Proof of The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space

Each result cited is universally quantified over the data in its own statement.

Throughout, $x$ ranges over $\mathbb{R}^{dN}$, and $1\le dN$ by N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level, so that Confining Potentials on Euclidean Space, the notation of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation and the results cited below on Euclidean open sets may be read with $dN$ in place of the dimension; $\mathbb{R}^{dN}$ and $\mathbb{R}^{d}$ are open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. The block index is $b(k,i)=(k-1)d+i$ of Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions, read with $q=d$ as in N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles. We write $\bar{N}=\iota_{\mathbb{R}}(N)$ for the image of $N$ under the canonical map of The Canonical Map from the Natural Numbers to a Field, and $2t=t+t$ for real $t$.

Step 0 (Two facts about $\bar{N}$). For every $c\in\mathbb{R}$, $\sum_{k=1}^{N}c=\sum_{k=1}^{N}(c\cdot1)=c\,\bar{N}$ by claim 3 of Properties of Finite Sums and The Canonical Map from the Natural Numbers to a Field; call this (F1). Moreover $1\in[N]$ by claim 4 of Properties of the Order on the Natural Numbers, and every summand $1$ of $\bar{N}=\sum_{k=1}^{N}1$ is nonnegative by claim 1 of Elementary Arithmetic in an Ordered Field, so claim 6 of Properties of Finite Sums gives $1\le\bar{N}$, and hence $0\le\bar{N}$; call this (F2).

Step 1 (Coordinate functions and block maps). For $l\in[dN]$ let $\pi_{l}:\mathbb{R}^{dN}\to\mathbb{R}$, $\pi_{l}(x)=x_{l}$, be the $l$th coordinate function; it is smooth on $\mathbb{R}^{dN}$ by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. For $j,l\in[dN]$ and $x\in\mathbb{R}^{dN}$, moving the $j$th coordinate of $x$ by a real $h\ne0$ changes $\pi_{l}$ by $h$ if $j=l$ and by $0$ if $j\ne l$, so every difference quotient in Partial Derivative on a Euclidean Open Set equals $1$ if $j=l$ and $0$ if $j\ne l$; the defining condition therefore holds with this value for every $\varepsilon$ (with $\delta=1$), and by Uniqueness of the Partial Derivative on a Euclidean Open Set we get $\partial_{j}\pi_{l}(x)=1$ if $j=l$ and $\partial_{j}\pi_{l}(x)=0$ if $j\ne l$. By Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks the $i$th coordinate function of $\mathfrak{p}_{k}$ is $\pi_{b(k,i)}$, for $k\in[N]$ and $i\in[d]$. Hence $\mathfrak{p}_{k}$ is smooth on $\mathbb{R}^{dN}$ by claim 1 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, and in particular of class $C^{1}$ and $C^{2}$ there by Smooth Map on a Euclidean Open Set.

Step 2 (A chain-rule formula). Let $G:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{d}$ and let $k\in[N]$. We show that $G\circ\mathfrak{p}_{k}$ is of class $C^{1}$ on $\mathbb{R}^{dN}$ and that, for all $x\in\mathbb{R}^{dN}$, $l\in[N]$ and $i\in[d]$,

The first assertion is claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ (with the order $1$ in place of its $k$, $F=\mathfrak{p}_{k}$, and the open set $\mathbb{R}^{d}$ and the map $G$ in the roles of its $V$ and $G$). By claim 1 of the same theorem and Step 1,

and the $r$th summand is $\partial_{r}G(\mathfrak{p}_{k}(x))$ if $b(k,r)=b(l,i)$ and $0$ otherwise. By the uniqueness in Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection, $b(k,r)=b(l,i)$ holds exactly when $k=l$ and $r=i$. If $l=k$, every summand with $r\ne i$ is therefore $0$, and claim 7 of Properties of Finite Sums gives the value $\partial_{i}G(\mathfrak{p}_{k}(x))$; if $l\ne k$, every summand is $0$, and the same claim (with the index $i$) gives $0$. This proves (2.1).

Step 3 (Class $C^{2}$ and derivatives of the sum). Since $V$ is of class $C^{2}$ on $\mathbb{R}^{d}$ by Confining Potentials on Euclidean Space §confining and $\mathfrak{p}_{k}$ is of class $C^{2}$ by Step 1, each $V\circ\mathfrak{p}_{k}$ is of class $C^{2}$ on $\mathbb{R}^{dN}$ by claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$. For $n\in[N]$ let $S_{n}:\mathbb{R}^{dN}\to\mathbb{R}$, $S_{n}(x)=\sum_{k=1}^{n}V(\mathfrak{p}_{k}(x))$; by claim 1 of Properties of Finite Sums, $S_{1}=V\circ\mathfrak{p}_{1}$ and $S_{n+1}=S_{n}+V\circ\mathfrak{p}_{n+1}$ whenever $n+1\in[N]$, and $S_{N}=V_{N}$. We show by induction on $n$ (Principle of Induction for the Natural Numbers, applied to the set of $n\in\mathbb{N}$ such that $n\notin[N]$ or the following assertion holds for $n$) that each $S_{n}$ is of class $C^{2}$ on $\mathbb{R}^{dN}$ and satisfies (3.1) below for every $j\in[dN]$ and $x\in\mathbb{R}^{dN}$. Base case $n=1$: $S_{1}=V\circ\mathfrak{p}_{1}$ is of class $C^{2}$, and both sums in (3.1) have the single summand $k=1$, so (3.1) holds by claim 1 of Properties of Finite Sums. Inductive step: suppose the assertion holds for $n$ and $n+1\in[N]$. Then $S_{n+1}=S_{n}+V\circ\mathfrak{p}_{n+1}$ is of class $C^{2}$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set; by claim 1 of that lemma $\partial_{j}S_{n+1}=\partial_{j}S_{n}+\partial_{j}(V\circ\mathfrak{p}_{n+1})$, and applying the same claim to these two functions, which are of class $C^{1}$ by clause 2 of C^k Maps on a Euclidean Open Set, $\partial_{j}\partial_{j}S_{n+1}=\partial_{j}\partial_{j}S_{n}+\partial_{j}\partial_{j}(V\circ\mathfrak{p}_{n+1})$. Inserting (3.1) for $n$ and using the recursion of claim 1 of Properties of Finite Sums for the families $(\partial_{j}(V\circ\mathfrak{p}_{k})(x))_{k\in[n+1]}$ and $(\partial_{j}\partial_{j}(V\circ\mathfrak{p}_{k})(x))_{k\in[n+1]}$ gives (3.1) for $n+1$. Thus, for every $n\in[N]$, $j\in[dN]$ and $x\in\mathbb{R}^{dN}$,

Taking $n=N$, $V_{N}$ is of class $C^{2}$ on $\mathbb{R}^{dN}$, so $DV_{N}(x)$ and $\Delta V_{N}(x)$ are defined.

Step 4 (The gradient). Fix $x$, $l\in[N]$ and $i\in[d]$. By (3.1), (2.1) with $G=V$ (of class $C^{1}$ by claim 2 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map) and claim 7 of Properties of Finite Sums (only the summand $k=l$ can be nonzero), $\partial_{b(l,i)}V_{N}(x)=\partial_{i}V(\mathfrak{p}_{l}(x))$. By Gradient of a Real-Valued Function on a Euclidean Open Set, the left side is the coordinate of index $b(l,i)$ of $DV_{N}(x)$ and the right side is the $i$th coordinate of $DV(\mathfrak{p}_{l}(x))$, which by Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration is the coordinate of index $b(l,i)$ of $[DV(\mathfrak{p}_{1}(x)),\dots,DV(\mathfrak{p}_{N}(x))]$. Every index in $[dN]$ is of the form $b(l,i)$ by Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection, so the two points of $\mathbb{R}^{dN}$ have the same coordinates and are equal, points being tuples by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces. This is the gradient formula of The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space §regularity.

Step 5 (The Laplacian). Fix $x$, $l\in[N]$, $i\in[d]$ and $k\in[N]$. By clause 2 of C^k Maps on a Euclidean Open Set (with the scalar convention of its clause 3), $\partial_{i}V$ is of class $C^{1}$ on $\mathbb{R}^{d}$. By (2.1) with $G=V$, the function $\partial_{b(l,i)}(V\circ\mathfrak{p}_{k})$ on $\mathbb{R}^{dN}$ equals $(\partial_{i}V)\circ\mathfrak{p}_{k}$ if $l=k$, and is the zero function if $l\ne k$. In the first case (2.1) with $G=\partial_{i}V$ gives $\partial_{b(k,i)}\partial_{b(k,i)}(V\circ\mathfrak{p}_{k})(x)=\partial_{i}\partial_{i}V(\mathfrak{p}_{k}(x))$, in the notation of clause 4 of C^k Maps on a Euclidean Open Set; in the second case every difference quotient of the zero function is $0$, so $\partial_{b(l,i)}\partial_{b(l,i)}(V\circ\mathfrak{p}_{k})(x)=0$ by Partial Derivative on a Euclidean Open Set and Uniqueness of the Partial Derivative on a Euclidean Open Set. By (3.1) and claim 7 of Properties of Finite Sums, $\partial_{b(l,i)}\partial_{b(l,i)}V_{N}(x)=\partial_{i}\partial_{i}V(\mathfrak{p}_{l}(x))$. Hence, by The Laplacian of a Twice Continuously Differentiable Function §laplacian in dimension $dN$, Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §sums, and The Laplacian of a Twice Continuously Differentiable Function §laplacian in dimension $d$,

With Steps 3 and 4 this proves The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space §regularity.

Step 6 (Convexity). $\mathbb{R}^{dN}$ is convex, as recorded in the preamble of Confining Potentials on Euclidean Space (read in dimension $dN$). Let $x,y\in\mathbb{R}^{dN}$, $t\in\mathbb{R}$ with $0\le t\le1$, and $k\in[N]$. Since $\mathfrak{p}_{k}$ is linear by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear, clauses 1 and 2 of Linear Map give $\mathfrak{p}_{k}(tx+(1-t)y)=t\,\mathfrak{p}_{k}(x)+(1-t)\,\mathfrak{p}_{k}(y)$, and convexity of $V$ on $\mathbb{R}^{d}$ (Confining Potentials on Euclidean Space §confining, Convex Real-Valued Function on a Convex Subset of $\mathbb{R}^n$) gives

Thus $V\circ\mathfrak{p}_{k}$ is convex on $\mathbb{R}^{dN}$ by Convex Real-Valued Function on a Convex Subset of $\mathbb{R}^n$. With the partial sums $S_{n}$ of Step 3, induction on $n$ using claim 2 of Affine Functions, Sums, Nonnegative Multiples and Pointwise Suprema of Convex Functions shows that every $S_{n}$ is convex on $\mathbb{R}^{dN}$; in particular $V_{N}=S_{N}$ is.

Step 7 (A lower bound for $V$ and the estimate (E)). By Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §growth there is $v_{0}\in\mathbb{R}$ with $v_{0}\le V(y)$ for every $y\in\mathbb{R}^{d}$. Put $m=|v_{0}|$, so $0\le m$ by claim 1 of Properties of the Absolute Value in an Ordered Field and $-m\le v_{0}$ by claim 3 there. Let $y\in\mathbb{R}^{d}$. Then $-m\le V(y)$, so $0\le V(y)+m$ by claim 3 of Elementary Arithmetic in an Ordered Field, whence $|V(y)+m|=V(y)+m$ by Absolute Value in an Ordered Field; also $|-m|=|m|=m$ by claim 2 of Properties of the Absolute Value in an Ordered Field and Absolute Value in an Ordered Field. The triangle inequality (claim 5 there) applied to $V(y)=(V(y)+m)+(-m)$ gives

Applying this with $y=\mathfrak{p}_{k}(x)$ and summing, claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, claim 2 of Properties of Finite Sums, (F1), and $V_{N}(x)\le|V_{N}(x)|$ (claim 3 of Properties of the Absolute Value in an Ordered Field) give

Step 8 (Superquadratic growth). Let $M\in\mathbb{R}$ be positive. By claim 1 of Elementary Order Arithmetic in an Ordered Field, $0+M<M+M$, so $0<2M$ by claim 2 there. By Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §minorant applied with $2M$ there is $C'\in\mathbb{R}$ with $2M\lVert y\rVert^{2}-C'\le V(y)$ for every $y\in\mathbb{R}^{d}$. The choices are made in this order: $M$, then $C'$, then $B=\bar{N}|C'|$ and $K=1+BM^{-1}$, where $M^{-1}$ exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field. By (F2), claim 1 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field, $0\le B$ and $0\le BM^{-1}$, hence $1\le K$ and $BM^{-1}\le K$ by claim 3 of Elementary Arithmetic in an Ordered Field (as $K-1=BM^{-1}$ and $K-BM^{-1}=1$ are nonnegative, the latter by claim 1 there), and $0<K$ by claims 6 and 2 of Elementary Order Arithmetic in an Ordered Field. For every $x$, claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, claims 2 and 3 of Properties of Finite Sums, (F1) and Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product give

the last step because $C'\le|C'|$ (claim 3 of Properties of the Absolute Value in an Ordered Field) and $0\le\bar{N}$ (claim 5 of Elementary Arithmetic in an Ordered Field). Now let $K\le\lVert x\rVert$. Since $0\le K$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $K^{2}\le\lVert x\rVert^{2}$; since $1\le K$ and $0\le K$, claim 5 of Elementary Arithmetic in an Ordered Field gives $K=K\cdot1\le K\cdot K$. Hence $BM^{-1}\le\lVert x\rVert^{2}$, and multiplying by $0\le M$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $B\le M\lVert x\rVert^{2}$. Therefore $V_{N}(x)\ge M\lVert x\rVert^{2}+(M\lVert x\rVert^{2}-B)\ge M\lVert x\rVert^{2}$ by claim 3 of Elementary Arithmetic in an Ordered Field. So $K$ is a positive real with $M\lVert x\rVert^{2}\le V_{N}(x)$ whenever $K\le\lVert x\rVert$, which is condition Confining Potentials on Euclidean Space §superquadratic for $V_{N}$.

Step 9 (Slope bound). By Confining Potentials on Euclidean Space §slope fix $C\in\mathbb{R}$ with $\lVert DV(y)\rVert\le C(1+|V(y)|)$ for every $y\in\mathbb{R}^{d}$, and put $A=\bar{N}+2\bar{N}m$ and $C_{N}=|C|A$. Fix $x$ and write $g_{k}=DV(\mathfrak{p}_{k}(x))$, $s_{k}=\lVert g_{k}\rVert$ and $S=\sum_{k=1}^{N}s_{k}$. By Step 4, $DV_{N}(x)=[g_{1},\dots,g_{N}]$, whose $k$th particle is $g_{k}$ by Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration, so Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product gives $\lVert DV_{N}(x)\rVert^{2}=\sum_{k=1}^{N}s_{k}s_{k}$. Each $s_{k}$ is nonnegative (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$), so $0\le S$ and $s_{k}\le S$ by claims 5 and 6 of Properties of Finite Sums, and $s_{k}s_{k}\le s_{k}S$ by claim 5 of Elementary Arithmetic in an Ordered Field; summing with claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers and claim 3 of Properties of Finite Sums gives $\lVert DV_{N}(x)\rVert^{2}\le S\cdot S$, hence $\lVert DV_{N}(x)\rVert\le S$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Next, $0\le1+|V(\mathfrak{p}_{k}(x))|$ by claims 1 and 2 of Elementary Arithmetic in an Ordered Field and claim 1 of Properties of the Absolute Value in an Ordered Field, and $C\le|C|$ by claim 3 there, so claim 5 of Elementary Arithmetic in an Ordered Field gives $s_{k}\le|C|\,(1+|V(\mathfrak{p}_{k}(x))|)$. Summing, and using claims 2 and 3 of Properties of Finite Sums, (F1), (E) and $0\le|C|$,

Since $1\le\bar{N}\le A$ by (F2) and $0\le2\bar{N}m$, and $0\le|V_{N}(x)|$, claim 5 of Elementary Arithmetic in an Ordered Field gives $|V_{N}(x)|\le A\,|V_{N}(x)|$, so $A+|V_{N}(x)|\le A(1+|V_{N}(x)|)$ and $\lVert DV_{N}(x)\rVert\le C_{N}(1+|V_{N}(x)|)$. This is condition Confining Potentials on Euclidean Space §slope for $V_{N}$, with the constant $C_{N}$.

Step 10 (Curvature). Let $\varepsilon\in\mathbb{R}$ be positive, choose $C_{\varepsilon}$ for $V$ by Confining Potentials on Euclidean Space §curvature, and then put $C'_{\varepsilon}=2\varepsilon\bar{N}m+\bar{N}C_{\varepsilon}$. For every $x$, Step 5, claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, claims 2 and 3 of Properties of Finite Sums, (F1), and (E) multiplied by $0\le\varepsilon$ (claim 5 of Elementary Arithmetic in an Ordered Field) give

This is condition Confining Potentials on Euclidean Space §curvature for $V_{N}$.

Step 11 (Conclusion). By Steps 3 and 6, $V_{N}$ is of class $C^{2}$ and convex on $\mathbb{R}^{dN}$, and by Steps 8, 9 and 10 it satisfies conditions (a), (b) and (c) of Confining Potentials on Euclidean Space §confining in dimension $dN$, its gradient and Laplacian being those computed in Steps 4 and 5. Hence $V_{N}$ is a confining potential on $\mathbb{R}^{dN}$, which is The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space §confining.