial _x {{\bar{\rho }}}_f\right) , \end{aligned}$$
(17)
where we have assumed all particles move to the right. The obstacle density up to order \(\gamma ^2\) is given by
$$\begin{aligned} \rho _f(x,t)=1&- \frac{\gamma }{\delta \eta } \bigg [ {\bar{\rho }}_g(x)-\big [M_{2\delta }*{\bar{\rho }}_g\big ](x) \bigg ]-\frac{\gamma ^2}{\eta }\partial _t \partial _x^2 {{\bar{\rho }}}_g. \end{aligned}$$
(18)
For \(\delta \rightarrow 0\) and using only terms up to order \(\gamma \), (18) simplifies to
$$\begin{aligned} \rho _f(x,t)=1 + \frac{\gamma }{\eta }\partial _x^2{{\bar{\rho }}}_g. \end{aligned}$$
(19)
Proof of Theorem 1
For the coarse-graining of the kinetic SPP equation (4b), we refer to previous work (Degond and Motsch 2008; Degond et al. 2015). We note that the obstacle density enters the SPP equation solely through its macroscopic density \(\rho _f(x,t)\) via the interaction operator \(\nabla _x{\bar{\rho }}_f\), which has a structure analogous to the SPP self-repulsion term, hence analogous techniques can be applied.
To derive an expression for the obstacle density \(\rho _f(x,t)\), we formulate and prove the following Theorem:
Theorem 2
Let \(\rho _A\equiv 1\) and f(x, y, t) fulfil (9) with \({\mathcal {A}}_y(f)\) defined in (11). Let \(\gamma \ll 1\) and expand f(x, y, t) as
$$\begin{aligned} f(x,y,t)=f_0(x,y,t)+\gamma f_1(x,y,t)+\gamma ^2 f_2(x,y,t)+{\mathcal {O}}(\gamma ^3). \end{aligned}$$
(20)
Then the macroscopic densities defined by
$$\begin{aligned} \rho _{f_i}(x,t)=\int f_i(x,y,t) \,\mathrm{d}y \end{aligned}$$
(21)
satisfy
$$\begin{aligned} \rho _{f_0}(x)&=1\nonumber \\ \rho _{f_1}(x)&=-\mathrm{div} ({{\tilde{v}}})-\delta \frac{1}{2}\varDelta _x \mathrm{div} ({{\tilde{v}}})+ {\mathcal {O}}(\delta ^2),\nonumber \\ \rho _{f_2}(x)&=\frac{1}{2}\nabla _x \cdot \left[ {{\tilde{v}}}\, \mathrm{div}{({{\tilde{v}}})}-({{\tilde{v}}}\cdot \nabla _x){{\tilde{v}}}\right] +\partial _t \mathrm{div}{({{\tilde{v}}})}+ {\mathcal {O}}(\delta ), \end{aligned}$$
(22)
as \(\delta \rightarrow 0\). We use the notation \(\text {div}= \nabla _x \cdot \) for the divergence of a vector field.
Proof
In the following, we drop the t-dependence of most terms to increase readability. We obtain the following equations for the three highest orders of \(\gamma \)
$$\begin{aligned}&{\mathcal {A}}_y(f_0)=0, \end{aligned}$$
(23a)
$$\begin{aligned}&{\mathcal {A}}_y(f_1)=\partial _tf_0+\nabla _x \cdot ({{\tilde{v}}}(x)f_0), \end{aligned}$$
(23b)
$$\begin{aligned}&{\mathcal {A}}_y(f_2)=\partial _tf_1 +\nabla _x \cdot ({{\tilde{v}}}(x)f_1). \end{aligned}$$
(23c)
Let us note that (23a), (23b), and (23c) can be recast as follows: Given a function h find \(\psi \) (in a suitable functional space) such that
$$\begin{aligned} {\mathcal {A}}_y( \psi ) = h \, . \end{aligned}$$
(24)
Due to the conservation of mass property of \({\mathcal {A}}_y\), i.e. \(\int {\mathcal {A}}_y \,\mathrm{d}x = 0\), a necessary condition to warranty the existence of a solution of (24) is \(\int h \,\mathrm{d}x = 0\). It can be shown that the operator \({\mathcal {A}}_y\) has compact resolvent on a suitable functional space and its kernel is generated by \(M_{\delta }(x-y)\), given in (12). The most important properties of the Gaussian \(M_\delta \), that we will use repeatedly are
$$\begin{aligned} \int M_{\delta }(z) \,\mathrm{d}z=1, \qquad \int z M_{\delta }(z) \,\mathrm{d}z=0, \quad \nabla _z M_{\delta }(z)=-\frac{z}{\delta }M_{\delta }(z). \end{aligned}$$
Hence, we can obtain a complete characterisation of the solutions of (24) via the Fredholm alternative, namely, for any function h such that \(\int h \,\mathrm{d}x = 0\) there exists a unique solution \(\psi \) up to an element of the kernel of \({\mathcal {A}}_y\). For a proof of this result, consult (Aceves-Sanchez et al. 2019).
Let us start by considering (23a), we search for a solution \(f_0\) such that \(\int f_0 ( x, y) \,\mathrm{d}x = 1\); hence, according to the results obtained for (24), the unique solution is given as
$$\begin{aligned} f_0(x,y)= M_{\delta }(x-y), \end{aligned}$$
(25)
where \(M_{\delta }\) is defined in (12). For the remaining two equations, we require the following scaling condition to hold, which ensures that the average mass is one,
$$\begin{aligned} \int f_i(x,y,t)\,\mathrm{d}x=0,\qquad i=1,2. \end{aligned}$$
(26)
Step 1: Rescaling Next we define the functions \(h_1(\sigma , y,t)\) and \(h_2(\sigma , y,t)\) as
$$\begin{aligned} f_1(x,y,t)&=\frac{1}{\sqrt{\delta }}\, M_\delta (x-y)\,h_1\left( \frac{x-y}{\sqrt{\delta }},y,t\right) ,\\ f_2(x,y,t)&=\frac{1}{\delta }\, M_\delta (x-y)\,h_2\left( \frac{x-y}{\sqrt{\delta }},y,t\right) . \end{aligned}$$
This turns (23b) and (23c) into equations for \(h_1(\sigma ,y,t)\) and \(h_2(\sigma ,y,t)\). Defining \({\mathcal {B}}\) as the operator
$$\begin{aligned}&{\mathcal {B}}(h)=\varDelta _\sigma h-\sigma \cdot \nabla _\sigma h, \end{aligned}$$
(27)
we obtain, after tedious but straightforward computations, the following relationships
$$\begin{aligned} {\mathcal {B}}(h_1)&=\sqrt{\delta }\,\text {div}({{\tilde{v}}})|_{y+\sqrt{\delta }\sigma }-\sigma \cdot {{\tilde{v}}}|_{y+\sqrt{\delta }\sigma },\nonumber \\ {\mathcal {B}}(h_2)&=\sqrt{\delta }\left( \partial _th_1 + h_1\,\text {div}({{\tilde{v}}})|_{y+\sqrt{\delta }\sigma }\right) +{{\tilde{v}}}|_{y+\sqrt{\delta }\sigma }\cdot (\nabla _\sigma h_1-\sigma \cdot h_1). \end{aligned}$$
(28)
There are several advantages to this scaling: Firstly, the operator \({\mathcal {B}}\) is the generator of the Ornstein–Uhlenbeck stochastic process (a consequence of using \(\sigma =(x-y)/\sqrt{\delta }\)) and we can use its well-known properties directly without having to scale by \(\delta \). Secondly, we have removed the Gaussian \(M_{\delta }\) from the equation (it cancelled). Finally, additionally scaling \(f_1\) and \(f_2\) by \(1/\sqrt{\delta }\) and \(1/\delta \), respectively, turns out to be the correct choice when calculating the densities.
Before we proceed to the next step, we need to collect a number of properties of \({\mathcal {B}}\), all of which are well known and stated in “Appendix A.2”.
Step 2: Expansion in terms of the obstacle noise \(\delta \). The next step involves expansion of the right-hand sides of (28), \(h_1\) and \(h_2\) with respect to \(\delta \), i.e.
$$\begin{aligned} h_1(\sigma ,y,t)&=h_1^0(\sigma ,y,t)+\sqrt{\delta }h_1^1(\sigma ,y,t)+\delta h_1^2(\sigma ,y,t)+{\mathcal {O}}(\delta ^{3/2}),\\ h_2(\sigma ,y,t)&=h_2^0(\sigma ,y,t)+\sqrt{\delta }h_2^1(\sigma ,y,t)+\delta h_2^2(\sigma ,y,t)+{\mathcal {O}}(\delta ^{3/2}). \end{aligned}$$
This yields as equations for \(h_1^0\), \(h_1^1\) and \(h_1^2\)
$$\begin{aligned} {\mathcal {B}}\big (h_1^0\big )&=-{{\tilde{v}}}_k \sigma _k,\\ {\mathcal {B}}\big (h_1^1\big )&=\partial _i {{\tilde{v}}}_i- \sigma _k \sigma _i \partial _k {{\tilde{v}}}_i, \\ {\mathcal {B}}\big (h_1^2\big )&=\sigma _k \partial _{ki}{{\tilde{v}}}_i - \frac{1}{2}\sigma _k \sigma _i \sigma _j \partial _{ij}{{\tilde{v}}}_k. \end{aligned}$$
Note that we have used the Einstein’s summation convention and that now \({{\tilde{v}}}\) and its derivatives are all evaluated at (y, t). Here partial derivatives are understood to act on the spatial variable, i.e. \(\partial _i {{\tilde{v}}}:= \frac{\partial }{\partial y_i} {{\tilde{v}}}(y,t)\). The advantage of this procedure is the following: Now the right-hand sides are low-order polynomials in \(\sigma \) and since \({\mathcal {B}}\) only acts on \(\sigma \), the equations can be solved explicitly by rewriting the right-hand sides in terms of the Hermite basis and using P2 of Lemma 4 in “Appendix A.2”. This procedure yields the explicit solutions
$$\begin{aligned} h_1^0(\sigma ,y,t)&={{\tilde{v}}}_i{\mathcal {H}}_{e_i},\nonumber \\ h_1^1(\sigma ,y,t)&=\frac{1}{2}\partial _k {{\tilde{v}}}_j {\mathcal {H}}_{e_k+e_j},\nonumber \\ h_1^2(\sigma ,y,t)&=\frac{1}{2}\left[ \partial _{ii}{{\tilde{v}}}_k\, {\mathcal {H}}_{e_k}+\frac{1}{3}\partial _{ij}{{\tilde{v}}}_k\, {\mathcal {H}}_{e_k+e_i+e_j}\right] , \end{aligned}$$
(29)
where \({\mathcal {H}}\) are the tensor Hermite polynomials defined in Lemma 4 in “Appendix A.2”. Note that \({{\tilde{v}}}\) and all its derivatives are evaluated at (y, t) and \({\mathcal {H}}\) at \(\sigma \).
As equations for \(h_2^0\) and \(h_2^1\), we obtain
$$\begin{aligned} {\mathcal {B}}(h_2^0)&={{\tilde{v}}}\cdot \left( \nabla _\sigma h_1^0-\sigma h_1^0\right) ,\\ {\mathcal {B}}(h_2^1)&=\partial _th_1^0+\partial _i {{\tilde{v}}}_i\,h_1^0 +(\sigma _k\partial _k{{\tilde{v}}}) \cdot \left( \nabla _\sigma h_1^0-\sigma h_1^0\right) + {{\tilde{v}}}\cdot \left( \nabla _\sigma h_1^1-\sigma h_1^1\right) . \end{aligned}$$
As above \({{\tilde{v}}}\) and its derivatives are all evaluated at (y, t). Using the solutions for \(h_1^0\), \(h_1^1\) and \(h_1^2\) given in (29), we can solve the equations for \(h_2^0\) and \(h_2^1\) in the same fashion, yielding the explicit expressions
$$\begin{aligned} h_2^0(\sigma ,y,t)&=\frac{1}{2}{{\tilde{v}}}_k{{\tilde{v}}}_j {\mathcal {H}}_{e_k+e_j},\nonumber \\ h_2^1(\sigma ,y,t)&=\left( -\partial _t{{\tilde{v}}}_k + {{\tilde{v}}}_i \partial _i {{\tilde{v}}}_k\right) \, {\mathcal {H}}_{e_k}+\frac{1}{2}{{\tilde{v}}}_i\partial _k{{\tilde{v}}}_j\, {\mathcal {H}}_{e_k+e_i+e_j}. \end{aligned}$$
(30)
Note that the solutions fulfil the scaling condition (26) since it holds that
$$\begin{aligned} \int M_1(\sigma )h_i^j(\sigma ,y,t)\,\mathrm{d}\sigma =0,\quad i=1,2,\quad j=0,1,2. \end{aligned}$$
(31)
Step 3: Calculating the macroscopic moments of the obstacle density With the preparation of the two steps above, the calculation of the obstacle densities
$$\begin{aligned} \rho _{f_i}(x,t)=\int f_i(x,y,t) \,\mathrm{d}y, \end{aligned}$$
and consequently its contribution to the SPP equation becomes relatively simple. The procedure and calculations are described in “Appendix A.3”. This yields (22) as claimed. \(\square \)
Explicit Solution for \(f_1\) The above outlined procedure works for any given external velocity \({{\tilde{v}}}(x,t)\), i.e. it allows to include other influences as well. For example, in future work we plan to use the derivation strategy to include the description of a fluid in which the obstacles and SPPs are immersed in. However, for this model, we can use the fact that \({{\tilde{v}}}(x,t)\) is in fact a conservative vector field. This allows to solve the first-order equation (23b) for \(f_1(x,y,t)\) directly. This is covered in the following lemma, where the t dependence has been suppressed for notational convenience.
Lemma 2
Let \({{\tilde{v}}}(x)\) be a conservative vector field, i.e. there exists a scalar function V(x), such that \(\nabla _x V={{\tilde{v}}}\), then we can write the solution to (23b) as
$$\begin{aligned} f_1(x,y)=M_{\delta }(x-y)\frac{1}{\delta }\left[ V(x)-\left( M_{\delta }*V\right) (y)\right] \end{aligned}$$
Proof
By direct calculation, we see that
$$\begin{aligned}&(x-y)f_1+\delta \nabla _x f_1 = M_\delta (x-y){{\tilde{v}}}(x), \end{aligned}$$
which shows that \(f_1\) is indeed a solution to (23b). Finally, we have to verify the normalisation condition (26)
$$\begin{aligned} \int f_1(x,y) \,\mathrm{d}x&=\frac{1}{\delta }\int M_{\delta }(x-y)\left[ V(x)-\left( M_{\delta }*V\right) (y)\right] \,\mathrm{d}x\\&=\frac{1}{\delta }\left[ \int M_{\delta }(x-y)V(x)\,\mathrm{d}x-\left( M_{\delta }*V\right) (y)\right] =0, \end{aligned}$$
which finishes the proof. \(\square \)
The above Lemma is applicable for this model of SPP–obstacle interactions since we have that
$$\begin{aligned} {{\tilde{v}}}(x,t)=-\frac{1}{\eta }\nabla _x {\bar{\rho }}_g(x,t), \end{aligned}$$
i.e. we can use Lemma 2 with \(V(x,t)=-\frac{1}{\eta } {\bar{\rho }}_g(x,t)\). We consequently find
$$\begin{aligned} f_1(x,y,t)&=- M_{\delta }(x-y)\frac{1}{\delta \eta }\left[ {\bar{\rho }}_g(x,t)-(M_{\delta }*{\bar{\rho }}_g)(y,t)\right] . \end{aligned}$$
From this, we can calculate
$$\begin{aligned} \rho _{f_1}(x,t)&=- \frac{1}{\delta \eta } \bigg [ {\bar{\rho }}_g(x)-\left( M_{2\delta }*{\bar{\rho }}_g\right) (x) \bigg ]. \end{aligned}$$
(32)
Remark 4
Note that since
$$\begin{aligned} - \frac{1}{\delta \eta } \bigg [ {\bar{\rho }}_g(x)-\left( M_{2\delta }*{\bar{\rho }}_g\right) (x) \bigg ]&=\frac{1}{\eta }\left[ \varDelta _x {\bar{\rho }}_g(x)+\frac{\delta }{2}\varDelta _x^2 {\bar{\rho }}_g\right] +{\mathcal {O}}(\delta ^2), \end{aligned}$$
we see that this is consistent with (22), but contains more information about the \({\mathcal {O}}(\delta ^2)\) term.
The Macroscopic Obstacle Density Collecting the results of Theorem 2 and Lemma 2 and using the definition of \({{\tilde{v}}}\) given in (10), we find that the maximum order of approximation of the obstacle density we can now write is given in (14) as claimed. This finishes the proof of Theorem 1. \(\square \)