Sedimentation of random suspensions and the effect of hyperuniformity

Abstract

This work is concerned with the mathematical analysis of the bulk rheology of random suspensions of rigid particles settling under gravity in viscous fluids. Each particle generates a fluid flow that in turn acts on other particles and hinders their settling. In an equilibrium perspective, for a given ensemble of particle positions, we analyze both the associated mean settling speed and the velocity fluctuations of individual particles. In the 1970s, Batchelor gave a proper definition of the mean settling speed, a 60-year-old open problem in physics, based on the appropriate renormalization of long-range particle contributions. In the 1980s, a celebrated formal calculation by Caflisch and Luke suggested that velocity fluctuations in dimension $d=3$ should diverge with the size of the sedimentation tank, contradicting both intuition and experimental observations. The role of long-range self-organization of suspended particles in form of hyperuniformity was later put forward to explain additional screening of this divergence in steady-state observations. In the present contribution, we develop the first rigorous theory that allows to justify all these formal calculations of the physics literature.

Appendix 1. Functional-analytic version of hyperuniformity

The present appendix is devoted to a more detailed discussion and motivation of the hyperuniformity assumptions (Hyp) and (Hyp$^+$) introduced in Sect. 2.1. Pioneered by Lebowitz [47, 54] in the physical literature for Coulomb systems, the notion of hyperuniformity for a point process ${\mathcal {P}}$ on ${\mathbb {R}}^d$ was first coined and theorized by Torquato and Stillinger [73] (see also [32, 71]) as the suppression of density fluctuations. More precisely, while for a Poisson point process one has $\mathrm {Var}\left[ \sharp ({\mathcal {P}}\cap {B_R})\right] \propto |B_R|$, the process ${\mathcal {P}}$ is said to be hyperuniform if instead

$$\begin{aligned} \lim _{R\uparrow \infty } \frac{\mathrm {Var}\left[ \sharp ({\mathcal {P}}\cap {B_R})\right] }{|B_R|}=0. \end{aligned}$$

(A.1)

Typically, this concerns processes for which number fluctuations are a boundary effect, that is, $\mathrm {Var}\left[ \sharp ({\mathcal {P}}\cap {B_R})\right] \lesssim |\partial B_R|$. Hyperuniformity can be interpreted as a hidden form of order on large scales and has been observed in various types of physical and biological systems, see e.g. [71, 73]. For Coulomb gases, rigorous results on the hyperuniformity of the Gibbs state have been recently obtained in [9, 53, 68]. The simplest examples of hyperuniform processes are given by perturbed lattices, e.g. ${\mathcal {P}}:=\{z+U_z:z\in {\mathbb {Z}}^d\}$ where the lattice points in ${\mathbb {Z}}^d$ are pertubed by iid random variables $\{U_z\}_{z\in {\mathbb {Z}}^d}$ (this model however only enjoys discrete stationarity due to the underlying lattice structure; see [65, 77] for refined properties).

Alternatively, hyperuniformity is known to be equivalent to the vanishing of the structure factor in the small-wavenumber limit, that is,

$$\begin{aligned}\lim _{k\rightarrow 0}S(k)=0,\end{aligned}$$

where the structure factor is defined as the Fourier transform $S(k):={{\widehat{h}}}_{2}(k)$ of the total pair correlation function $h_2$, cf. Definition 2.1. If the pair correlation function is integrable, this can equivalently be written as

$$\begin{aligned} S(0)=\int _{{\mathbb {R}}^d}h_2=0. \end{aligned}$$

(A.2)

The advantage of this reformulation in terms of the structure factor S(k) is that the latter can be directly observed in diffraction experiments. This property of vanishing structure factor is reminiscent of crystals, and indeed hyperuniform processes share crystalline properties on large scales, although they can be statistically isotropic like gases, thereby leading to a new state of matter [72].

In the spirit of (A.2), in our periodized setting, for a family $\{{\mathcal {P}}_L\}_{L\ge 1}$ of random point processes ${\mathcal {P}}_L$ on $Q_L$, we consider the following slightly relaxed definition of hyperuniformity, cf. (Hyp),

$$\begin{aligned} \sup _{L\ge 1}\,L^2\,\Big |\int _{Q_L}h_{2,L}\Big |\,<\, \infty , \end{aligned}$$

(A.3)

which is viewed as the approximate vanishing of the corresponding structure factors at 0 in the limit $L\uparrow \infty$. The precise rate $O(L^{-2})$ is chosen in view of Lemma A.2 below. As claimed, such a definition of hyperuniformity in terms of structure factors implies the suppression of density fluctuations in the following sense.

Lemma A.1

(Density fluctuations [32, 73]) Let a family $\{{\mathcal {P}}_L\}_{L\ge 1}$ of random point processes ${\mathcal {P}}_L$ on $Q_L$ be hyperuniform in the sense of (A.3). Then, for all $1\le R\le L$,

$$\begin{aligned}\mathrm {Var}\left[ \sharp ({\mathcal {P}}_L \cap {B_R})\right] \,\lesssim \, \rho _L^2|B_R|\,\Big (L^{-2}+\int _{Q_L}\big (1\wedge \tfrac{|x|_L}{R}\big )\,|g_{2,L}(x)|\,dx\Big ).\end{aligned}$$

In particular, provided that the pair correlation function $g_{2,L}$ has fast enough decay in the sense of $\sup _{L\ge 1}\int _{Q_L}|x|_L|g_{2,L}(x)|\,dx<\infty$, we deduce for all $1\le R\le L$,

$$\begin{aligned}\mathrm {Var}\left[ \sharp ({\mathcal {P}}_L \cap {B_R})\right] \,\lesssim \,\rho _L^2|\partial B_R|.\end{aligned}$$

Proof

Number fluctuations are computed as follows,

$$\begin{aligned} \mathrm {Var}\left[ \sharp ({\mathcal {P}}_L \cap {B_R})\right] \,=\,{\text {Var}}\bigg [{\sum _n\mathbb {1}_{B_R}(x_{n,L})}\bigg ]= & {} \rho _L^2\iint _{B_R\times B_R}h_{2,L}(x-y)\,dxdy\\= & {} \rho _L^2\int _{Q_L}|B_R(-x)\cap B_R|\,h_{2,L}(x)\,dx. \end{aligned}$$

Hence, decomposing

$$\begin{aligned}|B_R(-x)\cap B_R|=|B_R|-|B_R\setminus B_R(-x)|=|B_R|+\big (1\wedge \tfrac{|x|_L}{R}\big )\,O(|B_R|),\end{aligned}$$

where the last summand is a continuous function that vanishes at $x=0$, we deduce

$$\begin{aligned}&\bigg |\frac{1}{|B_R|}\mathrm {Var}\left[ \sharp ({\mathcal {P}}_L \cap {B_R})\right] -\rho _L^2\int _{Q_L}h_{2,L}\bigg | \\&\quad \,\lesssim \,\rho _L^2\int _{Q_L}\big (1\wedge \tfrac{|x|_L}{R}\big )\,|g_{2,L}(x)|\,dx, \end{aligned}$$

and the conclusion follows. $\square$

Recall that a Poisson point process ${\mathcal {P}}=\{x_n\}_n$ on ${\mathbb {R}}^d$ satisfies $\mathrm {Var}\left[ \sum _n\zeta (x_n)\right] \propto \int _{{\mathbb {R}}^d}|\zeta |^2$ for all $\zeta \in C^\infty _c({\mathbb {R}}^d)$, and that similarly any point process ${\mathcal {P}}$ with integrable pair correlation function satisfies

$$\begin{aligned} {\text {Var}}\bigg [{\sum _n\zeta (x_n)}\bigg ]\,=\,\rho ^2\iint _{{\mathbb {R}}^d\times {\mathbb {R}}^d}\zeta (x)\zeta (y)\,h_2(x-y)\,dxdy\,\lesssim \,\rho ^2\int _{{\mathbb {R}}^d}|\zeta |^2. \end{aligned}$$

(A.4)

The suppression of density fluctuations under hyperuniformity is naturally expected to lead to an improved version of such a variance estimate. Indeed, given an independent copy $\{x_{n}'\}_n$ of ${\mathcal {P}}=\{x_{n}\}_n$, we may represent

$$\begin{aligned}{\text {Var}}\bigg [\sum _n \zeta (x_{n})\bigg ]\,=\,{\mathbb {E}}{\mathbb {E}}'\bigg [\frac{1}{2}\Big (\sum _n \zeta (x_{n})-\sum _n \zeta (x_{n}')\Big )^2\bigg ],\end{aligned}$$

and the suppression of density fluctuations would formally allow to locally couple the random point sets $\{x_{n}'\}_n$ and $\{x_{n}\}_n$, only comparing points of the two realizations one to one locally, which would ideally translate into the gain of a derivative: for all $\zeta \in C^\infty _c({\mathbb {R}}^d)$,

$$\begin{aligned} {\text {Var}}\bigg [\sum _n \zeta (x_{n})\bigg ]\,\lesssim \,\rho ^2\int _{{\mathbb {R}}^d}|\nabla \zeta |^2. \end{aligned}$$

(A.5)

Indeed, provided that the pair correlation function has fast enough decay, it can be checked that hyperuniformity (A.1) is equivalent to this improved variance inequality (A.5). In our periodized setting, a rigorous statement is as follows.

Lemma A.2

(Functional characterization of hyperuniformity) Consider a family $\{{\mathcal {P}}_L\}_{L\ge 1}$ of random point processes ${\mathcal {P}}_L=\{x_{n,L}\}_n$ on $Q_L$ and assume that the pair correlation function $g_{2,L}$ has fast enough decay in the sense of

$$\begin{aligned}\sup _{L\ge 1}\int _{Q_L}|x|_L^2|g_{2,L}(x)|\,dx<\infty .\end{aligned}$$

Then $\{{\mathcal {P}}_L\}_{L\ge 1}$ is hyperuniform in the sense of (A.3) if and only if for all $L\ge 1$ and $\zeta \in C_{{\text {per}}}^\infty (Q_L)$ we have

$$\begin{aligned} {\text {Var}}\bigg [{\sum _n\zeta (x_{n,L})}\bigg ]\,\lesssim \,\rho _L^2\int _{Q_L}|\nabla \zeta |^2+L^{-2}\rho _L^2\int _{Q_L}|\zeta |^2. \end{aligned}$$

(A.6)

In particular, the latter implies for all $\zeta \in C_{{\text {per}}}^\infty (Q_L)$ with ${\mathbb {E}}\left[ \sum _n\zeta (x_{n,L}) \right] =\rho _L\int _{Q_L}\zeta =0$,

$$\begin{aligned}{\text {Var}}\bigg [{\sum _n\zeta (x_{n,L})}\bigg ]\,\lesssim \,\rho _L^2\int _{Q_L}|\nabla \zeta |^2.\end{aligned}$$

Proof

By the definition of the total pair correlation function $h_{2,L}$, cf. Definition 2.1, recall that

$$\begin{aligned}{\text {Var}}\bigg [{\sum _n\zeta (x_{n,L})}\bigg ]=\rho _L^2\iint _{Q_L\times Q_L}\zeta (x)\zeta (y)\,h_{2,L}(x-y)\,dxdy.\end{aligned}$$

Choosing $\zeta =1$, the variance inequality (A.6) yields

$$\begin{aligned}\Big |\int _{Q_L}h_{2,L}\Big |\lesssim L^{-2},\end{aligned}$$

that is, our definition (A.3) of hyperuniformity. It remains to prove the converse implication. Recomposing the square, the above identity for the variance takes the form

$$\begin{aligned} {\text {Var}}\bigg [{\sum _n\zeta (x_{n,L})}\bigg ]= & {} -\frac{1}{2}\rho _L^2\iint _{Q_L\times Q_L}|\zeta (x)-\zeta (y)|^2\,g_{2,L}(x-y)\,dxdy\\&+\Big (\rho _L^2\int _{Q_L}|\zeta |^2\Big )\Big (\int _{Q_L}h_{2,L}\Big ). \end{aligned}$$

Using the decay of correlations to estimate the first right-hand side term, and hyperuniformity (A.3) to estimate the last one, the variance inequality (A.6) follows.

While the above variance inequality is restricted to linear functionals $Y_L=\sum _n\zeta (x_{n,L})$ of the point process, the analysis of nonlinear multibody interactions requires a corresponding tool for general nonlinear functionals. For general functionals $Y=Y({\mathcal {P}})$ of a Poisson point process ${\mathcal {P}}$ with unit intensity on ${\mathbb {R}}^d$, the following variance inequality is known to hold [52, 76],

$$\begin{aligned}\mathrm {Var}\left[ Y({\mathcal {P}})\right] \le {\mathbb {E}}\bigg [\int _{{\mathbb {R}}^d}\big (\partial ^{{\text {add}}}_yY({\mathcal {P}})\big )^2dy\bigg ],\qquad \partial ^{{\text {add}}}_yY({\mathcal {P}}):=Y({\mathcal {P}}\cup \{y\})-Y({\mathcal {P}}),\end{aligned}$$

where the difference operator $\partial ^{{\text {add}}}$ is known as the add-one-point operator. More general versions of this type of functional inequality have been considered in the literature as a convenient quantification of nonlinear mixing in order to cover various classes of examples. In this spirit, our improved mixing assumption (Mix$^+$) in Sect. 2.1 is formulated in terms of the multiscale variance inequality (2.2) of [20, 21]. As shown in [21, Section 3], this covers most examples of interest in materials science [70], including for instance (periodized) hardcore Poisson processes and random parking processes. Applied to a linear functional $Y_L=\sum _n\zeta (x_{n,L})$, this variance inequality (2.2) clearly reduces to (A.4), so that (2.2) can indeed be viewed as a nonlinear version of (A.4).

In the hyperuniform setting, as number fluctuations are suppressed, the add-one-point operator in the above or the general oscillation in (Mix$^+$) could be intuitively replaced by a suitable “move-point” operator, only allowing to locally move points of the process, but not add or remove any. A general version of this idea is formalized as the improved hyperuniformity assumption (Hyp$^+$) in Sect. 2.1, cf. (2.3). Again, applied to a linear functional $Y_L=\sum _n\zeta (x_{n,L})$, this new variance inequality (2.3) clearly reduces to (A.6), so that (2.3) can be viewed as a nonlinear version. We believe that this new functional inequality is of independent interest.

Example A.3

(Perturbed lattices) We briefly show that assumption (Hyp$^+$) in terms of the hyperuniform multiscale variance inequality (2.3) is not empty. For that purpose, we consider the simplest example of a hyperuniform process on $Q_L$, that is, the perturbed lattice ${\mathcal {P}}_L:=\{z+U_z:z\in {\mathbb {Z}}^d\cap Q_L\}$, where the lattice points in ${\mathbb {Z}}^d \cap Q_L$ are perturbed by iid random variables $\{U_z\}_{z\in {\mathbb {Z}}^d\cap Q_L}$, say with values in the unit ball B. This model ${\mathcal {P}}_L$ is easily checked to satisfy the following stronger version of the variance inequality (2.3): for all $\sigma ({\mathcal {P}}_L)$-measurable random variables $Y({\mathcal {P}}_L)$,

$$\begin{aligned} \mathrm {Var}\left[ Y({\mathcal {P}}_L)\right] \,\le \,\frac{1}{2}{\mathbb {E}}\bigg [\int _{Q_L}\Big (\partial ^{{\text {mov}}}_{{\mathcal {P}}_L,B_{1+\sqrt{d}/2}(z)}Y({\mathcal {P}}_L)\Big )^2\,dz\bigg ], \end{aligned}$$

(A.7)

where we recall that the move-point derivative $\partial ^{{\text {mov}}}$ is defined in (Hyp$^+$). This is indeed a direct consequence of the Efron-Stein inequality [27] for the iid sequence $\{U_z\}_{z\in {\mathbb {Z}}^d\cap Q_L}$ in the following form: for ${\mathcal {P}}_{L,z}:=\{y+U_y\}_{y:y\ne z}\cup \{z+U'_z\}$ with $\{U'_z\}_z$ an iid copy of $\{U_z\}_z$,

$$\begin{aligned} \mathrm {Var}\left[ Y({\mathcal {P}}_L)\right] \,\le \,\frac{1}{2}{\mathbb {E}}\bigg [\sum _{z\in {\mathbb {Z}}^d\cap Q_L} \big (Y({\mathcal {P}}_L)-Y({\mathcal {P}}_{L,z})\big )^2\bigg ], \end{aligned}$$

while $Y({\mathcal {P}}_L)-Y({\mathcal {P}}_{L,z})$ can be bounded by $\partial ^{{\text {mov}}}_{{\mathcal {P}},B(z)}Y({\mathcal {P}})$, thus leading to (A.7).