Proof of The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians

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

Throughout, $N$ is read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background §numbers; it is positive and $N^{-1}$ exists and is positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and $\sum_{k=1}^{N}t=tN$ for $t\in\mathbb{R}$ by claim 3 of Properties of Finite Sums and The Canonical Map from the Natural Numbers to a Field. The block maps $\mathfrak{p}_{k}$, product maps, diagonal points, tensor powers and one-particle marginals are read with $q=d$ as fixed in N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles, and the same instantiation $q=d$ is used for Entropy on the Configuration Space: Tensorization, Subadditivity over the Particles, and the One-Particle Marginal and Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals, each stated for an arbitrary natural number $q$. Each $\mathfrak{p}_{k}$ is linear and Borel by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear; $V$, $\nabla V$ and the functions $\partial_{i}\partial_{i'}V$ ($i,i'\in[d]$) are Borel by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §regularity; and compositions of Borel maps are Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. The constant $\tfrac{\sigma^{2}}{2}$ is positive: $0<\sigma\sigma$ by claim 5 of Elementary Order Arithmetic in an Ordered Field, $0<2^{-1}$ by claim 8 there, and the product of these is positive by claim 5 there.

By The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space §pair read at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level), with the confining potential $V_{N}$ on $\mathbb{R}^{dN}$ and gradient map $\nabla V_{N}:x\mapsto DV_{N}(x)$: $\mathcal{D}_{N}$ is the set of $P\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{dN})$ for which $V_{N}$ is integrable with respect to $P$, $\mathcal{E}_{N}(P)=\tfrac{\sigma^{2}}{2}\mathrm{Ent}(P)+\int_{\mathbb{R}^{dN}}V_{N}\,dP$, $\mathcal{D}_{N,\Sigma}$ is the set of $P\in\mathcal{D}_{N}\cap\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{dN})$ with $\int_{\mathbb{R}^{dN}}\lVert\nabla V_{N}\rVert^{2}\,dP<\infty$, and $\Sigma_{N}(P)=\nabla V_{N}+\tfrac{\sigma^{2}}{2}\xi_{P}$ in $T_{P}\subseteq L^{2}(P;\mathbb{R}^{dN})$. The particle-level pair is given by the same formulas with $V$ and $d$. Every space $L^{2}(\rho;\mathbb{R}^{r})$ below is that of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields, a real vector space under the operations $[\eta]+[\eta']=[\eta+\eta']$ and $t[\eta]=[t\eta]$ on classes of representatives, by The Space of Square-Integrable Random Vectors §classes.

Step 1 (Linearity of product fields). Let $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $g,g'\in L^{2}(P^{[1]};\mathbb{R}^{d})$ and $t\in\mathbb{R}$. We show $(g+tg')^{\oplus}=g^{\oplus}+t\,g'^{\oplus}$ in $L^{2}(P;\mathbb{R}^{dN})$. Let $\tilde g,\tilde g'$ be Borel representatives of $g,g'$. Then $\tilde g+t\tilde g'$ is a Borel representative of $g+tg'$ by The Space of Square-Integrable Random Vectors §classes, and for every $x\in\mathbb{R}^{dN}$, by Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map and Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear,

Taking classes, Product Fields and the Projection onto One-Particle Tangent Fields §product-field (which allows any Borel representative) and The Space of Square-Integrable Random Vectors §classes give the claim.

Step 2 (The gradient of $V_{N}$ is a product map). For every $x\in\mathbb{R}^{dN}$, The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space §regularity and Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map (with $h=\nabla V$ and $p=d$) give

so the gradient map $\nabla V_{N}$ is the product map $(\nabla V)^{\oplus}$. Hence, by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §product-maps with $g=\nabla V$, for every $P\in\mathcal{P}(\mathbb{R}^{dN})$

Moreover, if $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ and $\int\lVert\nabla V\rVert^{2}\,dP^{[1]}<\infty$, then $\nabla V$ is a Borel representative of its class in $L^{2}(P^{[1]};\mathbb{R}^{d})$, and by Product Fields and the Projection onto One-Particle Tangent Fields §product-field the class of $\nabla V_{N}=(\nabla V)^{\oplus}$ in $L^{2}(P;\mathbb{R}^{dN})$ is the product field of the class $\nabla V$; both are written $(\nabla V)^{\oplus}$ below.

Step 3 (Integrals of $V_{N}$). By The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal, $P^{[1]}$ is the measure $A_{P}$ of Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average, whose final assertion gives: a Borel $f:\mathbb{R}^{d}\to\mathbb{R}$ is integrable with respect to $P^{[1]}$ exactly when each $f\circ\mathfrak{p}_{k}$ is integrable with respect to $P$, and then

Clause 2 of Linearity and Monotonicity of the Lebesgue Integral, extended to finite sums by induction on the number of summands through claim 1 of Properties of Finite Sums, shows that a finite sum of integrable functions is integrable with integral the sum of the integrals; we call this (L).

(3a) Let $P\in\mathcal{P}(\mathbb{R}^{dN})$ be such that $V$ is integrable with respect to $P^{[1]}$. Then each $V\circ\mathfrak{p}_{k}$ is integrable with respect to $P$ and (3.1) holds for $f=V$; since $V_{N}=\sum_{k=1}^{N}V\circ\mathfrak{p}_{k}$ pointwise, (L) shows that $V_{N}$ is integrable with respect to $P$ and, multiplying (3.1) by $N$,

(3b) Let $P\in\mathcal{P}(\mathbb{R}^{dN})$ be such that $V_{N}$ is integrable with respect to $P$. We show that $V$ is integrable with respect to $P^{[1]}$. 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 $y\in\mathbb{R}^{d}$. For $k\in[N]$ put $w_{k}=V\circ\mathfrak{p}_{k}-v_{0}$, a Borel function which is nonnegative by claim 3 of Elementary Arithmetic in an Ordered Field. Pointwise, $\sum_{k=1}^{N}w_{k}=V_{N}-Nv_{0}$ by claims 2 and 3 of Properties of Finite Sums, so claim 6 there gives $w_{k}\le V_{N}-Nv_{0}$ for every $k$. The constant $Nv_{0}$ is bounded and Borel, hence integrable with respect to $P$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, so $V_{N}-Nv_{0}$ is integrable by clause 2 of Linearity and Monotonicity of the Lebesgue Integral; being nonnegative, its integral in $[0,\infty]$ is finite. By the monotonicity in clause 1 of Linearity and Monotonicity of the Lebesgue Integral, $\int w_{k}\,dP\le\int(V_{N}-Nv_{0})\,dP<\infty$, so the nonnegative Borel function $w_{k}$ is integrable, and $V\circ\mathfrak{p}_{k}=w_{k}+v_{0}$ is integrable by clause 2 there. By the final assertion of Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average, $V$ is integrable with respect to $P^{[1]}$.

Step 4 (Tensor powers: the penalty). Let $\mu\in\mathcal{D}$, so $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ and $V$ is integrable with respect to $\mu$, and put $Q=\mu^{\otimes N}$. By Entropy on the Configuration Space: Tensorization, Subadditivity over the Particles, and the One-Particle Marginal §tensor, $Q\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{dN})$ and $\mathrm{Ent}(Q)=N\,\mathrm{Ent}(\mu)$. By Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §marginal-of-tensor, $Q^{[1]}=\mu$, so (3a) shows that $V_{N}$ is integrable with respect to $Q$ with $\int V_{N}\,dQ=N\int V\,d\mu$. Hence $Q\in\mathcal{D}_{N}$ and, by the field axioms,

Step 5 (Tensor powers: the score). Let moreover $\mu\in\mathcal{D}_{\Sigma}$, so that also $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and $\int\lVert\nabla V\rVert^{2}\,d\mu<\infty$. By Scores on the Configuration Space: the Score of a Tensor Power is the Product Field of the Score, and the Score of the One-Particle Marginal is the Projection of the Score §tensor, $Q\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{dN})$ and $\xi_{Q}=(\xi_{\mu})^{\oplus}$, the product field being formed for $Q$, whose one-particle marginal is $\mu$. By (2.1) with $P=Q$, $\int\lVert\nabla V_{N}\rVert^{2}\,dQ=N\int\lVert\nabla V\rVert^{2}\,d\mu$, a real number, hence finite. With Step 4, $Q\in\mathcal{D}_{N,\Sigma}$. By Step 2 the class of $\nabla V_{N}$ in $L^{2}(Q;\mathbb{R}^{dN})$ is $(\nabla V)^{\oplus}$, and by Step 1 (with $g=\nabla V$, $g'=\xi_{\mu}$, $t=\tfrac{\sigma^{2}}{2}$),

the last equality by The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space §pair, whose combination is formed in the linear subspace $T_{\mu}$ of $L^{2}(\mu;\mathbb{R}^{d})$ with the operations of the latter. Steps 4 and 5 prove The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §tensor.

Step 6 (One-particle marginals: the penalty). Let $P\in\mathcal{D}_{N}$. By Entropy on the Configuration Space: Tensorization, Subadditivity over the Particles, and the One-Particle Marginal §marginal, $P^{[1]}\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ and $N\,\mathrm{Ent}(P^{[1]})\le\mathrm{Ent}(P)$. By (3b), $V$ is integrable with respect to $P^{[1]}$, so $P^{[1]}\in\mathcal{D}$, and (3a) gives $\int V_{N}\,dP=N\int V\,dP^{[1]}$. Since $0\le\tfrac{\sigma^{2}}{2}$, claim 5 of Elementary Arithmetic in an Ordered Field gives $\tfrac{\sigma^{2}}{2}N\,\mathrm{Ent}(P^{[1]})\le\tfrac{\sigma^{2}}{2}\mathrm{Ent}(P)$, and adding the common real number $\int V_{N}\,dP=N\int V\,dP^{[1]}$ to both sides (claim 3 of Elementary Arithmetic in an Ordered Field, the difference being unchanged) yields

Step 7 (One-particle marginals: the score). Let moreover $P\in\mathcal{D}_{N,\Sigma}$, so $P\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{dN})$ and $\int\lVert\nabla V_{N}\rVert^{2}\,dP<\infty$. By Scores on the Configuration Space: the Score of a Tensor Power is the Product Field of the Score, and the Score of the One-Particle Marginal is the Projection of the Score §marginal, $P^{[1]}\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and $\xi_{P^{[1]}}=\Pi_{P}(\xi_{P})$. By Step 2, Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map and Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product, $\lVert\nabla V_{N}(x)\rVert^{2}=\sum_{k=1}^{N}\lVert\nabla V(\mathfrak{p}_{k}(x))\rVert^{2}$ for every $x\in\mathbb{R}^{dN}$, a sum of nonnegative reals, so claim 6 of Properties of Finite Sums gives $\lVert\nabla V\rVert^{2}\circ\mathfrak{p}_{k}\le\lVert\nabla V_{N}\rVert^{2}$ pointwise for every $k\in[N]$. These functions are nonnegative and Borel, so the monotonicity in clause 1 of Linearity and Monotonicity of the Lebesgue Integral gives $\int\lVert\nabla V\rVert^{2}\circ\mathfrak{p}_{k}\,dP\le\int\lVert\nabla V_{N}\rVert^{2}\,dP<\infty$; hence each $\lVert\nabla V\rVert^{2}\circ\mathfrak{p}_{k}$ is integrable with respect to $P$, and by the final assertion of Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average the Borel function $\lVert\nabla V\rVert^{2}$ is integrable with respect to $P^{[1]}$, that is, $\int\lVert\nabla V\rVert^{2}\,dP^{[1]}<\infty$. With Step 6, $P^{[1]}\in\mathcal{D}_{\Sigma}$, and the class $\nabla V$ lies in $T_{P^{[1]}}$ by The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space §pair. By Step 2, $\Sigma_{N}(P)=(\nabla V)^{\oplus}+\tfrac{\sigma^{2}}{2}\xi_{P}$. Since $\Pi_{P}$ is linear by The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §contraction, and $\Pi_{P}((\nabla V)^{\oplus})=\nabla V$ by The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §product with $g=\nabla V\in T_{P^{[1]}}$,

Steps 6 and 7 prove The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §marginal.

Step 8 (The Hessian of $V_{N}$). 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, with $q=d$; by its bijection clause, $b(k,i)=b(l,r)$ holds exactly when $k=l$ and $i=r$. The sets $\mathbb{R}^{d}$ and $\mathbb{R}^{dN}$ are open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Fix $x\in\mathbb{R}^{dN}$.

(8a) First partials. For $l\in[N]$ and $i'\in[d]$, the coordinate of index $b(l,i')$ of $DV_{N}(x)$ is $\partial_{b(l,i')}V_{N}(x)$ by Gradient of a Real-Valued Function on a Euclidean Open Set; by Step 2 and Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration it is the $i'$th coordinate of $DV(\mathfrak{p}_{l}(x))$, which is $\partial_{i'}V(\mathfrak{p}_{l}(x))$ by Gradient of a Real-Valued Function on a Euclidean Open Set. Thus $\partial_{b(l,i')}V_{N}=(\partial_{i'}V)\circ\mathfrak{p}_{l}$ as functions on $\mathbb{R}^{dN}$.

(8b) Second partials. For $j,m\in[dN]$ let $\pi_{m}(y)=y_{m}$ be the $m$th coordinate function of $\mathbb{R}^{dN}$, smooth by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. Moving the $j$th coordinate by a real $h\ne0$ changes $\pi_{m}$ by $h$ if $j=m$ and by $0$ otherwise, so every difference quotient of Partial Derivative on a Euclidean Open Set is $1$, respectively $0$, and by Uniqueness of the Partial Derivative on a Euclidean Open Set $\partial_{j}\pi_{m}(x)=1$ if $j=m$ and $\partial_{j}\pi_{m}(x)=0$ otherwise. By Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks the $r$th coordinate function of $\mathfrak{p}_{l}$ is $\pi_{b(l,r)}$, so $\mathfrak{p}_{l}$ is smooth by claim 1 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, hence of class $C^{1}$ by Smooth Map on a Euclidean Open Set. The function $G=\partial_{i'}V$ is of class $C^{1}$ on $\mathbb{R}^{d}$ by clauses 2 and 3 of C^k Maps on a Euclidean Open Set, $V$ being of class $C^{2}$ by Confining Potentials on Euclidean Space §confining. For $k\in[N]$ and $i\in[d]$, claim 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ and (8a) give

The $r$th summand is $\partial_{r}G(\mathfrak{p}_{l}(x))$ if $b(l,r)=b(k,i)$, that is if $l=k$ and $r=i$, and $0$ otherwise. By claim 7 of Properties of Finite Sums the sum is $\partial_{i}\partial_{i'}V(\mathfrak{p}_{k}(x))$ if $l=k$ and $0$ if $l\ne k$. By Hessian Matrix of a C^2 Function, in dimensions $dN$ and $d$,

(8c) Diagonal directions. Let $a\in\mathbb{R}^{d}$. By Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §diagonal and Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration, the coordinate of index $b(l,i')$ of $a^{\oplus}$ is $a_{i'}$. For $k\in[N]$ and $i\in[d]$, Matrix-Vector Product, Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §sums and (8.1) give

where the inner sums with $l\ne k$ have only zero summands and so vanish by claim 7 of Properties of Finite Sums, the same claim then reducing the outer sum to its summand $l=k$. Since every index in $[dN]$ is some $b(k,i)$ and points are tuples (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces), $\mathfrak{p}_{k}\bigl(D^{2}V_{N}(x)a^{\oplus}\bigr)=D^{2}V(\mathfrak{p}_{k}(x))\,a$ by Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks; also $\mathfrak{p}_{k}(a^{\oplus})=a$ by Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration. Hence Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product gives

Step 9 (Quadratic forms of entrywise integrals). Let $m,n\in\mathbb{N}$, $\rho\in\mathcal{P}(\mathbb{R}^{m})$, let $M$ assign to each $y\in\mathbb{R}^{m}$ a real $n\times n$ matrix $M(y)$ whose entries $y\mapsto M(y)_{jj'}$ are integrable with respect to $\rho$, let $H$ be the $n\times n$ matrix with entries $H_{jj'}=\int M_{jj'}\,d\rho$, and let $z\in\mathbb{R}^{n}$. For every real $n\times n$ matrix $A$, Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, Matrix-Vector Product and claim 3 of Properties of Finite Sums give $z\cdot(Az)=\sum_{j=1}^{n}\sum_{j'=1}^{n}z_{j}z_{j'}A_{jj'}$. Applying this to $A=M(y)$ and to $A=H$, (L) together with clause 2 of Linearity and Monotonicity of the Lebesgue Integral for the constant factors $z_{j}z_{j'}$ shows that $y\mapsto z\cdot(M(y)z)$ is integrable with respect to $\rho$ and

Step 10 (Diagonal translation Hessians). Let $P\in\mathcal{D}_{N}$ and $a\in\mathbb{R}^{d}$. By Step 6, $P^{[1]}\in\mathcal{D}$, so $H_{\mathcal{E}}(P^{[1]})$ is defined. The result The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty adopts The Intrinsic Calculus on the Wasserstein Space: Standing Notation, which takes its dimension from The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §data, so by N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level its clause The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §pair applies at the configuration level to the confining potential $V_{N}$ (The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space §confining): each entry of $D^{2}V_{N}$ is integrable with respect to $P$ and $H_{\mathcal{E}_{N}}(P)_{jj'}=\int(D^{2}V_{N})_{jj'}\,dP$. At the particle level the same clause gives that each entry of $D^{2}V$ is integrable with respect to $P^{[1]}$ and $H_{\mathcal{E}}(P^{[1]})_{ii'}=\int(D^{2}V)_{ii'}\,dP^{[1]}$. Put

By (9.1), $a^{\oplus}\cdot(H_{\mathcal{E}_{N}}(P)a^{\oplus})=\int\Phi\,dP$, and $\varphi$ is integrable with respect to $P^{[1]}$ with $a\cdot(H_{\mathcal{E}}(P^{[1]})a)=\int\varphi\,dP^{[1]}$; $\varphi$ is Borel, being by Step 9 a finite sum of real multiples of the Borel functions $\partial_{i}\partial_{i'}V$ (Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §regularity, Hessian Matrix of a C^2 Function). By the final assertion of Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average, each $\varphi\circ\mathfrak{p}_{k}$ is integrable with respect to $P$ and (3.1) holds for $f=\varphi$. By (8.2), $\Phi=\sum_{k=1}^{N}\varphi\circ\mathfrak{p}_{k}$ pointwise, so by (L) and (3.1) multiplied by $N$,

This is The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §hessian.