# Variable-moment fluid closures with Hamiltonian structure

J. W. Burby  Los Alamos National Laboratory, Theoretical Division, Los Alamos,
NM 87545, USA<EMAIL_ADDRESS>
###### Abstract

Based on ideas due to Scovel-Weinstein, I present a general framework for
constructing fluid moment closures of the Vlasov-Poisson system that exactly
preserve that system’s Hamiltonian structure. Notably, the technique applies
in any space dimension and produces closures involving arbitrarily-large
finite collections of moments. After selecting a desired collection of
moments, the Poisson bracket for the closure is uniquely determined. Therefore
data-driven fluid closures can be constructed by adjusting the closure
Hamiltonian for compatibility with kinetic simulations.

##  1 Introduction

The simplest non-dissipative kinetic model for a single-component plasma in
$N$ space dimensions is the Vlasov-Poisson system,

$\displaystyle\partial_{t}f+m^{-1}p_{i}\,\partial_{q^{i}}f-e\,m^{-1}\partial_{q^{i}}V\partial_{p_{i}}f=0,\quad
f:\mathbb{R}_{q}^{N}\times\mathbb{R}_{p}^{N}\rightarrow\mathbb{R}$
$\displaystyle-\Delta V=4\pi e\int f\,dp-4\pi en_{0},\quad
V:\mathbb{R}_{q}^{N}\rightarrow\mathbb{R}.$

Here $f=f(q,p)$ denotes the Vlasov distribution function and $V=V(q)$ is the
electrostatic potential. Particles that comprise the sole plasma species have
charge $e$, mass $m$, position $q=(q^{1},\dots,q^{N})$, and momentum
$p=(p_{1},\dots,p_{N})$. A uniform neutralizing background $n_{0}$ ensures
overall charge neutrality. The explicit time dependence of $f$ and $V$ has
been suppressed. This single-component model generalizes simply to allow for
multiple plasma species.

Kinetic plasma models, like the Vlasov-Poisson system, are formally equivalent
to infinite hierarchies of hydrodynamic equations for the fluid moments. The
degree-$k$ fluid moment $\bm{M}^{k}=M_{i_{1}\dots
i_{k}}\,dq^{i_{1}}\otimes\dots\otimes dq^{i_{k}}$ is a symmetric tensor whose
components are given by the momentum space integrals

$\displaystyle M_{i_{1}\dots i_{k}}(q)=\int p_{i_{1}}\dots
p_{i_{k}}\,f(q,p)\,dp,\quad i_{\ell}=1,\dots,N,\quad k=0,1,2,\dots.$

In general, the evolution equation for the degree-$k$ fluid moment couples to
the degree-$(k+1)$ fluid moment, a phenomenon that epitomizes the so-called
moment closure problem. For example, evolution equations for the first three
fluid moments $(M^{0},M^{1},M^{2})=(n,\bm{P},\mathbb{S})$ are

$\displaystyle\partial_{t}n+m^{-1}\nabla\cdot\bm{P}=0$ (1)
$\displaystyle\partial_{t}\bm{P}+m^{-1}\nabla\cdot\mathbb{S}=-e\,n\,\nabla{V}$
(2)
$\displaystyle\partial_{t}\mathbb{S}+m^{-1}\nabla\cdot\mathcal{T}=-e(\nabla{V}\bm{P}+\bm{P}\nabla{V}),$
(3)

where $\mathcal{T}=M^{3}$ denotes the degree-$3$ fluid moment. Any
prescription for expressing the degree-$(k+1)$ fluid moment as a function of
lower-degree fluid moments is known as a fluid closure. Upon introducing a
fluid closure, the rate of change of the first $k$ fluid moments becomes a
function of only those first $k$ fluid moments. Thus, it becomes feasible to
solve a finite collection of hydrodynamic equations to find time evolution of
the first $k$ fluid moments. Of course, there is no guarantee that moment
evolution determined by a fluid closure bears any relation to moment evolution
determined by the parent kinetic model, unless the fluid closure is chosen
wisely.

Traditional methods for developing fluid closures involve some combination of
asymptotic expansions[1, 2, 3, 4] and intuitive guesswork. More recently,
advances in machine learning[5, 6] have accelerated the development of data-
driven fluid closures, as well as other data-driven reduced-order models[7].
But fluid closures built using black-box neural networks are known to deliver
poor results when employed over large simulation time intervals. There is
therefore a growing effort to incorporate inductive bias into data-driven
fluid closures, with the hope of improving long-term predictions. For example,
Huang _et al._[8, 9] and Porteous _et al._[10] developed variable-moment
neural-network-based fluid closures that automatically enforce hyperbolicity
of the ensuing moment evolution equations.

This Article presents a framework for building variable-moment fluid closures
that automatically enjoy an important structural property inherent to non-
dissipative kinetic plasma models: Hamiltonian structure. The framework
involves two essential components. (1) Specification of a Poisson bracket for
the first $m_{0}$ fluid moments, where $m_{0}$ is any integer greater than
$1$. (2) Specification of the plasma system energy as a function of the first
$m_{0}$ fluid moments. The form of the Poisson bracket only depends on the the
number of moments $m_{0}$ included in a desired closure. On the other hand,
the form of the energy function, or Hamiltonian $\mathcal{H}$, is free to
specify. Parameterizing $\mathcal{H}$ using a neural network as in Hamiltonian
neural networks[11], or a library of candidate terms[12, 13], offers a novel
path toward data-driven closure modeling for non-dissipative kinetic plasmas.
As presented here, the framework applies specifically to fluid closures of the
Vlasov-Poisson system in $N$ space dimensions. However, similar ideas apply to
other non-dissipative kinetic plasma models, such as the Vlasov-Maxwell
system[14, 15, 16].

The framework still applies in the presence of dissipation caused by particle
collisions since a dissipative fluid closure should respect the
metriplectic[17] structure of collisional kinetic theory. Notably,
metriplectic dynamical systems always decompose as the sum of a Hamiltonian
part and a dissipative part; the framework developed here applies to the
Hamiltonian part.

The Hamiltonian structure underlying the _infinite_ hierarchy of fluid moments
for the Vlasov-Poisson system was first identified by Gibbons[18]. However,
finding true Poisson brackets for finite, but arbitrarily large collections of
fluid moments for the Vlasov-Poisson system represents an outstanding
theoretical challenge. The Poisson bracket for the closure framework presented
here provides the first complete solution. In the context of slab drift
kinetic models, Tassi[19, 20, 21] previously found Hamiltonian fluid moment
closures with any number of fluid moments. But the methods underlying this
result fail to extend to other dissipation-free kinetic models in any obvious
manner. In the Vlasov-Poisson context, Holm and Tronci[22] found Hamiltonian
fluid moment closures that include arbitrarily-large collections of fluid
moments that _do not_ include the density moment. Otherwise, the best
available partial solutions for Vlasov-Poisson work in one space dimension
$(N=1)$, where the tensorial nature of fluid moments does not appear, and fall
into two categories. The first category contains Hamiltonian moment closures
derived by way of direct imposition of the Jacobi identity — Jacobi closures.
The second category contains closures based on exact solutions of the parent
kinetic model that may be parameterized by fluid moments — exact reductions.

At a schematic level, derivations of Jacobi closures work by first
substituting an unknown closure relation into the Poisson bracket for the
parent kinetic model, and then finding conditions the closure relation needs
to satisfy in order for the closure bracket to satisfy the Jacobi identity.
These conditions take the form of nonlinear partial differential equations
(PDEs) that, in general, do not admit solutions or cannot be solved in closed
form. However, there are notable exceptions. Tassi[23] studied Jacobi closures
for drift kinetics that include two moments. Chandre and Shadwick[24] found a
Jacobi closure that accommodates all fluid moments with degree $k<5$ and
ensures stability of uniform solutions. This extends the earlier result[25] of
Perin _et al._ that also found Hamiltonian closures with $k<5$, but without
stability guarantees. The feasibility of finding Jacobi closures was first
established[26] by Perin _et al._ through a Jacobi-closure derivation of the
earliest Hamiltonian fluid model due to Morrison and Greene[27]. Jacobi
closures involving moments with degree $k<2$ are automatic; the algebra of
functionals that depend only on density and momentum density closes under the
Vlasov-Maxwell Poisson bracket. The same algebraic closure property is not
automatic for models that include moments with degrees $k\geq 2$. de Guillebon
and Chandre[28] dramatically highlighted this last point by revealing an
earlier fluid moment model[29] as a false-positive; a claimed Hamiltonian
fluid closure including moments with degree $k=2$ that was provably not
Hamiltonian. Jacobi closures that include fluid moments of arbitrarily large
degree have not yet been discovered.

Derivations of Hamiltonian moment closures based on exact reductions work like
the classical variation of parameters method. After postulating a parametric
form for the distribution function $f$, evolution equations for the parameters
are derived that ensure a time-evolving parametric $f$ solves the kinetic
equation exactly. This procedure only works for very special parametric $f$.
For example, a distribution function of the form
$f(q,p)=n(q)\,\delta(p-\bm{P}(q)/n(q))$, where $n$ is a positive function and
$\bm{P}$ is a vector field, solves the Vlasov-Poisson system if and only if
$(n,\bm{P})$ satisfies the Euler equations with a particular barotropic
equation of state — the exact reduction. The parameters for this
$\delta$-closure are just the fluid moments with degrees $k<2$; the Poisson
bracket among these parameters induced by the Vlasov-Poisson bracket recovers
the $k<2$ closure bracket mentioned above. Perin _et al._[30] initiated a
program aimed at finding moment closures with arbitrarily high-degree moments
by re-parameterizing a multiple waterbag exact reduction. (Perin _et al._[31]
also studied waterbags in the drift kinetic context.) Simple explicit formulas
express the fluid moments of multiple waterbags as polynomials in waterbag
parameters. Whenever these polynomial relationships can be inverted to give
waterbag parameters in terms of fluid moments a Hamiltonian moment closure
results. However, due to the general insolubility of large-degree polynomial
equations, the inversion has only been demonstrated for a single waterbag.
Like Jacobi closures, Hamiltonian closures based on exact reductions have yet
to accommodate fluid moments of arbitrarily large degree.

The Hamiltonian moment closures presented here fall into the exact reduction
category. The relevant parametric form for $f$ is given by

$\displaystyle f(q,p)$
$\displaystyle=n(q)\,\delta(p+d\psi(q))-\mu_{i_{1}}(q)\,\partial_{p_{i_{1}}}\delta(p+d\psi(q))$
$\displaystyle+\frac{1}{2}\,\mu_{i_{1}\,i_{2}}(q)\,\partial^{2}_{p_{i_{1}\,i_{2}}}\delta(p+d\psi(q))+\dots+\frac{(-1)^{m_{0}}}{m_{0}!}\mu_{i_{1}\dots
i_{m_{0}}}(q)\,\partial^{m_{0}}_{p_{i_{1}}\dots
p_{i_{m_{0}}}}\delta(p+d\psi(q)).$ (4)

Here, $n$ denotes the density, the gradient of the scalar function $\psi$
determines the center of the distribution function, and $\mu_{i_{1}\dots
i_{k}}$ denotes the degree-$k$ centered fluid moment. Section 5 contains a
detailed exposition. A simple upper-triangular transformation, Eq. (20),
relates the first $m_{0}$ centered fluid moments to the first $m_{0}$ standard
fluid moments. Hamiltonian closures based on this form of $f$ therefore
accommodate arbitrarily high-degree moments by increasing the integer $m_{0}$.
This feature of the closures, together with their applicability in every space
dimension $N$, sets them apart from all previous Hamiltonian fluid closures
that currently appear in the literature.

The ansatz (4) can reproduce the first $m_{0}$ moments of _any_ distribution.
The Poisson bracket associated with this exact reduction may therefore be used
to model the first $m_{0}$ moments of any distribution. In particular, it is
not necessary to assume that the distributions modeled by the new closures are
exactly of the form (4). This observation forms the basis for the proposed
closure modeling approach, described with greater detail in Section 6: fix the
Poisson bracket and incorporate all modeling choices into the form of the
Hamiltonian $\mathcal{H}$. However, it is important to keep in mind that the
theory developed here only guarantees the existence of closures that apply to
$f$ close to the form (4). Applications of the new closures to, e.g., bump-on-
tail distributions should be considered exploratory efforts.

Demonstrating that (4) comprises an exact reduction and uncovering the
corresponding Poisson bracket structure for general $m_{0}$ and $N$ requires
considerable bookkeeping technology. Fortunately, Scovel and Weinstein[32]
developed the necessary general machinery in the context of distributions $f$
localized near a point $z=(q,p)$ in single-particle phase space. (Channell[33]
provides additional insights into elements of the Scovel-Weinstein
construction.) Such distributions, which are relevant to self-consistent beam
dynamics in particle accelerators, also give rise to a moment closure problem,
where the moments are redefined as the phase space integrals

$\displaystyle m^{\nu_{1}\dots\nu_{k}}=\int z^{\nu_{1}}\dots
z^{\nu_{k}}f(z)dz,\quad\nu_{\ell}=1,\dots,2N.$

In order to find a finite-dimensional Poisson bracket that describes dynamics
of the first $m_{0}$ phase space moments, Scovel-Weinstein developed a
reduction strategy that can be applied to any Lie-Poisson Hamiltonian system
whose associated Lie algebra $\mathfrak{g}$ decomposes as a sum of subalgebras
$\mathfrak{g}=\mathfrak{b}+\mathfrak{s}$. (The Lie bracket on $\mathfrak{g}$
need not factor as the sum of brackets from $\mathfrak{b}$ and
$\mathfrak{s}$.) For the beam problem, $\mathfrak{g}$ comprises the phase
space functions with multiplication provided by the usual single-particle
Poisson bracket, $\mathfrak{b}$ contains the affine phase space functions, and
$\mathfrak{s}$ is the collection of phase space functions that vanish at
$(q,p)=(0,0)$, together with their first partial derivatives. For fluid
moments, the Lie algebra $\mathfrak{g}$ remains the same, but the subalgebras
$\mathfrak{b},\mathfrak{s}$ differ substantially. Section 2 summarizes the Lie
theory necessary to apply Scovel-Weinstein theory to fluid moments, and in
particular defines the subalgebras $\mathfrak{b}$ and $\mathfrak{s}$. Section
3 precisely states the elements of Scovel-Weinstein theory that enable
subsequent development of the new closure method in Sections 4, 5, and 6.

The following notation is used in what follows without further comment. If
$V=\mathbb{R}^{d}$ is any finite-dimensional vector space then $C(V)$ contains
smooth functions on $V$ with mild growth (e.g. at most polynomial grown) at
infinity. The dual space $C^{*}(V)$ contains rapidly decaying distributional
functions on $V$. The duality pairing between $\kappa\in C(V)$ and $\rho\in
C^{*}(V)$ is $\langle\rho,\kappa\rangle=\int_{V}\kappa(v)\,\rho(v)\,dv$, where
$dv$ denotes the Lebesgue measure on $\mathbb{R}^{d}$.

##  2 Lie theory for the contactomorphism group

A complete understanding of the Hamiltonian moment closures described in this
Article requires familiarity with some of the rich structure of the
contactomorphism group $G$. This Section aims to provide a self-contained
introduction to $G$ and its properties with greatest relevance to fluid
moments. For this Section and Section 3 only, readers should be familiar with
the basic mathematical theory of Lie groups at the level of Chapter 4 in the
book[34] by Abraham and Marsden. Marsden and Weinstein[15] were first to
recognize that $G$ is the Lie group that integrates the Lie algebra underlying
the Vlasov equation. Subsequently, Gay-Balmaz and Tronci[35] used structural
properties of contactomorphisms to show that the integrable Bloch-Iserles
system is geodesic flow on a subgroup of $G$. Gay-Balmaz and Tronci[36, 37]
also provide additional exposition concerning the contactomorphism group as
part of their ongoing efforts in quantum-classical hybrid modeling.

Let $Q\ni q$ be an $N$-dimensional vector space representing the configuration
space for an $N$-degree of freedom Hamiltonian system. The corresponding
momentum phase space is $P=Q\times Q^{*}\ni(q,p)$, where $Q^{*}$ denotes the
dual vector space to $Q$. The cotangent projection map is $\pi:P\rightarrow
Q:(q,p)\mapsto q$. Each $p\in Q^{*}$ is a linear mapping
$p:Q\rightarrow\mathbb{R}$. Application of the linear map $p\in Q^{*}$ to a
vector $q\in Q$ will be denoted $\langle p,q\rangle\in\mathbb{R}$. The
canonical $1$-form $\vartheta$ on $P$ is defined by $\vartheta=\langle
p,dq\rangle$. The corresponding canonical symplectic form is the $2$-form
$\omega=-d\vartheta$ given by exterior differentiation of $\vartheta$. If
$h:P\rightarrow\mathbb{R}$ is any smooth function on $P$, its partial
derivatives $\delta h/\delta q$ and $\delta h/\delta p$ are defined by the
directional derivative formula

$\displaystyle\frac{d}{d\epsilon}\bigg{|}_{0}h(q+\epsilon\,\delta
q,p+\epsilon\,\delta p)=\left\langle\frac{\delta h}{\delta q},\delta
q\right\rangle+\left\langle\delta p,\frac{\delta h}{\delta p}\right\rangle.$

Note that the value of $\delta h/\delta q$ at a particular $(q,p)\in P$ lies
in $Q^{*}$, while $\delta h/\delta p$ lies in $Q$. The Poisson bracket
associated with the canonical symplectic form $\omega$ is given on pairs of
functions $f,g:P\rightarrow\mathbb{R}$ by

$\displaystyle\\{f,g\\}=\left\langle\frac{\delta f}{\delta q},\frac{\delta
g}{\delta p}\right\rangle-\left\langle\frac{\delta g}{\delta q},\frac{\delta
f}{\delta p}\right\rangle.$

A (strict) contactomorphism $g=(\Phi,\varphi)$ is a pair comprising a
diffeomorphism $\Phi:P\rightarrow P$ and a smooth function
$\varphi:P\rightarrow\mathbb{R}$ that together satisfy the identity

$\displaystyle d\varphi+\Phi^{*}\vartheta=\vartheta.$

The product of a pair of contactomorphisms $g_{1},g_{2}$, denoted
$g_{1}g_{2}$, is

$\displaystyle
g_{1}g_{2}=(\Phi_{1}\circ\Phi_{2},\Phi_{2}^{*}\varphi_{1}+\varphi_{2}).$

When equipped with this product, the set of all contactomorphisms comprise a
group known as the (strict) contactomorphism group.

The contactomorphisms $b=(\Phi,\varphi)$ such that

$\displaystyle\Phi(q,p)=\bigg{(}q,p-\frac{\delta\psi}{\delta
q}\bigg{)},\quad\varphi(q,p)=\psi(q),\quad\psi\in C(Q).$

form a subgroup $B$, known as the group of fiber translations. Note that if
$\tau_{\psi}:P\rightarrow P$ denotes the mapping
$(q,p)\mapsto(q,p-\delta\psi/\delta q)$, then every fiber translation may be
written as $b=(\tau_{\psi},\pi^{*}\psi)$ for some unique $\psi$. The mapping
$I_{B}:B\rightarrow C(Q):(\tau_{\psi},\pi^{*}\psi)\mapsto\psi$, from $B$ to
the Abelian group of smooth functions on configuration space is a group
isomoprhism.

Let $Z=\\{(q,p)\in P\mid p=0\\}$ denote the zero section in phase space. The
submanifold $Z$ contains the static system states. The contactomorphisms
$s=(\Phi,\varphi)$ such that $\Phi(Z)=Z$ and $\varphi(Z)=0$ form a Lie
subgroup $S\subset G$. In this work, $S$ will be referred to as the group of
isostatic contactomorphisms. The isostatic contactomorphisms complement the
fiber translations in the following sense.

######  Theorem 1.

The product mapping

$\displaystyle m:B\times S\rightarrow G:(b,s)\mapsto bs,$

restricts to a diffemorphism from a neighborhood of $(e,e)\in B\times S$ to a
neighborhood of $e\in G$. In particular, for each $g\in G$ sufficiently close
to the identity there is a unique fiber translation $b$ and a unique isostatic
contactomorphism $s$ such that $g=bs$.

###### Proof.

The proof proceeds by constructing the inverse of $m_{0}$ on an appropriate
neighborhood $U_{G}$ of the identity in $G$. In particular, given $g\in G$ the
key step is finding a cooresponding $b=b(g)$ such that $s(g)=b(g)^{-1}g\in S$.
The inverse of $m_{0}$ is then $m^{-1}(g)=(b(g),s(g))$, since
$m(b(g),s(g))=b(g)s(g)=b(g)(b(g)^{-1}g)=g$.

Observe that the zero section $Z$ is a graph over $Q$. Let $U_{G}\subset G$
denote the set of contactomorphisms $g=(\Phi,\varphi)$ such that $\Phi(Z)$ is
a graph over $Q$. Given $g\in U_{G}$, the graph $\Phi(Z)$ is a Lagrangian
submanifold of $P$. There is therefore a function
$\psi:Q\rightarrow\mathbb{R}$ on configuration space such that $\Phi(Z)$ is
the graph of (minus) the differential of $\psi$,

$\Phi(Z)=\\{(q,p)\in P\mid p=-\delta\psi/\delta q\\}.$

In other words, $\psi$ is (minus) a generating function for the Lagrangian
submanifold $\Phi(Z)$. A generating function $\psi$ for $\Phi(Z)$ determines a
fiber translation $b=(\tau_{\psi},\pi^{*}\psi)$, which very nearly furnishes
the $b=b(g)$ mentioned above. However, there is more to do because
$(\tau_{\psi},\pi^{*}\psi)^{-1}g$ is not necessarily in $S$. Moreover, the
fiber translation $(\tau_{\psi},\pi^{*}\psi)$ is not uniquely determined by
$g$. In fact, $\psi^{\prime}=\psi+c$, $c\in\mathbb{R}$, also generates
$\Phi(Z)$, but determines a different fiber translation
$(\tau_{\psi^{\prime}},\pi^{*}\psi^{\prime})=(\tau_{\psi},\pi^{*}\psi+c)$.

The following argument shows that for each $g\in U_{G}$ there is a unique
generating function $\psi=\psi(g)$ for $\Phi(Z)$ such that
$s=(\tau_{\psi},\pi^{*}\psi)^{-1}g\in S$. Let $\iota_{Q}:Q\rightarrow
P:q\mapsto(q,0)$ denote the map that embeds $Q$ in $P$ as $Z$, and set
$(\Phi_{s},\varphi_{s})=s=(\tau_{\psi}^{-1}\circ\Phi,-\Phi^{*}\pi^{*}\psi+\varphi)$.
As mentioned already, there exists a (non-unique) $\psi$ that generates
$\Phi(Z)$. Since $\Phi(Z)$ is the graph of $-\delta\psi/\delta q$, the mapping
$\tau_{\psi}^{-1}$ maps the Lagrangian submanifold $\Phi(Z)$ back to $Z$, i.e.
$\Phi_{s}(Z)=\tau_{\psi}^{-1}(\Phi(Z))=Z$. Equivalently, there is a
diffeomorphism $\eta_{s}:Q\rightarrow Q$ such that
$\Phi_{s}\circ\iota_{Q}=\iota_{Q}\circ\eta_{s}$. Since $s$ is a
contactomorphism, $d\varphi_{s}+\Phi_{s}^{*}\vartheta=\vartheta$. Pulling back
this identity along $\iota_{Q}$ implies

$0=\iota_{Q}^{*}\vartheta=\iota_{Q}^{*}(d\varphi_{s}+\Phi_{s}^{*}\vartheta)=d\iota_{Q}^{*}\varphi_{s}+(\Phi_{s}\circ\iota_{Q})^{*}\vartheta=d\iota_{Q}^{*}\varphi_{s}+(\iota_{Q}\circ\eta_{s})^{*}\vartheta=d\iota_{Q}^{*}\varphi_{s}+\eta_{s}^{*}\iota_{Q}^{*}\vartheta=d\iota_{Q}^{*}\varphi_{s}.$

This means $\varphi_{s}(q,0)=\varphi(q,0)-\psi(\eta_{s}(q))$ is constant as a
function of $q$. Eliminate the ambiguity in the definition of $\psi$ by
requiring that $\varphi_{s}(q,0)=0$. Once this is done,
$b=b(g)=(\tau_{\psi(g)},\pi^{*}\psi(g))$ is uniquely determined by $g$ and
$b^{-1}g\in S$, as desired. The inverse of $m_{0}$ on $U_{G}$ is now
$m^{-1}(g)=(b(g),s(g))$, with $s(g)=b(g)^{-1}g$. ∎

The Lie algebra $\mathfrak{g}=T_{e}G$ of infinitesmial contactomorphisms
comprises pairs $(X,\Sigma)$, where $X$ is a vector field on $P$,
$\Sigma:P\rightarrow\mathbb{R}$ is a function on $P$, and
$d\Sigma+\mathcal{L}_{X}\vartheta=0.$ The adjoint action of $G$ and Lie
bracket on $\mathfrak{g}$ are given by

$\displaystyle\text{Ad}_{g}(X,\Sigma)$
$\displaystyle=(\Phi_{*}X,\Phi_{*}(\Sigma+\mathcal{L}_{X}\varphi)),\quad
g=(\Phi,\varphi),$ $\displaystyle[(X_{1},\Sigma_{1}),(X_{2},\Sigma_{2})]$
$\displaystyle=(-[X_{1},X_{2}],-\mathcal{L}_{X_{1}}\Sigma_{2}+\mathcal{L}_{X_{2}}\Sigma_{1}).$

The linear map $I_{\mathfrak{g}}:\mathfrak{g}\rightarrow
C(P):(X,\Sigma)\mapsto\Sigma+\vartheta(X)$, from infinitesimal
contactomorphisms to phase space functions under Poisson bracket is a Lie
algebra isomorphism. This observation recovers the well-known fact that the
contactomorphism group integrates the Lie algebra of phase space functions
equipped with the Poisson bracket.

The Lie subalgebra $\mathfrak{b}=T_{e}B\subset\mathfrak{g}$ of infinitesimal
fiber translations comprises pairs $(X,\Sigma)\in\mathfrak{g}$, where
$\Sigma=\pi^{*}\sigma$ for some $\sigma\in C(Q)$ and $X=X_{\pi^{*}\sigma}$.
The Lie bracket on $\mathfrak{b}$ vanishes because $B$ is Abelian. When
identifying $\mathfrak{g}$ with phase space functions _via_
$I_{\mathfrak{g}}$, $\mathfrak{b}$ corresponds to $p$-independent functions.
Denote the image Lie subalgebra of $\mathfrak{b}$ in
$(C(P),\\{\cdot,\cdot\\})$ as

$\displaystyle C_{\mathfrak{b}}(P)=\\{h\in C(P)\mid\delta h/\delta p=0\\}.$

The Lie subalgebra $\mathfrak{s}=T_{e}S\subset\mathfrak{g}$ of isostatic
infinitesimal contactomorphisms comprises pairs $(X,\Sigma)\in\mathfrak{g}$
such that $X$ is tangent to $Z$ and $\Sigma(Z)=0$. (In fact, the latter
condition implies the former, and so can be taken as the sole property
characterizing $\mathfrak{s}$ as a subspace of $\mathfrak{g}$.) The image Lie
subalgebra $I_{\mathfrak{g}}(\mathfrak{s})$ in $(C(P),\\{\cdot,\cdot\\})$ is

$\displaystyle C_{\mathfrak{s}}(P)=\\{h\in C(P)\mid h(Z)=0\\}.$

Theorem 1 implies that, as a vector space, the Lie algebra of infinitesimal
contactomorphisms is the direct sum of $\mathfrak{b}$ and $\mathfrak{s}$,
$\mathfrak{g}=\mathfrak{b}+\mathfrak{s}$. The corresponding statement in
$C(P)$, namely $C(P)=C_{\mathfrak{b}}(P)+C_{\mathfrak{s}}(P)$, may be
understood more simply as a byproduct of Taylor’s theorem with remainder.

The dual space $\mathfrak{g}^{*}$ comprises linear functions
$\mu:\mathfrak{g}\rightarrow\mathbb{R}$. Since
$\mathfrak{g}=\mathfrak{b}+\mathfrak{s}$, the duals to $\mathfrak{b}$ and
$\mathfrak{s}$ are subspaces of $\mathfrak{g}^{*}$ given by

$\displaystyle\mathfrak{b}^{*}=\\{\mu\in\mathfrak{g}^{*}\mid\langle\mu,\mathfrak{s}\rangle=0\\},\quad\mathfrak{s}^{*}=\\{\mu\in\mathfrak{g}^{*}\mid\langle\mu,\mathfrak{b}\rangle=0\\}.$

Similarly, the dual space to $C(P)$, $C^{*}(P)$, contains linear functions
${f}:C(P)\rightarrow\mathbb{R}$, and the duals $C_{\mathfrak{b}}^{*}(P)$,
$C_{\mathfrak{s}}^{*}(P)$, are the subspaces of $C^{*}(P)$ given by

$\displaystyle C_{\mathfrak{b}}^{*}(P)=\\{{f}\in
C^{*}(P)\mid\langle{f},C_{\mathfrak{s}}(P)\rangle=0\\},\quad
C_{\mathfrak{s}}^{*}(P)=\\{{f}\in
C^{*}(P)\mid\langle{f},C_{\mathfrak{b}}(P)\rangle=0\\}.$

The coadjoint action of $G$ on $\mathfrak{g}^{*}$ is given by
$\text{Ad}^{*}_{g}:\mathfrak{g}^{*}\rightarrow\mathfrak{g}^{*}:(X^{*},\varphi^{*})\mapsto(\text{Ad}_{g^{-1}})^{*}(X^{*},\varphi^{*})$.
The corresponding coadjoint action of $G$ on $C^{*}(P)$ is

$\displaystyle\text{Ad}^{*}_{g}{f}=\Phi_{*}{f},\quad{f}\in C^{*}(P),\quad
g=(\Phi,\varphi)\in G.$

Provided the space of phase space functions is restricted to smooth functions
with slow growth at infinity, the dual $C^{*}(P)$ contains all of the typical
smooth Vlasov densities. It includes more exotic functionals as well, such as
Dirac $\delta$-functions and their derivatives.

##  3 Scovel-Weinstein theory

The (infinite-dimensional) Hamiltonian structure underlying the Vlasov-Poisson
system is dual to the Lie algebra $C(P)$ of phase space functions under the
standard (finite-dimensional) Poisson bracket. The problem of finding reduced
models, in particular reduced fluid models, for Vlasov-Poisson may therefore
be approached at the level of the algebra $C(P)$. Adopting this point of view
requires addressing how the Lie algebra $C(P)$ might be “reduced." For the
purposes of this Article, any reduction should somehow encode the objects dual
to the first $m_{0}$ fluid moments, namely the phase space functions $h(q,p)$
that are polynomial in momentum $p$ with degree at most $m_{0}$.

Plausible reductions of $C(P)$ include proper Lie subalgebras: subspaces of
$C(P)$ that close under Poisson bracket. For example, the phase space
functions that only depend on the configuration variable $q$ (i.e. degree-$0$
momentum polynomials) comprise an Abelian Lie subalgebra dual to the space of
density moments. More generally, the momentum polynomials with degree at most
$1$ close under Poisson bracket because

$\displaystyle\left\\{\bigg{(}c_{1}(q)+c_{1}^{i}(q)p_{i}\bigg{)},\bigg{(}c_{2}(q)+c_{2}^{i}(q)p_{i}\bigg{)}\right\\}=-\bigg{(}c_{1}^{i}(q)\partial_{i}c_{2}(q)-c_{2}^{i}(q)\partial_{i}c_{1}(q)\bigg{)}-\bigg{(}c_{1}^{j}(q)\partial_{j}c_{2}^{i}(q)-c_{2}^{j}(q)\partial_{j}c_{1}^{i}(q)\bigg{)}p_{i}.$

However, by itself, subalgebra reduction of $C(P)$ provides insufficient
flexibility to accommodate fluid moments with degree between $0$ and $m_{0}$
when $m_{0}>1$. This follows from the failure of momentum polynomials with
degree at most $2$ to close under Poisson bracket.

Quotient algebra reduction offers a more nuanced alternative to subalgebra
reduction. Instead of attempting to encapsulate moments in a small subalgebra,
quotient algebra reduction aims to find _large_ ideals $\mathcal{I}$ in $C(P)$
such that the quotient space $C_{\mathcal{I}}(P)=C(P)/\mathcal{I}$ encodes
fluid moments. Here, ideals are subalgebras with an additional absorbing
property: the bracket of functions in $\mathcal{I}$ with _arbitrary_ functions
in $C(P)$ must be contained in $\mathcal{I}$. The subspace $\mathcal{I}$ being
an ideal ensures that the quotient $C_{\mathcal{I}}(P)$ inherits a Lie algebra
structure from $C(P)$. Unfortunately, as for subalgebra reduction, quotient
algebra reduction cannot accommodate the fluid moments with degree between $0$
and $m_{0}$ by itself. As described in detail within the proof of Theorem 2,
it _can_ successfully encode moments with degree between $1$ and $m_{0}$.

The paper[32] by Scovel and Weinstein devises an alternative Lie-theoretic
reduction strategy that hybridizes subalgebra reduction and quotient algebra
reduction. Developed originally for application to phase space moments,
Scovel-Weinstein theory adapts to the fluid moment context in straightforward
fashion. In fact, their methods apply any time a Lie group $G$ admits a
decomposition by Lie subgroups $B,S\subset G$. Here, decomposition means that
for each $g\in G$ there are unique $b\in B$ and $s\in S$ such that $g=bs$. In
the context of fluid moments, the group $G$ is the contactomorphism group, $B$
is the group of fiber translations, and $S$ is the group of isostatic
contactomorphisms. Each of these groups is described with some detail in
Section 2. The technical heart of Scovel-Weinstein theory may be described as
follows.

Suppose that $G$ is a Lie group that admits a decomposition by Lie subgroups
$B,S\subset G$, $G=BS$. The Lie algebra $\mathfrak{g}$ then admits a
corresponding direct sum decomposition
$\mathfrak{g}=\mathfrak{b}+\mathfrak{s}$ by Lie subalgebras. Likewise, the
dual space to $\mathfrak{g}$ admits the direct sum decomposition
$\mathfrak{g}^{*}=\mathfrak{b}^{*}+\mathfrak{s}^{*}$. The Scovel-Weinstein map
$\Gamma:(B\times\mathfrak{b}^{*})\times\mathfrak{s}^{*}\rightarrow\mathfrak{g}^{*}$
is given by

$\displaystyle\Gamma(b,b^{*},s^{*})=\text{Ad}^{*}_{b}(b^{*}+s^{*}).$

Here $\text{Ad}^{*}_{g}=(\text{Ad}_{g^{-1}})^{*}$ denotes the (left) coadjoint
action of $G$ on $\mathfrak{g}^{*}$. Let $\\{\cdot,\cdot\\}_{T^{*}B}$ denote
the Poisson bracket on $B\times\mathfrak{b}^{*}$ given by pulling back the
canonical Poisson bracket on $T^{*}B$ along the left trivialization map. Let
$\\{\cdot,\cdot\\}_{\mathfrak{s}^{*}}$ denote the Lie-Poisson bracket on
$\mathfrak{s}^{*}$. Finally, let $\\{\cdot,\cdot\\}_{\mathfrak{g}^{*}}$ denote
the Lie-Poisson bracket on $\mathfrak{g}^{*}$. The central result due to
Scovel-Weinstein is the following. When
$(B\times\mathfrak{b}^{*})\times\mathfrak{s}^{*}$ is equipped with the evident
product Poisson manifold structure, the Scovel-Weinstein map $\Gamma$ is a
Poisson map. In other words, the direct product Poisson bracket is compatible
with the Lie-Poisson bracket on $\mathfrak{g}$ according to

$\displaystyle\\{\Gamma^{*}f,\Gamma^{*}g\\}_{T^{*}B}+\\{\Gamma^{*}f,\Gamma^{*}g\\}_{\mathfrak{s}^{*}}=\Gamma^{*}\\{f,g\\}_{\mathfrak{g}^{*}},\quad
f,g\in C(\mathfrak{g}^{*}).$

Note that the space $(B\times\mathfrak{b}^{*})\times\mathfrak{s}^{*}$ is
larger that $\mathfrak{g}^{*}$ by the factor $B$. This extra variable $b$ is
the price paid for “untangling" the Lie-Poisson structure on
$\mathfrak{g}^{*}$ in terms of standard Poisson structures associated with
$\mathfrak{b}^{*}$ and $\mathfrak{s}^{*}$. Scovel-Weinstein advocate the
following interpretation of $\Gamma$ and its source and target spaces. In the
space $(B\times\mathfrak{b}^{*})\times\mathfrak{s}^{*}$ the factor
$\mathfrak{g}_{0}^{*}=\mathfrak{b}^{*}\times\mathfrak{s}^{*}$ represents a
“reference" copy of $\mathfrak{g}^{*}$ sitting at the identity $e$ in the
group $B$. The mapping $\Gamma$ advects the reference space
$\mathfrak{g}_{0}^{*}$ from $e$ to $b$.

This advection interpretation warrants further elaboration. Along a solution
of the Vlasov-Poisson system, the distribution function $f(q,p)$ is advected
by canonical transformations that prescribe the motion of individual particles
in phase space. These canonical transformations naturally reside with the
contactomorphism group $G$. The Scovel-Weinstein map $\Gamma$ may therefore be
interpreted as a general way of identifying special system motions advected by
a subgroup $B\subset G$.

Now suppose that the Lie subalgebra $\mathfrak{s}$ contains an ideal
$\mathcal{I}\subset\mathfrak{s}$. Note that ideals in $\mathfrak{s}$ are not
necessarily ideals in $\mathfrak{g}$! The quotient algebra
$\mathfrak{s}_{\mathcal{I}}=\mathfrak{s}/\mathcal{I}$ then inherits a Lie
algebra structure from $\mathfrak{s}$ by requiring that the quotient map
$\Pi_{\mathcal{I}}:\mathfrak{s}\rightarrow\mathfrak{s}_{\mathcal{I}}$ is a Lie
algebra homomorphism. If
$s_{1}+\mathcal{I},s_{2}+\mathcal{I}\in\mathfrak{s}_{\mathcal{I}}$ are
elements of the quotient space their Lie bracket is

$\displaystyle[s_{1}+\mathcal{I},s_{2}+\mathcal{I}]=[s_{1},s_{2}]+\mathcal{I}.$

Scovel-Weinstein highlight that since $\Pi_{\mathcal{I}}$ is a Lie algebra
homomorphism, the dual map
$\Pi_{\mathcal{I}}^{*}:\mathfrak{s}^{*}_{\mathcal{I}}\rightarrow\mathfrak{s}^{*}$
is a Poisson map. Therefore the map

$\displaystyle\text{id}\times\Pi_{\mathcal{I}}^{*}:(B\times\mathfrak{b}^{*})\times\mathfrak{s}^{*}_{\mathcal{I}}\rightarrow(B\times\mathfrak{b}^{*})\times\mathfrak{s}^{*},$

is Poisson as well, relative to the product Poisson manifold structures on the
source and target spaces. It follows that there is a second Poisson map
related to $\Gamma$ _via_ the quotient algebra construction. The reduced
Scovel-Weinstein map $\Gamma_{\mathcal{I}}$ is the Poisson map

$\displaystyle\Gamma_{\mathcal{I}}:(B\times\mathfrak{b}^{*})\times\mathfrak{s}^{*}_{\mathcal{I}}\rightarrow\mathfrak{g}^{*}$

given as the composition
$\Gamma_{\mathcal{I}}=\Gamma\circ[\text{id}\times\Pi_{\mathcal{I}}^{*}]$.
Theorem 3 below represents a restatement of the Poisson map property for
$\Gamma_{\mathcal{I}}$ within the fluid moment context.

##  4 Poisson bracket on the space of truncated fluid moment vectors

This Section uses Poisson-geometric and Lie-theoretic methods to identify a
Poisson bracket on a space that encodes density, fluid momentum, pressure, and
higher fluid moments up to some truncation order $m_{0}$. The precise
relationship between the Poisson bracket constructed here and the Lie Poisson
bracket on the space of distribution functions $f\in C^{*}(P)$ will be
described in Section 5.

A degree-$k$ fluid moment $M^{k}$, $k\in\\{0,1,2,\dots\\}$, is a symmetric,
covariant, degree-$k$ tensor field on configuration space $Q$. (Geometrically,
fluid moments are naturally tensor densities[38] instead of ordinary tensors,
but here these two kinds of objects are identified using the standard flat
metric on configuration space.) If $\\{q^{i}\\}_{i=1,\dots,N}$ denotes the
standard coordinate system on $Q=\mathbb{R}^{N}$, $M^{k}$ may be written as

$\displaystyle M^{k}=M_{i_{1}\,\dots\,i_{k}}\,dq^{i_{1}}\otimes\dots\otimes
dq^{i_{k}},\quad M_{i_{1}\,\dots\,i_{k}}\in C^{*}(Q).$

Any phase space distribution $f=f(q,p)$ that decays sufficiently fast as
$|p|\rightarrow\infty$ determines fluid moments of each degree by the formula

$\displaystyle M_{i_{1}\dots i_{k}}(q)=\int p_{i_{1}}\dots
p_{i_{k}}\,f(q,p)\,dp.$

Conversely, under certain conditions a sequence of fluid moments $M^{k}$,
$k=0,\dots,\infty$, determines a distribution $f$. This is true, for example,
when the moment generating function associated with the sequence $M^{k}$
exists. A fluid moment vector with degree between $n\geq 0$ and $m\geq n$ is a
tuple $\bm{M}=(M^{n},M^{n+1},\dots,M^{m})$ whose $i^{\text{th}}$ entry is a
fluid moment of degree $i$. Denote the set of all such fluid moment vectors
$\mathcal{M}_{n}^{m_{0}}$. Note that $\mathcal{M}_{k}^{k}$ is the vector space
of degree-$k$ fluid moments, and that

$\displaystyle\mathcal{M}_{n}^{m_{0}}=\bigoplus_{k=n}^{m_{0}}\mathcal{M}_{k}^{k}.$

Also note that elements of $\mathcal{M}_{0}^{0}$ are just fluid densities
$n\in C^{*}(Q)$.

Although specifying moments of each degree $k=0,\dots,\infty$ formally
specifies a unique phase space distribution, specifying only the moments with
degree $k\leq m_{0}$ does not. Additional data must be supplied to recover a
distribution function from a fluid moment vector in $\mathcal{M}_{0}^{m_{0}}$.
In this work that additional data will comprise a scalar configuration space
function $\psi\in C(Q)$ whose derivative $d\psi(q)$ specifies the “center" of
the distribution $f(q,p)$ at $q$. The precise goal of this Section is
constructing a Poisson bracket on the space

$\displaystyle P_{m_{0}}=C(Q)\times\mathcal{M}_{0}^{m_{0}}=C(Q)\times
C^{*}(Q)\times\mathcal{M}_{1}^{m_{0}}\ni(\psi,n,\bm{\mu}),$

for any non-negative integer $m_{0}$. Here the symbol $\bm{\mu}$ is used in
place of $\bm{M}$ to emphasize that the moment vector $\bm{\mu}$ contains
_centered moments_.

The Poisson bracket on $P_{m_{0}}$ originates from a Lie algebra structure on
the objects dual to fluid moment vectors: fluid comoment vectors. A fluid
comoment $C^{k}$ of degree $k\in\\{0,1,2,\dots\\}$ is a symmetric,
contravariant, degree-$k$ tensor field on configuration space $Q$. If
$\\{e_{i}\\}_{i\in\\{1,\dots,N\\}}$ denotes the standard basis for $Q$, evey
degree-$k$ fluid comoment may be expressed using index notation as

$\displaystyle C^{k}=C^{i_{1}\,i_{2}\,\dots\,i_{k}}e_{i_{1}}\otimes
e_{i_{2}}\otimes\dots\otimes e_{i_{k}},\quad C^{i_{1}\,i_{2}\,\dots\,i_{k}}\in
C(Q),$

where the array of component functions $C^{i_{1}\,i_{2}\,\dots\,i_{k}}$ is
symmetric under index permutation. A fluid comoment vector with degree between
$n\geq 0$ and $m\geq n$ is a tuple $\bm{C}=(C^{n},C^{n+1},\dots,C^{m})$ whose
$i^{\text{th}}$ entry is a fluid comoment of degree $i$. The vector space of
all fluid comoment vectors with degree between $n$ and $m_{0}$ is
$\mathcal{C}_{n}^{m_{0}}$. Fluid comoments with degrees $k=0,1,2$ will be
denoted using the special symbols $\sigma,\bm{w},\mathbb{Q}$, respectively.
The duality pairing between fluid moment vectors $\mathcal{M}_{n}^{m_{0}}$ and
fluid comoment vectors $\mathcal{C}_{n}^{m_{0}}$ is given by

$\displaystyle\langle\bm{M},\bm{C}\rangle=\sum_{k=n}^{m_{0}}\int M_{i_{1}\dots
i_{k}}(q)\,C^{i_{1}\dots i_{k}}(q)\,dq.$

By way of this pairing, the fluid moment vectors arise as the dual space to
fluid comoment vectors,
$\mathcal{M}_{n}^{m_{0}}=(\mathcal{C}_{n}^{m_{0}})^{*}$.

Fluid comoment vectors with degree between $1$ and $m_{0}\geq 1$ comprise a
Lie algebra. This result is striking because the comoment vectors with degree
between $0$ and $m_{0}$ do not comprise a Lie algebra! The following Theorem
exhibits the relevant Lie bracket and proves that it satisfies the Jacobi
identity.

######  Theorem 2.

Let $m_{0}\geq 1$ be an integer. The space $\mathcal{C}_{1}^{m_{0}}$ of fluid
comoment vectors with degree between $1$ and $m_{0}$ is a Lie algebra. The Lie
bracket between pairs of comoment vectors $\bm{C}_{1},\bm{C}_{2}$ is given by

$\displaystyle[\bm{C}_{1},\bm{C}_{2}]_{m_{0}}$
$\displaystyle=(B^{1},B^{2},\dots,B^{m_{0}})$ $\displaystyle B^{k}$
$\displaystyle=\sum_{k_{1}+k_{2}-1=k}\bigg{(}k_{2}\,\partial_{q^{I}}C_{1}^{(i_{1}\dots
i_{k_{1}}}C_{2}^{i_{k_{1}+1}\dots
i_{k})I}-k_{1}\,\partial_{q^{I}}C_{2}^{(i_{1}\dots
i_{k_{2}}}C_{1}^{i_{k_{2}+1}\dots
i_{k})I}\bigg{)}\,e_{i_{1}}\otimes\dots\otimes e_{i_{k}},$ (5)

where the sum is over all integer pairs $(k_{1},k_{2})$ with $k_{1},k_{2}\geq
1$ and $k_{1}+k_{2}-1=k$.

######  Remark 1.

This formula for the Lie bracket on $\mathcal{C}_{1}^{m_{0}}$ reproduces the
indicial form of the so-called Schouten concomitant[39, 38] between symmetric
contravariant tensors, modulo degree $m_{0}+1$. This observation is consistent
with the method of proof given below since the Schouten concomitant is the
unique Lie algebra structure on symmetric tensors that makes the map sending
symmetric tensors to momentum polynomials a Lie algebra homomorphism. However,
note that the symmetric contravariant tensors with degree between $1$ and
$m_{0}$ are _not_ closed under the Schouten concomitant. Instead, these
tensors possess a quotient Lie algebra structure. A similar quotient algebra
structure was first noticed by Channell[33] in the context of phase space
moments. Partially motivated by Channell’s observations, Holm and Tronci[22]
found the Lie-Poisson bracket associated with (5) without identifying its
corresponding Lie-algebraic origin. Therefore, Theorem 2 may be formulated in
the following alternative manner: the Poisson bracket for fluid moments with
degree between $1$ and $m_{0}$ presented in Holm and Tronci’s Section 2.3 is a
Lie Poisson bracket for the Lie algebra whose bracket is given by (2).

###### Proof.

First observe that for any non-negative integers $n_{1}\leq n_{2}$ there is a
linear isomorphism $I_{n_{1}}^{n_{2}}:\mathcal{C}_{n_{1}}^{n_{2}}\rightarrow
C_{n_{1}}^{n_{2}}(P)$ between the space $\mathcal{C}_{n_{1}}^{n_{2}}$ and the
space $C_{n_{1}}^{n_{2}}(P)$ of momentum polynomials on phase space $P$ with
degree between $n_{1}$ and $n_{2}$. If $\bm{C}\in\mathcal{C}_{n_{1}}^{n_{2}}$
then the corresponding momentum polynomial $I_{n_{1}}^{n_{2}}(\bm{C})=c\in
C_{n_{1}}^{n_{2}}(P)$ is

$\displaystyle
c(q,p)=\sum_{k=n_{1}}^{n_{2}}p_{i_{1}}\,\dots\,p_{i_{k}}C^{i_{1}\,\dots\,i_{k}}(q).$

Note the the sum over comoment degree is denoted explicitly, while the sums
over tensor indices are implicit. The proof now proceeds by examining momentum
polynomials in place of comoment vectors.

Let $C_{\mathfrak{s}}(P)$ denote the smooth phase space functions that vanish
when $p=0$. Note that $C_{1}^{\ell}(P)\subset C_{\mathfrak{s}}(P)$ for each
integer $\ell\geq 1$. The space $C_{\mathfrak{s}}(P)$ a Lie subalgebra of the
Lie algebra of phase space functions under Poisson bracket. This follows
immediately from the discussion of Lie subgroups of contactomorphisms in
Section 2. Alternatively it can be verified directly as follows. Suppose
$s_{1},s_{2}\in C_{\mathfrak{s}}(P)$. The Poisson bracket of these functions
is

$\displaystyle\\{s_{1},s_{2}\\}=\partial_{q^{i}}s_{1}\,\partial_{p_{i}}s_{2}-\partial_{q^{i}}s_{2}\,\partial_{p_{i}}s_{1}.$

The sums on the right-hand-side each vanish when $p=0$ because
$(\partial_{q^{i}}s_{\ell})(q,0)=\partial_{q^{i}}(s_{\ell}(q,0))=\partial_{q^{i}}(0)=0$,
$\ell=1,2$. Therefore $\\{s_{1},s_{2}\\}\in C_{\mathfrak{s}}(P)$, as claimed.

For each integer $\ell\geq 0$ let $\mathcal{I}_{\ell}\subset
C_{\mathfrak{s}}(P)$ denote the phase space functions that vanish together
with their first $\ell$ momentum derivatives when $p=0$. If
$r\in\mathcal{I}_{\ell}$ and $s\in C_{\mathfrak{s}}(P)$ the Poisson bracket
$\\{r,s\\}$ is

$\displaystyle\\{r,s\\}=\partial_{q^{i}}r\,\partial_{p_{i}}s-\partial_{q^{i}}s\,\partial_{p_{i}}r.$

Differentiating the first sum with respect to momentum $\ell$ times or less
produces a function that vanishes when $p=0$ because $\partial_{q_{i}}r$
vanishes together with its first $\ell$ momentum derivatives when $p=0$.
Differentiating the second sum with respect to momentum fewer than $\ell$
times likewise produces a function that vanishes when $p=0$ because
$\partial_{p_{i}}r$ vanishes when $p=0$ together with its first $\ell-1$
momentum derivatives. The only contribution to the $\ell^{\text{th}}$-order
momentum derivative of the second sum that does not vanish at $p=0$ due to
vanishing of $\partial_{p_{i}}r$ is a product of $\ell$ momentum derivatives
of $\partial_{p_{i}}r$ with $\partial_{q^{i}}s$; $\ell^{\text{th}}$-order
derivatives of $\partial_{p_{i}}r$ need not vanish when $p=0$. But this
product vanishes when $p=0$ because $\partial_{q^{i}}s$ vanishes when $p=0$.
Therefore $\\{r,s\\}\in\mathcal{I}_{\ell}$. In other words
$\mathcal{I}_{\ell}$ is an ideal inside the Lie algebra $C_{\mathfrak{s}}(P)$
for each integer $\ell\geq 0$.

The quotient space
$C_{\mathfrak{s}_{m_{0}}}(P)=C_{\mathfrak{s}}(P)/\mathcal{I}_{m_{0}}$ inherits
the structure of a Lie algebra because $\mathcal{I}_{m_{0}}$ is an ideal. The
Lie bracket is given by Poisson bracket mod $\mathcal{I}_{m_{0}}$. That is, if
$s_{1}+\mathcal{I}_{m_{0}},s_{2}+\mathcal{I}_{m_{0}}\in
C_{\mathfrak{s}_{m_{0}}}(P)$ then the Lie bracket is

$\displaystyle[s_{1}+\mathcal{I}_{m_{0}},s_{2}+\mathcal{I}_{m_{0}}]=\\{s_{1},s_{2}\\}+\mathcal{I}_{m_{0}}.$

There is a canonical isomorphism that sends each equivalence class in
$C_{\mathfrak{s}_{m}}(P)$ to its unique representative that is a momentum
space polynomial with degree between $1$ and $m_{0}$. Equivalently,
$C_{\mathfrak{s}_{m_{0}}}(P)\approx C_{1}^{m_{0}}(P)$. Therefore
$C_{1}^{m_{0}}(P)$ is a Lie algebra with Lie bracket given by

$\displaystyle[c_{1},c_{2}]=\\{c_{1},c_{2}\\}\text{ mod }p^{m+1},\quad
c_{1},c_{2}\in C_{1}^{m_{0}}(P).$

Here $\text{ mod }p^{m+1}$ denotes setting all polynomial coefficients with
degree greater than $m_{0}$ equal to $0$. In particular, if $c_{1}\in
C_{1}^{m_{0}}(P)$ is homogeneous of degree $k_{1}\leq m_{0}$ and $c_{2}\in
C_{1}^{m_{0}}(P)$ is homogeneous of degree $k_{2}\leq m_{0}$,

$\displaystyle c_{1}(q,p)=p_{i_{1}}\dots p_{i_{k_{1}}}C_{1}^{i_{1}\dots
i_{k_{1}}},\quad c_{2}(q,p)=p_{i_{1}}\dots p_{i_{k_{2}}}C_{2}^{i_{1}\dots
i_{k_{2}}},$

the Lie bracket $[c_{1},c_{2}]$ is given by

$\displaystyle[c_{1},c_{2}]=\begin{cases}0\text{ if }k_{1}+k_{2}-1>m_{0}\\\
\bigg{(}k_{2}\,\partial_{q^{I}}C_{1}^{(i_{1}\dots
i_{k_{1}}}C_{2}^{i_{k_{1}+1}\dots
i_{k_{1}+k_{2}-1})I}-k_{1}\,\partial_{q^{I}}C_{2}^{(i_{1}\dots
i_{k_{2}}}C_{1}^{i_{k_{2}+1}\dots i_{k_{1}+k_{2}-1})I}\bigg{)}p_{i_{1}}\dots
p_{i_{k_{1}+k_{2}-1}}\text{ otherwise.}\end{cases}$ (6)

Here round braces around a grouping of indices denotes averaging over all
permutations of that grouping.

Finally, since $\mathcal{C}_{1}^{m_{0}}$ is isomorphic to $C_{1}^{m_{0}}(P)$,
there is a unique Lie algebra structure on comoment vectors
$\mathcal{C}_{1}^{m_{0}}$ that makes $I_{1}^{m_{0}}$ an isomorphism of Lie
algebras. In symbols, the bracket is
$[\bm{C}_{1},\bm{C}_{2}]_{m_{0}}=(I_{1}^{m_{0}})^{-1}[I_{1}^{m_{0}}(\bm{C}_{1}),I_{1}^{m_{0}}(\bm{C}_{2})]$.
Directly computing the right-hand-side of this formula with the aid of Eq. (6)
reproduces Eq. (5), as claimed.

∎

The Lie algebra structure on the comoment vectors $\mathcal{C}_{1}^{m_{0}}$
with degree between $1$ and $m_{0}$ implies that the dual space
$(\mathcal{C}_{1}^{m_{0}})^{*}$ is a Poisson manifold with a Lie-Poisson
bracket. But this dual space coincides with the space of moment vectors with
degree between $1$ and $m_{0}$,
$(\mathcal{C}_{1}^{m_{0}})^{*}=\mathcal{M}_{1}^{m_{0}}$. The corresponding
Lie-Poisson bracket $\\{\cdot,\cdot\\}_{1}^{m_{0}}$ on
$\mathcal{M}_{1}^{m_{0}}$ comes very close to providing the sought-after
Poisson structure on $P_{m_{0}}$. However, the degree-$0$ moments, those that
correspond to density, are conspicuously missing from
$\mathcal{M}_{1}^{m_{0}}$. The scalar field $\psi\in C(Q)$ that specifies the
distribution center is also not contained in $\mathcal{M}_{1}^{m_{0}}$. This
shortcoming of the bracket on $\mathcal{M}_{1}^{m_{0}}$, as well as the
analogous bracket on phase space moments with degree between $2$ and $m_{0}$,
has been recognized previously[33, 22].

Further progress comes upon noticing that the space of fluid densities $n\in
C^{*}(Q)$ is dual to the space of scalars $\psi\in C(Q)$. This means there is
a natural Poisson bracket $\\{\cdot,\cdot\\}_{0}$ on the space $C(Q)\times
C^{*}(Q)$ given on pairs of functionals $F,G$ by

$\displaystyle\\{F,G\\}_{0}=\left\langle\frac{\delta
F}{\delta\psi},\frac{\delta G}{\delta n}\right\rangle-\left\langle\frac{\delta
G}{\delta\psi},\frac{\delta F}{\delta n}\right\rangle.$

It follows that there is a natural Poisson bracket on each of the factors in
$P_{m_{0}}=\bigg{(}C(Q)\times C^{*}(Q)\bigg{)}\times\mathcal{M}_{1}^{m_{0}}$.
The simplest Poisson bracket on $P_{m_{0}}$ that incorporates the Lie Poisson
bracket on $\mathcal{M}_{1}^{m_{0}}$ is therefore the direct sum

$\displaystyle\\{F,G\\}_{m_{0}}$
$\displaystyle=\\{F,G\\}_{0}+\\{F,G\\}_{1}^{m_{0}}$
$\displaystyle=\left\langle\frac{\delta F}{\delta\psi},\frac{\delta G}{\delta
n}\right\rangle-\left\langle\frac{\delta G}{\delta\psi},\frac{\delta F}{\delta
n}\right\rangle+\left\langle\bm{\mu},\left[\frac{\delta
F}{\delta\bm{\mu}},\frac{\delta G}{\delta\bm{\mu}}\right]\right\rangle.$ (7)

The Jacobi identity for this bracket is automatic since the direct sum of
Poisson manifolds is again a Poisson manifold. Equation (7) provides the
Poisson bracket for the Hamiltonian fluid moment closures presented in this
Article. Its relevance to moment dynamics will be established in Section 5,
where it will become clear that this bracket naturally addresses the
shortcoming of the Lie-Poisson bracket on $\mathcal{M}_{1}^{m_{0}}$ in
isolation. Note that, for each $m_{0}$, this bracket accommodates all fluid
moments with degree less than $m_{0}$, _including the density moment_. Also
note that there is no restriction on the dimension of configuration space $N$.

When $m_{0}=0$, the bracket $\\{\cdot,\cdot\\}_{m_{0}}$ is given by

$\displaystyle\\{F,G\\}_{0}$ $\displaystyle=\left\langle\frac{\delta
F}{\delta\psi},\frac{\delta G}{\delta n}\right\rangle-\left\langle\frac{\delta
G}{\delta\psi},\frac{\delta F}{\delta n}\right\rangle,$ (8)

which provides the appropriate Poisson bracket for Hamiltonian fluid closures
involving $\psi$ and $n$ only. This is also the bracket for the Madelung
transform of the Schrodinger equation.

When $m_{0}=1$, with $\mu^{1}=\bm{P}_{0}$, the bracket
$\\{\cdot,\cdot\\}_{m_{0}}$ is given by

$\displaystyle\\{F,G\\}_{1}$ $\displaystyle=\left\langle\frac{\delta
F}{\delta\psi},\frac{\delta G}{\delta n}\right\rangle-\left\langle\frac{\delta
G}{\delta\psi},\frac{\delta F}{\delta
n}\right\rangle-\int\bm{P}_{0}\cdot\bigg{(}\frac{\delta
F}{\delta\bm{P}_{0}}\cdot\nabla\frac{\delta G}{\delta\bm{P}_{0}}-\frac{\delta
G}{\delta\bm{P}_{0}}\cdot\nabla\frac{\delta F}{\delta\bm{P}_{0}}\bigg{)}\,dq,$
(9)

which provides the appropriate Poisson bracket for Hamiltonian fluid closures
involving $\psi$ and fluid moments with degree $k<2$.

When $m_{0}=2$, with $\mu^{2}=\mathbb{S}_{0}$, the bracket
$\\{\cdot,\cdot\\}_{m_{0}}$ is given by

$\displaystyle\\{F,G\\}_{2}$ $\displaystyle=\left\langle\frac{\delta
F}{\delta\psi},\frac{\delta G}{\delta n}\right\rangle-\left\langle\frac{\delta
G}{\delta\psi},\frac{\delta F}{\delta
n}\right\rangle-\int\bm{P}_{0}\cdot\bigg{(}\frac{\delta
F}{\delta\bm{P}_{0}}\cdot\nabla\frac{\delta G}{\delta\bm{P}_{0}}-\frac{\delta
G}{\delta\bm{P}_{0}}\cdot\nabla\frac{\delta F}{\delta\bm{P}_{0}}\bigg{)}\,dq$
$\displaystyle-\int\mathbb{S}_{0}:\bigg{(}\frac{\delta
F}{\delta\bm{P}_{0}}\cdot\nabla\frac{\delta
G}{\delta\mathbb{S}_{0}}-\frac{\delta
G}{\delta\bm{P}_{0}}\cdot\nabla\frac{\delta
F}{\delta\mathbb{S}_{0}}\bigg{)}\,dq-\int\mathbb{S}_{0}:\bigg{(}\frac{\delta
F}{\delta\mathbb{S}_{0}}\cdot\nabla\frac{\delta
G}{\delta\bm{P}_{0}}-\frac{\delta
G}{\delta\mathbb{S}_{0}}\cdot\nabla\frac{\delta
F}{\delta\bm{P}_{0}}\bigg{)}_{s}\,dq,$ (10)

where $A_{s}=A+A^{T}$ for a dyad $A$.

##  5 Hamiltonian dynamics of localized momentum distributions

Extending the observations of Channell[33], Scovel-Weinstein[32] provide a
technical device that can be used to relate the product Poisson bracket on
$P_{m_{0}}=C(Q)\times C^{*}(Q)\times\mathcal{M}_{1}^{m_{0}}$ described in
Section 4 to the Lie-Poisson bracket on $C^{*}(P)$ that underlies the Vlasov-
Poisson system. The following Theorem represents an application of that
device, which was originally developed for treating phase space moments, to
fluid moments.

######  Theorem 3.

Let $\\{\cdot,\cdot\\}_{0}$ denote the canonical Poisson bracket on the space
$C(Q)\times C^{*}(Q)$. Let $\\{\cdot,\cdot\\}_{1}^{m_{0}}$ denote the Lie-
Poisson bracket on $\mathcal{M}_{1}^{m_{0}}$ provided by Theorem 2. Equip the
space $\bigg{(}C(Q)\times C^{*}(Q)\bigg{)}\times\mathcal{M}_{1}^{m_{0}}$ with
the product Poisson bracket
$\\{\cdot,\cdot\\}_{m}=\\{\cdot,\cdot\\}_{0}+\\{\cdot,\cdot\\}_{1}^{m_{0}}$.
The mapping $\Gamma_{m_{0}}:C(Q)\times
C^{*}(Q)\times\mathcal{M}_{1}^{m_{0}}\rightarrow C^{*}(P)$ given by

$\displaystyle\Gamma_{m_{0}}(\psi,n,\bm{\mu})(q,p)$
$\displaystyle=n(q)\,\delta(p+d\psi(q))-\mu_{i_{1}}(q)\,\partial_{p_{i_{1}}}\delta(p+d\psi(q))$
$\displaystyle+\frac{1}{2}\,\mu_{i_{1}\,i_{2}}(q)\,\partial^{2}_{p_{i_{1}\,i_{2}}}\delta(p+d\psi(q))+\dots+\frac{(-1)^{m_{0}}}{m!}\mu_{i_{1}\dots
i_{m_{0}}}(q)\,\partial^{m_{0}}_{p_{i_{1}}\dots
p_{i_{m_{0}}}}\delta(p+d\psi(q)).$ (11)

is a Poisson map.

######  Remark 2.

The right-hand-side of Eq. (11) recovers the ansatz (4) for the novel exact
reduction of Vlasov-Poisson dynamics mentioned in the introduction. The fact
that the map (11) is Poisson justifies the claim from Section 4 that the
bracket $\\{\cdot,\cdot\\}_{m_{0}}$ repairs the shortcoming of the Hamiltonian
fluid moment closures due to Holm and Tronci[22] that do not include the
density moment. It is interesting to note that Chong[40] previously
constructed a Poisson map that sends phase space distributions into fluid-like
quantities; that Poisson map appears to be unrelated to this one.

###### Proof.

Let $G$ be a Lie group with Lie subgroups $B,S\subset G$ such that the
multiplication map $B\times S\rightarrow G:(b,s)\mapsto bs$ is a
diffeomorphism. As summarized in Section 3, Scovel-Weinstein proved that

$\displaystyle\Gamma(b,b_{0}^{*},s_{0}^{*})=\text{Ad}^{*}_{b}(b_{0}^{*}+s_{0}^{*}),$

defines a Poisson map from the product Poisson manifold
$(B\times\mathfrak{b}^{*})\times(\mathfrak{s}^{*})$ to the Lie-Poisson space
$\mathfrak{g}^{*}$. Here $\text{Ad}^{*}_{g}$ denotes the (left) coadjoint
action of $G$ on $\mathfrak{g}^{*}$. Let $G$ be the contactomorphism group,
$B\subset G$ the fiber translations, and $S\subset G$ the isostatic
contactomorphisms, as defined in Section 2. Identify the fiber translations
$B$ with the Abelian group of scalar functions $\psi\in C(Q)$. Identify the
Lie algebras $\mathfrak{b},\mathfrak{s},\mathfrak{g}$, with configuration
space scalars $\sigma\in C(Q)\approx\mathfrak{b}$, phase space functions $s\in
C_{\mathfrak{s}}(P)\approx\mathfrak{s}$ that vanish when $p=0$, and general
phase space functions $h\in C(P)\approx\mathfrak{g}$. The dual spaces are then
(distributional) configuration space functions $n\in
C^{*}(Q)\approx\mathfrak{b}^{*}$, (distributional) phase space functions
$\tilde{f}_{0}\in C^{*}_{\mathfrak{s}}(P)$ with zero fluid density, and
general (distributional) phase space functions $f\in
C^{*}(P)\approx\mathfrak{g}^{*}$. The corresponding duality pairings are the
usual $L^{2}$ integration pairings. Under these identifications, the Poisson
map $\Gamma:C(Q)\times C^{*}(Q)\times C^{*}_{\mathfrak{s}}(P)\rightarrow
C^{*}(P)$ becomes

$\displaystyle\Gamma(\psi,n,\tilde{f}_{0})(q,p)$
$\displaystyle=\tau_{\psi*}(\pi^{*}n\,\delta(p)+\tilde{f}_{0})(q,p)=n(q)\,\delta(p+d\psi(q))+\tilde{f}_{0}(q,p+d\psi(q)).$

Note that $C(Q)\times C^{*}(Q)\times C^{*}_{\mathfrak{s}}(P)$ is endowed with
the product Poisson structure corresponding to the canonical poisson bracket
on $C(Q)\times C^{*}(Q)$ and the Lie Poisson bracket on
$C^{*}_{\mathfrak{s}}(P)$.

As shown in the proof of Theorem 2, the phase space functions
$r\in\mathcal{I}_{m_{0}}$ that vanish along with their first $m_{0}$ momentum
derivatives when $p=0$ form an ideal in the Lie algebra $C_{\mathfrak{s}}(P)$.
Therefore the quotient space
$C_{\mathfrak{s}}(P)/\mathcal{I}_{m_{0}}=C_{\mathfrak{s}_{m_{0}}}(P)$ has a
natural Lie algebra structure and the quotient map
$\Pi_{m_{0}}:C_{\mathfrak{s}}(P)\rightarrow C_{\mathfrak{s}_{m_{0}}}(P)$ is a
Lie algebra homomorphism. Also from the proof of Theorem 2, the quotient space
$C_{\mathfrak{s}_{m_{0}}}(P)$ is naturally isomorphic as a Lie algebra to the
space of comoment vectors $\mathcal{C}_{1}^{m_{0}}$ with degree between $1$
and $m_{0}$. (The Lie algebra structure on $\mathcal{C}_{1}^{m_{0}}$ is the
one provided by Theorem 2.) Under this identification,
$\Pi_{m_{0}}:C_{\mathfrak{s}}(P)\rightarrow\mathcal{C}_{1}^{m_{0}}$ is the map
that assigns to each $s\in C_{\mathfrak{s}}(P)$ its weighted momentum Taylor
series coefficients with degrees between $1$ and $m_{0}$:

$\displaystyle\Pi_{m_{0}}(s)=\bigg{(}\partial_{p_{i_{1}}}s(q,0)\,e_{i_{1}},\frac{1}{2}\partial^{2}_{p_{i_{1}}\,p_{i_{2}}}s(q,0)\,e_{i_{1}}\otimes
e_{i_{2}},\dots,\frac{1}{m!}\partial^{m_{0}}_{p_{i_{1}}\dots
p_{i_{m_{0}}}}s(q,0)\,e_{i_{1}}\otimes\dots\otimes e_{i_{m_{0}}}\bigg{)}.$

Since $\Pi_{m_{0}}$ is a Lie algebra homomorphism, the adjoint
$\Pi_{m_{0}}^{*}:\mathcal{M}_{1}^{m_{0}}\rightarrow C^{*}_{\mathfrak{s}}(P)$
is a Poisson mapping between Lie-Poisson spaces. A simple direct calculation
leads to the explicit formula

$\displaystyle\Pi_{m_{0}}^{*}(\bm{\mu})(q,p)=-\mu_{i_{1}}(q)\,\partial_{p_{i_{1}}}\delta(p)+\frac{1}{2}\,\mu_{i_{1}\,i_{2}}(q)\,\partial^{2}_{p_{i_{1}\,i_{2}}}\delta(p)+\dots+\frac{(-1)^{m_{0}}}{m_{0}!}\mu_{i_{1}\dots
i_{m_{0}}}(q)\,\partial^{m_{0}}_{p_{i_{1}}\dots p_{i_{m_{0}}}}\delta(p).$

Since $\Pi_{m_{0}}^{*}$ and $\Gamma$ are Poisson maps, the composition

$\displaystyle\Gamma_{m_{0}}:C(Q)\times
C^{*}(Q)\times\mathcal{M}_{1}^{m_{0}}\xrightarrow{\text{id}_{C(Q)\times
C^{*}(Q)}\times\Pi_{m_{0}}^{*}}C(Q)\times C^{*}(Q)\times
C^{*}_{\mathfrak{s}}(P)\xrightarrow{\Gamma}C^{*}(P),$

is a Poisson map as well. Evaluating the composition using the above formulas
shows that $\Gamma_{m_{0}}$ is the map given in the Theorem statement. ∎

Theorem 3 is quite powerful due to a construction[41] known as
“collectivisation." Given a Poisson mapping $\Gamma:\Lambda\rightarrow Z$
between Poisson manifolds $\Lambda,Z$, every Hamiltonian function $h$ on $Z$
gives rise to a corresponding Hamiltonian function $H=\Gamma^{*}h$ on
$\Lambda$ known as the “collective" Hamiltonian. Remarkably, if $\lambda(t)$
is any integral curve for the collective Hamiltonian vector field $X_{H}$ then
the image curve $z(t)=\Gamma(\lambda(t))$ is an integral curve for the
Hamiltonian vector field $X_{h}$. This follows from the simple chain of
identities for arbitrary $q\in C(Z)$:

$\displaystyle\frac{d}{dt}q(z(t))=\frac{d}{dt}q(\Gamma(\lambda(t)))=\frac{d}{dt}(\Gamma^{*}q)(\lambda(t))=\\{\Gamma^{*}q,\Gamma^{*}h\\}_{\Lambda}(\lambda(t))=(\Gamma^{*}\\{q,h\\}_{Z})(\lambda(t))=\\{q,h\\}_{Z}(z(t)).$

In the context of fluid moments, the collectivisation construction and Theorem
3 have the following implication. Let

$\displaystyle\mathcal{H}(f)=\int\frac{1}{2m}|p|^{2}f\,dq\,dp+\int\frac{1}{2}e\,\hat{V}(n)\,n\,dq$
(12)

denote the Hamiltonian for the Vlasov-Poisson system. Here
$\hat{V}:C^{*}(Q)\rightarrow C(Q)$ is the self-adjoint linear operator that
sends a density $n$ to the corresponding solution $V$ of Poisson’s equation,
$-\Delta V=4\pi e\,\widetilde{n}$, where $\widetilde{n}$ denotes the number
density less its average. Pulling back $\mathcal{H}$ along the Poisson map
$\Gamma_{m_{0}}$ produces a collective Hamiltonian $\mathcal{H}_{m_{0}}$ on
$P_{m_{0}}=C(Q)\times C^{*}(Q)\times\mathcal{M}_{1}^{m_{0}}$. If
$(\psi(t),n(t),\bm{\mu}(t))$ is any solution of the Hamiltonian system on
$P_{m_{0}}$ defined by $\mathcal{H}_{m_{0}}$ then
$f(t)=\Gamma_{m_{0}}(\psi(t),n(t),\bm{\mu}(t))$ is an exact solution of the
Vlasov-Poisson system. In other words, Theorem 3 produces exact solutions of
Vlasov-Poisson with any finite number $m_{0}$ of fluid moments!

When $m_{0}=0$, the space $P_{m_{0}}$ is just $P_{0}=C(Q)\times
C^{*}(Q)\ni(\psi,n)$, with the standard canonical Poisson bracket (8). Pulling
back the physical Hamiltonian $\mathcal{H}(f)$ along $\Gamma_{0}$ leads to the
collective Hamiltonian

$\displaystyle\mathcal{H}_{0}(\psi,n)=\frac{1}{2m}\int|\nabla\psi|^{2}\,n\,dq+\frac{1}{2}\int
e\,\hat{V}(n)\,n\,dq.$

The functional derivatives of the collective Hamiltonian are given by

$\displaystyle\frac{\delta\mathcal{H}_{0}}{\delta\psi}=-m^{-1}\nabla\cdot(n\,\nabla\psi),\quad\frac{\delta\mathcal{H}_{0}}{\delta
n}=\frac{1}{2m}|\nabla\psi|^{2}+e\hat{V}(n).$

The corresponding equations of motion are therefore

$\displaystyle\partial_{t}n$ $\displaystyle=m^{-1}\nabla\cdot(n\,\nabla\psi)$
$\displaystyle\partial_{t}\psi$
$\displaystyle=\frac{1}{2m}|\nabla\psi|^{2}+e\hat{V}(n).$

Upon introducing the velocity variable $\bm{v}=-m^{-1}\nabla\psi$, these
equations may be re-written as

$\displaystyle\partial_{t}(m\,n\,\bm{v})+\nabla\cdot(m\,n\,\bm{v}\,\bm{v})=-e\,n\,\nabla\hat{V}(n),\quad\partial_{t}n+\nabla\cdot(n\,\bm{v})=0,\quad
m\,\bm{v}=-\nabla\psi,$

which reveals equivalence with Euler’s equations for a cold, irrotational
fluid. It is a well-known fact[42] that such irrotational flows correspond to
exact solutions of the Vlasov-Poisson system.

When $m_{0}=1$, with $\mu^{1}=\bm{P}_{0}$, the space $P_{m_{0}}$ becomes
$P_{1}=C(Q)\times C^{*}(Q)\times\mathcal{M}_{1}^{1}$ equipped with the Poisson
bracket in Eq. (9). Pulling back the physical Hamiltonian along $\Gamma_{1}$
leads to the collective Hamiltonian

$\displaystyle\mathcal{H}_{1}(\psi,n,\bm{P}_{0})=\int\frac{1}{2mn}|\bm{P}_{0}-n\nabla\psi|^{2}\,dq-\int\frac{1}{2mn}|\bm{P}_{0}|^{2}\,dq+\frac{1}{2}\int
e\,\hat{V}(n)\,n\,dq.$

The functional derivatives of the collective Hamiltonian are

$\displaystyle\frac{\delta\mathcal{H}_{1}}{\delta\psi}=m^{-1}\nabla\cdot(\bm{P}_{0}-n\nabla\psi),\quad\frac{\delta\mathcal{H}_{1}}{\delta
n}=\frac{1}{2m}|\nabla\psi|^{2}+e\hat{V}(n),\quad\frac{\delta\mathcal{H}_{1}}{\delta\bm{P}_{0}}=-m^{-1}\nabla\psi.$

The corresponding equations of motion are therefore

$\displaystyle\partial_{t}n$
$\displaystyle=-m^{-1}\nabla\cdot(\bm{P}_{0}-n\,\nabla\psi)$
$\displaystyle\partial_{t}\psi$
$\displaystyle=\frac{1}{2m}|\nabla\psi|^{2}+e\hat{V}(n)$
$\displaystyle\partial_{t}\bm{P}_{0}$
$\displaystyle=m^{-1}\bm{P}_{0}\cdot\nabla(\nabla\psi)+m^{-1}(\nabla\psi)\cdot\nabla\bm{P}_{0}+m^{-1}\nabla\cdot(\nabla\psi)\,\bm{P}_{0}.$

Distributions in the image of $\Gamma_{1}$ have momentum density $\bm{P}=\int
p\,f\,dp=\bm{P}_{0}-n\,\nabla\psi$ and momentum flux

$\displaystyle\mathbb{S}=m^{-1}\int
p\,p\,f\,dp=-m^{-1}\,\nabla\psi\,\bm{P}_{0}-m^{-1}\,\bm{P}_{0}\nabla\psi+m^{-1}\,n\,\nabla\psi\nabla\psi=-m^{-1}\,\nabla\psi\,\bm{P}-m^{-1}\,\bm{P}\nabla\psi-m^{-1}\,n\,\nabla\psi\nabla\psi.$

In terms of these variables, the equations of motion are instead

$\displaystyle\partial_{t}(mn)+\nabla\cdot\bm{P}=0,\quad\partial_{t}\bm{P}+\nabla\cdot\mathbb{S}=-e\,n\,\nabla\hat{V}(n),\quad\partial_{t}\psi$
$\displaystyle=\frac{1}{2m}|\nabla\psi|^{2}+e\hat{V}(n).$

These equations reveal that the exact reduction with $m_{0}=1$ again recovers
familiar hydrodynamic equations, but now with a non-trivial closure relation
for the momentum flux tensor $\mathbb{S}$. This Hamiltonian moment closure
appears to be new, but it would be interesting to compare it with previously-
derived Hamiltonian moment closures that include moments with degree $k<2$.

When $m_{0}=2$, with $\mu^{2}=\mathbb{S}_{0}$, the space $P_{m_{0}}=C(Q)\times
C^{*}(Q)\times\mathcal{M}_{1}^{2}$ is equipped with the Poisson bracket (10).
The collective Hamiltonian is

$\displaystyle\mathcal{H}_{2}(\psi,n,\bm{P}_{0},\mathbb{S}_{0})$
$\displaystyle=\frac{1}{2m}\int\text{Tr}(\mathbb{S}_{0})\,dq-\frac{1}{m}\int\bm{P}_{0}\cdot\nabla\psi\,dq+\frac{1}{2m}\int|\nabla\psi|^{2}\,n+\frac{1}{2}\int
e\,\hat{V}(n)\,n\,dq.$

The corresponding functional derivatives are

$\displaystyle\frac{\delta\mathcal{H}_{2}}{\delta\psi}=\frac{1}{m}\nabla\cdot(\bm{P}_{0}-n\nabla\psi),\quad\frac{\delta\mathcal{H}_{2}}{\delta
n}=\frac{1}{2m}|\nabla\psi|^{2}+e\,\hat{V}(n),\quad\frac{\delta\mathcal{H}_{2}}{\delta\bm{P}_{0}}=-\frac{1}{m}\nabla\psi,\quad\frac{\delta\mathcal{H}_{2}}{\delta\mathbb{S}_{0}}=\frac{1}{2m}\eta=\frac{1}{2m}\delta^{i_{1}\,i_{2}}\,e_{i_{1}}\otimes
e_{i_{2}}.$

The equations of motion are therefore given by

$\displaystyle\partial_{t}\psi$
$\displaystyle=\frac{1}{2}m|\bm{v}|^{2}+e\,\hat{V},\quad\bm{v}=-m^{-1}\nabla\psi$
$\displaystyle\partial_{t}n$
$\displaystyle=-m^{-1}\nabla\cdot(\bm{P}_{0}+m\,n\bm{v})$
$\displaystyle\partial_{t}\bm{P}_{0}$
$\displaystyle=-\bm{v}\cdot\nabla\bm{P}_{0}-\nabla\bm{v}\cdot\bm{P}_{0}-(\nabla\cdot\bm{v})\bm{P}_{0}-m^{-1}\nabla\cdot\mathbb{S}_{0}$
$\displaystyle\partial_{t}\bm{S}_{0}$
$\displaystyle=-\bm{v}\cdot\nabla\mathbb{S}_{0}-\nabla\bm{v}\cdot\mathbb{S}_{0}-\mathbb{S}_{0}\cdot\nabla\bm{v}-(\nabla\cdot\bm{v})\mathbb{S}_{0}.$

Note that these equations imply
$\partial_{t}\bm{v}+\bm{v}\cdot\nabla\bm{v}=-\frac{e}{m}\nabla\hat{V}$. Upon
introducing the conventional (non-centered) moments,

$\displaystyle\bm{P}$ $\displaystyle=\bm{P}_{0}+m\,n\,\bm{v}$
$\displaystyle\mathbb{S}$
$\displaystyle=\mathbb{S}_{0}+m\,(\bm{v}\,\bm{P}_{0}+\bm{P}_{0}\,\bm{v})+m^{2}\,n\,\bm{v}\,\bm{v},$

and performing a tedious explicit calculation, the evolution equations for
$(\psi,n,\bm{P}_{0},\mathbb{S}_{0})$ may be re-written as

$\displaystyle\partial_{t}\psi$
$\displaystyle=\frac{1}{2}m|\bm{v}|^{2}+e\,\hat{V},\quad\bm{v}=-m^{-1}\nabla\psi$
(13) $\displaystyle\partial_{t}n$ $\displaystyle=-m^{-1}\nabla\cdot\bm{P}$
(14) $\displaystyle\partial_{t}\bm{P}$
$\displaystyle=-m^{-1}\nabla\cdot\mathbb{S}-e\,n\,\nabla\hat{V}$ (15)
$\displaystyle\partial_{t}\mathbb{S}$
$\displaystyle=-m^{-1}\nabla\cdot\mathcal{T}-e\,(\nabla\hat{V}\bm{P}+\bm{P}\nabla\hat{V}).$
(16)

These evolution equations for $(n,\bm{P},\mathbb{S})$ exactly recover the
first three equations in the moment hierarchy (3) with a novel closure for the
third-degree fluid moment $\mathcal{T}=\int p\,p\,p\,f\,dp$ given by

$\displaystyle\mathcal{T}$
$\displaystyle=3m\,\bm{v}\odot\mathbb{S}_{0}+3m^{2}\,\bm{v}\bm{v}\odot\bm{P}_{0}+m^{3}\,n\,\bm{v}\bm{v}\bm{v}$
$\displaystyle=3m\,\bm{v}\odot\mathbb{S}-3m^{2}\,\bm{v}\bm{v}\odot\bm{P}+m^{3}\,n\,\bm{v}\bm{v}\bm{v}.$
(17)

Here $\odot$ denotes the symmetrized tensor product, e.g.
$3(\bm{v}\odot\mathbb{S}_{0})_{i_{1}\,i_{2}\,i_{3}}=v_{i_{1}}\,S_{i_{2}\,i_{3}}+v_{i_{2}}\,S_{i_{1}\,i_{3}}+v_{i_{3}}\,S_{i_{1}\,i_{2}}$.

The system (13)-(16) admits stationary homogeneous equilibria of the form
$\psi=0$, $n=n_{0}=\text{const.}$, $\bm{P}=0$,
$\mathbb{S}=\mathbb{S}_{0}=\text{const.}$ When $N=1$ and $\mathbb{S}_{0}\geq
0$, these equilibria are stable if and only if $\mathbb{S}_{0}=0$. When
$\mathbb{S}_{0}>0$ the linearized spectrum for Fourier mode $k$ contains a
pair of imaginary eigenvalues corresponding to oscillations at the Bohm-Gross
frequency,

$\displaystyle\omega^{2}=\omega_{p}^{2}\bigg{(}\frac{1}{2}+\frac{1}{2}\sqrt{1+4v_{\text{th}}^{2}\,k^{2}/\omega_{p}^{2}}\bigg{)},\quad\omega_{p}^{2}=\frac{4\pi\,e^{2}n_{0}}{m},\quad
v_{\text{th}}^{2}=\frac{\mathbb{S}_{0}}{m^{2}\,n_{0}},$

and a pair of real eigenvalues corresponding to exponential growth and decay
with rates

$\displaystyle\gamma^{2}=-\omega_{p}^{2}\bigg{(}\frac{1}{2}-\frac{1}{2}\sqrt{1+4v_{\text{th}}^{2}\,k^{2}/\omega_{p}^{2}}\bigg{)}\bigg{)}.$

(Curiously, equilibria with $\mathbb{S}_{0}<0$ are linearly stable!) So the
exact closure with $m_{0}=2$ already begins to reflect the well-known fact
that only certain homogeneous equilibria for the Vlasov-Poisson system, e.g.
monotone-decreasing distributions[43], are linearly stable. Note that the
condition $\mathbb{S}_{0}>0$ does not imply a stable distribution; there are
many linearly-unstable homogeneous Vlasov-Poisson equilibria with positive
second moment. Also note that the growth rate $\gamma=O(\sqrt{|k|})$ for large
$k$, implying ill-posedness of the initial value problem. Physical solutions
must therefore be found within the center manifolds[44, 2, 4] attached to such
equilibria, much as physical solutions of the Abraham-Lorentz-Dirac equation
for individual electrons are contained in a center manifold[45]. Further work
is needed to determine linear stability properties of homogeneous equilibria
with $N>1$, $m_{0}>2$. It is unclear whether the data-driven Hamiltonians
described in Section 6 will also require reduction to a center manifold to
describe physical dynamics.

##  6 A closure scheme for general distributions

The theory developed so far leads to a framework for constructing data-driven
fluid closures of the Vlasov-Poisson system with Hamiltonian structure. This
framework assumes availability of a legacy simulation code for the Vlasov-
Poisson system; either particle-in-cell (PIC) codes or continuum codes will
suffice. The framework presented here should be contrasted with the structure-
preserving reduced-order modeling techniques developed by Hesthaven,
Pagliantini, and Ripamonti[46, 47, 48]. Those reduced-order models assume the
reduced, data-driven dependent variables admit constant symplectic or Poisson
structures, or else rely on transformations to such variables. In contrast,
the method presented here assumes the reduced dependent variables admit the
non-constant Poisson structure provided by Scovel-Weinstein theory. In
addition, the reduced dependent variables used here are always simply related
to physical quantities of interest, namely the fluid moments.

Developing a closure that includes fluid moments up to degree-$m_{0}$ requires
a parametric energy function

$\displaystyle\mathcal{H}_{W}:C(Q)\times\mathcal{M}_{0}^{m_{0}}\rightarrow\mathbb{R}:(\psi,\bm{\mu})\mapsto\mathcal{H}_{W}(\psi,\bm{\mu}),$

and a _model_

$\displaystyle\mathcal{F}_{0}:\mathcal{M}_{0}^{m_{0}}\rightarrow C^{*}(P)$

that assigns a centered distribution $f_{0}=\mathcal{F}_{0}(\bm{\mu})$ to each
centered moment vector $\bm{\mu}\in\mathcal{M}_{0}^{m_{0}}$. This terminology
has been borrowed from Scovel-Weinstein[32], who introduced the notion of a
model for phase space moments. For consistency, the model distribution
function $\mathcal{F}_{0}(\bm{\mu})$ must have moments compatible with
$\bm{\mu}$:

$\displaystyle\mu^{k}_{i_{1}\,\dots\,i_{k}}=\int
p_{i_{1}}\,\dots\,p_{i_{k}}\mathcal{F}_{0}(\bm{\mu})\,dq,\quad
k=0,1,\dots,m_{0}.$

Given a field $\psi\in C(Q)$, the ordinary distribtion function $f$ associated
with $\mathcal{F}_{0}(\bm{\mu})$ is $f=\tau_{\psi*}\mathcal{F}_{0}(\bm{\mu})$.
To fully specify the closure, the variable $W$ that parameterizes the energy
must be adjusted to an optimal value $W^{*}$. The closure itself will then
comprise a Hamiltonian system on $C(Q)\times\mathcal{M}_{0}^{m_{0}}$ with
Hamiltonian $\mathcal{H}_{W^{*}}$ and Poisson bracket given by Eq. (7).

Finding the optimal parameter value $W^{*}$ proceeds as follows. Start by
selecting a finite collection $\mathcal{D}$ of samples
$(\psi,\bm{\mu})\in\mathcal{D}$ from the space
$C(Q)\times\mathcal{M}_{0}^{m_{0}}$, together with a timestep parameter
$\Delta t\in\mathbb{R}$. For each sample $(\psi,\bm{\mu})\in\mathcal{D}$
perform two computations. (1) Approximately evolve $(\psi,\bm{\mu})$ along the
Hamiltonian vector field
$X_{\mathcal{H}_{W}}=(\dot{\psi}_{\mathcal{H}_{W}},\dot{\bm{\mu}}_{\mathcal{H}_{W}})$
for $\Delta t$ seconds using the forward-Euler formula

$\displaystyle\psi(\Delta t)=\psi+\Delta
t\,\dot{\psi}_{\mathcal{H}_{W}}(\psi,\bm{\mu}),\quad\bm{\mu}(\Delta
t)=\bm{\mu}+\Delta t\,\dot{\bm{\mu}}_{\mathcal{H}_{W}}(\psi,\bm{\mu}).$

Then compute the corresponding candidate evolved moment vector
$\overline{\bm{M}}(\Delta t)=(\overline{M}^{0}(\Delta
t),\dots,\overline{M}^{m_{0}}(\Delta t))$, where the degree-$k$ candidate
moment is given by

$\displaystyle\overline{M}(\Delta t)_{i_{1}\,\dots\,i_{k}}=\int p_{i_{1}}\dots
p_{i_{k}}\tau_{\psi(\Delta t)*}\mathcal{F}_{0}(\bm{\mu}(\Delta t))\,dp.$ (18)

Note that $\overline{M}^{k}(\Delta t)$ depends explicitly on $W$. (2)
Reconstruct the distribution function
$f=\tau_{\psi*}\mathcal{F}_{0}(\bm{\mu})$. Use the legacy simulation code to
find the time evolution $f(\Delta t)$ of $f$. Then compute the corresponding
“ground-truth" evolved moment vector $\bm{M}(\Delta t)$, where the degree-$k$
ground-truth evolved moment is given by

$\displaystyle{M}_{i_{1}\,\dots\,i_{k}}=\int p_{i_{1}}\dots p_{i_{k}}f(\Delta
t)\,dp.$ (19)

With each sample $(\psi,\bm{\mu})\in\mathcal{D}$ now processed, construct the
mean-squared loss

$\displaystyle
S(W)=\frac{1}{|\mathcal{D}|}\sum_{(\psi,\bm{\mu})\in\mathcal{D}}\|\bm{M}(\Delta
t)-\overline{\bm{M}}(\Delta t)\|^{2},$

and find $W^{*}$ by solving the optimization problem

$\displaystyle W^{*}=\text{argmin}\,S(W).$

In this way, the energy parameters $W$ are chosen in order to maximize
agreement between moment evolution predicted by the Vlasov-Poisson system and
moment evolution predicted by the Hamiltonian hydrodynamic model for the field
$\psi$ and the centered moments $\bm{\mu}$.

Through application of the binomial formula for the degree-$k$ candidate
evolved moment

$\displaystyle\overline{M}^{k}(\Delta
t)=\sum_{\ell=0}^{k}(-1)^{\ell}\begin{pmatrix}k\\\
\ell\end{pmatrix}(\nabla\psi(\Delta t))^{\ell}\odot\mu^{k-\ell}(\Delta t),$
(20)

the $W$-dependence of the loss $S(W)$ can be made more explicit. For example,
when $m_{0}=1$, with $\mu^{0}=n$, $\mu^{1}\equiv\bm{P}_{0}$, the general
Hamiltonian vector field
$X_{\mathcal{H}}=(\dot{\psi}_{\mathcal{H}},\dot{n}_{\mathcal{H}},\dot{\bm{P}}_{0\,\mathcal{H}})$
is given by

$\displaystyle\dot{\psi}_{\mathcal{H}}$
$\displaystyle=\frac{\delta\mathcal{H}}{\delta n}$
$\displaystyle\dot{n}_{\mathcal{H}}$
$\displaystyle=-\frac{\delta\mathcal{H}}{\delta\psi}$
$\displaystyle\dot{\bm{P}}_{0\,\mathcal{H}}$
$\displaystyle=-\frac{\delta\mathcal{H}}{\delta\bm{P}_{0}}\cdot\nabla\bm{P}_{0}-\bigg{(}\nabla\frac{\delta\mathcal{H}}{\delta\bm{P}_{0}}\bigg{)}\cdot\bm{P}_{0}-\bigg{(}\nabla\cdot\frac{\delta\mathcal{H}}{\delta\bm{P}_{0}}\bigg{)}\bm{P}_{0}.$

The loss $S(W)$ with $m=1$ may therefore be written explicitly as

$\displaystyle S(W)$
$\displaystyle=\frac{1}{|\mathcal{D}|}\sum_{(\psi,\bm{\mu})\in\mathcal{D}}\int\left|n-\Delta
t\frac{\delta\mathcal{H}_{W}}{\delta\psi}-\int f(\Delta t)\,dp\right|^{2}\,dq$
$\displaystyle+\frac{1}{|\mathcal{D}|}\sum_{(\psi,\bm{\mu})\in\mathcal{D}}\int\bigg{|}\bm{P}_{0}-\Delta
t\frac{\delta\mathcal{H}_{W}}{\delta\bm{P}_{0}}\cdot\nabla\bm{P}_{0}-\Delta
t\,\bigg{(}\nabla\frac{\delta\mathcal{H}_{W}}{\delta\bm{P}_{0}}\bigg{)}\cdot\bm{P}_{0}-\Delta
t\bigg{(}\nabla\cdot\frac{\delta\mathcal{H}_{W}}{\delta\bm{P}_{0}}\bigg{)}\bm{P}_{0}$
$\displaystyle\hskip 130.0002pt-\left(n-\Delta
t\frac{\delta\mathcal{H}_{W}}{\delta\psi}\right)\nabla\left(\psi+\Delta
t\frac{\delta\mathcal{H}_{W}}{\delta n}\right)-\int p\,f(\Delta
t)\,dp\bigg{|}^{2}\,dq.$

##  7 Discussion

This Article has described a new method for developing fluid moment closures
of the Vlasov-Poisson system with Hamiltonian structure. These closures apply
in any number of space dimensions, and allow for collections of fluid moments
with degree between $0$ and $m_{0}$, where $m_{0}$ is any non-negative
integer. The closures work by introducing an irrotational velocity field
$\bm{v}$ that specifies the “center" of the distribution function in moment
space. The theory presented here leads to both novel exact solutions of the
Vlasov-Poisson system parameterized by arbitrarily-large collections of fluid
moments, as well as a general framework for finding data-driven fluid closures
with Hamiltonian structure.

Variable-moment data-driven fluid closures garner increasing attention in the
machine learning community. Recent results[10, 8, 9] indicate that it is
possible to construct such closures while strictly enforcing hyperbolicity.
The results presented in the present Article therefore suggest investigation
of the following intriguing question. Is it possible to develop expressive
variable-moment data-driven fluid closures that strictly enforce both
hyperbolicity _and_ Hamiltonian structure? Within the closure framework
presented here, such a constraint would limit the allowable forms of the
closure Hamiltonian $\mathcal{H}_{W}(\psi,n,\bm{\mu})$.

The fluid closures presented here always include all fluid moments up to and
including some maximum degree $m_{0}$. It would be interesting to determine
whether these closures can be specialized to only include a subset of these
moments. For example, are there subclosures with scalar pressure?

This Article has treated the electrostatic limit of collisionless plasma
kinetic theory. Variable-moment fluid closures with Hamiltonian structure
would also be interesting to develop for the electromagnetic problem, as
embodied by the Vlasov-Maxwell system. Forthcoming publications will reveal
that the Lie-theoretic methods used here transfer to the electromagnetic
problem in a mathematically and physically appealing fashion.

Each of the fluid closures developed here is associated with a certain direct-
sum Poisson bracket $\\{\cdot,\cdot\\}_{m_{0}}$. (See Eq. (7).) It would be
useful to identify Casimir functions for this bracket and use them to study
stability of solutions using the energy-Casimir method[49]. Finding the
Casimirs would also be a necessary step when including dissipation in the
fluid closures presented here using a metriplectic[17] bracket.

As mentioned in the introduction, Tassi[19, 20, 21] has found variable-moment
fluid closures for slab drift-kinetic systems. But the methods used by Tassi
to find closures are seemingly unrelated to the Lie-theoretic methods of
Scovel-Weinstein. Is it possible to find Tassi’s drift kinetic closures using
Scovel-Weinstein theory[32] as was done for the closures presented here? An
affirmative result might suggest a path toward variable-moment gyrofluid
closures for more intricate gyroaveraged kinetic theories such as full-$f$
gyrokinetics[50, 51, 52]. In a similar vein, it would be interesting to
determine whether the closures presented here can be understood as limits of
the multiple waterbag reductions described by Perin _et al._[30].

More broadly, this Article suggests that the methods developed by Scovel-
Weinstein[32] in the context of beam dynamics apply to a wider class of
dissipation-free statistical closure problems than originally envisioned.
There are many possible extensions. For example, consider the problem of
solving the functional Vlasov equation for a variational weakly-nonlinear wave
equation with random initial data. The algebra of functionals on the field-
theoretic phase space enjoys a canonical Poisson bracket. By decomposing this
algebra into affine functionals and functionals that vanish together with
their first functional derivatives for the vacuum state, Scovel-Weinstein
theory may be applied. The result will be Hamiltonian evolution equations for
a finite collection of field correlation functions. These non-dissipative
correlation dynamics do not seem to have been studied previously.

## Data availability

No datasets were generated or analysed during the current study.

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## Acknowledgements

Research reported in this Article was supported by a US DOE Mathematical
Multifaceted Integrated Capability Center (MMICC) grant. This Article benefits
from several helpful discussions with C. Chandre, A. Christlieb, P. Morrison,
E. Tassi, and C. Tronci.

## Author contributions statement

J.W.B. performed all work relevant to the preparation of this manuscript.

## Additional information

The author declares no competing interests.