Boundary-layer analysis of repelling particles pushed to an impenetrable barrier

To describe the gap in [GvMPS16], we first recall the corresponding setting. Consider $n+1$ many particles ($n\geq 1$) confined to the half-line

$$\Omega=[0,\infty).$$

We label their positions as $\mathbf{x}:=(x_{0},x_{1},\ldots,x_{n})$ and assume that they are ordered, i.e. $\mathbf{x}\in\Omega_{n}$, where

$$\Omega_{n}:=\{\mathbf{x}\in\mathbb{R}^{n+1}:0=x_{0}<x_{1}<\ldots<x_{n}\}.$$

The discrete (i.e. $n<\infty$) particle interaction energy is given by

$$E_{n}:\Omega_{n}\to[0,\infty),\qquad E_{n}(\mathbf{x}):=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=0}^{i-1}\gamma_{n}V(\gamma_{n}(x_{i}-x_{j}))+\frac{1}{n}\sum_{i=0}^{n}U(x_{i}),$$

(1)

where $V$ is an interaction potential, $U$ is a confining potential and $\gamma_{n}>0$ is a parameter. The double sum accounts for each pair of two particles. Figure 1 illustrates typical examples for $V$ and $U$. The assumptions and properties of $V$ and $U$ are roughly as follows. $U\in C^{1}(\Omega)$ with $\min_{\Omega}U=0$ and $U(x)\to\infty$ as $x\to\infty$. $V\in L^{1}(\mathbb{R})$ is normalized to $\int_{\mathbb{R}}V=1$, nonnegative, even and singular at $0$ with the bound

$$V(x)\leq C\left\{\begin{array}[]{ll}|x|^{-a}&\text{if }a>0\\ -\log|x|&\text{if }a=0\end{array}\right.\qquad\text{for all }x\in(0,\tfrac{1}{2})$$

(2)

for some $C>0$, where $a\in[0,1)$ is a parameter which bounds the strength of the singularity. We also assume that $V|_{(0,\infty)}$ is non-increasing and convex. We state the precise list of assumptions and resulting properties in Section 3. With respect to [GvMPS16], we put more assumptions on $V$, but allow for a general confining potential $U$ instead of the specific choice $U(x)=x$. This generalization of $U$ does not result in further complications. Instead, it clarifies the dependence of $U$ on the boundary layer.

One particular choice of $V$ which we have in mind is

$$V_{\operatorname{wall}}(x)=x\coth x-\log|2\sinh x|.$$

(3)

For this potential, $E_{n}$ is a model for the pile-up of dislocation walls at a lock. We refer to [GvMPS16] for the discussion of this model in the literature and its physical relevance. We show in Section 3 that it satisfies all the assumptions that we put on $V$.

The asymptotic behaviour of the parameter $\gamma_{n}$ in (1) as $n\to\infty$ plays a decisive role for the limiting energy $E$ of $E_{n}$ as $n\to\infty$. This can be expected from (1) by noting that the scaled potential

$$V_{\gamma}(x)=\gamma V(\gamma x),$$

(4)

which has unit integral for all $\gamma>0$, is squeezed to a delta-peak at $0$ as $\gamma\to\infty$. Hence, as $\gamma$ increases, the particle interactions become more localized. In [GPPS13] the $\Gamma$-limit of $E_{n}$ is obtained as $n\to\infty$. Depending on the asymptotic behavior of $\gamma_{n}$, five different limiting energies are obtained: two of these belong to the critical scaling regimes $\gamma_{n}\to\gamma>0$ and $\gamma_{n}/n\to\Gamma>0$ as $n\to\infty$, and the other three belong to the three regimes separated by these two critical regimes (the outer two regimes require a rescaling of $E_{n}$ and $\mathbf{x}$; we refer to [GPPS13] for the details).

In this paper we focus on the regime in between the two critical ones, i.e.

$$\lim_{n\to\infty}\gamma_{n}=\infty\quad\text{and}\quad\lim_{n\to\infty}\frac{\gamma_{n}}{n}=0.$$

(5)

In this regime the $\Gamma$-limit of $E_{n}$ (see [GPPS13, Thm. 7]) is given by

$$E:\mathcal{P}(\Omega)\to[0,\infty],\qquad E(\mu)=\frac{1}{2}\|\mu\|_{L^{2}(\Omega)}^{2}+\int_{\Omega}U(x)\,d\mu(x),$$

(6)

where $\mathcal{P}(\Omega)$ is the space of probability measures on $\Omega$. The $L^{2}$-norm is extended to measures (see (27) for details) and may be infinite. It is well-known (see, e.g. [KS80, Thm. 2.1] with minor modications to account for $U\notin L^{2}(\Omega)$) that the minimization problem of $E$ over $\mathcal{P}(\Omega)$ has a unique minimizer $\mu_{*}\in L^{2}(\Omega)\cap\mathcal{P}(\Omega)$ and that its density $\rho_{\ast}$ is characterized by

$$\left\{\begin{aligned} \rho_{\ast}+U&\geq C_{U}\quad\text{on }\Omega\\ \rho_{\ast}+U&=C_{U}\quad\text{on }\operatorname{supp}\rho_{\ast},\\ \end{aligned}\right.$$

(7)

where the constant $C_{U}>0$ is such that $\int_{\Omega}\rho_{\ast}(x)\,dx=1$. Obviously,

$$\rho_{\ast}=[C_{U}-U]^{+}.$$

(8)

Figure 2 illustrates $\rho_{\ast}$.

As pointed out in [GvMPS16], the $\Gamma$-convergence of $E_{n}$ is not completely satisfactory, because it does not detect any particle patterns on mesoscopic scales. For the scaling regime in (5), the numerical computations of the minimizer $\mathbf{x}_{*}$ in [GvMPS16] indicate that $O(n/\gamma_{n})$ particles are not distributed according to $\rho_{*}$; see Figure 3. The main result [GvMPS16, Thm. 1.1] captures the continuum boundary-layer profile according to which these $O(n/\gamma_{n})$ particles are distributed. This profile is obtained by firstly proving a first-order $\Gamma$-convergence result for the continuous counterpart of $E_{n}$ (see (9) below) and by secondly minimizing the first-order $\Gamma$-limit. While Figure 3 suggests strongly that the obtained boundary-layer profile accurately describes the discrete boundary layer, any proof for this observation was left open. This is the gap in [GvMPS16] which we aim to fill in this paper.

In order to describe this paper’s first-order $\Gamma$-convergence result which will fill the gap in [GvMPS16], we first recall that of [GvMPS16]. By [GPPS13, Thm. 5] the $\Gamma$-limit of $E_{n}$ in the regime $\gamma_{n}\to\gamma>0$ is given by

$$E^{\gamma}:\mathcal{P}(\Omega)\to[0,\infty],\qquad E^{\gamma}(\mu):=\frac{1}{2}\int_{\Omega}\int_{\Omega}V_{\gamma}(x-y)\,d\mu(y)d\mu(x)+\int_{\Omega}U(x)\,d\mu(x),$$

(9)

where $V_{\gamma}$ is defined in (4). The energy $E^{\gamma}$ is the continuous counterpart of $E_{n}$ which is considered in [GvMPS16]. It $\Gamma$-converges to $E$ as $\gamma\to\infty$ (see [GvMPS16, Thm. 2.1]). In particular, this means that there exists a sequence $(\mu^{\gamma})_{\gamma}$ such that

$$E^{\gamma}(\mu^{\gamma})-E(\rho_{*})=o(1)\qquad\text{as }\gamma\to\infty.$$

The idea of the authors of [GvMPS16] to upgrade this to a first-order $\Gamma$-convergence result was to characterize the $o(1)$-term. They predicted from a priori computations that this term is $O(1/\gamma)$, and that it is easier to replace $E(\rho_{*})$ by $E^{\gamma}(\rho_{*})$. This motivated them to consider the functional

$$F^{\gamma}(\mu):=\gamma\big{(}E^{\gamma}(\mu)-E^{\gamma}(\rho_{*})\big{)}.$$

They call $F^{\gamma}$ the boundary-layer energy. The first-order $\Gamma$-convergence of $E^{\gamma}$ is simply the (zeroth-order) $\Gamma$-convergence of $F^{\gamma}$.

For the $\Gamma$-convergence of $F^{\gamma}$ the topology needed to be chosen carefully. From the formal asymptotics in [Hal11] and their own numerical simulations the authors guessed that the width of the boundary layer is $O(1/\gamma)$. This motivated them to use the following spatial rescaling. For a measure $\mu\in\mathcal{P}(\Omega)$, let

$$\tilde{\mu}:=\gamma_{\to}\mu:=\gamma\;(\gamma\operatorname{id})_{\#}\mu.$$

(10)

The inverse scaling is given by

$$\mu:=\gamma_{\leftarrow}\tilde{\mu}:=\frac{1}{\gamma}\Big{(}\frac{1}{\gamma}\operatorname{id}\Big{)}_{\#}\tilde{\mu}.$$

Note that if $\mu$ has a density $\rho$, then the density of $\tilde{\mu}$ satisfies

$$\tilde{\rho}(x)=\rho(x/\gamma)=:\gamma_{\to}\rho(x).$$

Using this scaling, the authors of [GvMPS16] employed the following change of variables:

$$\nu^{\gamma}:=\tilde{\mu}-\tilde{\rho}_{*},\qquad\mu=\gamma_{\leftarrow}\nu^{\gamma}+\rho_{\ast}.$$

By subtracting $\rho_{*}$ the bulk behaviour gets separated from the boundary layer.

For the signed Radon measures $\nu^{\gamma}$ with total variation that growths linearly with $\gamma$, the authors used the vague topology. This topology is defined as follows on the space $\mathcal{M}(\Omega)$ of signed Radon measures on $\Omega$. A sequence $(\nu_{\varepsilon})_{\varepsilon>0}\subset\mathcal{M}(\Omega)$ converges to $\nu\in\mathcal{M}(\Omega)$ vaguely (denoted by $\nu_{\varepsilon}\stackrel{{\scriptstyle v}}{{\rightharpoonup}}\nu$) as $\varepsilon\to 0$ if

$$\int_{\Omega}\varphi\,d\nu_{\varepsilon}\xrightarrow{\varepsilon\to 0}\int_{\Omega}\varphi\,d\nu\qquad\text{for all }\varphi\in C_{c}(\Omega).$$

The main result [GvMPS16, Thm. 1.1] states that $F^{\gamma}$ $\Gamma$-converges with respect to the vague topology to a certain limiting boundary-layer energy $F$. This functional $F$ is defined on

$$\mathcal{A}=\Big{\{}\nu\in\mathcal{M}(\Omega)\mid\nu^{-}(dx)\leq\rho_{\ast}(0)dx,\ \sup_{x\geq 0}\nu^{+}([x,x+1])<\infty\Big{\}},$$

(11)

where $\nu^{+},\nu^{-}\geq 0$ are respectively the positive and negative part of $\nu$ such that $\nu=\nu^{+}-\nu^{-}$. While $\nu\in\mathcal{A}$ may have infinite total variation, we have that $\nu^{-}\in L^{\infty}(\Omega)$ and that the local bound on $\nu^{+}$ is translation invariant. For $\nu\in\mathcal{A}\cap L^{2}(\Omega)$,

$$F(\nu):=\frac{1}{2}\int_{\Omega}\int_{\Omega}V(x-y)\nu(y)\nu(x)\,dydx-\rho_{\ast}(0)\int_{-\infty}^{0}(V*\nu)(x)\,dx.$$

(12)

It is not obvious to extend this definition to $\nu\in\mathcal{A}$, because $\nu$ need not be of finite total variation. We recall this extension briefly in Section 5.1. Finally, [GvMPS16] noted that $F$ has a unique minimizer $\nu_{*}$ (existence follows from the $\Gamma$-convergence result in [GvMPS16] and uniqueness follows from the convexity of $\mathcal{A}$ and the strict convexity of $F$), and that the continuous boundary-layer profile is given by

$$\tilde{\rho}_{*}^{\gamma}:=\nu_{*}+\tilde{\rho}_{*},\qquad\rho_{*}^{\gamma}:=\gamma_{\leftarrow}\nu_{*}+\rho_{*}.$$

(13)

Figure 3 and all other numerical simulations performed in [GvMPS16] suggest that $\rho_{*}^{\gamma}$ gives a very good prediction for both the bulk and the boundary layer in the minimizer $\mathbf{x}_{*}$.

However, the match between $\mathbf{x}_{*}$ and $\rho_{*}^{\gamma}$ has only been observed and has not been proven. Hence, there is no guarantee that such a match extrapolates to any other choices for the potentials $U$ and $V$ and for the parameter $\gamma_{n}$. This motivates our aim to establish a first-order $\Gamma$-convergence result for the discrete energy $E_{n}$ instead of its continuous counterpart $E^{\gamma_{n}}$.

To establish a first-order $\Gamma$-convergence result for $E_{n}$, we follow largely the same setup as the one just described. In fact, from Figure 3 we expect the same limiting boundary-layer energy $F$. Also, there is a close connection between $E_{n}$ and $E^{\gamma_{n}}$, which can be seen as follows. Given $\mathbf{x}\in\Omega_{n}$, consider the corresponding empirical measure

$$\mu_{n}:=\frac{1}{n}\sum_{i=0}^{n}\delta_{x_{i}}\in\frac{n+1}{n}\mathcal{P}(\Omega).$$

(14)

Then, we can express $E_{n}(\mathbf{x})$ in terms of $\mu_{n}$ as

$$E_{n}(\mu_{n}):=\frac{1}{2}\iint_{\Delta^{c}}V_{\gamma_{n}}(x-y)\,d\mu_{n}(y)d\mu_{n}(x)+\int_{\Omega}U(x)\,d\mu_{n}(x),$$

(15)

where the diagonal

$$\Delta=\{(x,x)^{T}:x\in\mathbb{R}\}\subset\mathbb{R}^{2}$$

is removed from the integration domain to avoid self-interactions. Apart from removing the diagonal, the expressions for $E_{n}$ and $E^{\gamma_{n}}$ are the same. Yet, the removal of the diagonal and the difference in the admissible sets on which $E_{n}$ and $E^{\gamma_{n}}$ are defined are crucial. Indeed, $(x,y)\mapsto V_{\gamma_{n}}(x-y)$ concentrates around the diagonal as $n\to\infty$ and thus careful analysis is required.

Following the procedure from [GvMPS16], we consider the blown-up energy difference $\gamma_{n}[E_{n}(\mu_{n})-E^{\gamma_{n}}(\rho_{*})]$ and employ the following change of variables. For $\mu_{n}$ as in (14), we set

$$\nu_{n}:=\tilde{\mu}_{n}-\tilde{\rho}_{*},\qquad\mu_{n}=(\gamma_{n})_{\leftarrow}\nu_{n}+\rho_{\ast}.$$

(16)

Then, the discrete boundary-layer energy $F_{n}$ is defined on the admissible set

$$\mathcal{A}_{n}:=\bigg{\{}\nu_{n}\in\mathcal{M}(\Omega)\mid\nu_{n}^{-}=\tilde{\rho}_{*},\ \exists\,\mathbf{y}\in\Omega_{n}:\nu_{n}^{+}=\frac{\gamma_{n}}{n}\sum_{i=0}^{n}\delta_{y_{i}}\bigg{\}}$$

(17)

and given by

$$F_{n}(\nu_{n}):=\gamma_{n}\big{[}E_{n}\big{(}(\gamma_{n})_{\leftarrow}\nu_{n}+\rho_{\ast}\big{)}+E^{\gamma_{n}}(\rho_{\ast})\big{]}.$$

(18)

Note that if $\nu_{n}$ is constructed from $\mu_{n}$ by (16), then

$$F_{n}(\nu_{n})=\gamma_{n}\left(E_{n}(\mu_{n})-E^{\gamma_{n}}(\rho_{\ast})\right).$$

(19)

The main result of this paper in the following $\Gamma$-convergence result of $F_{n}$:

More precisely, the assumption on $\gamma_{n}$ is equivalent to

$$\lim_{n\to\infty}\gamma_{n}=\infty\quad\text{and}\quad\left\{\begin{aligned} \lim_{n\to\infty}\gamma_{n}n^{-\tfrac{1-a}{2-a}}&=0&&\text{if }0<a<1\\ \lim_{n\to\infty}\gamma_{n}\sqrt{\frac{\log n}{n}}&=0&&\text{if }a=0.\end{aligned}\right.$$

The proof of Theorem 1.1 is given in Section 6 with preliminaries in Sections 4 and 5. It follows the proof in [GvMPS16] with major modifications to allow for the discreteness. Here, we briefly describe the main features of Theorem 1.1 and focus in particular on these major modifications.

First, we recall from [GvMPS16] that the expression for $F$ in (12) arises naturally when the right-hand side in (18) is explicitly expressed in terms of $\nu_{n}$. In Section 4 we redo this computation, which in our case deals with the discrete setting and with a general confining potential $U$.

The main difficulty with respect to [GvMPS16] is that the diagonal $\Delta$ is removed from the integration domain (see (15)) and that the domain of $F_{n}$ is discrete (i.e. the degrees of freedom are empirical measures). To deal with this, we use essentially the particular regularization $V^{\beta}$ of $V$ constructed in [KvM21], which approximates $V$ from below as $\beta\to 0$. Using this regularization, we add and subtract the contribution of the diagonal. By adding the diagonal, we can apply similar arguments as those in [GvMPS16] to establish the liminf inequality. However, $\beta$ needs to be chosen carefully. If $\beta$ is too small, then the contribution of the diagonal is too large and may not vanish in the limit. On the other hand, if $\beta$ is too large, then we cannot control the error made by the replacement of $V$ by $V^{\beta}$. Balancing out these two errors results in the asymptotic upper bound on $\gamma_{n}$ in Theorem 1.1. This bound is a stronger requirement than in (5), which is sufficient for the (zeroth-order) $\Gamma$-convergence of $E_{n}$ to $E$.

Establishing the limsup inequality is also significantly more challenging than in [GvMPS16]. The discreteness of $\mathcal{A}_{n}$ forces us to discretize $\nu$, which was not necessary in the continuous setting in [GvMPS16]. Since we blow up the energy difference by the factor $\gamma_{n}$, we need to show that the discretization error is asymptotically smaller than $1/\gamma_{n}$. This is much more intricate than for the zeroth-order $\Gamma$-limit of $E_{n}$ (see [GPPS13, Thm. 7]), where it was sufficient to show that the discretization error simply vanishes as $n\to\infty$.

In conclusion, Theorem 1.1 extends its continuous counterpart [GvMPS16, Thm. 1.1] (i.e. the $\Gamma$-convergence of $F^{\gamma}$ to $F$) in two manners. First, on a minor note, it allows for a general confining potential $U$. This highlights the fact that the dependence of $F$ on $U$ is restricted to the single value $\rho_{\ast}(0)\geq 0$, which depends nonlocally on $U$ (see (8) and Figure 2). Second, on a major note, Theorem 1.1 considers the discrete energy $F_{n}$. As a consequence of Theorem 1.1, any sequence of minimizers $\nu_{*,n}$ of $F_{n}$ converges to $\nu_{*}$. Since this convergence happens on the mesoscopic scale of the boundary layer, this proves that $\rho_{*}^{\gamma_{n}}$ (see (13)) indeed describes the boundary layer which appears in $\mathbf{x}_{*}$. This gives the first theoretical motivation for the observations in Figure 3 and any other numerical computation in [GvMPS16] that fits to the regime of $\gamma_{n}$ assumed in Theorem 1.1. This fills the main gap that was left open in [GvMPS16].

Yet, the story is not complete; [GvMPS16] contains a number of conjectures sparked by numerical simulations to which Theorem 1.1 does not provide an answer. Here, we focus on the main limitation of Theorem 1.1, which is the upper bound on $\gamma_{n}$. Indeed, the numerical simulations in [GvMPS16] suggest that $\rho_{*}^{\gamma_{n}}$ is the correct boundary-layer profile for the whole regime of $\gamma_{n}$ in (5). However, [GvMPS16, Table 1] suggests that the anticipated scaling of the energy difference, i.e. $E_{n}(\mathbf{x}_{*})-E^{\gamma_{n}}(\rho_{\ast})\sim 1/\gamma_{n}$, ceases to hold at the upper bound on $\gamma_{n}$ in Theorem 1.1. Hence, this upper bound is not simply an artefact of our proof. Looking deeper into the proof in Section 6, it seems that this upper bound is caused by the contribution to $F_{n}$ from a narrow region around the diagonal in the double integral in (15). A more precise treatment of this diagonal region could perhaps reveal a contribution of the right-hand side in (19) which diverges to $\infty$ as $n\to\infty$. Specifying this contribution, subtracting it from $F_{n}$ and proving $\Gamma$-convergence of the resulting energy functional (provided that his is possible) would reveal that $\rho_{*}^{\gamma_{n}}$ remains the correct boundary layer profile beyond the upper bound on $\gamma_{n}$ in Theorem 1.1. Pursuing this direction is beyond our scope.

In Section 2 we set the notation. In Section 3 we state the precise assumptions on the potentials $V$ and $U$, and derive further properties that follow from these assumptions. In Section 4 we rewrite $F_{n}$ defined in (18) explicitly in terms of $\nu_{n}$, which will clarify the connection with the expression for $F$ in (12). In Section 5 we build the functional setting on which our proof of Theorem 1.1 relies. We also provide several a priori estimates. Finally, Section 6 is devoted to the proof of Theorem 1.1.