2.1 Newton/Liouville equations

Consider the function $H_N$ from equation (1.7) with $W\in C^\infty (\mathbb {R}^d)$ satisfying $W(-x) = W(x)$ . We recall from the introduction the rotation matrix $\mathbb {J}(x,v) = (-v,x)$ and the block-diagonal matrix $\mathbb {J}_N$ with diagonal entries $\mathbb {J}$ . The standard symplectic structure on $({\mathbb {R}}^{2d})^N$ is given by the form

(2.1)

where $\cdot $ denotes the Euclidean inner product on $({\mathbb {R}}^{2d})^N$ . We recall that the Hamiltonian vector field $X_{H_N}$ associated to $H_N$ is uniquely defined by the formula

(2.2) $$ \begin{align} \mathsf{d} H_N[\underline{z}_N](\delta\underline{z}_N) = \omega_N(X_{H_N}(\underline{z}_N), \delta\underline{z}_N), \qquad \forall \underline{z}_N,\delta\underline{z}_N\in ({\mathbb{R}}^{2d})^N. \end{align} $$

Set $\nabla _{z_j} = (\nabla _{x_j},\nabla _{v_j})$ , where $\nabla _{x_j} = ({\partial }_{x_j^1},\ldots ,{\partial }_{x_j^d})$ and $\nabla _{v_j}=({\partial }_{v_j^1},\ldots ,{\partial }_{v_j^d})$ . Writing $\nabla _{\underline {z}_N} = (\nabla _{z_1},\ldots ,\nabla _{z_N})$ , we compute from the property ${\mathbb {J}}^2=-\mathbb {I}$ together with the definition of the gradient that

(2.3) $$ \begin{align} \mathsf{d} H_N[\underline{z}_N](\delta\underline{z}_N) = \nabla_{\underline{z}_N}H_N(\underline{z}_N) \cdot \delta\underline{z}_N = - \mathbb{J}_N^2\nabla_{\underline{z}_N}H_N(\underline{z}_N) \cdot \delta\underline{z}_N = -\mathbb{J}_N\Big(\mathbb{J}_N\nabla_{\underline{z}_N}H_N(\underline{z}_N)\Big)\cdot\delta\underline{z}_N, \end{align} $$

which implies that $X_{H_N}(\underline {z}_N) = \mathbb {J}_N\nabla _{\underline {z}_N}H_N(\underline {z}_N)$ . Thus, the functional $H_N$ and the symplectic form $\omega _N$ together define the Hamiltonian equation of motion

(2.4) $$ \begin{align} \dot{\underline{z}}_N^t = X_{H_N}(\underline{z}_N^t), \end{align} $$

which is equivalent to equation (1.6). As is well known, the symplectic form $\omega _N$ induces a canonical Poisson bracket on $({\mathbb {R}}^{2d})^N$ by

(2.5)

referred to as the standard Poisson structure on $({\mathbb {R}}^{2d})^N$ . Thus, the symplectic formulation (2.4) of Newton’s second law of motion can be equivalently written in Poisson form as

(2.6)

To evaluate $N\rightarrow \infty $ limits, it is convenient to rescale the Poisson bracket and modify the Hamiltonian $H_N$ as follows:

(2.7)

with the subscript ‘New’ abbreviating Newton. Evidently, $\mathcal {H}_{New}$ depends on N, but we omit this dependence from our notation, as it will be clear from context. The addition of the term $W(0)$ in the Hamiltonian is harmless: It is a constant, and so it does not change the Hamiltonian vector field. Its inclusion reflects the fact that we do not need to exclude self-interaction since W is continuous at the origin. With these rescalings and translation, the Poisson formulation (2.6) becomes

(2.8)

For each $k \in {\mathbb {N}}$ , we define the set

(2.9)

In the sequel, we will use the shorthand . In other words, the space $\mathfrak {g}_k$ consists of smooth real-valued functions which are invariant under permutations of particle labels. We endow the set $\mathfrak {g}_k$ with the locally convex topology induced by the seminorms

(2.10)

where K above is compact and the supremum is taken over all multi-indices $\alpha \in ({\mathbb {N}}_0)^{2dk}$ with order at most n. We then regard $\mathfrak {g}_k$ as a real topological vector space, elements of which are our k-particle observables. We introduce a bracket on $\mathfrak {g}_k$ which will give the space the structure of a Lie algebra. For each $k \in {\mathbb {N}}$ , we define

(2.11)

where is the standard Poisson bracket on $({\mathbb {R}}^{2d})^k$ .

Next, for each $k\in {\mathbb {N}}$ , we define the real topological vector space $\mathfrak {g}_k^*$ to be the strong dual of $\mathfrak {g}_k$ . It can be characterized as follows:

(2.12) $$ \begin{align} \mathfrak{g}_k^* = \{\gamma\in \mathcal{C}^\infty(({\mathbb{R}}^{2d})^k)^* : \pi\# \gamma = \gamma,\ \forall \pi\in{\mathbb{S}}_k\}, \end{align} $$

where $\pi \#\gamma (f) = \gamma (f \circ \pi )$ for $f \in \mathcal {C}^\infty (({\mathbb {R}}^{2d})^k)$ . Using the isomorphism $\mathcal {C}^\infty (({\mathbb {R}}^{2d})^k)^*\cong \mathcal {E}'(({\mathbb {R}}^{2d})^k)$ , elements of $\mathfrak {g}_k^*$ , which we call k-particle states, are distributions on $({\mathbb {R}}^{2d})^k$ with compact support and which are invariant under the action of ${\mathbb {S}}_k$ (i.e., the permutation of particle labels). The space $\mathfrak {g}_k^*$ has the desirable property of being a reflexive, (DF) Montel space (see Lemma 4.1).

The canonical Lie–Poisson bracket induced by the Lie bracket gives $\mathfrak {g}_k^*$ the structure of a Poisson vector space in the precise sense of Definition 3.17. In fact, the space $\mathfrak {g}_k^*$ has stronger topological properties, namely it is a $k^\infty $ space (see Definition 3.18) that make it an example of a reflexive, locally convex Poisson vector space as defined in Definition 3.19. We will use these stronger topological properties to prove this Lie–Poisson assertion by appealing to the aforementioned ‘black box’ theorem of Glöckner recalled in Theorem 3.20 below.

Before stating the result, we record the following important observation. For any $\mathcal {G}\in \mathcal {C}^\infty (\mathfrak {g}_k^*)$ , we have by definition of the Gâteaux derivative that $\mathsf {d}\mathcal {G} \in \mathcal {C}^\infty (\mathfrak {g}_k^*; \mathfrak {g}_k^{**})$ . So, for any $\mu \in \mathfrak {g}_k^*$ we have that $\mathsf {d}\mathcal {G}[\mu ]\in \mathfrak {g}_k^{**}$ , that is $\mathsf {d}\mathcal {G}[\mu ]$ is a continuous linear functional on $\mathfrak {g}_k^*$ . Since we have the isomorphism $\mathfrak {g}_k^{**}\cong \mathfrak {g}_k$ , we are justified in making the identification

(2.13)

We then regard as an element in $\mathfrak {g}_k $ , and we denote the pairing of $\mu $ and as the ‘integral’

(2.14)

This identification will be made throughout this paper.

Proposition 2.2. For observables $\mathcal {G}, \mathcal {H} \in \mathcal {C}^\infty ( \mathfrak {g}_k^*)$ and k-particle state $\gamma \in \mathfrak {g}_k^*$ , we define the bracket

(2.15)

Then is a reflexive, locally convex Lie–Poisson space in the sense of Definition 3.19.

The reader may check that for any $N\geq 2$ , the function $H_N\in \mathfrak {g}_N$ , hence $\mathcal {H}_{New}\in \mathfrak {g}_N$ . Therefore, it makes sense to introduce the Liouville Hamiltonian functional

(2.16)

Evidently, $\mathcal {H}_{Lio}$ depends on N, though we omit this dependence from our notation. Being linear and continuous (by consequence of the separate continuity of the distributional pairing), $\mathcal {H}_{Lio} \in \mathcal {C}^\infty (\mathfrak {g}_N^*)$ . With $\mathcal {H}_{Lio}$ and Proposition 2.2, the Liouville equation may be written in Hamiltonian form. Proposition 2.3 stated below is the classical counterpart to the fact that the von Neumann equation from quantum mechanics is Hamiltonian (see [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20, pp. 17-18]).

Proposition 2.3. Let $I \subset {\mathbb {R}}$ be a compact interval and $N \in {\mathbb {N}}$ . Then $\gamma \in \mathcal {C}^\infty (I, \mathfrak {g}_N^*)$ is a solution to the Liouville equation (1.9) if and only if

(2.17) $$ \begin{align} \dot{\gamma} = X_{\mathcal{H}_{Lio}}(\gamma), \end{align} $$

where $X_{\mathcal {H}_{Lio}}$ is the unique Hamiltonian vector field generated by the Hamiltonian $\mathcal {H}_{Lio}$ with respect to the Lie-Poisson vector space .

Given a position-velocity configuration $\underline {z}_N = (z_1,\ldots ,z_N) \in ({\mathbb {R}}^{2d})^N$ , we can associate a symmetric probability measure on $({\mathbb {R}}^{2d})^N$ by defining

(2.18)

Evidently, the right-hand side is an element of $\mathfrak {g}_N^*$ . We call this assignment $\iota _{Lio}:({\mathbb {R}}^{2d})^N\rightarrow \mathfrak {g}_N^*$ the Liouville map. The reader may check from the invariance of $\mathcal {H}_{New}$ under the action of ${\mathbb {S}}_N$ that

(2.19) $$ \begin{align} \mathcal{H}_{Lio}(\iota_{Lio}(\underline{z}_N)) = \frac{1}{N!}\sum_{\pi\in{\mathbb{S}}_N} \mathcal{H}_{New}(z_{\pi(1)},\ldots,z_{\pi(N)}) = \mathcal{H}_{New}(\underline{z}_N). \end{align} $$

Moreover, the map $\iota _{Lio}$ is a morphism of Poisson vector spaces, implying that $\iota _{Lio}$ maps solutions of the Newtonian system (1.6) to solutions of the Liouville equation (1.9) (see Remark 4.5).

Proposition 2.4. The map $\iota _{Lio} \in \mathcal {C}^\infty (({\mathbb {R}}^{2d})^N, \mathfrak {g}_N^*)$ defines a morphism of the Poisson vector space into the Lie–Poisson space :

(2.20)

where $\iota _{Lio}^*$ denotes the pullback under $\iota _{Lio}$ .

2.2 The Lie algebra $\mathfrak {G}_N$ and Lie–Poisson space $\mathfrak {G}_N^*$

Building on the previous subsection, we transition to discussing finite hierarchies of observables and states.

For $N\in {\mathbb {N}}$ , we define the algebraic direct sum

(2.21)

and endow this vector space with the product topology (note that the direct sum is a direct product since the number of summands is finite). This turns $\mathfrak {G}_N$ into a locally convex real topological vector space. We refer to elements of $\mathfrak {G}_N$ as N-hierarchies of observables, alternatively observable N-hierarchies. The Lie brackets induce a Lie algebra structure on $\mathfrak {G}_N$ as follows.

For $N\in {\mathbb {N}}$ and $1\leq k\leq N$ , consider the map

(2.22)

where

(2.23)

and we are defining the set of length-k tuples drawn from $\{1,\ldots ,N\}$ by

(2.24)

In the sequel, we will use the tuple shorthand $\boldsymbol {j}_k$ and write $f_{\boldsymbol {j}_k}^{(k)}$ and $f^{(k)}(\underline {z}_{\boldsymbol {j}_k})$ . One can show that the maps $\epsilon _{k,N}$ are continuous, linear and therefore $\mathcal {C}^\infty $ , and that they are injective (see Lemmas 4.6 and 4.7, respectively). In words, the map $\epsilon _{k,N}$ embeds a k-particle observable in the space of N-particle observables. The maps $\epsilon _{k,N}$ have a filtration property (see Lemma 4.9) asserting that lies in the image of $\epsilon _{k,N}$ , and using this filtration property together with the injectivity of $\epsilon _{k,N}$ , we can define a Lie bracket on $\mathfrak {G}_N$ by

(2.25)

In fact, there is an explicit formula for (see equation (4.67)), which we do not state here. For N fixed, the maps $\{\epsilon _{k,N}\}_{k=1}^N$ also have the interesting property that they induce a Lie algebra homomorphism (see Proposition 4.17)

(2.26)

the dependence of $\iota _{\epsilon }$ on N being implicit. The map $\iota _{\epsilon }$ sends an N-hierarchy of observables to a single N-particle observable. After a series of lemmas establishing properties of these embedding maps $\epsilon _{k,N}$ , we arrive at our main result for the N-particle hierarchy Lie algebra.

If we define the real topological vector space $\mathfrak {G}_N^*$ as the strong dual of $\mathfrak {G}_N = \bigoplus _{k=1}^N \mathfrak {g}_k$ , then using the duality of direct sums and products [Reference Köthe and GarlingK6¨9, Proposition 2, §14, Chapter 3], we see that

(2.27) $$ \begin{align} \mathfrak{G}_N^* = \left( \bigoplus_{k=1}^N \mathfrak{g}_k\right)^* \cong \prod_{k=1}^N \mathfrak{g}_k^*, \end{align} $$

where the right-hand side is endowed with the product topology. The canonical Lie–Poisson bracket induced by the Lie bracket gives $\mathfrak {G}_N^*$ the structure of a Poisson vector space in the precise sense of Definition 3.17. Similar to $\mathfrak {g}_N^*$ , the space $\mathfrak {G}_N^*$ is a reflexive, locally convex Poisson vector space as defined in Definition 3.19. We will use Glöckner’s black box Theorem 3.20 to prove this assertion.

Theorem 2.6. For functionals $\mathcal {G}, \mathcal {H} \in \mathcal {C}^\infty ( \mathfrak {G}_N^*)$ and state N-hierarchy $\Gamma = (\gamma ^{(k)})_{k=1}^N\in \mathfrak {G}_N^*$ , we define the bracket

(2.28)

Then is a reflexive, locally convex Lie–Poisson space in the sense of Definition 3.19.

To show that the BBGKY hierarchy (1.11) is a Hamiltonian equation on the Poisson vector space , we introduce the N-particle BBGKY Hamiltonian functional

(2.29)

where

(2.30)

the dependence on N being implicit. Here, $|v_1|^2$ and $W(x_1-x_2), W(0)$ are viewed as functions on $({\mathbb {R}}^{2d})$ and $({\mathbb {R}}^{2d})^2$ , respectively. Note that ${\mathbf {W}}_{BBGKY}$ is indeed an element of $\mathfrak {G}_N$ by the assumption that $W \in \mathcal {C}^\infty ({\mathbb {R}}^d)$ . Tautologically, $\mathcal {H}_{BBGKY}$ is linear, and it is continuous by the separate continuity of the duality pairing; hence, $\mathcal {H}_{BBGKY}\in \mathcal {C}^\infty (\mathfrak {G}_N^*)$ . Interpreting the integrals as distributional pairings, we have

(2.31)

The following theorem, our main result for the BBGKY hierarchy, is the classical counterpart to [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20, Theorem 2.3] for the quantum BBGKY hierarchy.

Theorem 2.7. Let $I \subset {\mathbb {R}}$ be a compact interval and $N \in {\mathbb {N}}$ . Then $\Gamma \in \mathcal {C}^\infty (I, \mathfrak {G}_N^*)$ is a solution to the BBGKY hierarchy (1.11) if and only if

(2.32) $$ \begin{align} \dot{\Gamma} = X_{\mathcal{H}_{BBGKY}}(\Gamma), \end{align} $$

where $X_{\mathcal {H}_{BBGKY}}$ is the unique Hamiltonian vector field generated by the Hamiltonian $\mathcal {H}_{BBGKY}$ with respect to the Poisson vector space .

Returning to the homomorphism $\iota _{\epsilon }$ from equation (2.26), we can take its dual $\iota _{\epsilon }^*:\mathfrak {g}_N^*\rightarrow \mathfrak {G}_N^*$ . Analogous to the quantum setting (cf. [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20, Proposition 5.29]), this dual map is nothing but the marginal map $\gamma \mapsto (\gamma ^{(k)})_{k=1}^N$ and is Poisson morphism, facts shown in Proposition 4.18.

2.3 The Lie algebra $\mathfrak {G}_\infty $ and Lie–Poisson space $\mathfrak {G}_\infty ^*$

Having built up the necessary structure at the N-particle level, we transition to addressing the infinite-particle limit of our constructions. The natural inclusion map $\mathfrak {G}_N\subset \mathfrak {G}_M$ for any integers $M\geq N$ implies that one has the limiting topological vector space (a colimit of topological spaces ordered by inclusion)

(2.33)

Elements of $\mathfrak {G}_\infty $ are called observable $\infty $ -hierarchies, alternatively $\infty $ -hierarchies of observables. They take the form $F=(f^{(k)})_{k=1}^\infty $ , where $f^{(k)} \in \mathfrak {g}_k$ is the zero element for all $k\geq N+1$ , for some $N\in {\mathbb {N}}$ . Thus, given any $F,G\in \mathfrak {G}_\infty $ , by taking N sufficiently large, it makes sense to consider the Lie bracket . Our next result computes the limit of this expression as $N\rightarrow \infty $ and shows that is indeed a Lie algebra. Notably, the Lie bracket acquires a much simpler form than , as certain terms vanish as $N\rightarrow \infty $ . In contrast to the quantum setting of [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20], there are no technical difficulties involving compositions of distribution-valued operators to give meaning to .

Theorem 2.8. Let $F = (f^{(\ell )})_{\ell =1}^\infty , G = (g^{(j)})_{j=1}^\infty \in \mathfrak {G}_{\infty }$ . For each $k\in {\mathbb {N}}$ , define

(2.34)

where the limit is in the topology of $\mathfrak {G}_\infty $ , the wedge product $\wedge _1$ is defined by

(2.35)

and the k-particle symmetrization operator $\operatorname {\mathrm {Sym}}_k$ is defined byFootnote ⁴

(2.36)

Moreover, is a Lie algebra in the sense of Definition 3.15, and the bracket is boundedly hypocontinuous.

As with the N-particle setting, the next step is the dual problem of constructing a Lie–Poisson space from . We define the real topological vector space

(2.37)

equipped with the usual product topology, which is the strong dual of $\mathfrak {G}_\infty $ . Elements of $\mathfrak {G}_\infty ^*$ are called state $\infty $ -hierarchies, alternatively $\infty $ -hierarchies of states.

We want to construct a Lie–Poisson bracket over $\mathfrak {G}_\infty ^*$ similarly to as done in Theorem 2.6. Given any $\mathcal {F},\mathcal {G}\in \mathcal {C}^\infty (\mathfrak {G}_\infty ^*)$ and $\Gamma =(\gamma ^{(k)})_{k=1}^\infty \in \mathfrak {G}_\infty ^*$ , the continuous linear functionals $\mathsf {d}\mathcal {F}[\Gamma ],\mathsf {d}\mathcal {G}[\Gamma ]$ may be identified as elements of $\mathfrak {G}_\infty $ since $\mathfrak {G}_\infty ^{**} \cong \mathfrak {G}_\infty $ . Hence, is an element of $\mathfrak {G}_\infty $ , in particular only finitely many of its components are nonzero, and we are justified in defining

(2.38)

Here, we come to one of the main technical difficulties of the paper: We are unable to prove that . An essentially equivalent issue is that while we are able to show that a Hamiltonian vector field $X_{\mathcal {G}}$ exists, we are unable to show it is $\mathcal {C}^\infty $ as a map $\mathfrak {G}_\infty ^*\rightarrow \mathfrak {G}_\infty ^*$ . As remarked in Section 1.3, we cannot rely on Theorem 3.20, as done in the proof of Theorem 2.6, because we are unable to verify that $\mathfrak {G}_\infty ^*$ satisfies certain topological conditions, namely that it is a $k^\infty $ space (see Definition 3.18). Accordingly, we instead directly show that $\mathfrak {G}_\infty ^*$ admits a weak Lie–Poisson structure in the sense of Definition 3.23.

The key difference between a weak Lie–Poisson structure and Lie–Poisson structure is that, in the former, one specifies a unital subalgebra (with respect to pointwise product) ${\mathcal {A}}_\infty \subset \mathcal {C}^\infty (\mathfrak {G}_\infty ^*)$ , which must satisfy certain nondegeneracy conditions, as the ‘admissible’ functionals, in contrast to working with all the functionals in $\mathcal {C}^\infty (\mathfrak {G}_\infty ^*)$ . To this end, we choose ${\mathcal {A}}_{\infty }\subset \mathcal {C}^\infty (\mathfrak {G}_\infty ^*)$ to be the algebra generated with respect to pointwise product by the set

(2.39) $$ \begin{align} \{\mathcal{F}\in \mathcal{C}^\infty(\mathfrak{G}_\infty^*) : \mathcal{F}(\cdot) = \left\langle {F,\cdot} \right\rangle _{\mathfrak{G}_\infty-\mathfrak{G}_\infty^*} \, \enspace F\in \mathfrak{G}_\infty\} \cup \{\mathcal{F}\in \mathcal{C}^\infty(\mathfrak{G}_\infty^*) : \mathcal{F}(\cdot)\equiv C\in{\mathbb{R}}\}. \end{align} $$

Heuristically viewing the components of $\Gamma =(\gamma ^{(k)})_{k=1}^\infty $ as measures on $({\mathbb {R}}^{2d})^k$ , we call functionals of the form $\mathcal {F}(\cdot )= \left \langle {F,\cdot } \right \rangle _{\mathfrak {G}_\infty -\mathfrak {G}_\infty ^*}$ expectations. They are analogous to the ‘trace functionals’ of [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20]. In other words, the subalgebra ${\mathcal {A}}_{\infty }$ is generated by expectations and the constant functionals. The work [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20] employs the notion of a weak Poisson vector space at both the N-particle level for $\mathfrak {G}_N^*$ and the infinite-particle level for $\mathfrak {G}_\infty ^*$ ; while here, we only need this notion at the infinite-particle level. This is an advantage of the present work compared to [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20]. The motivation for this choice of algebra ${\mathcal {A}}_\infty $ is that expectation functionals have constant Gâteaux derivatives (see Remark 2.10 below). Since for fixed expectations $\mathcal {F},\mathcal {G}$ , the Gâteaux derivatives $\mathsf {d}\mathcal {F}[\Gamma ],\mathsf {d}\mathcal {G}[\Gamma ]$ have only finitely many nonzero components as elements in $\mathfrak {G}_\infty $ , uniformly in $\Gamma $ , this allows us then to directly check that the bracket is $\mathcal {C}^\infty $ , in fact it belongs to the subalgebra ${\mathcal {A}}_\infty $ , and also show that the the vector field $X_{\mathcal {G}}$ is $\mathcal {C}^\infty $ . This direct verification relies heavily on explicit formulae for the Poisson bracket and for the Hamiltonian vector field with respect to the bracket to show that these expressions reduce to finite sums of compositions of $\mathcal {C}^\infty $ maps.

Theorem 2.11. Let $\mathfrak {G}_\infty ^*$ be the strong dual of $\mathfrak {G}_\infty $ as given in equation (2.37). Define the bracket

(2.40)

and let ${\mathcal {A}}_\infty $ be as in equation (2.39). Then the triple is a weak Poisson vector space in the sense of Definition 3.23.

Having constructed a weak Poisson vector space for the infinite-particle setting, it makes sense to discuss Hamiltonian flows for $\infty $ -hierarchies. Our final result of this subsection is that the Vlasov hierarchy (1.12) is itself Hamiltonian, which is a new observation. The Vlasov hierarchy Hamiltonian functional is the expectation (cf. equations (2.29), (2.30) for the BBGKY Hamiltonian)

(2.41) $$ \begin{align} \mathcal{H}_{VlH}(\Gamma) = \left\langle {{\mathbf{W}}_{VlH}, \Gamma} \right\rangle _{\mathfrak{G}_\infty-\mathfrak{G}_\infty^*} \end{align} $$

generated by the observable $\infty $ -hierarchy

(2.42)

One immediately recognizes that ${\mathbf {W}}_{VlH}$ is the $N\rightarrow \infty $ limit of ${\mathbf {W}}_{BBGKY}$ in the topology of $\mathfrak {G}_\infty $ . Interpreting the integrals as distributional pairings, we can write, for $\Gamma =(\gamma ^{(k)})_{k=1}^\infty $ ,

(2.43) $$ \begin{align} \mathcal{H}_{VlH}(\Gamma) =\frac{1}{2}\int_{{\mathbb{R}}^{2d}}d\gamma^{(1)}(z)|v|^2 + \int_{({\mathbb{R}}^{2d})^2}d\gamma^{(2)}(z_1,z_2) W(x_1-x_2). \end{align} $$

In particular, the functional $\mathcal {H}_{VlH}$ belongs to the admissible algebra ${\mathcal {A}}_\infty $ introduced in equation (2.39). The next theorem asserts that the Vlasov hierarchy (1.12) is a Hamiltonian flow on , and it is the classical analogue of [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20, Theorem 2.10] for the Gross–Pitaevskii hierarchy.

Theorem 2.12. Let $I \subset {\mathbb {R}}$ be a compact interval. Then $\Gamma =(\gamma ^{(k)})_{k=1}^\infty \in \mathcal {C}^\infty (I, \mathfrak {G}_\infty ^*)$ is a solution to the Vlasov hierarchy (1.12) if and only if

(2.44) $$ \begin{align} \dot{\Gamma}= X_{\mathcal{H}_{VlH}}(\Gamma), \end{align} $$

where $X_{\mathcal {H}_{VlH}}$ is the unique Hamiltonian vector field generated by the Hamiltonian $\mathcal {H}_{VlH}$ with respect to the weak Lie–Poisson space .

2.4 From Vlasov hierarchy to Vlasov equation

Finally, we tie together the constituent results of the previous subsections to connect the Hamiltonian structure of the Vlasov hierarchy (1.12) to the Vlasov equation (1.1). This necessitates elaborating on the rigorous formulation of the Hamiltonian structure of the Vlasov equation (cf. [Reference Marsden and RatiuMR13, p. 329, 10.1(e)]). The Vlasov Hamiltonian functional is

(2.45)

In terms of ‘integrals’ (as before, understood rigorously as distributional pairings),

(2.46) $$ \begin{align} \mathcal{H}_{Vl}(\gamma) = \frac{1}{2}\int_{({\mathbb{R}}^d)^2}d\gamma(x,v)|v|^2+\int_{({\mathbb{R}}^d)^2}d\rho^{\otimes 2}(x_1,x_2)W(x_1-x_2), \end{align} $$

where is the density associated to $\gamma $ . Note that $\rho $ is well defined as a distribution, since for any test function $f\in \mathcal {C}^\infty ({\mathbb {R}}^d)$ , we can set

(2.47)

where $(f\otimes 1)(x,v) = f(x)$ for every $(x,v)\in ({\mathbb {R}}^d)^2$ . In contrast to the other Hamiltonian functionals we have seen so far, $\mathcal {H}_{Vl}$ is nonlinear, in fact quadratic, in the potential energy. Since $\mathcal {H}_{Vl}$ is multilinear in its argument $\gamma $ and continuous as a map from $\mathfrak {g}_1^*\rightarrow {\mathbb {R}}$ , it is straightforward to check that $\mathcal {H}_{Vl}\in \mathcal {C}^\infty (\mathfrak {g}_1^*)$ .

Proposition 2.13. Let $I \subset {\mathbb {R}}$ be a compact interval. Then $\gamma \in \mathcal {C}^\infty (I, \mathfrak {g}_1^*)$ is a solution to the Vlasov equation (1.1) if and only if

(2.48) $$ \begin{align} \dot{\gamma}= X_{\mathcal{H}_{Vl}}(\gamma), \end{align} $$

where $X_{\mathcal {H}_{Vl}}$ is the unique Hamiltonian vector field generated by the Hamiltonian $\mathcal {H}_{Vl}$ with respect to the Lie–Poisson space .

We connect the Vlasov hierarchy to the Vlasov equation, each as infinite-dimensional Hamiltonian systems, through the embedding

(2.49)

Here, $\gamma ^{\otimes k}$ denotes the usual k-fold tensor product of the distribution $\gamma $ . The geometric content of the map $\iota $ , which we call the trivial embedding or factorization map, is that it preserves the Poisson structures on $\mathfrak {g}_1^*$ and $\mathfrak {G}_\infty ^*$ , that is, it is a Poisson morphism in the sense of Definition 3.27.

Theorem 2.14. The map $\iota \in \mathcal {C}^\infty (\mathfrak {g}_1^*,\mathfrak {G}_\infty ^*)$ is a morphism of the Lie–Poisson space into the weak Lie–Poisson space :

(2.50)

Let us now explain why the results of this section constitute a rigorous derivation of the Hamiltonian structure for the Vlasov equation, as claimed in the title of the paper. The reader may check that (see also Remark 6.3)

(2.51) $$ \begin{align} \iota^* \mathcal{H}_{VlH} = \mathcal{H}_{Vl}, \end{align} $$

that is, the pullback of the Vlasov hierarchy Hamiltonian equals the Vlasov Hamiltonian. The identity (2.51) together with Theorems 2.12 and 2.14 then show that the Hamiltonian functional and Poisson bracket for the Vlasov equation are obtained via the pullback under the trivial embedding $\iota $ of the Hamiltonian functional and Poisson bracket for the Vlasov hierarchy; moreover, $\iota $ sends solutions of the Vlasov equation to special factorized solutions of the Vlasov hierarchy. Combined with the results of Section 2.2, which provide a geometric correspondence between Newton’s equations/Liouville equation and the BBGKY hierarchy, and Theorem 2.8, which allows us to take the infinite-particle limit of our N-particle geometric constructions, we arrive at a rigorous derivation of the Hamiltonian structure of the Vlasov equation directly from the Hamiltonian formulation of Newtonian mechanics.

Finally, as mentioned in Section 1.1, there is another way to derive the Vlasov equation from the Newtonian N-body problem (1.6) via the empirical measure. It is an interesting fact, which to our knowledge has not been previously observed, that the map $\iota _{EM}$ assigning a position-velocity configuration $\underline {z}_N \in ({\mathbb {R}}^{2d})^N$ to its empirical measure on ${\mathbb {R}}^{2d}$ is, in fact, a Poisson morphism (see Proposition 2.15 below). Since one also has $\iota _{EM}^*\mathcal {H}_{Vl}=\mathcal {H}_{New}$ (see Remark 6.1), this implies the previously mentioned fact that if $\underline {z}_N^t$ is a solution to equation (1.6), then the associated empirical measure $\mu _N^t$ is a weak solution to the Vlasov equation.

Proposition 2.15. The map

(2.52)

belongs to $\mathcal {C}^\infty (({\mathbb {R}}^{2d})^N, \mathfrak {g}_1^*)$ and defines a morphism of Poisson vector spaces.

2.5 Organization of paper

Let us close Section 2 with some comments on the organization of the remaining body of the article.

Section 3 contains background material on topological vector spaces, Lie algebras and (weak) Lie–Poisson vector spaces. The reader may wish to skip this section upon first reading and instead consult it as necessary during the reading of Sections 4 to 6.

Section 4 contains the N-particle setting results. The section is divided into several subsections, each building upon the previous one. Section 4.1 concerns the setting of the Newtonian system (1.6) and Liouville equation (1.9), proving Propositions 2.1 and 2.2 for $\mathfrak {g}_k$ and $\mathfrak {g}_k^*$ , respectively, Proposition 2.4 for $\iota _{Lio}$ , and Proposition 2.15 for $\iota _{EM}$ . Sections 4.2 and 4.3 concern the setting of the BBGKY hierarchy (1.11), proving Theorems 2.5 and 2.6 for $\mathfrak {G}_N,\mathfrak {G}_N^*$ , respectively. Finally, Section 4.4 concerns the operation of taking marginals, proving Proposition 4.18 for $\iota _{mar}$ .

Section 5 contains the infinite-particle setting results. As with Section 4, the section is divided into several subsections, each intended to build upon the previous one. Sections 5.1 and 5.2 are devoted to the proofs of Theorems 2.8 and 2.11 for $\mathfrak {G}_\infty ,\mathfrak {G}_\infty ^*$ , respectively. Section 5.3 contains the proof of Theorem 2.14 for the map $\iota $ .

Lastly, Section 6 contains the proofs of the Hamiltonian flows results Proposition 2.13 and Theorems 2.7 and 2.12, which assert that that the Vlasov equation, BBGKY hierarchy and Vlasov hierarchy, respectively, are Hamiltonian flows on their respective Lie–Poisson spaces given by Proposition 2.13 and Theorems 2.6 and 2.11. The section is broken into three subsections with Section 6.1 corresponding to the Vlasov equation, Section 6.2 to the BBGKY hierarchy and Section 6.3 to the Vlasov hierarchy.

3.1 Some function analysis facts

In this subsection, we review functional analytic notions which will be used throughout the rest of the paper. We begin by reviewing duality in topological vector spaces.

Definition 3.1. Let X be a topological vector space. We define $X^*$ to be the set of continuous linear functionals on X and endow it with the strong dual topology, which is given as follows. Let $\mathcal {A}$ be the set of bounded subsets of X. For each $A \in \mathcal {A}$ , we define the seminorm

(3.1)

Note that this is indeed a seminorm, since continuous linear operators are bounded. We define the topology of $X^*$ to be the one generated by the above seminorms. If the cannonical embedding

(3.2)

is an isomorphism between topological vector spaces, then we say that X is reflexive.

Definition 3.2. Let $X, Y$ be topological vector spaces, and let $F: X \rightarrow Y$ be a continuous linear map. We define the adjoint of F to be $F^* : Y^* \rightarrow X^*$ with

(3.3)

We continue with the necessary background in functional analysis by reviewing the concepts of barrelled, Montel and (DF) spaces following the presentation of [Reference Köthe and GarlingK6¨9, Reference KötheK7¨9].

In the above definition, a subset M of X is said to be absorbent if for every $x \in X$ , there exists a $\rho> 0$ such that $x \in \rho M$ ; it is said to be absolutely convex if for every $x,y \in M$ and $\alpha ,\beta \in {\mathbb {R}}$ with $|\alpha | + |\beta | \leq 1$ , the point $\alpha x + \beta y \in M$ . For the following, we recall that a locally convex topological vector space is Fréchet if it is metrizable and complete.

Next, we recall the notions of sequential spaces and k-spaces as presented in [Reference EngelkingEng89, pp. 53, 152].

That a sequential space is a fortiori a k-space is, perhaps well known. For the sake of completeness, we present a proof of this fact in the next proposition, which will be crucially used in Section 4.3.

Proof. Assume that there exists a nonclosed set $A \subset X$ which satisfies $K \cap A$ is closed in A for every compact K. Since A is not closed, and X is a sequential space, we must have that A is not sequentially closed. So, there exists some sequence $(x_i)_{i=1}^\infty $ in A that converges to a point $x \in X \setminus A$ . Note that the set $ \{ x_i : i \in {\mathbb {N}} \} \cup \{x\}$ is compact, and so

(3.4) $$ \begin{align} A \cap (\{ x_i : i \in {\mathbb{N}} \} \cup \{x\}) =\{ x_i : i \in {\mathbb{N}} \} \end{align} $$

is closed in A. However, since closed sets are sequentially closed and the sequence $x_i$ converges to x, it must be the case that $x \in A$ . This is a contradiction, so X is a k-space.

We are now ready to state a result of Webb [Reference WebbWeb68] which gives sufficient conditions for a topological vector space to be a sequential space.

We close this subsection by stating the notions of derivative and smooth function for infinite-dimensional spaces used in this work, which is that of the Gâteaux derivative. For more on calculus in the setting of topological vector spaces, we refer to the lecture notes of Milnor [Reference MilnorMil84].

In the remainder of the paper, we write $\mathcal {C}(X)$ (similarly, $\mathcal {C}^n(X),\mathcal {C}^\infty (X)$ ) when the codomain is ${\mathbb {R}}$ , that is, the maps are real-valued functionals.

3.2 Lie algebras and Poisson vector spaces

We start this subsection by giving a precise definition of Lie algebra and Poisson vector space that we use in this paper. With these definitions in hand, we then present a result due to Glöckner [Reference GlocknerGlo09] which allows one to canonically construct a Lie–Poisson vector space from a Lie algebra, assuming certain topological conditions are met, as mentioned in Section 1.3. The use of Glöckner’s machinery is new to the present work compared to [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20].

The next definition introduces the notion of a possibly infinite-dimensional Poisson vector space, which is a natural extension of the finite-dimensional notion of a Poisson vector space, more generally Poisson manifold (e.g., see [Reference WeinsteinWei98]). Our usage is consistent with that of Glöckner [Reference GlocknerGlo09, Definition 4.2]. For other possible notions of a infinite-dimensional Poisson vector spaces, which are not appropriate for our purposes due to being restricted to the Banach category, we refer to [Reference Odzijewicz and RatiuOR03, Reference Odzijewicz and RatiuOR04].

Definition 3.17 (Poisson vector space)

Let X be a locally convex topological vector space, and

(3.7)

be a bilinear map. We say the pair is a Poisson vector space if it satisfies the following properties:

1. 1. is a Lie algebra in the sense of Definition 3.15 obeying the Leibniz rule:

   (3.8)

   

2. 2. For every $\mathcal {F} \in \mathcal {C}^\infty ( X)$ , there exists a smooth Hamiltonian vector field $X_{\mathcal {F}} : X \rightarrow X$ such that

   (3.9)

   

We now state the theorem of Glöckner [Reference GlocknerGlo09, Theorem 4.10] which will allow us to construct a Poisson vector space from a given Lie algebra in the N-particle setting (see Section 4.2). For the purposes of this paper, the reader may view this theorem as a ‘black box’. But to use this black box, certain topological conditions need to be satisfied. Namely, [Reference GlocknerGlo09] works in the context of $k^\infty $ spaces, a class of topological vector spaces introduced in that paper. Accordingly, we shall start this portion of the exposition by recalling the definition of this class of spaces, as well as the notion of reflexive locally convex Poisson vector spaces which we also need.

Equipped with Definitions 3.18 and 3.19, we are now prepared to state the following result from [Reference GlocknerGlo09], the statement of which has been tailored to our setting.

Theorem 3.20 [Reference GlocknerGlo09, Theorem 4.10]

Let E be a reflexive locally convex Poisson space in the sense of Definition 3.19 such that its dual $E^*$ is equipped with a hypocontinuous bracket . For $\mathcal {F}, \mathcal {G} \in \mathcal {C}^\infty ( E)$ , the Lie–Poisson bracket is defined by the expression

(3.10)

where $ \left \langle {\cdot ,\cdot } \right \rangle _{E^*-E}$ denotes the duality pairing. The pair , called a Lie–Poisson space, is a Poisson vector space in the sense of Definition 3.17.

The space $\mathfrak {G}_\infty ^*$ is a nontrivial countably infinite product of (DF) spaces and as such is not a (DF) space itself (see [Reference Schaefer and WolffSW99, p. 196]). Therefore, Theorem 3.13 is not applicable, which renders verification of the assumptions of Theorem 3.20 out of reach. To overcome this obstacle, we need a weaker notion of a Poisson vector space than assumed in Theorem 3.20. Namely, we need to restrict to a proper subalgebra $\mathcal {A}$ of functionals in $\mathcal {C}^\infty (E)$ for which smoothness of the Poisson bracket and Hamiltonian vector field can be verified. To this end, we use, as in the previous work [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20], the framework of weak Poisson vector spaces due to Neeb et al. [Reference Neeb, Sahlmann, Thiemann and DobrevNST14].

Remark 3.26. The property (3.12) uniquely characterizes the Hamiltonian vector field. Indeed, if $X_{\mathcal {F}},\tilde {X}_{\mathcal {F}}$ are two $\mathcal {C}^\infty $ vector fields obeying equation (3.12), then given $x\in X$ ,

(3.13) $$ \begin{align} \forall \mathcal{G}\in{\mathcal{A}}, \qquad ({({X_{\mathcal{F}}-\tilde{X}_{\mathcal{F}}})(\mathcal{G})})(x) = \mathsf{d}\mathcal{G}[x]({X_{\mathcal{F}}(x)-\tilde{X}_{\mathcal{F}}(x)}). \end{align} $$

Applying 2 with $v= X_{\mathcal {F}}(x)-\tilde {X}_{\mathcal {F}}(x)$ , we conclude that $X_{\mathcal {F}}(x)=\tilde {X}_{\mathcal {F}}(x)$ .

Finally, we need the notion of a morphism between (weak) Poisson vector spaces.

Definition 3.27. Let be Poisson vector spaces in the sense of Definition 3.17. We say that a $\mathcal {C}^\infty $ map $T: E_1\rightarrow E_2$ is a morphism of Poisson vector spaces if

(3.14)

where $T^*$ denotes the pullback under T. Suppose now that are weak Poisson vector spaces in the sense of Definition 3.23. We say that a $\mathcal {C}^\infty $ map $T:E_1\rightarrow E_2$ is a morphism of weak Poisson vector spaces if for any $\mathcal {F},\mathcal {G}\in {\mathcal {A}}_2$ , $T^*\mathcal {F}, T^*\mathcal {G}\in {\mathcal {A}}_1$ and equation (3.14) holds with $\mathcal {C}^\infty (E_2)$ replaced by ${\mathcal {A}}_2$ .

4.1 N-particle Newton/Liouville equations

The goal of this subsection is to establish the Hamiltonian structure of the Newtonian system (1.6) and the Liouville equation (1.9), as well as to connect the two structures through a Poisson morphism. Since the Newtonian system is classical, we leave the proofs of the statements concerning it in Section 2.1 as simple exercises for the reader.

We recall from equations (2.9) and (2.11) the definitions of the space $\mathfrak {g}_k$ and the bracket . Our first task is to prove Proposition 2.1 asserting that is a Lie algebra.

Proof of Proposition 2.1

Since our bracket is defined as a scalar multiple of the standard Poisson bracket, the algebraic properties 1-3 are satisfied, so it only remains to check continuity. It suffices to show that the multiplication and differentiation maps

(4.1) $$ \begin{align} M: \mathfrak{g}_k \times \mathfrak{g}_k \rightarrow \mathfrak{g}_k, \qquad (f,g) \mapsto fg \end{align} $$

and

(4.2) $$ \begin{align} \partial^\alpha: \mathfrak{g}_k \rightarrow \mathfrak{g}_k, \qquad f \mapsto \partial^\alpha f \end{align} $$

are continuous for each multi-index $\alpha \in {\mathbb {N}}^{2dk}$ . But this follows because is just a linear combination of compositions of $M,\partial ^\alpha $ .

We first show that M is continuous. Since the spaces $\mathfrak {g}_k$ are Fréchet, it suffices to show that M is sequentially continuous. To this end, let $(f_j, g_j) \rightarrow (f,g)$ be a convergent sequence in $\mathfrak {g}_k \times \mathfrak {g}_k$ . Note that for any compact set $K \subset ({\mathbb {R}}^{2d})^k$ and $n \in {\mathbb {N}}$ , we have

(4.3)

where the constant $C_{K,n}$ depends only on the set K and the index n. Using the Leibniz rule and triangle inequality, we now estimate

(4.4)

where in the last line we have used the bound (4.3). Since the last line converges to $0$ as $j \rightarrow \infty $ and $K,n$ were arbitrary, we have shown that $f_j g_j \rightarrow fg$ in $\mathfrak {g}_k$ . Thus, M is continuous.

Fix a multi-index $\alpha $ . To show that $\partial ^\alpha $ is continuous, let $f_j \rightarrow f$ in $\mathfrak {g}_k$ and calculate

(4.5) $$ \begin{align} \rho_{K,n}(\partial^\alpha (f_j - f)) = \sup_{|\gamma|\leq n} \Vert \partial^\gamma \partial^\alpha (f_j - f) \Vert_{L^\infty(K)} \leq \sup_{|\beta| \leq n + |\alpha|} \Vert \partial^\beta( f_j -f) \Vert_{L^\infty(k)} = \rho_{K,n+|\alpha|} (f_j -f). \end{align} $$

The right-hand side converges to $0$ as $j \rightarrow \infty $ , which shows that the operator $\partial ^\alpha $ is continuous.

Now, recall the definitions of the space $\mathfrak {g}_k^*$ and the bracket from equations (2.12) and (2.15), respectively. Our next task is to prove Proposition 2.2, asserting that is a Lie–Poisson space. To this end, we need the following technical lemma alluded to in Section 2.1.

Proof. We first prove that $\mathfrak {g}_k$ is Montel. The proof is an adaptation to symmetric functions of the argument that the space $\mathcal {C}^\infty ( {\mathbb {R}}^n)$ , for any $n\in {\mathbb {N}}$ , is a Montel space (see [Reference SchwartzSch66, Theorem VII, §2, Chapter 3]). We reproduce it here for the reader’s convenience.

First, we fix an equivalent sequence of seminorms on $\mathfrak {g}_k$ which give the same topology. Let $K_n$ be a compact exhaustion of $({\mathbb {R}}^{2d})^k$ , that is, let $\{K_n\}_{n=1}^\infty $ be an increasing sequence of compact sets such that $\bigcup _{n=1}^\infty K_n = ({\mathbb {R}}^{2d})^k$ . Then define the seminorms

(4.6)

These seminorms are equivalent to those given in equation (2.10), as the reader may check. This implies that $\mathfrak {g}_k$ is indeed a Fréchet space and hence a barrelled space by Lemma 3.5. We will now show that $\mathfrak {g}_k$ satisfies the Heine–Borel property, that is, that bounded closed subsets are compact. Note that since $\mathfrak {g}_k$ is a metric space, it suffices to show that bounded closed sets are sequentially compact. To this end, let $B \subset \mathfrak {g}_k$ be a bounded, closed set, and let $\{f_k \}_{k=1}^\infty \subset B$ . Then by definition of bounded, there exist constants $C_n> 0$ such that

(4.7) $$ \begin{align} \sup_{k\in{\mathbb{N}}} \Vert \partial^\alpha f_k \Vert_{L^\infty(K_n)} \leq C_n, \quad \forall |\alpha |\leq n. \end{align} $$

We will be using the convention that subsequences of $\{f_k\}$ are still denoted by $\{f_k\}$ . We take subsequences and diagonalize in the following way:

1. 1. Apply Arzelà-Ascoli and diagonalize with respect to $K_n$ to get a subsequence $f_k \rightarrow f$ locally uniformly in $L^\infty $ .

2. 2. Apply Arzelà—Ascoli again and diagonalize with respect to each $|\alpha | \leq n$ to get a further subsequence $\partial ^\alpha f_k \rightarrow \partial ^\alpha f$ locally uniformly in $L^\infty $ .

3. 3. Now, we can conclude that $f_k \rightarrow f$ in $\mathfrak {g}_k$ . Hence, since B is closed in $\mathfrak {g}_k$ , we have proved that $f \in B$ .

Hence, the space $\mathfrak {g}_k$ satisfies the Heine–Borel property. Since we noted above that $\mathfrak {g}_k$ is also barrelled, we conclude that it is a Montel space (recall Definition 3.6).

Finally, we are ready to conclude the proof of our lemma. By invoking Lemma 3.7, we now have that $\mathfrak {g}_k^*$ is Montel and that $\mathfrak {g}_k$ is reflexive. The fact that $\mathfrak {g}_k^*$ is $(DF)$ follows from Lemma 3.9 since $\mathfrak {g}_k$ is a Fréchet space.

We now have the necessary ingredients to prove Proposition 2.2.

We next turn to proving Proposition 2.4, asserting that the Liouville map $\iota _{Lio}:({\mathbb {R}}^{2d})^N\rightarrow \mathfrak {g}_N^*$ is a morphism of Poisson vector spaces. To do this, we need the following technical lemma computing the Gâteaux derivatives of $\iota _{Lio}$ .

Lemma 4.2. It holds that $\iota _{Lio}\in \mathcal {C}^\infty (({\mathbb {R}}^{2d})^N, \mathfrak {g}_N^*)$ and for every $n\in {\mathbb {N}}$ , $\underline {z}_N,\underline {w}_N^1,\ldots ,\underline {w}_N^n\in ({\mathbb {R}}^{2d})^N$ ,

(4.8) $$ \begin{align} \mathsf{d}^n\iota_{Lio}[\underline{z}_N](\underline{w}_N^1,\ldots,\underline{w}_N^n) = (-1)^n\frac{1}{N!}\sum_{\pi\in{\mathbb{S}}_N}\sum_{\substack{0\leq n_1,\ldots,n_N\leq n \\ n_1+\cdots+n_N = n}} \sum_{\mathcal{I}} \bigotimes_{j=1}^N \left({\nabla^{\otimes n_j}\delta_{z_{\pi(j)}} : \bigotimes_{k=1}^{n} w_{\pi(j)}^{i_k^j}}\right), \end{align} $$

where the summation $\sum _{\mathcal {I}}$ is over all tuples

(4.9) $$ \begin{align} \mathcal{I} = (\mathbf{i}^1,\ldots,\mathbf{i}^N), \qquad \mathbf{i}^j := (i_1^j,\ldots,i_n^j) \in \{0,1\}^n \ \text{with} \ i_1^j+\cdots+i_n^j = n_j \end{align} $$

and $w_{\pi (j)}^0$ denotes the factor in the tensor product is vacuous. Here, $\nabla ^{\otimes n_j}\delta _{z_{\pi (j)}} : \bigotimes _{k=1}^n w_{\pi (j)}^{i_k^j}$ is the distribution in $\mathcal {E}'({\mathbb {R}}^{2d})$ defined

(4.10) $$ \begin{align} \forall \varphi\in \mathcal{C}^\infty({\mathbb{R}}^{2d}), \qquad \left\langle {\varphi,\nabla^{\otimes n_j}\delta_{z_{\pi(j)}} : \bigotimes_{k=1}^n w_{\pi(j)}^{i_k^j} } \right\rangle = (-1)^{n_j}\nabla^{\otimes n_j}\varphi(z_{\pi(j)}) : \bigotimes_{k=1}^n w_{\pi(j)}^{i_k^j}, \end{align} $$

with $\nabla ^{\otimes n_j}\varphi = ({\partial }_{x^{\alpha _1}}{\partial }_{v^{\beta _1}}\cdots {\partial }_{x^{\alpha _{n_j}}}{\partial }_{v^{\beta _{n_j}}}\varphi )_{\alpha _1,\beta _1,\ldots ,\alpha _{n_j},\beta _{n_j}=1}^d$ and $:$ denoting the tensor inner product.

Remark 4.3. Specializing the identity (4.8) to $n=1$ , we obtain

(4.11)

Proof of Proposition 2.4

Let $\mathcal {F},\mathcal {G}\in \mathcal {C}^\infty (\mathfrak {g}_N^*)$ , and set . By the chain rule, we have the identity

(4.12) $$ \begin{align} \sum_{i=1}^N \nabla_{z_i} F(\underline{z}_N) \cdot w_i =\mathsf{d} F[\underline{z}_N](\underline{w}_N)=\mathsf{d}\mathcal{F}[\iota_{Lio}(\underline{z}_N)]\Big(\mathsf{d}\iota_{Lio}[\underline{z}_N](\underline{w}_N)\Big), \qquad \forall \underline{w}_N \in ({\mathbb{R}}^{2d})^N. \end{align} $$

Identifying $\mathsf {d}\mathcal {F}[\iota _{Lio}(\underline {z}_N)]$ as an element of $\mathfrak {g}_N$ and using equation (4.11), the preceding right-hand side equals

(4.13) $$ \begin{align} & \left\langle {\mathsf{d}\mathcal{F}[\iota_{Lio}(\underline{z}_N)],\mathsf{d}\iota_{Lio}[\underline{z}_N](\underline{w}_N)} \right\rangle _{\mathfrak{g}_N-\mathfrak{g}_N^*} \nonumber\\ &\quad = \frac{1}{N!}\sum_{\pi\in{\mathbb{S}}_N}\sum_{i=1}^N w_{\pi(i)}\cdot\nabla_{z_\pi(i)}\mathsf{d}\mathcal{F}[\iota_{Lio}(\underline{z}_N)](z_{\pi(1)},\ldots,z_{\pi(N)}). \end{align} $$

Since $\mathsf {d}\mathcal {F}[\iota _{Lio}(\underline {z}_N)]$ is symmetric with respect to exchange of particle labels, the right-hand side simplifies to

(4.14) $$ \begin{align} \sum_{i=1}^N w_i\cdot\nabla_{z_i}\mathsf{d}\mathcal{F}[\iota_{Lio}(\underline{z}_N)](z_1,\ldots,z_N). \end{align} $$

Returning to our starting identity (4.12), the arbitrariness of $\underline {w}_N$ and the uniqueness of the gradient field imply that

(4.15) $$ \begin{align} \nabla_{z_i}F(z_1,\ldots,z_N) = \nabla_{z_i}\mathsf{d}\mathcal{F}[\iota_{Lio}(\underline{z}_N)](z_1,\ldots,z_N), \qquad \forall 1\leq i\leq N. \end{align} $$

With this identity, we compute

(4.16)

Since is symmetric with respect to exchange of particle labels, the last line may be rewritten as

(4.17)

which is exactly what we needed to show.

Using a similar argument, we can also prove that the empirical measure map $\iota _{EM}$ from equation (2.52) is also a Poisson morphism. This then proves Proposition 2.15. First, a technical lemma, analogous to Lemma 4.2, for the Gâteaux derivatives of $\iota _{EM}$ . We leave the proof to the reader.

Lemma 4.4. It holds that $\iota _{EM} \in \mathcal {C}^\infty (({\mathbb {R}}^{2d})^N, \mathfrak {g}_1^*)$ and for every $n\in {\mathbb {N}}$ , $\underline {z}_N, \underline {w}_N^1,\ldots ,\underline {w}_N^n\in ({\mathbb {R}}^{2d})^N$ ,

(4.18) $$ \begin{align} \mathsf{d}^n\iota_{EM}[\underline{z}_N](\underline{w}_N^1,\ldots,\underline{w}_N^n) = \frac1N \sum_{j=1}^N (-1)^n\nabla^{\otimes n}\delta_{z_j} : \bigotimes_{k=1}^{n} w_{j}^{k}. \end{align} $$

Proof of Proposition 2.15

Let $\mathcal {F},\mathcal {G}\in \mathcal {C}^\infty (\mathfrak {g}_1^*)$ , and set and , which belong to $\mathcal {C}^\infty (({\mathbb {R}}^{2d})^N)$ . By the chain rule, we have the identity

(4.19) $$ \begin{align} \sum_{i=1}^N \nabla_{z_i} F(\underline{z}_N) \cdot w_i =\mathsf{d}\mathcal{F}[\iota_{EM}(\underline{z}_N)](\mathsf{d}\iota_{EM}[\underline{z}_N](\underline{w}_N)), \qquad \forall \underline{w}_N \in ({\mathbb{R}}^{2d})^N. \end{align} $$

Identifying $\mathsf {d}\mathcal {F}[\iota _{EM}(\underline {z}_N)]$ as an element of $\mathfrak {g}_1$ and using the identity (4.18) specialized to $n=1$ , the preceding right-hand side equals

(4.20) $$ \begin{align} \left\langle {\mathsf{d}\mathcal{F}[\iota_{EM}(\underline{z}_N)],\mathsf{d}\iota_{EM}[\underline{z}_N](\underline{w}_N)} \right\rangle _{\mathfrak{g}_1-\mathfrak{g}_1^*} = \frac{1}{N}\sum_{j=1}^N w_{j}\cdot(\nabla_{z}\mathsf{d}\mathcal{F}[\iota_{EM}(\underline{z}_N)])(z_j). \end{align} $$

Returning to our starting identity (4.19), the arbitrariness of $\underline {w}_N$ and the uniqueness of the gradient field imply

(4.21) $$ \begin{align} \nabla_{z_i}F(z_1,\ldots,z_N) = \frac1N(\nabla_{z}\mathsf{d}\mathcal{F}[\iota_{EM}(\underline{z}_N)])(z_i), \qquad \forall 1\leq i\leq N. \end{align} $$

With this identity, we compute

(4.22)

But

(4.23)

which equals the final two lines of equation (4.22). Thus, the proof is complete.

We close this subsection by proving the Hamiltonian formulation of the Liouville equation as given by Proposition 2.3. The reader will recall from equation (2.16) the definition of the Liouville Hamiltonian functional $\mathcal {H}_{Lio}$ .

Proof of Proposition 2.3

We remark that since $\mathcal {H}_{Lio}$ is a linear functional, it is trivial that we have the identification $\mathsf {d}\mathcal {H}_{Lio}[\gamma ] = \mathcal {H}_{New}\in \mathfrak {g}_N$ for every $\gamma \in \mathfrak {g}_N^*$ . To find a formula for the Hamiltonian vector field $X_{\mathcal {H}_{Lio}}$ with respect to the Poisson bracket , we compute for any $\mathcal {F}\in \mathcal {C}^\infty (\mathfrak {g}_N^*)$ and $\gamma \in \mathfrak {g}_N^*$ ,

(4.24)

where the ultimate line follows from unpacking the definition of and integration by parts. Since the second entry of the pairing $ \left \langle {\cdot ,\cdot } \right \rangle _{\mathfrak {g}_N-\mathfrak {g}_N^*}$ in the last line satisfies the characterizing property of the Hamiltonian vector field (recall that $\mathcal {F}$ was arbitrary), the uniqueness of the vector field implies that

(4.25) $$ \begin{align} X_{\mathcal{H}_{Lio}}(\gamma) &= -N\sum_{i=1}^N\left(\operatorname{{\mathrm{div}}}_{x_i}(\nabla_{v_i}\mathcal{H}_{New} \gamma) - \operatorname{{\mathrm{div}}}_{v_i}(\nabla_{x_i}\mathcal{H}_{New}\gamma)\right) \nonumber\\ &= -\sum_{i=1}^N \left(v_i\cdot \nabla_{x_i}\gamma - \frac{2}{N}\sum_{\substack{1\leq j\leq N}}\nabla W(x_i-x_j)\cdot\nabla_{v_i} \gamma\right), \end{align} $$

where the second line follows from the product rule and the fact that $\nabla _{x_i}\nabla _{v_i}\mathcal {H}_{New} = \nabla _{v_i}\nabla _{x_i}\mathcal {H}_{New} = 0$ . Thus, we have shown that equation (1.9) is equivalent to

(4.26) $$ \begin{align} \dot{\gamma} = X_{\mathcal{H}_{Lio}}(\gamma), \end{align} $$

exactly as desired.

Remark 4.5. Together, Propositions 2.3 and 2.15 imply that the Liouville map $\iota _{Lio}$ sends solutions of the Newtonian N-particle system (1.6) to solutions of the N-particle Liouville equation (1.9). Indeed, if $\underline {z}_N \in \mathcal {C}^\infty (I; ({\mathbb {R}}^{2d})^N)$ is a solution to equation (1.6) on some interval I, define for every $t\in I$ . Using that $\iota _{Lio}^* \mathcal {H}_{Lio} = \mathcal {H}_{New}$ , which is easy to check from the symmetry of $\mathcal {H}_{New}$ , Proposition 2.4 implies

(4.27)

Since $\iota _{Lio}^*\mathcal {F}\in \mathcal {C}^\infty (({\mathbb {R}}^{2d})^N)$ by Lemma 4.2 and the chain rule, one has that

(4.28)

Since $\mathcal {F}\in \mathcal {C}^\infty (\mathfrak {g}_N^*)$ was arbitrary, the claim follows.

4.2 Lie algebra $\mathfrak {G}_N$ of N particle observables

In this subsection, we transition to discussing N-hierarchies, with the goal of proving Theorem 2.5, which asserts that is a Lie algebra in the sense of Definition 3.15. As sketched in Section 2.2, we accomplish this task through a series of lemmas.

The starting point is the introduction of the maps $\epsilon _{k,N}:\mathfrak {g}_k \rightarrow \mathfrak {g}_N$ for $N \geq k\geq 1$ , which in turn will be used to define a Lie bracket on the space of N-particle hierarchies of observables. For the reader’s benefit, we recall from Section 2.2 the definition of $\epsilon _{k,N}$ :

(4.29)

where and

(4.30)

For example, if $k =1$ , $N =2$ , and $f^{(1)} \in \mathcal {C}^\infty ({\mathbb {R}}^{2d})$ , then as a function we have

(4.31) $$ \begin{align} \epsilon_{1,2}(f^{(1)})(z_1,z_2) = \frac{1}{2} \left(f^{(1)}(z_1) + f^{(1)}(z_2)\right). \end{align} $$

In the sequel, it will be convenient to use the tuple shorthand $\mathbf {j}_{k} = (j_1,\ldots ,j_k)\in P_k^N$ , similarly $f_{\boldsymbol {j}_k}^{(k)}$ and $\underline {z}_{\boldsymbol {j}_k}$ .

The next two lemmas show that each map $\epsilon _{k,N}$ is continuous, linear and injective (cf. [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20, Lemmas 5.3 and 5.4]). In particular, the first two properties imply that $\epsilon _{k,N} \in \mathcal {C}^\infty (\mathfrak {g}_k,\mathfrak {g}_N)$ .

Proof. Linearity follows directly from the definition. For continuity, note that the spaces $\mathfrak {g}_k, \mathfrak {g}_N$ are Fréchet, so it suffices to show that for any sequence $(f_j)_{j=1}^\infty \subset \mathfrak {g}_k$ with $f_j \rightarrow f \in \mathfrak {g}_k$ , we have $\epsilon _{k,N} (f_j) \rightarrow \epsilon _{k,N}(f)$ in $\mathfrak {g}_N$ . By linearity, we may assume that $f_j \rightarrow 0$ . Now, for any compact set $K \subset ({\mathbb {R}}^{2d})^N$ and $j \in {\mathbb {N}}$ , we estimate using triangle inequality

(4.32) $$ \begin{align} \rho_{K,n} (\epsilon_{k,N}(f_j)) \leq \frac{1}{|P_k^N|} \sum_{ \boldsymbol{p}_k \in P_k^N} \sup_{|\alpha| \leq n} \|\partial^\alpha (f_{j})_{ \boldsymbol{p}_k}\|_{L^\infty(K)} \leq \rho_{K,n}(f_j), \end{align} $$

which converges to $0$ as $j \rightarrow \infty $ . Since $K, n$ were arbitrary, we have that $\epsilon _{k,N} (f_j) \rightarrow 0$ in $\mathfrak {g}_N$ .

Proof. Fix $1\leq k\leq N$ . To prove injectivity, we will show the contrapositive statement: If $f^{(k)} \neq 0$ , then $\epsilon _{k,N}(f^{(k)}) \neq 0$ . The argument presented below is a ‘classical version’ of the argument used to prove [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20, Lemma 5.4].

We introduce a parameter $n\in {\mathbb {N}}_0$ with $n<k$ . We say that $f^{(k)}$ has property $\mathbf {P}_n$ if the following holds: If $n=0$ , then there exists a point $z_0\in {\mathbb {R}}^{2d}$ such that

(4.33) $$ \begin{align} f^{(k)}(z_0,\ldots,z_0) \neq 0 \end{align} $$

and if $n\geq 1$ , then there exist points $z_0,\ldots ,z_n \in {\mathbb {R}}^{2d}$ such that

(4.34) $$ \begin{align} f^{(k)}(z_0^{\times k-n},z_1,\ldots,z_n) \neq 0, \end{align} $$

where and the Cartesian product is understood as vacuous when $n=k$ . Observe that $f^{(k)}$ always has property $\mathbf {P}_{k-1}$ , since $f^{(k)}$ is nonzero by assumption, therefore there exist points $z_0,\ldots ,z_{k-1}\in {\mathbb {R}}^{2d}$ such that $f^{(k)}(z_0,\ldots ,z_{k-1})\neq 0$ . We define the integer $n_{\min }$ by

(4.35)

To avoid confusion over notation, we first dispense with the trivial case $n_{\min } = 0$ . The definition of $\mathbf {P}_0$ implies that there exists a point $z_0\in {\mathbb {R}}^{2d}$ such that $f^{(k)}(z_0,\ldots ,z_0) \neq 0$ . It then follows from the definition of $f_{(j_1,\ldots ,j_k)}^{(k)}$ that $f_{(j_1,\ldots ,j_k)}^{(k)}(z_0^{\times N}) = f^{(k)}(z_0^{\times k})$ for each tuple $(j_1,\ldots ,j_k)\in P_k^N$ . Hence,

(4.36) $$ \begin{align} \epsilon_{k,N}(f^{(k)})(z_0^{\times N}) = f^{(k)}(z_0^{\times k}) \neq 0. \end{align} $$

We next consider the case $1\leq n_{\min }<k$ . The definition of $\mathbf {P}_{n_{\min }}$ implies that there exist points $z_0,\ldots ,z_{n_{\min }} \in {\mathbb {R}}^{2d}$ such that

(4.37) $$ \begin{align} f^{(k)}(z_0^{\times k-n_{\min}}, z_1,\ldots,z_{n_{\min}}) \neq 0. \end{align} $$

We claim that $\epsilon _{k,N}(f^{(k)})(z_0^{\times N-n_{\min }},z_1,\ldots ,z_{n_{\min }}) \neq 0$ . To see this, we observe from unpacking the definition of $\epsilon _{k,N}(f^{(k)})$ that

(4.38) $$ \begin{align} \epsilon_{k,N}(f^{(k)})(z_0^{\times N-n_{\min}},z_1,\ldots,z_{n_{\min}}) = C_{k,N}\sum_{\mathbf{j}_k \in P_{k}^N} f_{(j_1,\ldots,j_k)}^{(k)}(z_0^{\times N-n_{\min}}, z_1,\ldots,z_{n_{\min}}). \end{align} $$

For each $\mathbf {j}_k=(j_1,\ldots ,j_k) \in P_k^N$ , we can use the symmetry of $f^{(k)}$ to write

(4.39) $$ \begin{align} f_{(j_1,\ldots,j_k)}^{(k)}(z_0^{\times N-n_{\min}}, z_1,\ldots,z_{n_{\min}}) = f^{(k)}(w_1,\ldots,w_k), \end{align} $$

where for some $r\leq k$ , $w_1=\cdots =w_r = z_0$ and for $r<i\leq k$ , the $w_i$ are distinct elements of $\{z_1,\ldots ,z_{n_{\min }}\}$ . If $r> k-n_{\min }$ , then by definition of $n_{\min }$ , $f^{(k)}(w_1,\ldots ,w_k) = 0$ . Since only $n_{\min }$ coordinates of $(z_0^{\times N-n_{\min }}, z_1,\ldots ,z_{n_{\min }})$ are not equal to $z_0$ , we must have $r=k-n_{\min }$ . But in this case, the symmetry of $f^{(k)}$ implies

(4.40) $$ \begin{align} f^{(k)}(w_1,\ldots,w_k) = f^{(k)}(z_0^{\times k-n_{\min}},z_1,\ldots,z_{n_{\min}}) \neq 0, \end{align} $$

by choice of the points $z_0,z_1,\ldots ,z_{n_{\min }}$ . Therefore, we have shown that

(4.41) $$ \begin{align} C_{k,N}\sum_{\mathbf{j}_k \in P_{k}^N} f_{(j_1,\ldots,j_k)}^{(k)}(z_0^{\times N-n_{\min}}, z_1,\ldots,z_{n_{\min}}) &= C_{k,N}' f^{(k)}(w_1,\ldots,w_k) \nonumber\\ &= C_{k,N}'f^{(k)}(z_0^{\times k-n_{\min}},z_1,\ldots,z_{n_{\min}}) \nonumber\\ &\neq 0, \end{align} $$

where $C_{k,N}'$ is some other combinatorial factor depending on $k,N$ . This then implies $\epsilon _{k,N}(f^{(k)}) \neq 0$ , completing the proof of injectivity.

We now present a technical lemma which will be applied to prove Lemma 4.9 for the filtration property. It shows the commutativity of the following diagram.

Proof. Let $f \in \mathfrak {g}_a$ . Then by definition of $\epsilon _{a,b}$ and $\epsilon _{b,N}$ ,

(4.42) $$ \begin{align} \epsilon_{b,N} (\epsilon_{a,b}(f)) (\underline{z}_N) & = \frac{1}{|P_b^N|} \sum_{\mathbf{m}_b \in P_b^N} \epsilon_{a,b}(f)({\underline{z}}_{(m_1,m_2,\dots,m_b)}) \nonumber\\ & = \frac{1}{|P_b^N||P_a^b|} \sum_{\mathbf{m}_b \in P_b^N} \sum_{\mathbf{n}_a \in P_a^b} f(\underline{z}_{(m_{n_1},m_{n_2},\dots,m_{n_a})}). \end{align} $$

Fix an a-tuple $\mathbf {l}_a = (l_1,\ldots ,l_a) \in P_a^N$ . Let $\mathscr {A}_{\mathbf {l}_a}$ denote the set of b-tuples $\mathbf {m}_b\in P_b^N$ such that $\{l_1,\ldots ,l_a\}\subset \{m_1,\ldots ,m_b\}$ . Any element in $\mathbf {m}_b\in \mathscr {A}_{\mathbf {l}_a}$ is a permutation of $(l_1,\ldots ,l_a,j_1,\ldots ,j_{b-a})$ for some choice $1\leq j_1<\cdots < j_{b-a}\leq N$ not in $\{l_1,\ldots ,l_a\}$ . There are ${N-a \choose {b-a}}$ such choices $(j_1,\ldots ,j_{b-a})$ , and $b!$ permutations of b letters, hence

(4.43) $$ \begin{align} |\mathscr{A}_{\mathbf{l}_a}| = {N-a\choose b-a}b! = \frac{b!(N-a)!}{(N-b)!(b-a)!}. \end{align} $$

We also see that, given $\mathbf {m}_b \in \mathscr {A}_{\mathbf {l}_a}$ , there is a unique choice of indices $n_1,\ldots ,n_a\in \{1,\ldots ,b\}$ such that $(m_{n_1},\ldots ,m_{n_a}) = (l_1,\ldots ,l_a)$ . Let us denote this unique choice by $\mathbf {n}_{a,\mathbf {l}_a}$ . We write

(4.44) $$ \begin{align} \sum_{\mathbf{m}_b \in P_b^N}\sum_{\mathbf{n}_a \in P_a^b} (\cdots) = \sum_{\mathbf{l}_a \in P_a^N} \sum_{\substack{(\mathbf{m}_b,\mathbf{n}_a) \in P_b^N\times P_a^b \\ (m_{n_1},\ldots,m_{n_a}) = (l_1,\ldots,l_a)}} (\cdots) = \sum_{\mathbf{l}_a \in P_a^N} \sum_{(\mathbf{m}_b,\mathbf{n}_a) \in \mathscr{A}_{\mathbf{l}_a} \times \{\mathbf{n}_{a,\mathbf{l}_a}\}}(\cdots). \end{align} $$

For any $(\mathbf {m}_b,\mathbf {n}_a) \in \mathscr {A}_{\mathbf {l}_a} \times \{\mathbf {n}_{a,\mathbf {l}_a}\}$ , we have $f(\underline {z}_{(m_{n_1},\ldots ,m_{n_a})})=f(\underline {z}_{\mathbf {l}_a})$ . Returning to equation (4.42), this identity and equation (4.43) imply

(4.45) $$ \begin{align} \epsilon_{b,N} (\epsilon_{a,b}(f)) (\underline{z}_N) &= \frac{1}{|P_b^N||P_a^b|} \frac{b!(N-a)!}{(N-b)!(b-a)!}\sum_{\mathbf{l}_a \in P_a^N} f(\underline{z}_{\mathbf{l}_a}) \nonumber \\ &= \frac{(N-a)!}{N!}\sum_{\mathbf{l}_a \in P_a^N} f(\underline{z}_{\mathbf{l}_a}) \nonumber\\ &=\epsilon_{a,N}(f)(\underline{z}_N), \end{align} $$

where the penultimate line follows from simplification of the combinatorial factor in the first line and the ultimate line is tautological. Hence, the lemma is proved.

Lemma 4.9. ( $\epsilon _{k,N}$ filters) Let $N \in {\mathbb {N}}$ , and let $1 \leq \ell , j \leq N$ . Then for any $f^{(\ell )} \in \mathfrak {g}_\ell $ and $g^{(j)} \in \mathfrak {g}_j$ , there exists a unique $h^{(k)} \in \mathfrak {g}_k$ such that

(4.46)

given by

(4.47)

where and for $1\leq r\leq \min (\ell , j)$ , $f^{(\ell )}\wedge _r g^{(j)} \in \mathcal {C}^\infty (({\mathbb {R}}^{2d})^{\ell +j-r})$ is defined by

(4.48)

Proof of Lemma 4.9

Injectivity of the $\epsilon _{k,N}$ operators shows uniqueness. The case where $\ell + j -1 \geq N$ is trivial since the operator $\epsilon _{N,N}$ is the identity on $\mathfrak {g}_N$ . So, assume that $ \ell + j -1 < N$ , in which case $k=\ell +j-1$ . From the definition (4.29) of $\epsilon _{k,N}$ , we obtain the following equality of functions on $({\mathbb {R}}^{2d})^N$ :

(4.49)

To avoid any confusion over notation, we remind the reader that, here, $f_{\boldsymbol {m}_\ell }^{(\ell )}, g_{\boldsymbol {n}_j}^{(j)}$ are regarded as functions on $({\mathbb {R}}^{2d})^N$ , and therefore, for instance,

(4.50) $$ \begin{align} \nabla_{x_i}f_{\boldsymbol{m}_\ell}^{(\ell)}(\underline{z}_N) &= \lim_{h\rightarrow 0} \frac{f_{\boldsymbol{m}_\ell}^{(\ell)}(\underline{z}_N + h e_{i,x}) - f_{\boldsymbol{m}_\ell}^{(\ell)}(\underline{z}_N)}{h}, \end{align} $$

where $e_{i,x}$ denotes the basis vector in $({\mathbb {R}}^{2d})^N$ for the $x_i$ variable. Note that if $i\notin \{m_1,\ldots ,m_\ell \}\cap \{n_1,\ldots ,n_j\}$ , then $\nabla _{z_i}f_{\boldsymbol {m}_\ell }^{(\ell )} = 0$ or $\nabla _{z_i}g_{\boldsymbol {n}_j}^{(j)}=0$ . Thus,

(4.51) $$ \begin{align} \sum_{i=1}^N (\cdots) = \sum_{ \substack{ i \in \{m_1,\dots,m_\ell\} \cap \{n_1,\dots n_j\}}} (\cdots). \end{align} $$

In particular, if the cardinality of the intersection equals r, then there are only r indices i for which the expression inside the parentheses is possibly nonzero.

Let $ 1 \leq r \leq \min (\ell , j)$ . Let us count the number of elements in the set

(4.52) $$ \begin{align} \{(\boldsymbol{m}_\ell,\boldsymbol{n}_j)\in P_{\ell}^N \times P_j^N : |\{m_1,\ldots,m_\ell\}\cap \{n_1,\ldots,n_j\}| = r\}. \end{align} $$

For a positive integer q, let $\mathscr {P}_{q}^N$ denote the subsets of $\{1,\ldots ,N\}$ with cardinality q. As the reader may check, there is a bijection between the set (4.52) and the set of tuples

(4.53)

In words, $A_{o,r}$ is the set of r overlaps between the elements of $\boldsymbol {m}_\ell $ and $\boldsymbol {n}_j$ ; $A_{opos,m}, A_{opos,n}$ are the sets of indices corresponding to the placements of the overlaps in $\boldsymbol {m}_\ell ,\boldsymbol {n}_j$ , respectively; $A_{no,m},A_{no,n}$ are the remaining sets of elements in $\boldsymbol {m}_\ell ,\boldsymbol {n}_j$ , respectively, which do not overlap (and therefore which are disjoint from $A_{o,r}$ ); $\pi _{r,m},\pi _{r,n}$ are permutations of the sets $A_{opos,m}, A_{opos,n}$ ; and $\tau _{\ell -r,m},\tau _{j-r,n}$ are permutations of the sets $\{1,\ldots ,N\}\setminus A_{opos,m}, \{1,\ldots ,N\}\setminus A_{opos,n}$ , respectively. We note that

(4.54) $$ \begin{align} (A_{o,r}, A_{opos,m}, A_{no,m}, \pi_{r,m},\tau_{\ell-r,m}) \mapsto \boldsymbol{m}_\ell \end{align} $$

(4.55) $$ \begin{align} (A_{o,r}, A_{opos,n}, A_{no,n}, \pi_{r,n}, \tau_{j-r,n}) \mapsto \boldsymbol{n}_j \end{align} $$

define bijections.

Fix an overlap set $A_{o,r}$ , and fix nonoverlap sets $A_{no,m},A_{no,n}$ . If

(4.56) $$ \begin{align} \boldsymbol{m}_\ell \leftrightarrow (A_{o,r},A_{opos,m}, A_{no,m},\pi_{r,m},\tau_{\ell-r,m}), \end{align} $$

(4.57) $$ \begin{align} \boldsymbol{m}_\ell' \leftrightarrow (A_{o,r},A_{opos,m}', A_{no,m},\pi_{r,m}',\tau_{\ell-r,m}'), \end{align} $$

where $\leftrightarrow $ denotes the bijection, then the symmetry of $f^{(\ell )}$ implies $f_{\boldsymbol {m}_\ell }^{(\ell )} = f_{\boldsymbol {m}_\ell '}^{(\ell )}$ . Similarly, if

(4.58) $$ \begin{align} \boldsymbol{n}_j \leftrightarrow (A_{o,r}, A_{opos,n},A_{no,n},\pi_{r,n},\tau_{j-r,n}), \end{align} $$

(4.59) $$ \begin{align} \boldsymbol{n}_j' \leftrightarrow (A_{o,r},A_{opos,n}',A_{no,n},\pi_{r,n}',\tau_{j-r,n}'), \end{align} $$

then the symmetry of $g^{(j)}$ implies $g_{\boldsymbol {n}_j}^{(j)} = g_{\boldsymbol {n}_j'}^{(j)}$ . Since $|\mathscr {P}_{r}^\ell | = {\ell \choose r}$ , $|\mathscr {P}_{r}^j| = {j\choose r}$ and $|{\mathbb {S}}_{r}|=r!$ , it follows that

(4.60) $$ \begin{align} \sum_{\substack{(\boldsymbol{m}_\ell,\boldsymbol{n}_j) \in P_\ell^N\times P_j^N \\ |\{m_1,\dots,m_\ell\} \cap \{n_1,\dots n_j\}|=r}} \sum_{i\in \{m_1,\ldots,m_\ell\}\cap \{n_1,\ldots,n_j\}} \Big( \nabla_{x_i} f_{\boldsymbol{m}_\ell}^{(\ell)} \cdot\nabla_{v_i} g_{\boldsymbol{n}_j}^{(j)} - \nabla_{x_i} g_{\boldsymbol{n}_j}^{(j)} \cdot\nabla_{v_i} f_{\boldsymbol{m}_\ell}^{(\ell)}\Big)\nonumber\\ = {j\choose r}{\ell\choose r}r!\sum_{\substack{\mathcal{T}\\ A_{opos,m}=A_{opos,n} = \{1,\ldots,r\} \\ \pi_{r,n}=\pi_{r,m}}} \sum_{i=1}^r\Big( \nabla_{x_i} f_{\boldsymbol{m}_\ell}^{(\ell)} \cdot\nabla_{v_i} g_{\boldsymbol{n}_j}^{(j)} - \nabla_{x_i} g_{\boldsymbol{n}_j}^{(j)} \cdot\nabla_{v_i} f_{\boldsymbol{m}_\ell}^{(\ell)}\Big). \end{align} $$

Since there is a bijection between tuples $\boldsymbol {p}_{\ell +j-r} \in P_{\ell +j-r}^N$ and tuples

(4.61)

we conclude upon relabeling that equation (4.49) equals

(4.62) $$ \begin{align} \frac{N}{|P_\ell^N| |P_j^N|} \sum_{r = 1}^{\min( \ell,j)} {j\choose r}{\ell\choose r}r! \sum_{\boldsymbol{p}_{\ell+j-r}} \sum_{i=1}^r \Big(\nabla_{x_{p_i}} f_{\boldsymbol{p}_\ell}^{(\ell)} \cdot\nabla_{v_{p_i}} g_{(\boldsymbol{p}_r,\boldsymbol{p}_{\ell+1;\ell+j-r})}^{(j)} - \nabla_{x_{p_i}} g_{\boldsymbol{p}_j}^{(j)} \cdot\nabla_{v_{p_i}} f_{(\boldsymbol{p}_r,\boldsymbol{p}_{j+1;j+\ell-r})}^{(\ell)}\Big), \end{align} $$

where we have used the shorthand $(\boldsymbol {p}_r,\boldsymbol {p}_{\ell +1;\ell +j-r}) = (p_1,\ldots ,p_r,p_{\ell +1},\ldots ,p_{\ell +j-r})$ (similarly when $\ell $ and j are swapped). Note that for , the preceding sum is vacuous.

Using the $\wedge _r$ wedge product notation and the fact that the sum over $\mathbf {p}_{\ell +j-r}\in P_{\ell +j-r}^N$ is invariant under the ${\mathbb {S}}_{\ell +j-r}$ action, we can rewrite the expression (4.62) as

(4.63)

where to obtain the final line, we have used Lemma 4.8 with $a = \ell + j -r, b = \ell + j -1$ and have used the linearity of $\epsilon _{\ell +j-1,N}$ . Observing

(4.64) $$ \begin{align} \frac{N}{|P_\ell^N| |P_j^N|} \frac{N!}{(N-\ell-j+r)!} = \frac{(N-\ell)!(N-j)!}{(N-1)!(N-\ell-j+r)!}, \end{align} $$

we arrive at the stated assertion of the lemma. This completes the proof in all cases of $\ell ,j$ .

With Lemma 4.9 in hand, we can show that the expression (2.25) for is well defined. Indeed, fix $1\leq k<N$ , given $F=(f^{(\ell )})_{\ell =1}^N, G=(g^{(j)})_{j=1}^N\in \mathfrak {G}_N$ , Lemma 4.9 yields, for any $1\leq \ell ,j\leq N$ satisfying $\ell +j-1 = k$ ,

(4.65)

Summing over $\ell ,j$ such that $\ell +j-1=k$ and applying $\epsilon _{k,N}^{-1}$ to both sides, we arrive at equation (2.25). In fact, we have also obtained an explicit formula for :

(4.66)

If $k=N$ , then we should modify the preceding reasoning to sum over $\ell +j-1\geq N$ . In all cases of k, we obtain

(4.67)

Remark 4.11. Note that the constant

(4.68)

satisfies $\lim _{N \rightarrow \infty }C_{j \ell N r}= \lim _{N \rightarrow \infty }N^{1-r} = \mathbf {1}_{r = 1}$ . This observation will be used in Sections 5.1 and 5.2 to evaluate $N\rightarrow \infty $ limits of N-particle Lie and Lie–Poisson brackets.

We now have all the ingredients to prove Theorem 2.5.

Proof of Theorem 2.5

We first consider the algebraic part, which amounts to checking properties 1-3 from Definition 3.15. This part is similar to the algebraic portion of the proof of [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20, Proposition 2.1], but we present again the computations in an effort to make the present article self-contained.

The first two properties are a ready consequence of the definition of . For the third property, let $F,G,H\in \mathfrak {G}_N$ . We need to show that

(4.69)

Since $\epsilon _{k,N}$ is injective, it suffices to show that $\epsilon _{k,N}$ applied to the k-th component of the left-hand side of the preceding identity equals the zero element of $\mathfrak {g}_N$ . We only consider the case $1\leq k<N$ and leave the $k=N$ case as an exercise for the reader. Using the definition of the Lie bracket and bilinearity, we have the identities

(4.70)

(4.71)

(4.72)

Since $[\cdot ,\cdot ]_{\mathfrak {g}_N}$ is a Lie bracket and therefore satisfies the Jacobi identity, it follows that for fixed integers $1\leq \ell _1,\ell _2,\ell _3\leq N$ ,

(4.73)

Hence,

(4.74)

We now consider the analytic part, which amounts to checking the continuity of $[\cdot ,\cdot ]_{\mathfrak {G}_N}$ . We wish to show that

(4.75)

is continuous, for which it suffices to show that for each $k\in \{1,\ldots ,N\}$ , the map

is continuous. By the $\mathfrak {G}_N$ Lie bracket formula given in equation (4.67) and the continuity of $\epsilon _{k,N}$ from Lemma 4.6, the continuity of $(F,G)\mapsto [F,G]_{\mathfrak {G}_N}$ is then reduced to proving continuity of the map

(4.76) $$ \begin{align} \mathcal{C}^\infty(({\mathbb{R}}^{2d})^\ell) \times \mathcal{C}^\infty(({\mathbb{R}}^{2d})^j) \rightarrow \mathcal{C}^\infty(({\mathbb{R}}^{2d})^{\ell+j-r}), \qquad (f^{(\ell)},g^{(j)})\mapsto f^{(\ell)} \wedge_r g^{(j)}, \end{align} $$

where $\wedge _r$ is defined in equation (4.48). Since $\wedge _r$ is a linear combination of products of derivatives, it is continuous by a similar argument to the analytic part of the proof of Proposition 2.1.

We conclude this subsection with what we believe is an interesting fact relating the mappings $\epsilon _{k,N}$ with the notion of taking the marginal of a distribution, the latter notion being important to obtaining Hamiltonian vector field formulae in Sections 4.3 and 5.2. For each $1 \leq k < N$ , define the k-particle marginal mapping

(4.77) $$ \begin{align} \int_{({\mathbb{R}}^{2d})^{N-k}} d\underline{z}_{k+1;N} : \mathfrak{g}_N^* \rightarrow \mathfrak{g}_k^*, \end{align} $$

where, for $\gamma \in \mathfrak {g}_N^*$ , $\int _{({\mathbb {R}}^{2d})^{N-k}}d\underline {z}_{k+1;N}\gamma $ is the unique element in $\mathfrak {g}_k^*$ satisfying

(4.78)

We additionally define the mapping $\int dz_{N+1, N} : \mathfrak {g}_N^* \rightarrow \mathfrak {g}_N^*$ to be the identity map. Sometimes, we use the alternative notation $\int _{({\mathbb {R}}^{2d})^{N-k}}d\gamma (\cdot ,\underline {z}_{k+1;N})$ . Our duality result for the maps $\epsilon _{k,N}$ and $\int _{({\mathbb {R}}^{2d})^{N-k}}d\underline {z}_{k+1;N}$ is the following proposition.

Proposition 4.12. For $1 \leq k \leq N$ , the map $\int dz_{k+1,N}: \mathfrak {g}_N^*\rightarrow \mathfrak {g}_{k}^*$ is a continuous linear operator and we have the operator equality

(4.79) $$ \begin{align} \epsilon_{k,N}^* = \int d\underline{z}_{k+1;N}. \end{align} $$

Proof. Let $\gamma \in \mathfrak {g}_N^*$ and $\phi \in \mathfrak {g}_k$ . We calculate

(4.80) $$ \begin{align} \left\langle {\phi, \epsilon_{k,N}^*(\gamma)} \right\rangle _{\mathfrak{g}_k-\mathfrak{g}_k^*} = \left\langle {\epsilon_{k,N}(\phi), \gamma} \right\rangle _{\mathfrak{g}_N-\mathfrak{g}_N^*} = \left\langle {\frac{1}{|P_k^N|} \sum_{\mathbf{m}_k \in P_k^N} \phi_{\mathbf{m}_k}, \gamma} \right\rangle _{\mathfrak{g}_N-\mathfrak{g}_N^*} \end{align} $$

and

(4.81)

where . For each k-tuple $\mathbf {m}_k \in P_k^N$ , define the set

(4.82)

which has cardinality $(N-k)!$ . If $\sigma \in A(\mathbf {m}_k) \cap A(\mathbf {m}_k')$ , then $\mathbf {m}_k' = (\sigma (1),\ldots ,\sigma (k)) = \mathbf {m}_k$ , which implies that the sets $A(\mathbf {m}_k)$ are pairwise disjoint. Given a permutation $\sigma \in {\mathbb {S}}_N$ , set $\mathbf {m}_k = (\sigma (1),\ldots ,\sigma (k))$ . Then trivially, $\sigma \in A(\boldsymbol {m}_k)$ and we have shown the partition

(4.83) $$ \begin{align} \bigsqcup_{\mathbf{m}_k \in P_k^N} A(\mathbf{m}_k) = {\mathbb{S}}_N. \end{align} $$

Hence, we have

(4.84) $$ \begin{align} \sum_{\sigma \in {\mathbb{S}}_N } (\phi \otimes 1^{\otimes (N-k)})_\sigma &= \sum_{\mathbf{m}_k \in P_k^N} \sum_{\sigma \in A(\mathbf{m}_k)}(\phi \otimes 1^{\otimes (N-k)})_\sigma \nonumber\\ &= \sum_{\mathbf{m}_k \in P_k^N} \sum_{\sigma \in A(\mathbf{m}_k)} \phi_{\mathbf{m}_k} \nonumber\\ &=(N-k)! \sum_{\mathbf{m}_k \in P_k^N} \phi_{\mathbf{m}_k}. \end{align} $$

Recalling that $|P_k^N| = N!/(N-k)!$ , we have shown that equation (4.80) equals equation (4.81). This equality of operators proves the continuity of $\int _{({\mathbb {R}}^{2d})^{N-k}} d\underline {z}_{k+1;N}$ by Proposition 3.3 and Lemma 4.6.

4.3 Lie–Poisson space $\mathfrak {G}_N^*$ of N-hierarchies of states

We turn to proving Theorem 2.6, which asserts that there is a well-defined Lie–Poisson structure on the dual space $\mathfrak {G}_N^*$ of N-hierarchies of states. Later, in Section 6.2, we will use this Lie–Poisson structure to demonstrate a Hamiltonian formulation of the BBGKY hierarchy.

We start with a technical lemma, which is a straightforward consequence of Lemma 4.1.

Proof. Since each $\mathfrak {g}_k$ is reflexive by Lemma 4.1, $\mathfrak {G}_N$ is also reflexive since, by using once again [Reference Köthe and GarlingK6¨9, Proposition 2 §14, Chapter 3], we have the chain of isomorphisms

(4.85) $$ \begin{align} \mathfrak{G}_N^{**} = \left(\bigoplus_{k=1}^N \mathfrak{g}_k \right)^{**} \cong \left( \prod_{k=1}^N \mathfrak{g}_k^* \right)^{*} \cong \bigoplus_{k=1}^N \mathfrak{g}_k^{**} \cong \bigoplus_{k=1}^N \mathfrak{g}_k \cong \mathfrak{G}_N. \end{align} $$

Proof of Theorem 2.6

The proof is similar to that of Proposition 2.2 for $\mathfrak {g}_k^*$ , except now we will apply Theorem 3.20 with $E = \mathfrak {G}_N^*$ . First, note that by Lemma 4.14, Theorem 2.5 and equation (2.14), the bracket on $E^* = \mathfrak {G}_N^{**} $ is continuous, a fortiori hypocontinuous. It therefore suffices to show that E is a $k^\infty $ -space in the sense of Definition 3.18. To show this, note that for $n \in {\mathbb {N}}$ ,

(4.86) $$ \begin{align} (\mathfrak{G}_N^*)^n \cong \prod_{j=1}^n \prod_{k=1}^N \mathfrak{g}_k^*, \end{align} $$

which shows that $(\mathfrak {G}_N^*)^n$ is a finite product of (DF) Montel spaces $\mathfrak {g}_k^*$ (recall Lemma 4.1), hence $(DF)$ Montel itself. Therefore, Theorem 3.13 implies that $(\mathfrak {G}_N^*)^n$ is a sequential space, which in turn implies that $(\mathfrak {G}_N^*)^n$ is a k-space by Proposition 3.12. Since $n \in {\mathbb {N}}$ was arbitrary, it follows that $\mathfrak {G}_N^*$ is a $k^\infty $ -space.

Abstractly, Theorem 2.6 tells us that for any functional $\mathcal {G}\in \mathcal {C}^\infty (\mathfrak {G}_N^*)$ , there exists a unique Hamiltonian vector field $X_{\mathcal {G}} \in \mathcal {C}^\infty (\mathfrak {G}_N^*, \mathfrak {G}_N^*)$ characterized by the property that

(4.87)

For applications, in particular as it pertains to the BBGKY and Vlasov hierarchies (see Sections 6.2 and 6.3) and evaluating limits as $N\rightarrow \infty $ (see Section 5.1), it is useful to have an explicit formula for the Hamiltonian vector field $X_{\mathcal {G}}$ . We provide such a formula with the next proposition.

Proposition 4.15. If $\mathcal {G}\in \mathcal {C}^\infty (\mathfrak {G}_N^*)$ , then we have the following formula for the Hamiltonian vector field $X_{\mathcal {G}}$ with respect to the bracket : for $1\leq \ell \leq N$ and any $\Gamma =(\gamma ^{(k)})_{k=1}^N\in \mathfrak {G}_N^*$ ,

(4.88)

where , , and .

Before proceeding to the proof of Proposition 4.15, some explanation regarding the well definedness of the expression (4.88) is in order. First, thanks to the identification $\mathfrak {g}_j\cong \mathfrak {g}_j^{**}$ , each $\mathsf {d}\mathcal {G}[\Gamma ]^{(j)} \in \mathfrak {g}_j$ is a symmetric element of the test function space $\mathcal {C}^\infty (({\mathbb {R}}^{2d})^j)$ . Thus, the symmetrized function $\sum _{\mathbf {a}_r\in P_r^\ell } \mathsf {d}\mathcal {G}[\Gamma ]_{(\mathbf {a}_r,\ell +1,\ldots ,\ell +j-r)}^{(j)}$ is an element of $\mathfrak {g}_k$ , that is, a symmetric element of $\mathcal {C}^\infty (({\mathbb {R}}^{2d})^k)$ . Although $\gamma ^{(k)}\in \mathfrak {g}_k^*$ , that is, a symmetric distribution on $\mathcal {C}^\infty (({\mathbb {R}}^{2d})^k)$ , is not a function, the usual rules for multiplication of a distribution by a test function and differentiation of a distribution show that the Poisson bracket above is again a well-defined element of $\mathfrak {g}_k^*$ . Hence, by Proposition 4.12, we can take its $\ell $ -particle marginal, which is symbolically denoted in equation (4.88) by the integration over the last $k-\ell $ coordinates, to obtain an element of $\mathfrak {g}_\ell ^*$ . Thus, the right-hand side in equation (4.88) is nothing but a linear combination of elements in $\mathfrak {g}_\ell ^*$ , hence itself an element of $\mathfrak {g}_\ell ^*$ . With these clarifications, we turn to the proof of Proposition 4.15.

Proof of Proposition 4.15

To increase the transparency of our computations, it is convenient to use integral notation, instead of distributional pairings, throughout the proof. By definition of the $\mathfrak {G}_N^*$ Poisson bracket (2.28),

(4.89)

By the formula (4.67) for the $\mathfrak {G}_N$ Lie bracket,

(4.90)

where $C_{\ell jNr}$ is as above. To compactify the notation, let us set and , the dependence on $\Gamma $ being implicit. By definition of $\epsilon _{\ell +j-r,k}$ and using the invariance of the summation $\sum _{\boldsymbol {p}_{\ell +j-r}\in P_{\ell +j-r}^k}$ under the ${\mathbb {S}}_{\ell +j-r}$ action, we have that

(4.91)

where the ultimate line follows from the definition (4.48) of $\wedge _r$ . Thus, setting , we arrive at the identity

(4.92)

By introducing in the second term of the last line the relabeling $\boldsymbol {q}_{\ell +j-r}$ of $\mathbf {p}_{\ell +j-r}$ according to

(4.93)

we see that

(4.94) $$ \begin{align} \sum_{\mathbf{p}_{\ell+j-r}\in P_{\ell+j-r}^{k}}\sum_{i=1}^r\int_{({\mathbb{R}}^{2d})^k}d\gamma^{(k)}(\underline{z}_k) \Big(\nabla_{x_{p_i}}f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_{k})\cdot \nabla_{v_{p_i}}g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k) \nonumber\\ -\nabla_{x_{p_i}}g_{\mathbf{p}_j}^{(j)}(\underline{z}_k)\cdot \nabla_{v_{p_i}}f_{(\mathbf{p}_r,\mathbf{p}_{j+1;j+\ell-r})}^{(\ell)}(\underline{z}_k) \Big)\nonumber\\ = \sum_{\mathbf{p}_{\ell+j-r}\in P_{\ell+j-r}^{k}}\sum_{i=1}^r\int_{({\mathbb{R}}^{2d})^k}d\gamma^{(k)}(\underline{z}_k) \Big(\nabla_{x_{p_i}}f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_{k})\cdot \nabla_{v_{p_i}}g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k) \nonumber\\ -\nabla_{x_{p_i}}g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k)\cdot \nabla_{v_{p_i}}f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k)\Big). \end{align} $$

Integrating by parts (in the distributional sense) the $\nabla _{x_{p_i}}$ , we have that

(4.95) $$ \begin{align} & \int_{({\mathbb{R}}^{2d})^k} d\gamma^{(k)}(\underline{z}_k)\nabla_{x_{p_i}}f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k)\cdot \nabla_{v_{p_i}}g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k) \nonumber\\ &\quad = - \int_{({\mathbb{R}}^{2d})^k}d\gamma^{(k)}(\underline{z}_k) f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k) \operatorname{{\mathrm{div}}}_{x_{p_i}}\nabla_{v_{p_i}}g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k) \nonumber\\ &\qquad -\int_{({\mathbb{R}}^{2d})^k}d\nabla_{x_{p_i}}\gamma^{(k)}(\underline{z}_k)\cdot f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k)\nabla_{v_{p_i}} g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k). \end{align} $$

Similarly, integrating by parts the $\nabla _{v_{p_i}}$ ,

(4.96) $$ \begin{align} & -\int_{({\mathbb{R}}^{2d})^k} d\gamma^{(k)}(\underline{z}_k)\nabla_{v_{p_i}}f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k)\cdot \nabla_{x_{p_i}}g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k) \nonumber\\ &\quad = \int_{({\mathbb{R}}^{2d})^k} d\gamma^{(k)}(\underline{z}_k)f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k)\operatorname{{\mathrm{div}}}_{v_{p_i}}\nabla_{x_{p_i}}g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k)\nonumber\\ &\qquad + \int_{({\mathbb{R}}^{2d})^k} d\nabla_{v_{p_i}}\gamma^{(k)}(\underline{z}_k) \cdot f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k) \nabla_{x_{p_i}}g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k). \end{align} $$

Since $g^{(j)}$ is locally $\mathcal {C}^\infty $ , we have the equality $\operatorname {{\mathrm {div}}}_{x_{p_i}}\nabla _{v_{p_i}}g_{(\mathbf {p}_r,\mathbf {p}_{\ell +1;\ell +j-r})}^{(j)} = \operatorname {{\mathrm {div}}}_{v_{p_i}}\nabla _{x_{p_i}}g_{(\mathbf {p}_r,\mathbf {p}_{\ell +1;\ell +j-r})}^{(j)}$ . Therefore,

(4.97) $$ \begin{align} & \int_{({\mathbb{R}}^{2d})^k}d\gamma^{(k)}(\underline{z}_k) \Big(\nabla_{x_{p_i}}f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k)\cdot \nabla_{v_{p_i}}g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k) - \nabla_{v_{p_i}}f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k)\cdot \nabla_{x_{p_i}}g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k)\Big)\nonumber\\ &\quad =-\int_{({\mathbb{R}}^{2d})^k}d\nabla_{x_{p_i}}\gamma^{(k)}(\underline{z}_k)\cdot f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k)\nabla_{v_{p_i}} g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k) \nonumber\\ &\qquad + \int_{({\mathbb{R}}^{2d})^k} d\nabla_{v_{p_i}}\gamma^{(k)}(\underline{z}_k) \cdot f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k) \cdot \nabla_{x_{p_i}}g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k). \end{align} $$

Let us make the change of variable $\underline {z}_k\mapsto \underline {w}_k$ according to

(4.98) $$ \begin{align} \underline{w}_k = (w_1,\ldots,w_k) = (z_{p_1},\ldots,z_{p_{\ell+j-r}}, z_{m_1},\ldots,z_{m_{k-(\ell+j-r)}}), \end{align} $$

where $m_1<\cdots <m_{k-(\ell +j-r)}$ is the increasing ordering of the set $\{1,\ldots ,k\} \setminus \{p_1,\ldots ,p_{\ell +j-r}\}$ . Write $w_i=(y_i,u_i)$ . Since $\gamma ^{(k)}$ is symmetric with respect to exchange of particle labels,

(4.99) $$ \begin{align} & -\int_{({\mathbb{R}}^{2d})^k}d\nabla_{x_{p_i}}\gamma^{(k)}(\underline{z}_k)\cdot f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k)\nabla_{v_{p_i}} g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k) \nonumber\\ &\qquad + \int_{({\mathbb{R}}^{2d})^k} d\nabla_{v_{p_i}}\gamma^{(k)}(\underline{z}_k) \cdot f_{\mathbf{p}_\ell}^{(\ell)}(\underline{z}_k) \nabla_{x_{p_i}}g_{(\mathbf{p}_r,\mathbf{p}_{\ell+1;\ell+j-r})}^{(j)}(\underline{z}_k) \nonumber\\ &\quad = - \int_{({\mathbb{R}}^{2d})^k}d\nabla_{y_i}\gamma^{(k)}(\underline{w}_k)\cdot f_{1;\ell}^{(\ell)}(\underline{w}_k)\nabla_{u_i}g_{(1;r, \ell+1;\ell+j-r)}^{(j)}(\underline{w}_k) \nonumber\\ &\qquad + \int_{({\mathbb{R}}^{2d})^k} d\nabla_{u_i}\gamma^{(k)}(\underline{w}_k)\cdot f_{1;\ell}^{(\ell)}(\underline{w}_k)\nabla_{y_i}g_{(1;r,\ell+1;\ell+j-r)}^{(j)}(\underline{w}_k). \end{align} $$

Next, since we have the partition

(4.100) $$ \begin{align} \bigsqcup_{k=1}^N \{(\ell,j)\in\{1,\ldots,N\}^2 : \min(\ell+j-1,N) = k\} = \{1,\ldots,N\}^2, \end{align} $$

we can interchange the order of summation over k and summation over $\ell ,j$ in the right-hand side of equation (4.92) to obtain

(4.101)

By the distributional Fubini–Tonelli theorem, the sum of the integrals in the right-hand side may be re-expressed as

(4.102) $$ \begin{align} \int_{({\mathbb{R}}^{2d})^{\ell}}d\underline{w}_\ell f^{(\ell)}(\underline{w}_\ell)\Bigg(\int_{({\mathbb{R}}^{2d})^{k-\ell}} d\underline{w}_{\ell+1;k}\Big(\nabla_{u_i}\gamma^{(k)}(\underline{w}_k)\cdot\nabla_{y_i}g_{(1;r,\ell+1;\ell+j-r)}^{(j)}(\underline{w}_k)\nonumber\\ -\nabla_{y_i}\gamma^{(k)}(\underline{w}_k)\cdot \nabla_{u_i}g_{(1;r,\ell+1;\ell+j-r)}^{(j)}(\underline{w}_k)\Big) \Bigg), \end{align} $$

where the inner integral should be understood as the $\ell $ -particle marginal (recall Proposition 4.12) of the distribution given by the integrand. Going forward, we return to the original $z,x,v$ notation.

We claim that we can rewrite the preceding right-hand side more concisely in terms of the canonical Poisson bracket on $({\mathbb {R}}^{2d})^k$ . Indeed, consider the expression

(4.103)

If $\beta \in \{\ell +1,\ldots ,k\}$ , then integrating by parts to swap $\nabla _{x_\beta }$ and $\nabla _{v_{\beta }}$ , we see that the resulting summand is zero. Therefore, only coordinates $\beta \in \{1,\ldots ,\ell \}$ produce a nonzero contribution. Similarly, for $\beta \in \{1,\ldots ,\ell \} \setminus \{a_1,\ldots ,a_r\}$ ,

(4.104) $$ \begin{align} \nabla_{x_\beta}g_{(\mathbf{a}_r,\ell+1,\ldots,\ell+j-r)}^{(j)} = \nabla_{v_\beta}g_{(\mathbf{a}_r,\ell+1,\ldots,\ell+j-r)}^{(j)} = 0. \end{align} $$

Therefore, by relabeling the sum over $\beta $ , we have the equality

(4.105) $$ \begin{align} &\sum_{\beta=1}^k \int_{({\mathbb{R}}^{2d})^{k-\ell}}d\underline{z}_{\ell+1;k}\Big(\nabla_{x_\beta}g_{(\mathbf{a}_r,\ell+1,\ldots,\ell+j-r)}^{(j)}\cdot\nabla_{v_\beta}\gamma^{(k)} - \nabla_{v_\beta}g_{(\mathbf{a}_r,\ell+1,\ldots,\ell+j-r)}^{(j)}\cdot\nabla_{x_\beta}\gamma^{(k)}\Big)\nonumber\\ &\quad =\sum_{i=1}^r \int_{({\mathbb{R}}^{2d})^{k-\ell}}d\underline{z}_{\ell+1;k}\Big(\nabla_{x_{a_i}}g_{(\mathbf{a}_r,\ell+1,\ldots,\ell+j-r)}^{(j)}\cdot\nabla_{v_{a_i}}\gamma^{(k)} - \nabla_{v_{a_i}}g_{(\mathbf{a}_r,\ell+1,\ldots,\ell+j-r)}^{(j)}\cdot\nabla_{x_{a_i}}\gamma^{(k)}\Big). \end{align} $$

We claim that

(4.106) $$ \begin{align} \sum_{\mathbf{a}_r\in P_r^\ell}\sum_{i=1}^r \int_{({\mathbb{R}}^{2d})^\ell}d\underline{z}_\ell f^{(\ell)}(\underline{z}_{\ell})\int_{({\mathbb{R}}^{2d})^{k-\ell}}d\underline{z}_{\ell+1;k}\Big(\nabla_{x_{a_i}}g_{(\mathbf{a}_r,\ell+1,\ldots,\ell+j-r)}^{(j)}\cdot\nabla_{v_{a_i}}\gamma^{(k)} \nonumber\\ - \nabla_{v_{a_i}}g_{(\mathbf{a}_r,\ell+1,\ldots,\ell+j-r)}^{(j)}\cdot\nabla_{x_{a_i}}\gamma^{(k)}\Big)(\underline{z}_k)\nonumber\\ ={\ell\choose r}r! \sum_{i=1}^r\int_{({\mathbb{R}}^{2d})^\ell}d\underline{z}_\ell f^{(\ell)}(\underline{z}_{\ell})\int_{({\mathbb{R}}^{2d})^{k-\ell}}d\underline{z}_{\ell+1;k}\Big(\nabla_{x_{i}}g_{(1,\ldots,r,\ell+1,\ldots,\ell+j-r)}^{(j)}\cdot\nabla_{v_{i}}\gamma^{(k)}\nonumber\\ - \nabla_{v_{i}}g_{(1,\ldots,r,\ell+1,\ldots,\ell+j-r)}^{(j)}\cdot\nabla_{x_{i}}\gamma^{(k)}\Big)(\underline{z}_k). \end{align} $$

Indeed, let $m_1<\cdots <m_{\ell -r}$ denote the increasing ordering of the set $\{1,\ldots ,\ell \}\setminus \{a_1,\ldots ,a_r\}$ , and consider the change of variable

(4.107) $$ \begin{align} \underline{w}_k = (w_1,\ldots,w_k) = (z_{a_1},\ldots,z_{a_r}, z_{m_1},\ldots,z_{m_{\ell-r}},z_{\ell+1},\ldots,z_{k}). \end{align} $$

Then since $f^{(\ell )}$ is symmetric with respect to permutation of particle labels,

(4.108) $$ \begin{align} f^{(\ell)}(z_1,\ldots,z_\ell) = f^{(\ell)}(w_1,\ldots,w_\ell). \end{align} $$

Similarly,

(4.109) $$ \begin{align} \ \ \ \nabla_{v_{a_i}}\gamma^{(k)}(\underline{z}_k) &= \nabla_{u_i}\gamma^{(k)}(\underline{w}_k),\qquad \end{align} $$

(4.110) $$ \begin{align} \ \ \ \kern-0.2pt\nabla_{x_{a_i}}\gamma^{(k)}(\underline{z}_k) &= \nabla_{y_i}\gamma^{(k)}(\underline{w}_k),\qquad \end{align} $$

(4.111) $$ \begin{align} \!\nabla_{x_{a_i}}g_{(\mathbf{a}_r,\ell+1,\ldots,\ell+j-r)}^{(j)}(\underline{z}_k) &= \nabla_{y_i}g_{(1,\ldots,r,\ell+1,\ldots,\ell+j-r)}^{(j)}(\underline{w}_k), \end{align} $$

(4.112) $$ \begin{align} \nabla_{v_{a_i}}g_{(\mathbf{a}_r,\ell+1,\ldots,\ell+j-r)}^{(j)}(\underline{z}_k) &= \nabla_{u_i}g_{(1,\ldots,r,\ell+1,\ldots,\ell+j-r)}^{(j)}(\underline{w}_k). \end{align} $$

Making the change of variable $\underline {z}_k\mapsto \underline {w}_k$ in the first two lines of equation (4.106) and recalling that $|P_{r}^\ell | = {\ell \choose r}r!$ , we arrive at the desired conclusion. Therefore, we have shown that

(4.113)

which, upon recalling that $f^{(\ell )} = \mathsf {d}\mathcal {F}[\Gamma ]^{(\ell )}$ and $g^{(j)} = \mathsf {d}\mathcal {G}[\Gamma ]^{(j)}$ , implies that

(4.114)

completing the proof of the lemma.

4.4 Marginals

We close this section with the observation that the operation of forming an N-hierarchy of marginals from an N-particle distribution defines a map which is a Poisson morphism. The material in this subsection closely parallels that of [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20, Section 5.3] for the quantum setting, as the underlying algebraic structure is the same. Therefore, we shall be brief in our remarks.

Proposition 4.17 below states that there is a linear homomorphism of Lie algebras $\mathfrak {G}_{N}\rightarrow \mathfrak {g}_{N}$ induced by the embeddings $\{\epsilon _{k,N}\}_{k=1}^N$ . The proof carries over verbatim from [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20, Proposition 5.28].

Proposition 4.17. For any $N\in {\mathbb {N}}$ , the map $\iota _{\epsilon }: \mathfrak {G}_{N} \rightarrow \mathfrak {g}_{N}$ defined by

(4.115)

is a continuous linear homomorphism of Lie algebras.

The dual of a Lie algebra homomorphism is automatically a Poisson morphism between the induced Lie–Poisson structures [Reference Marsden and RatiuMR13, Proposition 10.7.2].Footnote ⁶ Therefore, by showing that the map $\iota _{mar} : \mathfrak {g}_N^* \rightarrow \mathfrak {G}_N^*$ defined by

(4.116)

is the dual of the map $\iota _{\epsilon }$ , it follows that $\iota _{mar}$ is a Poisson morphism. The proof is completely analogous to that of [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20, Proposition 5.29], replacing the trace pairing by the duality pairing $ \left \langle {\cdot ,\cdot } \right \rangle _{\mathfrak {G}_N-\mathfrak {G}_N^*}$ ; therefore, we omit the details.

5.1 The Lie algebra $\mathfrak {G}_\infty $ of observable $\infty $ -hierarchies

We recall from equation (2.33) that $\mathfrak {G}_\infty = \bigoplus _{k=1}^\infty \mathfrak {g}_k$ equipped with the locally convex direct sum topology, and we can identify $\mathfrak {G}_N$ as a subspace via inclusion. We also recall that any element $F\in \mathfrak {G}_\infty $ belongs to $\mathfrak {G}_N$ for all N sufficiently large. The goal of this subsection is to prove Theorem 2.8, asserting that for any $F,G\in \mathfrak {G}_\infty $ , exists and defines a Lie bracket for $\mathfrak {G}_\infty $ . In contrast to the quantum setting (see [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20, Section 6.2]), the limit $\mathfrak {G}_\infty $ of the spaces $\mathfrak {G}_N$ is large enough to contain all $\infty $ -hierarchies of observables of interest. This is a technical advantage of the classical setting versus the quantum setting.

Proof of Theorem 2.8

We first show the limit (2.34). Fix $F=(f^{(\ell )})_{\ell =1}^\infty , G=(g^{(j)})_{j=1}^\infty \in \mathfrak {G}_\infty $ and $k\in {\mathbb {N}}$ . Let $M_0 \in {\mathbb {N}}$ be such that $f^{(\ell )}=g^{(j)}=0$ for $\min (\ell ,j)>M_0$ . Note that for $k=\ell +j-1\geq 2M_0$ , we must have $\max (\ell ,j)>M_0$ , implying

(5.1) $$ \begin{align} f^{(\ell)}\wedge_r g^{(j)} = 0, \qquad 1\leq r\leq \min(\ell,j). \end{align} $$

For any $N\geq M_0$ , $F,G$ can be identified as elements of $\mathfrak {G}_N$ by projection onto the first N components, without any loss of information. For $1\leq k\leq N$ , the formula (4.67) gives

(5.2)

where we recall that $C_{\ell jNr} = \frac {(N-\ell )!(N-j)!}{(N-1)! (N-\ell -j +r)!}$ and $r_0 =\max (1,\ell +j-N)$ . Suppose $N\geq 2M_0$ . Then if $\ell +j-N>1$ , we have $\max (\ell ,j)>M_0+1$ , implying either $f^{(\ell )}=0$ or $g^{(j)}=0$ , hence $\operatorname {\mathrm {Sym}}_{\ell +j-r}(f^{(\ell )}\wedge _r g^{(j)})=0$ . So there is no harm in starting the summation at $r=1$ in the right-hand side of equation (5.2). More importantly, we note that for any $N\geq 2M_0$ and $2M_0\leq k\leq N$ , we have . By Remark 4.11, we have $\lim _{N\rightarrow \infty } C_{\ell j Nr} = \mathbf {1}_{r=1}$ . Using that there are only finitely many terms in the right-hand side of equation (5.2) independent of N, we compute, for fixed k,

(5.3)

Observing that $\ell +j-1=k$ and therefore $\epsilon _{\ell +j-1,k}$ is the identity map $\mathfrak {g}_k\rightarrow \mathfrak {g}_k$ , we arrive at equation (2.34).

Next, we turn to showing that is a Lie algebra in the sense of Definition 3.15. This step, which is algebraic, consists of the verification that the bracket satisfies properties 1-3. The argument is essentially identical to that for the quantum case (cf. [Reference Mendelson, Nahmod, Pavlović, Rosenzweig and StaffilaniMNP⁺20, proof of Proposition 2.7]), therefore we omit the details.

Finally, we turn to the analytic matter of showing that is boundedly hypocontinuous, which is unique to this work. First, note that the sum defining is finite: For k fixed, there are k pairs $(\ell ,j) \in {\mathbb {N}}^2$ satisfying $\ell +j-1=k$ . Furthermore, our remarks at the beginning of the proof give that for $k>2M_0$ . Let $\mathcal {B} \subset \mathfrak {G}_\infty $ be bounded, and let $\mathsf {P}_{\ell }$ be the projection onto the $\ell $ -th component of $\mathfrak {G}_{\infty }$ . Then there exists an $N \in {\mathbb {N}}$ such that $\mathsf {P}_\ell (\mathcal {B}) = \{ 0 \}$ for $\ell> N$ and $\mathsf {P}_\ell (\mathcal {B})$ is bounded in $\mathfrak {g}_\ell $ for every $1\leq \ell \leq N$ (see [Reference Köthe and GarlingK6¨9, (4), p. 213]). Equicontinuity of now follows from the the equicontinuity of $ \wedge _1$ .

5.2 The weak Lie–Poisson space $\mathfrak {G}_\infty ^*$ of state $\infty $ -hierarchies

The objective of this subsection is to prove Theorem 2.11, asserting that the Lie bracket constructed in Theorem 2.8 induces a well-defined weak Lie–Poisson structure on $\mathfrak {G}_\infty ^*$ in the sense of Definition 3.23, if we choose equation (2.39) as our unital subalgebra ${\mathcal {A}}_\infty \subset \mathcal {C}^\infty (\mathfrak {G}_\infty ^*)$ .

The reader will recall Remark 2.10 that any expectation in ${\mathcal {A}}_\infty $ has constant Gâteaux derivative. We record below the following observation on the structure of elements of ${\mathcal {A}}_\infty $ , which will be crucial to verification of the weak Poisson properties 1–3 from Definition 3.23.

Remark 5.1. By definition, any element $\mathcal {F}\in {\mathcal {A}}_\infty $ takes the form

(5.4) $$ \begin{align} \mathcal{F} = \sum_{m=0}^\infty C_m\sum_{a=0}^{n_m}\mathcal{F}_{m1a}\cdots \mathcal{F}_{mma}, \end{align} $$

where, for each $m\in {\mathbb {N}}_0$ , $n_m\in {\mathbb {N}}_0$ , $\mathcal {F}_{m1a} = \left \langle {F_{m1a},\cdot } \right \rangle _{\mathfrak {G}_\infty -\mathfrak {G}_\infty ^*},\ldots ,\mathcal {F}_{mma}= \left \langle {F_{mma},\cdot } \right \rangle _{\mathfrak {G}_\infty -\mathfrak {G}_\infty ^*}$ for some $F_{1ma},\ldots ,F_{mma}\in \mathfrak {G}_\infty $ and $(C_m)_{m=0}^\infty $ are real coefficients such that there exists $M\in {\mathbb {N}}$ for which $C_m = 0$ if $m>M$ . In words, $\mathcal {F}$ is a linear combination of finite products of expectations. This observation will be quite useful in the sequel, as invocation of some form of linearity will allow us to verify certain identities under the assumption that $\mathcal {F}$ is just a finite product of expectations.

We break the proof of Theorem 2.11, which entails the verification of the properties 1–3, into a series of lemmas. We begin with the following technical lemma for the Gâteaux derivative of .

Lemma 5.2. If $\mathcal {G}_1=\mathcal {G}_{1,1}\cdots \mathcal {G}_{1,n_1}$ and $\mathcal {G}_2 = \mathcal {G}_{2,1}\cdots \mathcal {G}_{2,n_2}$ are the product of $n_1$ and $n_2$ expectations in ${\mathcal {A}}_{\infty }$ , respectively, then through the isomorphism $\mathfrak {G}_\infty ^{**} \cong \mathfrak {G}_\infty $ , the Gâteaux derivative at the point $\Gamma \in \mathfrak {G}_\infty ^*$ may be identified with

(5.5)

In particular, if $\mathcal {G}_1,\mathcal {G}_2$ are expectations, that is $\mathcal {G}_q(\Gamma ) = \left \langle {\mathsf {d}\mathcal {G}_q[0],\Gamma } \right \rangle _{\mathfrak {G}_\infty -\mathfrak {G}_\infty ^*}$ , then may be identified with the element

(5.6)