Theorems Examples of Nonconvex Interaction Energies with Multiple Local Minimisers and Concentration Attributions

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Statement

Let $n\in\mathbb N$, and let $\mathcal P_2(\mathbb R^n)$ denote the set of Borel probability measures on $\mathbb R^n$ with finite second moment, equipped with the quadratic Wasserstein distance $W_2$. For an even Borel function $W:\mathbb R^n\to(-\infty,\infty]$, define

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\begin{align*} \mathcal E_W[\rho]:=\frac{1}{2}\int_{\mathbb R^n}\int_{\mathbb R^n}W(x-y)\,d\rho(x)\,d\rho(y) \end{align*}

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whenever this integral is well-defined as an element of $(-\infty,\infty]$. For $a\in\mathbb R^n$, let $\tau_a:\mathbb R^n\to\mathbb R^n$ be the map $x\mapsto x+a$, and, for $\rho\in\mathcal P_2(\mathbb R^n)$, define

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\begin{align*} \mathcal O(\rho):=\{(\tau_a)_\#\rho:a\in\mathbb R^n\}. \end{align*}

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For $\sigma,\rho\in\mathcal P_2(\mathbb R^n)$, define

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\begin{align*} \operatorname{dist}_{W_2}(\sigma,\mathcal O(\rho)):=\inf_{a\in\mathbb R^n}W_2(\sigma,(\tau_a)_\#\rho). \end{align*}

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A measure $\rho\in\mathcal P_2(\mathbb R^n)$ with $\mathcal E_W[\rho]<\infty$ is called a strict $W_2$-local minimiser of $\mathcal E_W$ modulo translations if there exists $r>0$ such that, for every $\sigma\in\mathcal P_2(\mathbb R^n)$ for which $\mathcal E_W[\sigma]$ is well-defined and

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\begin{align*} 0<\operatorname{dist}_{W_2}(\sigma,\mathcal O(\rho))<r, \end{align*}

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one has

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\begin{align*} \mathcal E_W[\sigma]>\mathcal E_W[\rho]. \end{align*}

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The functional $\mathcal E_W$ is displacement convex on $\mathcal P_2(\mathbb R^n)$ if, for every constant-speed $W_2$-geodesic $(\rho_t)_{t\in[0,1]}$ such that $\mathcal E_W[\rho_t]$ is well-defined for every $t\in[0,1]$,

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\begin{align*} \mathcal E_W[\rho_t]\le (1-t)\mathcal E_W[\rho_0]+t\mathcal E_W[\rho_1] \end{align*}

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for every $t\in[0,1]$.

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Assume the following finite-cluster realisation input for one-dimensional interaction energies: for every $d_0,d_1\in(0,\infty)$ with $d_0\ne d_1$, there exists an even function $W\in C^\infty(\mathbb R;\mathbb R)$ with at most polynomial growth such that, for each $i\in\{0,1\}$, the probability measure

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\begin{align*} \alpha_i:=\frac{1}{2}\delta_0+\frac{1}{2}\delta_{d_i} \end{align*}

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is a strict $W_2$-local minimiser of $\mathcal E_W$ modulo translations, and such that $\mathcal E_W$ is not displacement convex on $\mathcal P_2(\mathbb R)$.

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Then the following phenomena occur.

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1. There exist an even function $W\in C(\mathbb R;\mathbb R)$ with at most polynomial growth and measures $\rho_0,\rho_1\in\mathcal P_2(\mathbb R)$ such that $\mathcal E_W$ is not displacement convex on $\mathcal P_2(\mathbb R)$, $\rho_1\notin\mathcal O(\rho_0)$, and both $\rho_0$ and $\rho_1$ are strict $W_2$-local minimisers of $\mathcal E_W$ modulo translations. 2. Let $0<s<n$, and define $W:\mathbb R^n\to[-\infty,0)$ by

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\begin{align*} W(x):=-|x|^{-s} \end{align*}

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for $x\in\mathbb R^n_0:=\mathbb R^n\setminus\{0\}$ and by $W(0):=-\infty$. Then $\mathcal E_W$ is not bounded below on the class of absolutely continuous compactly supported probability measures in $\mathcal P_2(\mathbb R^n)$. More precisely, there exists a sequence $(\rho_k)_{k=1}^{\infty}\subset\mathcal P_2(\mathbb R^n)$ such that each $\rho_k$ has a compactly supported density with respect to $\mathcal L^n$, $\rho_k\rightharpoonup\delta_0$ weakly as Borel probability measures, and

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\begin{align*} \mathcal E_W[\rho_k]\to-\infty. \end{align*}

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3. There exist an even function $W\in C^\infty(\mathbb R;\mathbb R)$ with at most polynomial growth and measures $\rho_0,\sigma_0\in\mathcal P_2(\mathbb R)$ such that $\mathcal E_W$ is not displacement convex on $\mathcal P_2(\mathbb R)$, $\sigma_0\notin\mathcal O(\rho_0)$, both $\rho_0$ and $\sigma_0$ are strict $W_2$-local minimisers of $\mathcal E_W$ modulo translations, and the constant curves $\rho:[0,\infty)\to\mathcal P_2(\mathbb R)$, $t\mapsto\rho_0$, and $\sigma:[0,\infty)\to\mathcal P_2(\mathbb R)$, $t\mapsto\sigma_0$, are energy-dissipating Wasserstein gradient-flow solutions for $\mathcal E_W$ in the following metric-slope sense: they are locally absolutely continuous curves in $(\mathcal P_2(\mathbb R),W_2)$ and, for every $0\le s\le t$,

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\begin{align*} \mathcal E_W[\mu_t]+\frac{1}{2}\int_s^t |\mu'|^2(r)\,d\mathcal L^1(r)+\frac{1}{2}\int_s^t |\partial\mathcal E_W|^2(\mu_r)\,d\mathcal L^1(r)\le \mathcal E_W[\mu_s], \end{align*}

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where $\mu$ is either $\rho$ or $\sigma$, $|\mu'|$ is the metric derivative, and $|\partial\mathcal E_W|$ is the descending metric slope

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\begin{align*} |\partial\mathcal E_W|(\nu):=\limsup_{\eta\to\nu}\frac{(\mathcal E_W[\nu]-\mathcal E_W[\eta])^+}{W_2(\nu,\eta)} \end{align*}

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with the limit superior taken over $\eta\in\mathcal P_2(\mathbb R)$, $\eta\ne\nu$, for which $\mathcal E_W[\eta]$ is well-defined. These two stationary solutions have different limits as $t\to\infty$.

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Thus the absence of displacement convexity is compatible, in these examples, with multiple strict local minimisers modulo translations, concentration through an attractive singularity, and non-uniqueness of stationary long-time limits.

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