# On a Theorem by Schlenk

Yannis<EMAIL_ADDRESS>MINT Learning Institute at ETH
Zürich

###### Abstract

In this paper we prove a generalisation of Schlenk’s theorem about the
existence of contractible periodic Reeb orbits on stable, displaceable
hypersurfaces in symplectically aspherical, geometrically bounded, symplectic
manifolds, to a forcing result for contractible twisted periodic Reeb orbits.
We make use of holomorphic curve techniques for a suitable generalisation of
the Rabinowitz action functional in the stable case in order to prove the
forcing result. As in Schlenk’s theorem, we derive a lower bound for the
displacement energy of the displaceable hypersurface in terms of the action
value of such periodic orbits. The main application is a forcing result for
noncontractible periodic Reeb orbits on quotients of certain symmetric star-
shaped hypersurfaces. Either there exist two geometrically distinct
noncontractible periodic Reeb orbits or the period of the noncontractible
periodic Reeb orbit is small. This theorem can be applied to many physical
systems including the Hénon–Heiles Hamiltonian and Stark–Zeeman systems.
Further applications include a new proof of the well-known fact that the
displacement energy is a relative symplectic capacity on $\mathbb{R}^{2n}$ and
that the Hofer metric is indeed a metric.

###### Contents

  1. 1 Introduction
  2. 2 Results
    1. 2.1 Preliminaries on Twisted Stable Hypersurfaces
    2. 2.2 A Forcing Theorem for Twisted Periodic Reeb Orbits
  3. 3 Applications
    1. 3.1 The Hofer Distance and Relative Symplectic Capacities
    2. 3.2 Physical Systems and the Mañé Critical Value
  4. 4 Proof of Theorem 2.2
  5. Acknowledgements

##  1 Introduction

In [Bäh23], a generalisation of Rabinowitz–Floer homology was constructed.
Rabinowitz–Floer homology is the Morse–Bott homology in the sense of Floer
associated with the Rabinowitz action functional introduced by Kai Cieliebak
and Urs Frauenfelder in 2009. The main application of this generalisation was
to prove an existence result for noncontractible periodic Reeb orbits on
quotients of certain symmetric star-shaped hypersurfaces in $\mathbb{C}^{n}$,
$n\geq 2$. More precisely, let $\Sigma\subseteq\mathbb{C}^{n}$ be a compact
and connected star-shaped hypersurface invariant under the rotation

$\varphi\colon\mathbb{C}^{n}\to\mathbb{C}^{n},\quad\varphi(z_{1},\dots,z_{n})\mathrel{\mathop{\mathchar
58\relax}}=\left(e^{2\pi ik_{1}/m}z_{1},\dots,e^{2\pi ik_{n}/m}z_{n}\right)$

for some even $m\geq 2$ and $k_{1},\dots,k_{n}\in\mathbb{Z}$ coprime to $m$.
Then $\Sigma/\mathbb{Z}_{m}$ admits a noncontractible periodic Reeb orbit
generating the fundamental group
$\pi_{1}(\mathbb{S}^{2n-1}/\mathbb{Z}_{m})\cong\mathbb{Z}_{m}$. For a proof
see [Bäh23, Theorem 1.2] and [ALM22, Theorem 1.1] for the more general result,
removing the restriction of $m$ being even. The existence of noncontractible
periodic Reeb orbits on lens spaces is extremely relevant and attracts much
attention in celestial mechanics as mentioned in [LZ22, Introduction] or
[HS16]. We quickly recall the setup for the proof of this result. Let
$(W,\lambda)$ be a connected Liouville domain with connected boundary
$\partial W$ and consider a Liouville automorphism
$\varphi\in\operatorname{Aut}(W,\lambda)$, that is,
$\varphi\in\operatorname{Diff}(W)$ is of finite order and there exists a
unique function $f_{\varphi}\in C^{\infty}(\operatorname{Int}W)$ such that
$\varphi^{*}\lambda-\lambda=df_{\varphi}$. The main step was to construct a
homology theory for the _twisted Rabinowitz action functional_

$\mathscr{A}^{H}_{\varphi}\colon\mathscr{L}_{\varphi}M\times\mathbb{R}\to\mathbb{R},\qquad\mathscr{A}^{H}_{\varphi}(\gamma,\tau)\mathrel{\mathop{\mathchar
58\relax}}=\int_{0}^{1}\gamma^{*}\lambda-\tau\int_{0}^{1}H(\gamma(t))dt-
f_{\varphi}(\gamma(0))$

on the completion $(M,\lambda)$ of $(W,\lambda)$, where

$\mathscr{L}_{\varphi}M\mathrel{\mathop{\mathchar 58\relax}}=\\{\gamma\in
C^{\infty}(\mathbb{R},M)\mathrel{\mathop{\mathchar
58\relax}}\gamma(t+1)=\varphi(\gamma(t))\>\forall t\in\mathbb{R}\\}$

denotes the _twisted loop space of $M$ and $\varphi$_. Twisted loops play a
significant role in physical systems with symmetries, see for example [CFV23,
Section 6.2]. Consider the chain complex
$\operatorname{RFC}^{\varphi}(\partial W,M)$ generated by a suitable Morse
function on the critical manifold
$\operatorname{Crit}(\mathscr{A}^{H}_{\varphi})$, where

$(\gamma,\tau)\in\operatorname{Crit}(\mathscr{A}^{H}_{\varphi})\qquad\Leftrightarrow\qquad\begin{cases}\gamma\in\mathscr{L}_{\varphi}\partial
W,\\\ \dot{\gamma}(t)=\tau R(\gamma(t))\>\forall t\in\mathbb{R},\end{cases}$

with $R\in\mathfrak{X}(\partial W)$ denoting the Reeb vector field. We then
define twisted Rabinowitz-Floer homology as the Morse–Bott homology with
coefficients in $\mathbb{Z}_{2}$ by

$\operatorname{RFH}^{\varphi}(\partial W,M)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{HM}(\mathscr{A}^{H}_{\varphi})=\frac{\ker\partial\colon\operatorname{RFC}^{\varphi}(\partial
W,M)\to\operatorname{RFC}^{\varphi}(\partial
W,M)}{\operatorname{im}\partial\colon\operatorname{RFC}^{\varphi}(\partial
W,M)\to\operatorname{RFC}^{\varphi}(\partial W,M)},$

where the boundary map $\partial$ counts twisted negative gradient flow lines
modulo two with respect to a suitable $d\lambda$-compatible almost complex
structure on $M$. This homology theory has the following crucial properties.

  1. 1.

The semi-infinite dimensional Morse–Bott homology
$\operatorname{RFH}^{\varphi}(\partial W,M)$ is well-defined. Moreover,
twisted Rabinowitz–Floer homology is invariant under twisted homotopies of
Liouville domains.

  2. 2.

Twisted Rabinowitz–Floer homology is indeed a generalisation of the standard
Rabinowitz–Floer homology $\operatorname{RFH}(\partial W,M)$ defined in
[CF09], as

$\operatorname{RFH}^{\operatorname{id}_{W}}(\partial
W,M)\cong\operatorname{RFH}(\partial W,M).$

  3. 3.

If $\partial W$ is simply connected and does not admit any nonconstant twisted
periodic Reeb orbits, then

$\operatorname{RFH}^{\varphi}_{*}(\partial
W,M)\cong\operatorname{H}_{*}(\operatorname{Fix}(\varphi|_{\partial
W});\mathbb{Z}_{2}).$

Note that $\operatorname{Fix}(\varphi)$ is a symplectic submanifold of $M$ by
[MS17, Lemma 5.5.7].

  4. 4.

If $\partial W$ is displaceable by a compactly supported Hamiltonian
symplectomorphism in the completion $(M,\lambda)$, then

$\operatorname{RFH}^{\varphi}(\partial W,M)\cong 0.$

For a proof see [Bäh23, Theorem 1.1]. Note that there are two possible ways
for proving property 4: either one shows that the norm of the gradient of a
perturbed version of the twisted Rabinowitz action functional is uniformly
bounded from below as in [CF09, Lemma 3.9], or one generalises leaf-wise
intersection points following [AF10]. A direct consequence of properties 3 and
4 is the following observation as in [CF09, Corollary 1.5]. Suppose that
$\partial W$ is Hamiltonianly displaceable in the completion $(M,\lambda)$ and
simply connected. If $\operatorname{Fix}(\varphi|_{\partial
W})\neq\varnothing$, then $\partial W$ does admit a twisted periodic Reeb
orbit. Indeed, if there does not exist any twisted periodic Reeb orbit on the
boundary $\partial W$, we compute using property 3

$\operatorname{RFH}^{\varphi}(\partial
W,M)\cong\operatorname{H}(\operatorname{Fix}(\varphi|_{\partial
W});\mathbb{Z}_{2})=\bigoplus_{j\geq
0}\operatorname{H}_{j}(\operatorname{Fix}(\varphi|_{\partial
W});\mathbb{Z}_{2})\neq 0,$

contradicting property 4. However, if $\operatorname{Fix}(\varphi|_{\partial
W})=\varnothing$, then one cannot directly conclude the existence of a twisted
periodic Reeb orbit on $\partial W$. This is for example the case for the
rotation $\varphi\colon\mathbb{C}^{n}\to\mathbb{C}^{n}$ from the beginning. So
the best one can hope for is some kind of forcing result to hold. More
precisely, if we know that there exists a sufficiently well-behaved twisted
periodic Reeb orbit, then this _forces_ the existence of another one. The
above observation is already a forcing result, as
$\operatorname{Fix}(\varphi|_{\partial W})$ is precisely the set of all
constant twisted periodic Reeb orbits on the contact manifold $\partial W$.

##  2 Results

###  2.1 Preliminaries on Twisted Stable Hypersurfaces

###### Definition 2.1 (Stable Hypersurface, [CFP10, p. 1774]).

Let $(M,\omega)$ be a connected symplectic manifold. A _stable hypersurface_
in $(M,\omega)$ is a compact and connected hypersurface $\Sigma\subseteq M$
such that the following conditions hold:

  1. 1.

$\Sigma$ is separating, that is, $M\setminus\Sigma$ consists of two connected
components $M^{\pm}$, where $M^{-}$ is bounded and $M^{+}$ is unbounded.

  2. 2.

There exists a vector field $X$ in a neighbourhood of $\Sigma$ such that $X$
is outward-pointing to $\Sigma\cup M^{-}$ and
$\ker\omega|_{\Sigma}\subseteq\ker L_{X}\omega|_{\Sigma}$.

We write $(\Sigma,\omega|_{\Sigma},\lambda)$ for a stable hypersurface, where
the stabilising form $\lambda\in\Omega^{1}(\Sigma)$ is defined by
$\lambda\mathrel{\mathop{\mathchar 58\relax}}=i_{X}\omega|_{\Sigma}$.

###### Definition 2.2 (Twisted Stable Hypersurface).

Let $(\Sigma,\omega|_{\Sigma},\lambda)$ be a stable hypersurface in a
connected symplectic manifold $(M,\omega)$ and
$\varphi\in\operatorname{Symp}(M,\omega)$. We say that $\Sigma$ is _twisted_
by $\varphi$, if $\varphi(\Sigma)=\Sigma$, $\varphi$ is of finite order and
$\varphi^{*}X=X$.

###### Example 2.1 (Star-Shaped Hypersurfaces).

Consider the Liouville automorphism

$\varphi\colon\mathbb{C}^{n}\to\mathbb{C}^{n},\qquad\varphi(z_{1},\dots,z_{n})\mathrel{\mathop{\mathchar
58\relax}}=\left(e^{2\pi ik_{1}/m}z_{1},\dots,e^{2\pi ik_{n}/m}z_{n}\right)$

for $m\geq 2$ an integer and $k_{1},\dots,k_{n}\in\mathbb{Z}$ coprime to $m$.
Let $f\in C^{\infty}(\mathbb{S}^{2n-1})$ be a positive function such that
$f\circ\varphi=f$. Then the star-shaped hypersurface

$\Sigma_{f}=\\{f(z)z\mathrel{\mathop{\mathchar
58\relax}}z\in\mathbb{S}^{2n-1}\\}\subseteq\mathbb{C}^{n}$

is a contact manifold with $\varphi$-invariant contact form
$\lambda|_{\Sigma_{f}}$, where

$\lambda\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\sum_{j=1}^{n}(y_{j}dx_{j}-x_{j}dy_{j})=\frac{i}{4}\sum_{j=1}^{n}(\overline{z}_{j}dz_{j}-z_{j}d\overline{z}_{j})$

with complex coordinates $z_{j}=x_{j}+iy_{j}$. Indeed, by [FK18, Lemma
12.2.2], we have that

$X_{H_{f}}|_{\Sigma_{f}}\in\ker
d\lambda|_{\Sigma_{f}}\qquad\text{and}\qquad\lambda(X_{H_{f}})|_{\Sigma_{f}}=1$

for the defining Hamiltonian function

$H_{f}\colon\mathbb{C}^{n}\setminus\\{0\\}\to\mathbb{R},\qquad
H_{f}(z)\mathrel{\mathop{\mathchar
58\relax}}=\frac{\|z\|^{2}}{f(z/\|z\|^{2})}-1.$

Hence $(\Sigma_{f},\lambda|_{\Sigma_{f}})$ is a contact manifold as the
Liouville vector field

$X\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\sum_{j=1}^{n}\left(x_{j}\frac{\partial}{\partial
x_{j}}+y_{j}\frac{\partial}{\partial
y_{j}}\right)\in\mathfrak{X}(\mathbb{R}^{2n})$

satisfies $i_{X}d\lambda=\lambda$ and is outward-pointing as

$dH_{f}(X)|_{\Sigma_{f}}=d\lambda(X,X_{H_{f}})|_{\Sigma_{f}}=\lambda(X_{H_{f}})|_{\Sigma_{f}}=1.$

Finally, we conclude that

$X_{H_{f}}|_{\Sigma_{f}}=R_{f}\in\mathfrak{X}(\Sigma_{f})$

is the Reeb vector field.

###### Example 2.2 (Magnetic Torus, [CFP10, Section 6.1]).

Let $\mathbb{T}^{n}$ be the standard flat torus for $n\geq 2$ and let
$J\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ be an antisymmetric nonzero linear
map. Define $\rho\in\Omega^{2}(\mathbb{T}^{n})$ by setting
$\rho(\cdot,\cdot)\mathrel{\mathop{\mathchar
58\relax}}=\langle\cdot,J\cdot\rangle$ and denote by $\omega_{\rho}=dp\wedge
dq+\pi^{*}\rho$ the magnetic symplectic form on
$T^{*}\mathbb{T}^{n}\cong\mathbb{T}^{n}\times\mathbb{R}^{n}$. For an energy
value $k\in\mathbb{R}$ set $\Sigma_{k}\mathrel{\mathop{\mathchar
58\relax}}=H^{-1}(k)$ for the mechanical Hamiltonian function

$H(q,p)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\|p\|^{2}\qquad\forall(q,p)\in\mathbb{T}^{n}\times\mathbb{R}^{n}.$

Define $A\mathrel{\mathop{\mathchar 58\relax}}=(J|_{\operatorname{im}J})^{-1}$
and $\alpha\in\Omega^{1}(\operatorname{im}J)$ by

$\alpha_{x}(v)\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{2}\langle
x,Av\rangle.$

Then $\Sigma_{k}$ is a displaceable stable Hamiltonian manifold for every
$k>0$ by [CFP10, Proposition 6.3]. The stabilising form $\lambda$ on
$\Sigma_{k}$ is given by

$\lambda=f^{*}(pdq)+(\operatorname{pr}_{\parallel}\circ\operatorname{pr})^{*}\alpha,$
(1)

where

$\operatorname{pr}_{\perp}\colon\mathbb{R}^{n}\to\ker
J,\qquad\operatorname{pr}_{\parallel}\colon\mathbb{R}^{n}\to\operatorname{im}J,\qquad\operatorname{pr}\colon\mathbb{T}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$

denote the projections with respect to the orthogonal splitting

$\mathbb{R}^{n}=\ker J\oplus\operatorname{im}J,$

and

$f\colon\mathbb{T}^{n}\times\mathbb{R}^{n}\to\mathbb{T}^{n}\times\mathbb{R}^{n},\qquad
f(q,p)\mathrel{\mathop{\mathchar
58\relax}}=\left(q,\operatorname{pr}_{\perp}(p)\right).$

Let $\varphi\in\operatorname{Diff}(\mathbb{T}^{n})$ be an isometry of finite
order satisfying

$D\varphi\circ J=J\circ D\varphi$ (2)

and consider the cotangent lift

$D\varphi^{\dagger}\colon\mathbb{T}^{n}\times\mathbb{R}^{n}\to\mathbb{T}^{n}\times\mathbb{R}^{n},\qquad
D\varphi^{\dagger}(q,p)=\left(\varphi(q),\left(D\varphi^{-1}(q)\right)^{T}p\right).$

Then clearly $\varphi(\Sigma_{k})=\Sigma_{k}$ as $\varphi$ is an isometry and
$D\varphi^{\dagger}$ is of finite order as $\varphi$ is. Moreover, we have
that
$D\varphi^{\dagger}\in\operatorname{Symp}(T^{*}\mathbb{T}^{n},\omega_{\rho})$,
as $D\varphi^{\dagger}\in\operatorname{Symp}(T^{*}\mathbb{T}^{n},dp\wedge dq)$
and $D\varphi^{\dagger}$ preserves $\rho$ by assumption (2). Lastly, we have
that $\varphi^{*}\lambda=\lambda$ by considering formula (1) together with
assumption (2), and thus also $\varphi^{*}X=X$ by [Abb13, Proposition 4.2].

###### Definition 2.3 (Hofer Norm, [MS17, p. 466]).

Let $(M,\omega)$ be a symplectic manifold. Define the _Hofer norm_ of $F\in
C^{\infty}_{c}(M\times[0,1])$ by

$\|F\|\mathrel{\mathop{\mathchar 58\relax}}=\|F\|_{+}+\|F\|_{-},$

where

$\|F\|_{+}\mathrel{\mathop{\mathchar 58\relax}}=\int_{0}^{1}\max_{x\in
M}F_{t}(x)dt\qquad\text{and}\qquad\|F\|_{-}\mathrel{\mathop{\mathchar
58\relax}}=-\int_{0}^{1}\min_{x\in M}F_{t}(x)dt.$

###### Definition 2.4 (Displacement Energy, [MS17, p. 469]).

Let $(M,\omega)$ be a symplectic manifold. For a compact subset $A\subseteq M$
define the _displacement energy_ of $A$ by

$e(A)\mathrel{\mathop{\mathchar 58\relax}}=\inf_{\begin{subarray}{c}F\in
C^{\infty}_{c}(M\times[0,1])\\\ \varphi_{F}(A)\cap
A=\varnothing\end{subarray}}\|F\|,$

where $\varphi_{F}\mathrel{\mathop{\mathchar 58\relax}}=\phi^{X_{F}}_{1}$
denotes the time-1-map of the smooth flow of the time-dependent Hamiltonian
vector field $X_{F_{t}}$.

###### Example 2.3 ([BP08, p. 189]).

Let $M$ be a compact manifold without boundary. Then we have that
$e(M)=+\infty$ in $(T^{*}M,dp\wedge dq)$ for the zero-section $M$ in $T^{*}M$.
However, if $\rho\neq 0$ for a magnetic cotangent bundle
$(T^{*}M,\omega_{\rho})$ and $\chi(M)=0$ for the Euler-characteristic $\chi$
of $M$, then $e(M)<+\infty$ is finite. For more examples of nondisplaceable
hypersurfaces in cotangent bundles see [Mer14, Theorem 1.13].

###### Definition 2.5 (Symplectic Asphericity, [MS12, p. 302]).

A connected symplectic manifold $(M,\omega)$ is said to be _symplectically
aspherical_ , if

$\int_{\mathbb{S}^{2}}f^{*}\omega=0\qquad\forall f\in
C^{\infty}(\mathbb{S}^{2},M).$

Equivalently, $(M,\omega)$ is symplectically aspherical if and only if for the
de-Rham-homology class
$[\omega]|_{\pi_{2}(M)}=0\in\mathrm{H}_{\mathrm{dR}}^{2}(M;\mathbb{R})$.

###### Example 2.4 (Magnetic Torus).

The magnetic torus $(T^{*}\mathbb{T}^{n},\omega_{\rho})$ from Example 2.2 is
symplectically aspherical as $\omega_{\rho}=d\lambda_{\theta}$ is exact with

$\lambda_{\theta}\mathrel{\mathop{\mathchar
58\relax}}=pdq+\pi^{*}\theta,\qquad\theta_{q}(\cdot)=\frac{1}{2}\langle
q,J\cdot\rangle\in T^{*}_{q}\mathbb{T}^{n}$

for all $q\in\mathbb{T}^{n}$ by [CFP10, Lemma 6.2], where $\pi\colon
T^{*}\mathbb{T}^{n}\to\mathbb{T}^{n}$ denotes the projection. Alternatively,
the magnetic cotangent bundle $(T^{*}\mathbb{T}^{n},\omega_{\rho})$ is
symplectically aspherical as we have
$\pi_{2}(T^{*}\mathbb{T}^{n})\cong\pi_{2}(\mathbb{T}^{n})\times\pi_{2}(\mathbb{R}^{n})=0$.

###### Definition 2.6 (Contractible Twisted Loop Space).

Let $(M,\omega)$ be a symplectic manifold and
$\varphi\in\operatorname{Symp}(M,\omega)$ of finite order. A loop $v\in
C^{\infty}(\mathbb{T},M)$, $\mathbb{T}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{R}/\mathbb{Z}$, is said to be a _contractible twisted
periodic loop_ , if there exists $\gamma\in\mathscr{L}_{\varphi}M$ such that

$v(t)=\gamma(\operatorname{ord}(\varphi)t)\qquad\forall t\in\mathbb{T},$

and a filling $\bar{v}\in C^{\infty}(\mathbb{D},M)$ on the unit disc

$\mathbb{D}\mathrel{\mathop{\mathchar
58\relax}}=\\{z\in\mathbb{C}\mathrel{\mathop{\mathchar 58\relax}}|z|=1\\},$

such that $\bar{v}(e^{2\pi it})=v(t)$ for all $t\in\mathbb{T}$. We denote the
space of all contractible twisted periodic loops of $M$ and $\varphi$ by
$\Lambda_{\varphi}M$.

###### Definition 2.7 (Twisted Rabinowitz Action Functional).

Let $(\Sigma,\omega|_{\Sigma},\lambda)$ be a twisted stable hypersurface in a
symplectically aspherical symplectic manifold $(M,\omega)$. For a defining
Hamiltonian function $H$ for $\Sigma$ with $H\circ\varphi=H$, we define the
_twisted Rabinowitz action functional_

$\mathscr{A}^{H}_{\varphi}\colon\Lambda_{\varphi}M\times\mathbb{R}\to\mathbb{R},\qquad\mathscr{A}^{H}_{\varphi}(v,\tau)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\operatorname{ord}(\varphi)}\int_{\mathbb{D}}\bar{v}^{*}\omega-\tau\int_{0}^{1}H(v(t))dt.$

###### Remark 2.1 ($\operatorname{Crit}(\mathscr{A}^{H}_{\varphi})$).

Let $X\in\mathfrak{X}(\gamma)$ be a twisted variation, that is, $X$ is a
vector field along $\gamma$ and satisfies the condition

$X(t+1)=D\varphi(X(t))\qquad\forall t\in\mathbb{R}.$

Then a routine computation shows that

$(v,\tau)\in\operatorname{Crit}(\mathscr{A}^{H}_{\varphi})\qquad\Leftrightarrow\qquad\begin{cases}\gamma\in\mathscr{L}_{\varphi}\Sigma,\\\
\dot{\gamma}(t)=\tau R(\gamma(t))\>\forall t\in[0,1],\end{cases}$

where $R\in\mathfrak{X}(\Sigma)$ is the stable Reeb vector field. If $J$ is a
$\varphi$-invariant almost complex structure compatible with $\omega$, then
the gradient
$\operatorname{grad}_{J}\mathscr{A}^{H}_{\varphi}\in\mathfrak{X}(\Lambda_{\varphi}M\times\mathbb{R})$
with respect to the $L^{2}$-metric

$m\left((X,\eta),(Y,\sigma)\right)\mathrel{\mathop{\mathchar
58\relax}}=\int_{0}^{1}\omega(JX(t),Y(t))dt+\eta\sigma\qquad\forall(X,\eta),(Y,\sigma)\in
T_{(v,\tau)}\Lambda_{\varphi}M\times\mathbb{R},$

and $(v,\tau)\in\Lambda_{\varphi}M\times\mathbb{R}$, is given by

$\operatorname{grad}_{J}\mathscr{A}^{H}_{\varphi}|_{(v,\tau)}(t)=\begin{pmatrix}J(\dot{\gamma}(\operatorname{ord}(\varphi)t)-\tau
X_{H}(v(t)))\\\ \displaystyle-\int_{0}^{1}H\circ v\end{pmatrix}\qquad\forall
t\in\mathbb{T}.$

Hence $(u,\tau)\in C^{\infty}(\mathbb{R},\Lambda_{\varphi}M\times\mathbb{R})$
is a twisted negative gradient flow line, if the elliptic partial differential
equations or _twisted Rabinowitz–Floer equations_

$\partial_{s}u(s,t)+J\left(\partial_{t}\gamma(s,\operatorname{ord}(\varphi)t)-\tau(s)X_{H}(u(s,t))\right)=0\quad\text{and}\quad\partial_{s}\tau(s)=\int_{0}^{1}H(u(s,t))dt$

hold for all $(s,t)\in\mathbb{R}\times\mathbb{T}$.

###### Example 2.5 (Magnetic Torus).

Consider the displaceable twisted stable hypersurface
$\Sigma_{k}\subseteq(T^{*}\mathbb{T}^{n},\omega_{\rho},H)$ as in Example 2.2.
A point $(q,p)\in\Sigma_{k}$ gives rise to a twisted periodic Reeb orbit if
and only if

$\int_{0}^{\tau}e^{sJ}pds+q=\varphi(q),\quad e^{\tau
J}p=\left(D\varphi^{-1}(q)\right)^{T}p,\quad\text{and}\quad\|p\|^{2}=2k.$

A computation similar to [CFP10, p. 1843] shows

$\mathscr{A}^{H}_{\varphi}(v,\tau)=\operatorname{ord}(\varphi)k\tau.$

###### Definition 2.8 (Morse–Bott Component, [AF10, p. 86]).

Let $\mathscr{A}\colon\mathscr{E}\to\mathbb{R}$ be a smooth functional. A
subset $C\subseteq\operatorname{Crit}\mathscr{A}$ is called a _Morse–Bott
component_ , if

  1. 1.

$C$ is an action-constant submanifold of $\mathscr{E}$.

  2. 2.

$T_{x}C=\ker\operatorname{Hess}\mathscr{A}(x)$ for all $x\in C$ for the
Hessian $\operatorname{Hess}\mathscr{A}$ of $\mathscr{A}$.

###### Example 2.6 ($\operatorname{Fix}(\varphi|_{\Sigma})$).

Let $\Sigma$ be a twisted stable hypersurface in a symplectically aspherical
symplectic manifold. Then
$\operatorname{Fix}(\varphi|_{\Sigma})\subseteq\operatorname{Crit}\mathscr{A}^{H}_{\varphi}$
is a Morse–Bott component. Indeed, by [Bäh23, Proposition 2.23] we have that

$\ker\operatorname{Hess}\mathscr{A}^{H}_{\varphi}|_{(x,0)}\cong\ker(D\varphi_{x}-\operatorname{id}_{T_{x}\Sigma})=T_{x}\operatorname{Fix}(\varphi|_{\Sigma})$

for all $x\in\operatorname{Fix}(\varphi|_{\Sigma})$.

###### Definition 2.9 ([CFP10, p. 1768]).

A symplectic manifold $(M,\omega)$ is called _geometrically bounded_ , if
there exists an $\omega$-compatible almost complex structure $J$ and a
complete Riemannian metric such that the following conditions hold.

  1. 1.

There are constants $C_{0},C_{1}>0$ with

$\omega(Jv,v)\geq C_{0}\|v\|^{2}\qquad\text{and}\qquad|\omega(u,v)|\leq
C_{1}\|u\|\|v\|$

for all $u,v\in T_{x}M$ and $x\in M$.

  2. 2.

The sectional curvature of the metric is bounded above, and its injectivity
radius is bounded away from zero.

###### Example 2.7 ([CFP10, p. 1768]).

Magnetic cotangent bundles are geometrically bounded.

###  2.2 A Forcing Theorem for Twisted Periodic Reeb Orbits

Let $(W,\lambda)$ be a connected Liouville domain with connected boundary
$\Sigma\mathrel{\mathop{\mathchar 58\relax}}=\partial W$. Let $(M,\lambda)$ be
the completion of $(W,\lambda)$ and $\varphi\in\operatorname{Aut}(W,\lambda)$
a Liouville automorphism, that is, $\varphi\in\operatorname{Diff}(W)$ is a
diffeomorphism of finite order such that $\varphi^{*}\lambda=\lambda$. In this
setup, the kernel of the twisted Rabinowitz action functional
$\mathscr{A}^{H}_{\varphi}$ admits the canonical description

$\ker\operatorname{Hess}\mathscr{A}^{H}_{\varphi}|_{(v,\tau)}\cong\ker\left(D^{\xi}(\phi_{-\tau}^{R}\circ\varphi)|_{v(0)}-\operatorname{id}|_{\xi_{v(0)}}\right)\oplus\langle
R(v(0))\rangle$

by [Bäh23, Proposition 2.23], where $\xi\mathrel{\mathop{\mathchar
58\relax}}=\ker\lambda|_{\Sigma}$ denotes the contact distribution.

###### Definition 2.10 (Transversal Nondegeneracy, [FK18, Definition 7.3.1]).

Let $(M,\lambda)$ be the completion of a connected Liouville domain
$(W,\lambda)$. A contractible twisted periodic Reeb orbit
$(v,\tau)\in\operatorname{Crit}(\mathscr{A}_{\varphi}^{H})$ is said to be
_nondegenerate_ , if

$\ker\left(D^{\xi}(\phi_{-\tau}^{R}\circ\varphi)|_{v(0)}-\operatorname{id}|_{\xi_{v(0)}}\right)=\\{0\\}.$

###### Theorem 2.1.

Let $\Sigma\subseteq\mathbb{C}^{n}$, $n\geq 2$, be a compact and connected
star-shaped hypersurface invariant under the rotation

$\varphi\colon\mathbb{C}^{n}\to\mathbb{C}^{n},\quad\varphi(z_{1},\dots,z_{n})\mathrel{\mathop{\mathchar
58\relax}}=\left(e^{2\pi ik_{1}/m}z_{1},\dots,e^{2\pi ik_{n}/m}z_{n}\right)$

for some $m\geq 2$ and $k_{1},\dots,k_{n}\in\mathbb{Z}$ coprime to $m$. Assume
that there exists a nondegenerate noncontractible simple periodic Reeb orbit
$(\gamma_{0},\tau_{0})$ on $\Sigma/\mathbb{Z}_{m}$. Then there does exist a
noncontractible periodic Reeb orbit $(\gamma,\tau)$ on $\Sigma/\mathbb{Z}_{m}$
such that

$0<\tau-\tau_{0}\leq e(\Sigma).$

Consequently, we have two cases.

  1. 1.

If $\gamma$ is not an interate of $\gamma_{0}$, then $\Sigma/\mathbb{Z}_{m}$
admits two geometrically distinct noncontractible periodic Reeb orbits.

  2. 2.

If $\gamma$ is a $p$-fold iterate of $\gamma_{0}$, then the period $\tau_{0}$
is small in the sense of

$0<\tau_{0}\leq\frac{1}{p-1}e(\Sigma).$

###### Remark 2.2.

We cannot conclude the existence of two geometrically distinct noncontractible
periodic Reeb orbits as in [ALM22, Theorem 1.2] from Theorem 2.1 even under
the additional assumption that $\Sigma$ is dynamically convex. Indeed, the
sphere $\Sigma=\mathbb{S}^{2n-1}$ is dynamically convex by the
Hofer–Wysocki–Zehnder Theorem [FK18, Theorem 12.2.1], and both cases do occur
there.

###### Example 2.8 (The Hénon–Heiles Hamiltonian, [Sch20, Section 2]).

Consider the mechanical Hamiltonian function

$H\colon\mathbb{R}^{4}\to\mathbb{R},\qquad
H(q_{1},q_{2},p_{1},p_{2})\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\left(\|p\|^{2}+\|q\|^{2}\right)+q_{1}^{2}q_{2}-\frac{1}{3}q_{2}^{3}.$

This Hamiltonian function is known as the _Hénon–Heiles Hamiltonian_. On
$\mathbb{R}^{4}\cong\mathbb{C}^{2}$ consider the coordinates

$z\mathrel{\mathop{\mathchar 58\relax}}=q_{1}+iq_{2}\qquad\text{and}\qquad
w\mathrel{\mathop{\mathchar 58\relax}}=p_{1}+ip_{2}.$

Define

$\varphi\colon\mathbb{C}^{2}\to\mathbb{C}^{2},\qquad\varphi(z,w)\mathrel{\mathop{\mathchar
58\relax}}=e^{2\pi i/3}(z,w).$

We have that $\varphi^{*}\lambda=\lambda$ for

$\lambda=\frac{1}{2}(p_{1}dq_{1}-q_{1}dp_{1})+\frac{1}{2}(p_{2}dq_{2}-q_{2}dp_{2}).$

For every $0<k<\frac{1}{6}$, the regular energy surface $H^{-1}(k)$ contains a
strictly convex sphere-like component $\Sigma_{k}\cong\mathbb{S}^{3}$. The
resulting quotient $\Sigma_{k}/\mathbb{Z}_{3}$ is diffeomorphic to the lens
space $L(3,1)$, but not contactomorphic to it with the standard contact
distribution. Here we write $L(m,k_{2})$ for the lens space
$\mathbb{S}^{3}/\mathbb{Z}_{m}$ from Example 2.1 with $k_{1}=1$. Instead, the
quotient $\Sigma_{k}/\mathbb{Z}_{3}$ is contactomorphic to $L(3,2)$ with its
standard contact distribution. This is mainly due to the use of different
coordinates. By a shooting argument, one can show that there exist at least
two $\mathbb{Z}_{3}$-symmetric periodic orbits on $\Sigma_{k}$. In fact, by
[Sch20, Corollary 2.5], there exist infinitely many periodic orbits on
$\Sigma_{k}$.

###### Example 2.9 (Hill’s Lunar Problem, [FK18, Section 5.8]).

The mechanical Hamiltonian function $H\colon
T^{*}(\mathbb{R}^{2}\setminus\\{0\\})\to\mathbb{R}$ defined by

$H(q,p)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\|p\|^{2}-\frac{1}{\|q\|}+q_{1}p_{2}-q_{2}p_{1}-q_{1}^{2}+\frac{1}{2}q_{2}^{2}$

is called _Hill’s lunar Hamiltonian_. After Levi–Civita regularisation the
regularised Hill’s lunar Hamiltonian $K\colon
T^{*}\mathbb{R}^{2}\to\mathbb{R}$ is given by

$K(q,p)=\frac{1}{2}\left(\|p\|^{2}+\|q\|^{2}\right)+2\|q\|^{2}(q_{2}p_{1}-q_{1}p_{2})-4(q_{1}^{6}-3q_{1}^{4}q_{2}^{2}-3q_{1}^{2}q_{2}^{4}+q_{2}^{6}).$

For $k>0$ sufficiently small, the energy hypersurface $K^{-1}(k)$ admits at
least two periodic orbits by [LR11, Theorem 1] and contains a strictly convex
sphere-like component $\Sigma_{k}\cong\mathbb{S}^{3}$. On
$T^{*}\mathbb{R}^{2}\cong\mathbb{C}^{2}$ consider the coordinates

$z\mathrel{\mathop{\mathchar 58\relax}}=q_{1}+iq_{2}\qquad\text{and}\qquad
w\mathrel{\mathop{\mathchar 58\relax}}=p_{1}+ip_{2}$

and the rotation

$\varphi\colon\mathbb{C}^{2}\to\mathbb{C}^{2},\qquad\varphi(z,w)\mathrel{\mathop{\mathchar
58\relax}}=e^{\pi i/2}(z,w).$

Then $K$ is invariant under the rotation $\varphi$ and thus
$\Sigma/\mathbb{Z}_{4}$ is diffeomorphic to the lens space $L(4,1)$, but again
due to the choice of nonstandard coordinates not contactomorphic to it. It is
a delicate question in Contact Topology to decide the correct value of
$k_{2}\neq 1$, such that the obtained lens space $L(4,1)$ in Hill’s lunar
problem is contactomorphic to $L(4,k_{2})$. The tight contact structures on
the lens spaces $L(m,k_{2})$ are classified up to isotopy by [Hon00, Therorem
2.1], so in principle it should be possible to obtain the correct value of
$k_{2}$.

###### Example 2.10 (Stark–Zeeman Systems).

Planar Stark–Zeeman systems as in [CFK17] and [CFZ23] generalise many
important physical systems including the diamagnetic Kepler problem and the
restricted three body problem [Mor22]. By [CFK17, Corollary 1], for energy
values below the first critical value, the Moser regularised energy
hypersurfaces are diffeomorphic to the unit cotangent bundles
$S^{*}\mathbb{S}^{n}$. In particular, for $n=2$ we obtain
$S^{*}\mathbb{S}^{2}\cong\mathbb{RP}^{3}$, a real projective space.

Theorem 2.1 immediately follows from a more general result.

###### Theorem 2.2 (Forcing).

Let $\Sigma$ be a twisted stable displaceable hypersurface in a symplectically
aspherical, geometrically bounded, symplectic manifold $(M,\omega)$ for a
symplectomorphism $\varphi\in\operatorname{Symp}(M,\omega)$ of finite order
$\operatorname{ord}(\varphi)$ and suppose that $v_{0}$ is a contractible
twisted periodic Reeb orbit on $\Sigma$ belonging to a Morse–Bott component
$C$. Then there exists a contractible twisted periodic Reeb orbit $v\notin C$
such that

$\int_{\mathbb{D}}\bar{v}^{*}\omega-\int_{\mathbb{D}}\bar{v}_{0}^{*}\omega\leq\operatorname{ord}(\varphi)e(\Sigma).$

###### Remark 2.3.

The case $(M,\Sigma)=(\mathbb{C}^{n},\mathbb{S}^{2n-1})$ with the rotation

$\varphi\colon\mathbb{C}^{n}\to\mathbb{C}^{n},\quad\varphi(z_{1},\dots,z_{n})=\left(e^{2\pi
ik_{1}/m}z_{1},\dots,e^{2\pi ik_{n}/m}z_{n}\right),$

shows that the estimate in Theorem 2.2 is sharp.

Applying Theorem 2.2 to the Morse–Bott component
$\operatorname{Fix}(\varphi|_{\Sigma})$ from Example 2.6 yields the following
corollary.

###### Corollary 2.1.

Let $\Sigma$ be a twisted stable displaceable hypersurface in a symplectically
aspherical, geometrically bounded, symplectic manifold $(M,\omega)$ for
$\varphi\in\operatorname{Symp}(M,\omega)$ with
$\operatorname{Fix}(\varphi|_{\Sigma})\neq\varnothing$. Then there does exist
a nonconstant contractible twisted periodic Reeb orbit $v$ such that

$\int_{\mathbb{D}}\bar{v}^{*}\omega\leq\operatorname{ord}(\varphi)e(\Sigma).$

In particular, if we take $\varphi=\operatorname{id}_{M}$ in Corollary 2.1, we
recover Schlenk’s theorem as stated in [CFP10, Theorem 4.9] about the
existence of contractible closed characteristics on stable, displaceable
hypersurfaces with energy less or equal to the displacement energy of the
hypersurface. Schlenk proved this result in [Sch06, Theorem 1.1] using quite
different methods.

###### Example 2.11 (Magnetic Torus).

We can apply Theorem 2.2 and its Corollary 2.1 to the magnetic torus in
Example 2.2. Indeed, $(T^{*}\mathbb{T}^{n},\omega_{\rho})$ is geometrically
bounded by Example 2.7 and symplectically aspherical by Example 2.4. Moreover,
$\Sigma_{k}$ is stable and displaceable for every energy value $k>0$. Thus for
every contractible twisted periodic Reeb orbit $v_{0}$ belonging to a
Morse–Bott component, there does exist a contractible twisted periodic Reeb
orbit $v$ with

$\int_{0}^{1}v^{*}\lambda_{\theta}-\int_{0}^{1}v_{0}^{*}\lambda_{\theta}\leq\operatorname{ord}(\varphi)e(\Sigma).$

Further applications of Theorem 2.2 and its Corollary 2.1 are the content of
the next section. The proof of Theorem 2.2 is given in Section 4. It is also
the aim of future research to numerically investigate the Examples 2.8, 2.9
and 2.10, that is, finding upper bounds of the displacement energy and minimal
periods.

##  3 Applications

###  3.1 The Hofer Distance and Relative Symplectic Capacities

Computing the displacement energy 2.4 is usually quite difficult. Sometimes it
is possible to give upper bounds on the displacement energy as in [Iri14,
Theorem 1] or lower bounds as for any nonempty open subset $A\subseteq M$ of a
symplectic manifold $(M,\omega)$ we have $e(A)>0$ as in [BHS18, Theorem 1.1].
Corollary 2.1 has two immediate consequences. First, the existence of a
nonconstant contractible twisted periodic Reeb orbit on any twisted stable
displaceable hypersurface. Second, the existence of a lower bound for the
displacement energy via the action value of this critical point. If the
hypersurface is of contact type, this action value is precisely the period of
the parametrised periodic Reeb orbit. We illustrate the usefulness of the
second implication and give dynamical proofs of standard results. Recall, that
a _relative symplectic capacity on $\mathbb{R}^{2n}$_ is a map $c$ which
assigns to each subset $A\subseteq\mathbb{R}^{2n}$ a number
$c(A)\in[0,+\infty]$ such that the following three properties hold [MS17, p.
460].

  1. 1.

(_Relative Monotonicity_) If there exists a symplectomorphism $\psi$ of
$\mathbb{R}^{2n}$ such that $\psi(A)\subseteq B$, then $c(A)\leq c(B)$.

  2. 2.

(_Conformality_) $c(\lambda A)=\lambda^{2}c(A)$ for all
$\lambda\in\mathbb{R}$.

  3. 3.

(_Normalisation_) It holds that

$c(B^{2n}(r))=c(Z^{2n}(r))=\pi r^{2}\qquad\forall r>0,$

for the closed ball of radius $r$

$B^{2n}(r)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{(x,y)\in\mathbb{R}^{2n}\mathrel{\mathop{\mathchar
58\relax}}\|x\|^{2}+\|y\|^{2}\leq r^{2}\right\\},$

and the closed cylinder

$Z^{2n}(r)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{(x,y)\in\mathbb{R}^{2n}\mathrel{\mathop{\mathchar
58\relax}}x_{1}^{2}+y_{1}^{2}\leq r^{2}\right\\}.$

###### Proposition 3.1 ([MS17, Theorem 12.3.4]).

The displacement energy $e$ is a relative symplectic capacity on
$\mathbb{R}^{2n}$.

###### Proof.

Relative monotonicity and conformality are not hard to show. Moreover, by
relative monotonicity and [MS17, Exercise 12.3.7] we have that

$e(\partial B^{2n}(r))\leq e(B^{2n}(r))\leq e(Z^{2n}(r))\leq\pi
r^{2}\qquad\forall r>0.$

The periodic Reeb flow on $\partial B^{2n}(r)$ is given by

$\phi^{R_{r}}_{t}(z)=e^{-2it/r^{2}}z\qquad\forall z\in\partial B^{2n}(r).$

Hence the parametrised periodic Reeb orbits are $(\phi^{R_{r}}(z),\tau)$ with
$\tau\in\pi r^{2}\mathbb{Z}$. But Corollary 2.1 implies the existence of a
nonconstant closed periodic Reeb orbit $(v,\tau)$ on the contact hypersurface
$\partial B^{2n}(r)$ such that

$0<\tau=\int_{0}^{1}v^{*}\lambda\leq e(\partial B^{2n}(r))\leq\pi r^{2},$

where

$\lambda\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\sum_{j=1}^{n}(y_{j}dx_{j}-x_{j}dy_{j}).$

This is only possible for $\tau=\pi r^{2}$ and the statement follows. ∎

###### Proposition 3.2 ([BHS18, Theorem 1.1]).

For any subset $A\subseteq\mathbb{R}^{2n}$ with nonempty interior we have that
$e(A)>0$.

###### Proof.

If $A\subseteq M$ is not displaceable, we have that $e(A)=+\infty$ and thus
there is nothing to show. Moreover, if $A$ is not compact, we define

$e(A)\mathrel{\mathop{\mathchar 58\relax}}=\sup_{K\subseteq A\text{ $K$
compact}}e(K).$

So we can assume that $A$ is displaceable by a compactly supported Hamiltonian
symplectomorphism
$\varphi_{F}\in\operatorname{Ham}_{c}(\mathbb{R}^{2n},dy\wedge dx)$. As $A$ is
displaceable and has nonempty interior, there exists a closed ball $B(r)$ of
radius $r$ such that

$\varphi_{F}(B(r))\cap B(r)=\varnothing.$

Since the displacement energy is a normalised relative symplectic capacity by
Proposition 3.1, we conclude that

$e(A)\geq e(B(r))=\pi r^{2}>0.$

∎

###### Corollary 3.1 (Hofer Distance, [MS17, Theorem 12.3.3]).

On $\operatorname{Ham}_{c}(\mathbb{R}^{2n},dy\wedge dx)$ define the _Hofer
distance_

$\rho(\varphi_{0},\varphi_{1})\mathrel{\mathop{\mathchar
58\relax}}=\inf_{\varphi_{F}=\varphi_{1}\circ\varphi_{0}^{-1}}\|F\|.$

Then

$\rho(\varphi_{0},\varphi_{1})=0\quad\Rightarrow\quad\varphi_{0}=\varphi_{1}\qquad\forall\varphi_{0},\varphi_{1}\in\operatorname{Ham}_{c}(M,\omega),$

that is, the Hofer distance is a metric on
$\operatorname{Ham}_{c}(\mathbb{R}^{2n},dy\wedge dx)$.

###### Proof.

Let $\varphi\in\operatorname{Ham}_{c}(\mathbb{R}^{2n},dy\wedge dx)$ be not
equal to the identity. Thus there exists a set $A$ with nonempty interior such
that $\varphi(A)\cap A=\varnothing$. Thus Lemma 3.2 implies

$\rho(\varphi,\operatorname{id}_{\mathbb{R}^{2n}})\geq e(A)>0.$

This proves the statement. ∎

###### Remark 3.1.

In [LM95, Corollary 1.2], these results are generalised to arbitrary
symplectic manifolds.

###  3.2 Physical Systems and the Mañé Critical Value

###### Proposition 3.3.

Let $(T^{*}M,dp\wedge dq,H)$ be a Hamiltonian system for a compact
configuration space $M$ and define

$e_{0}(H)\mathrel{\mathop{\mathchar
58\relax}}=\inf\left\\{k\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\pi_{T^{*}M}\left(H^{-1}(k)\right)=M\right\\},$

where $\pi_{T^{*}M}\colon T^{*}M\to M$ denotes the projection. Suppose that
$\Sigma_{k}\mathrel{\mathop{\mathchar 58\relax}}=H^{-1}(k)$ with $k<e_{0}(H)$
is a $\varphi$-twisted stable regular energy surface admitting a contractible
twisted periodic Reeb orbit $(q_{0},p_{0})$ belonging to a Morse–Bott
component $C$. Then there exists a contractible twisted periodic Reeb orbit
$(q,p)\notin C$ such that

$\int_{0}^{1}p(t)\dot{q}(t)dt-\int_{0}^{1}p_{0}(t)\dot{q}_{0}(t)dt\leq\operatorname{ord}(\varphi)e(\Sigma_{c}).$

###### Proof.

We claim that $e(\Sigma_{k})<+\infty$ for all $k<e_{0}(H)$. In particular,
every energy hypersurface $\Sigma_{k}$ is displaceable in the geometrically
bounded and symplectically aspherical symplectic manifold $(T^{*}M,dp\wedge
dq)$ since $T^{*}M$ is an exact symplectic form with canonical Liouville form
$pdq$. As $k<e_{0}(H)$, we can displace $\Sigma_{k}$ into the missing fibres.
The explicit compactly supported Hamiltonian symplectomorphism achieving that
is constructed in [Con06, Proposition 8.2]. Hence if $\Sigma_{k}$ is twisted
stable and $k<e_{0}(H)$, we conclude the existence of such a contractible
periodic Reeb orbit from Theorem 2.2. ∎

###### Example 3.1 (Magnetic Torus).

Let $M$ be a compact manifold and $\theta\in\Omega^{1}(M)$. Then the map

$\varphi_{\theta}\colon(T^{*}M,dp\wedge
dq)\to(T^{*}M,\omega_{d\theta}),\qquad\varphi_{\theta}(q,p)\mathrel{\mathop{\mathchar
58\relax}}=(q,p-\theta_{q})$

is an exact symplectomorphism. Indeed, for every $(q,p)\in T^{*}M$ and $v\in
TT^{*}_{(q,p)}M$ we compute

$\displaystyle(\varphi^{*}_{-\theta}\lambda)_{(q,p)}(v)$
$\displaystyle=\lambda_{(q,p+\theta_{q})}\left(D\varphi_{-\theta}(v)\right)$
$\displaystyle=p\left(D\pi_{T^{*}M}(v)\right)+\theta_{q}\left(D\pi_{T^{*}M}(v)\right)$
$\displaystyle=(\lambda+\pi_{T^{*}M}^{*}\theta)_{(q,p)}(v),$

where $\lambda\in\Omega^{1}(T^{*}M)$ denotes the canonical Liouville form and
$\varphi_{-\theta}\circ\varphi_{\theta}=\operatorname{id}_{T^{*}M}$. A
mechanical Hamiltonian function

$H\colon(T^{*}M,\omega_{d\theta})\to\mathbb{R},\qquad
H(q,p)=\frac{1}{2}\|p\|_{m^{*}}^{2}+V(q),$

for some potential function $V\in C^{\infty}(M)$ is transformed under
$\varphi_{\theta}$ to a magnetic Hamiltonian function
$H_{\theta}=\varphi_{\theta}^{*}H$ given by

$H_{\theta}\colon(T^{*}M,dp\wedge dq)\to\mathbb{R},\qquad
H_{\theta}(q,p)=\frac{1}{2}\|p-\theta_{q}\|^{2}_{m^{*}}+V(q).$

In the case of a magnetic torus as in Example 2.11, we have that

$\theta_{q}(v)=\frac{1}{2}\langle
q,Jv\rangle\qquad\forall(q,v)\in\mathbb{T}^{n}\times\mathbb{R}^{n}.$

Thus if $k>0$, the intersection of $\Sigma_{k}=H^{-1}_{\theta}(k)$ with
$T_{q}^{*}\mathbb{T}^{n}$ is a sphere centred at $\theta_{q}$ for every
$q\in\mathbb{T}^{n}$. For more details see [AS19, Example 5.2]. Consequently,
we have that $e_{0}=0$ and Proposition 3.3 cannot be applied. Note that the
Mañé critical value $c$ is infinite in this case because a nonzero $\rho$ has
no bounded primitives in $\mathbb{R}^{n}$.

###### Remark 3.2.

In the setting of Proposition 3.3, if $H$ is a Tonelli Hamiltonian function,
that is, $H$ is strictly fibrewise convex and superlinear, then any stable
energy level of $H$ does contain a periodic Reeb orbit by [MP10]. See also
[Abb13, Theorem (iv)].

###### Remark 3.3.

The proof of Proposition 3.3 does not work for higher energy values in
general. This is due to a theorem of Will Merry in [Mer11, Theorem 1.1] and
[Mer11, Remark 1.7]. Let $H\in C^{\infty}(T^{*}M)$ be a Tonelli Hamiltonian
function. Define the Mañé critical value

$c\mathrel{\mathop{\mathchar
58\relax}}=\inf_{\theta}\sup_{q\in\widetilde{M}}\widetilde{H}(q,\theta_{q}),$

where the infimum is taken over all $1$-forms $\theta$ on the universal
covering manifold $\widetilde{M}$ with $d\theta=\widetilde{\rho}$, and
$\widetilde{H}\in C^{\infty}(T^{*}\widetilde{M})$ denotes the lift of $H$. We
always have that

$c\geq e_{0}(H).$

If $k>c$, then the Rabninowitz–Floer homology
$\operatorname{RFH}_{*}(\Sigma_{k},T^{*}M)$ of [CFP10] is well-defined and
does not vanish. In particular, $\Sigma_{k}$ is not displaceable. Thus we
cannot apply Theorem 2.2 in that case.

##  4 Proof of Theorem 2.2

The proof of Theorem 2.2 uses a method called a “homotopy of homotopies
argument”. Fix $\varepsilon>0$ and choose a Hamiltonian function $F\in
C^{\infty}_{c}(M\times[0,1])$ satisfying

$\textstyle F_{t}=0\>\forall
t\in[0,\frac{1}{2}],\qquad\|F\|<e(\Sigma)+\varepsilon\qquad\text{and}\qquad\varphi_{F}(\Sigma)\cap\Sigma=\varnothing.$

This is possible by definition of the displacement energy 2.4. Next we need to
carefully choose a twisted defining Hamiltonian function $H$ for the stable
hypersurface $\Sigma$. We postpone the construction of this Hamiltonian
function and explain the main idea of the proof. Choose a smooth family
$(\beta_{r})_{r\in[0,+\infty)}$ of cutoff functions $\beta_{r}\in
C^{\infty}(\mathbb{R},[0,1])$ such that

$\begin{cases}\beta_{r}(s)=0&|s|\geq r,\\\ \beta_{r}(s)=1&|s|\leq r-1,\\\
s\beta^{\prime}_{r}(s)\leq 0&\forall s\in\mathbb{R},\end{cases}$

for all $r\in[0,+\infty)$. Define a family of twisted Rabinowitz action
functionals

$\mathscr{A}_{r}\colon\Lambda_{\varphi}M\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$

by

$\mathscr{A}_{r}(v,\tau,s)\mathrel{\mathop{\mathchar
58\relax}}=\mathscr{A}^{H}_{\varphi}(v,\tau)-\beta_{r}(s)\int_{0}^{1}F_{t}(v(t))dt$

for all $r\in[0,+\infty)$. Note that
$\mathscr{A}_{0}=\mathscr{A}^{H}_{\varphi}$. For a suitable
$\varphi$-invariant $\omega$-compatible almost complex structure we consider
the moduli space

$\mathscr{M}\mathrel{\mathop{\mathchar 58\relax}}=\\{(u,\tau,r)\in
C^{\infty}(\mathbb{R},\mathscr{L}_{\varphi}M\times\mathbb{R})\times[0,+\infty)\mathrel{\mathop{\mathchar
58\relax}}(u,\tau,r)\text{ solution of \eqref{eq:moduli_space}}\\},$

where

$\begin{cases}\partial_{s}(u,\tau)=\operatorname{grad}\mathscr{A}_{r}|_{(u(s),\tau(s),s)}&\forall
s\in\mathbb{R},\\\
\displaystyle\lim_{s\to-\infty}(u(s),\tau(s))=(v_{0},\tau_{0}),\\\
\displaystyle\lim_{s\to+\infty}(u(s),\tau(s))\in C.\end{cases}$ (3)

Note that always $(v_{0},\tau_{0},0)\in\mathscr{M}$ and that such a
$\varphi$-invariant $\omega$-compatible almost complex structure always exists
by [MS17, Lemma 5.5.6]. The gradient $\operatorname{grad}\mathscr{A}_{r}$ of
$\mathscr{A}_{r}$ is taken with respect to the metric

$m\left((X,\eta),(Y,\sigma)\right)\mathrel{\mathop{\mathchar
58\relax}}=\int_{0}^{1}\omega(JX(t),Y(t))dt+\eta\sigma.$

###### Lemma 4.1.

If

$\mathscr{A}_{0}(v,\tau)>\|F\|+\mathscr{A}_{0}(v_{0},\tau_{0})\qquad\forall(v,\tau)\in\operatorname{Crit}(\mathscr{A}_{0})\setminus
C,$ (4)

then $\mathscr{M}$ is compact.

As a corollary of Lemma 4.1 we get Theorem 2.2. Indeed, the moduli space
$\mathscr{M}$ is the zero level set of a Fredholm section of a bundle over a
Banach manifold. As $v_{0}$ belongs to a Morse–Bott component, the Fredholm
section is regular at the point $v_{0}$, that is, the linearisation of the
gradient flow equation is surjective there. By compactness, we can therefore
perturb the Fredholm section to make it transverse. Hence $\mathscr{M}$ is a
compact smooth manifold with boundary consisting precisely of the point
$v_{0}$. See [MS12, Appendix A] for details. This is absurd, and we conclude
that there exists a critical point
$(v,\tau)\in\operatorname{Crit}(\mathscr{A}_{0})\setminus C$ such that

$\mathscr{A}_{0}(v,\tau)-\mathscr{A}_{0}(v_{0},\tau_{0})\leq\|F\|<e(\Sigma)+\varepsilon.$

As $\varepsilon>0$ was arbitrary, the statement follows since

$\mathscr{A}_{0}(v,\tau)=\frac{1}{\operatorname{ord}(\varphi)}\int_{\mathbb{D}}\bar{v}^{*}\omega\qquad\forall(v,\tau)\in\operatorname{Crit}(\mathscr{A}_{0}).$

We prove Lemma 4.1 in four steps.

_Step 1: If $(u,\tau,r)\in\mathscr{M}$, then $E(u,\tau)\leq\|F\|$ for the
energy_

$E(u,\tau)\mathrel{\mathop{\mathchar
58\relax}}=\int_{-\infty}^{+\infty}\|\partial_{s}(u,\tau)\|^{2}_{J}ds.$

We estimate

$\displaystyle E(u,\tau)$
$\displaystyle=\int_{-\infty}^{+\infty}\|\partial_{s}(u,\tau)\|_{J}^{2}ds$
$\displaystyle=\int_{-\infty}^{+\infty}d\mathscr{A}_{r}(\partial_{s}(u,\tau),s)ds$
$\displaystyle=\int_{-\infty}^{+\infty}\frac{d}{ds}\mathscr{A}_{r}(u,\tau,s)ds-\int_{-\infty}^{+\infty}(\partial_{s}\mathscr{A}_{r})(u,\tau,s)ds$
$\displaystyle=\lim_{s\to+\infty}\mathscr{A}_{r}(u,\tau,s)-\lim_{s\to-\infty}\mathscr{A}_{r}(u,\tau,s)-\int_{-\infty}^{+\infty}(\partial_{s}\mathscr{A}_{r})(u,\tau,s)ds$
$\displaystyle=\mathscr{A}_{0}(v,\tau)-\mathscr{A}_{0}(v_{0},\tau_{0})-\int_{-\infty}^{+\infty}(\partial_{s}\mathscr{A}_{r})(u,\tau,s)ds$
$\displaystyle=-\int_{-\infty}^{+\infty}(\partial_{s}\mathscr{A}_{r})(u,\tau,s)ds$
$\displaystyle=\int_{-\infty}^{+\infty}\dot{\beta}_{r}(s)\int_{0}^{1}F_{t}(u(s,t))dtds$
$\displaystyle\leq\|F\|_{+}\int_{-\infty}^{0}\dot{\beta}_{r}(s)ds-\|F\|_{-}\int_{0}^{+\infty}\dot{\beta}_{r}(s)ds$
$\displaystyle=\beta_{r}(0)(\|F\|_{-}+\|F\|_{+})$
$\displaystyle=\beta_{r}(0)\|F\|$ $\displaystyle\leq\|F\|,$

as $\mathscr{A}_{0}(v,\tau)=\mathscr{A}_{0}(v_{0},\tau_{0})$ since $C$ is
action-constant by definiton of a Morse–Bott component 2.8.

_Step 2: There exists $r_{0}\in\mathbb{R}$ such that $r\leq r_{0}$ for all
$(u,\tau,r)\in\mathscr{M}$._ Crucial is the existence of a constant $\delta>0$
such that

$\|\operatorname{grad}\mathscr{A}_{r}|_{(v,\tau,s)}\|_{J}\geq\delta\qquad\forall(v,\tau,s)\in\Lambda_{\varphi}M\times\mathbb{R}\times\mathbb{R}.$

This is proven along the lines of [CF09, Lemma 3.9]. With this inequality and
Step 1 we estimate

$\|F\|\geq
E(u,\tau)\geq\int_{-r}^{r}\|\operatorname{grad}\mathscr{A}_{r}|_{(u(s),\tau(s),s)}\|^{2}_{J}ds\geq
2r\delta^{2},$

and thus we can set

$r_{0}\mathrel{\mathop{\mathchar 58\relax}}=\frac{\|F\|}{2\delta^{2}}.$

_Step 3: There exists a constant $C>0$ such that $\|\tau\|_{\infty}\leq C$ for
all $(u,\tau,r)\in\mathscr{M}$._ This is a delicate estimate based on the
explicit construction of the defining Hamiltonian $H$ for $\Sigma$ as well as
an extension of the stabilising form $\lambda$. The bound on the Lagrange
multiplier is derived by comparing the twisted Rabinowitz action functional to
a different action functional. This modified version of the twisted Rabinowitz
action functional is obtained using a suitable extension of the
$\varphi$-invariant stabilising form $\lambda\in\Omega^{1}(\Sigma)$ to a
compactly supported form $\beta_{\lambda}\in\Omega^{1}(M)$. The precise
constructions can be found in [CFP10, Section 4.2.2]. Given $\beta_{\lambda}$,
we can define the auxiliary action functional

$\widehat{\mathscr{A}}_{0}\colon\Lambda_{\varphi}M\times\mathbb{R}\to\mathbb{R},\qquad\widehat{\mathscr{A}}_{0}(v,\tau)\mathrel{\mathop{\mathchar
58\relax}}=\int_{0}^{1}v^{*}\beta_{\lambda}-\tau\int_{0}^{1}H(v(t))dt.$

Moreover, we consider the bilinear form on the tangent bundle
$T\Lambda_{\varphi}M\times\mathbb{R}$

$\widehat{m}\left((X,\eta),(Y,\sigma)\right)\mathrel{\mathop{\mathchar
58\relax}}=\int_{0}^{1}d\beta_{\lambda}(JX(t),Y(t))dt+\eta\sigma.$

The main point in the choice of the $\varphi$-invariant $H\in C^{\infty}(M)$,
$\beta_{\lambda}\in\Omega^{1}(M)$ and the $\omega$-compatible
$\varphi$-invariant almost complex structure $J$ is to make sure, that the
properties

  1. 1.

$d\widehat{\mathscr{A}}_{0}(v,\tau)(X,\eta)=\widehat{m}\left(\operatorname{grad}\mathscr{A}_{0}(v,\tau),(X,\eta)\right)$,

  2. 2.

$(m-\widehat{m})\left((X,\eta),(X,\eta)\right)\leq 0$,

are true for all $(v,\tau)\in\Lambda_{\varphi}M\times\mathbb{R}$ and
$(X,\eta)\in T_{(v,\tau)}\Lambda_{\varphi}M\times\mathbb{R}$. These two
conditions ensure that the difference
$\mathscr{A}_{0}-\widehat{\mathscr{A}}_{0}$ is a Liapunov function for the
negative gradient flow lines of the twisted Rabinowitz action functional
$\mathscr{A}_{0}$. The uniform bound on the Lagrange multiplier $\tau$ now
follows from Steps 1 and 2. For details see [CFP10, p. 1808]. The only subtle
difference in our case is, that everything needs to be $\varphi$-invariant.
However, this is no problem as we explain now. The construction of $H$,
$\beta_{\lambda}$ and $J$ is based on the existence of a stable tubular
neighbourhood of $\Sigma$, that is, a pair $(\rho_{0},\psi)$ with $\rho_{0}>0$
and $\psi\colon(-\rho_{0},\rho_{0})\times\Sigma\hookrightarrow M$ an embedding
such that

$\psi|_{\\{0\\}\times\Sigma}=\iota_{\Sigma}\colon\Sigma\hookrightarrow
M\qquad\text{and}\qquad\psi^{*}_{\rho}\omega=\omega|_{\Sigma}+d(\rho\lambda).$

By [CFP10, Proposition 2.6 (a)], the space of stable tubular neighbourhoods of
$(\Sigma,\lambda)$ is nonempty. Using the quivariant Darboux–Weinstein Theorem
[VG84, Theorem 22.1], we get the existence of a stable tubular neighbourhood
$(\rho_{0},\psi)$, satisfying

$\varphi(\psi(\rho,x))=\psi(\rho,\varphi(x))\qquad\forall(\rho,x)\in(-\rho_{0},\rho_{0})\times\Sigma.$
(5)

Compare also [Bäh23, Equation (3.2)]. Hence the constructions [CFP10, p.
1791–1793] yield $\varphi$-invariant data $H$, $\beta_{\lambda}$ and $J$ due
to (5).

_Step 4: Proof of Lemma 4.1._ Let $(u_{k},\tau_{k},r_{k})$ be a sequence in
the moduli space $\mathscr{M}$. By Step 2 and Step 3, the sequences $(r_{k})$
and $(\tau_{k})$ are uniformly bounded. Thus $(u_{k},\tau_{k},r_{k})$ admits a
$C^{\infty}_{\mathrm{loc}}$-convergent subsequence by standard arguments
[MS12, Theorem B.4.2]. Indeed, the uniform $L^{\infty}$-bound on the sequence
$(u_{k})$ follows from the assumption that $(M,\omega)$ is geometrically
bounded and the uniform $L^{\infty}$-bound on the derivatives $(Du_{k})$
follows from the absence of bubbling as $(M,\omega)$ is symplectically
aspherical. In particular, there cannot exist a nonconstant $J$-holomorphic
sphere when the sequence of derivatives is unbounded [MS12, Section 4.2].
Denote the limit of this subsequence by $(u,\tau,r)$. This limit clearly
satisfies the first equation in (3), thus one only needs to check the
asymptotic conditions in (3). Again by compactness, $(u,\tau)$ converges to
critical points $(w_{\pm},\tau_{\pm})$ of $\mathscr{A}_{0}$ at its asymptotic
ends. We claim that

$\mathscr{A}_{r}(u(s),\tau(s),s)\in\left[-\|F\|+\mathscr{A}_{0}(v_{0},\tau_{0}),\|F\|+\mathscr{A}_{0}(v_{0},\tau_{0})\right]\qquad\forall
s\in\mathbb{R}.$ (6)

In particular,
$\mathscr{A}_{0}(w_{\pm},\tau_{\pm})\in\left[-\|F\|+\mathscr{A}_{0}(v_{0},\tau_{0}),\|F\|+\mathscr{A}_{0}(v_{0},\tau_{0})\right]$.
So if (6) holds, then by assumption (4) we conclude $(w_{\pm},\tau_{\pm})\in
C$ and $\mathscr{M}$ is indeed compact. It remains to prove (6). It is enough
to show

$\mathscr{A}_{r}(u_{k}(s),\tau_{k}(s),s)\in\left[-\|F\|+\mathscr{A}_{0}(v_{0},\tau_{0}),\|F\|+\mathscr{A}_{0}(v_{0},\tau_{0})\right]\qquad\forall
s\in\mathbb{R}$

for every $k\in\mathbb{N}$. As in the proof of [AF10, Lemma 2.8] we estimate

$\displaystyle 0$
$\displaystyle\leq\int^{+\infty}_{s_{0}}d\mathscr{A}_{r}(\partial_{s}(u_{k},\tau_{k}),s)ds$
$\displaystyle=\int^{+\infty}_{s_{0}}\frac{d}{ds}\mathscr{A}_{r}(u_{k},\tau_{k},s)ds-\int^{+\infty}_{s_{0}}(\partial_{s}\mathscr{A}_{r})(u_{k},\tau_{k},s)ds$
$\displaystyle=\lim_{s\to+\infty}\mathscr{A}_{r}(u_{k},\tau_{k},s)-\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})-\int^{+\infty}_{s_{0}}(\partial_{s}\mathscr{A}_{r})(u_{k},\tau_{k},s)ds$
$\displaystyle=\mathscr{A}_{0}(v,\tau)-\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})+\int^{+\infty}_{s_{0}}\dot{\beta}_{r}(s)\int_{0}^{1}F_{t}(u_{k}(s,t))dtds$
$\displaystyle\leq\mathscr{A}_{0}(v,\tau)-\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})+\int^{+\infty}_{-\infty}\|\dot{\beta}_{r}(s)F\|_{+}ds$
$\displaystyle\leq\mathscr{A}_{0}(v,\tau)-\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})+\|F\|$
$\displaystyle=\mathscr{A}_{0}(v_{0},\tau_{0})-\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})+\|F\|$

for all $s_{0}\in\mathbb{R}$. Similarly, we compute

$\displaystyle 0$
$\displaystyle\leq\int_{-\infty}^{s_{0}}d\mathscr{A}_{r}(\partial_{s}(u_{k},\tau_{k}),s)ds$
$\displaystyle=\int_{-\infty}^{s_{0}}\frac{d}{ds}\mathscr{A}_{r}(u_{k},\tau_{k},s)ds-\int_{-\infty}^{s_{0}}(\partial_{s}\mathscr{A}_{r})(u_{k},\tau_{k},s)ds$
$\displaystyle=\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})-\lim_{s\to-\infty}\mathscr{A}_{r}(u_{k},\tau_{k},s)-\int_{-\infty}^{s_{0}}(\partial_{s}\mathscr{A}_{r})(u_{k},\tau_{k},s)ds$
$\displaystyle=\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})-\mathscr{A}_{0}(v_{0},\tau_{0})+\int_{-\infty}^{s_{0}}\dot{\beta}_{r}(s)\int_{0}^{1}F_{t}(u_{k}(s,t))dtds,$

and thus we estimate

$\displaystyle\mathscr{A}_{r}(u_{k}(s_{0}),\tau_{k}(s_{0}),s_{0})$
$\displaystyle\geq\mathscr{A}_{0}(v_{0},\tau_{0})-\int_{-\infty}^{s_{0}}\dot{\beta}_{r}(s)\int_{0}^{1}F_{t}(u_{k}(s,t))dtds$
$\displaystyle\geq\mathscr{A}_{0}(v_{0},\tau_{0})-\int_{-\infty}^{+\infty}\|\dot{\beta}(s)F\|_{+}ds$
$\displaystyle\geq\mathscr{A}_{0}(v_{0},\tau_{0})-\|F\|.$

This shows the estimate (6) and so the proof of Lemma 4.1 and Theorem 2.2 is
complete.

## Acknowledgements

To Colin, Neil and Jil. In memory of Will J. Merry. A brilliant teacher and a
guiding light. Without him I would have never met Urs Frauenfelder, Kai
Cieliebak and Felix Schlenk.

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# Ion Kinetics and Neutron Generation Associated with Electromagnetic
Turbulence in Laboratory-scale Counter-streaming Plasmas

P. Liu  Institute for Fusion Theory and Simulation, School of Physics,
Zhejiang University, Hangzhou 310058, China  D. Wu<EMAIL_ADDRESS>Key
Laboratory for Laser Plasmas and School of Physics and Astronomy,
Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong
University, Shanghai 200240, China  T. X. Hu  Institute for Fusion Theory and
Simulation, School of Physics, Zhejiang University, Hangzhou 310058, China
Key Laboratory for Laser Plasmas and School of Physics and Astronomy,
Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong
University, Shanghai 200240, China  D. W. Yuan  Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100012, China  G. Zhao  Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China
Z. M. Sheng<EMAIL_ADDRESS>Institute for Fusion Theory and Simulation,
School of Physics, Zhejiang University, Hangzhou 310058, China  X. T. He
Institute for Fusion Theory and Simulation, School of Physics, Zhejiang
University, Hangzhou 310058, China  J. Zhang  Key Laboratory for Laser Plasmas
and School of Physics and Astronomy, Collaborative Innovation Center of IFSA
(CICIFSA), Shanghai Jiao Tong University, Shanghai 200240, China

###### Abstract

Electromagnetic turbulence and ion kinetics in counter-streaming plasmas hold
great significance in laboratory astrophysics, such as turbulence field
amplification and particle energization. Here, we quantitatively demonstrate
for the first time how electromagnetic turbulence affects ion kinetics under
achievable laboratory conditions (millimeter-scale interpenetrating plasmas
with initial velocity of $2000\ \mathrm{km/s}$, density of $4\times 10^{19}\
\mathrm{cm}^{-3}$, and temperature of $100\ \mathrm{eV}$) utilizing a recently
developed high-order implicit particle-in-cell code without scaling
transformation. It is found that the electromagnetic turbulence is driven by
ion two-stream and filamentation instabilities. For the magnetized scenarios
where an applied magnetic field of tens of Tesla is perpendicular to plasma
flows, the growth rates of instabilities increase with the strengthening of
applied magnetic field, which therefore leads to a significant enhancement of
turbulence fields. Under the competition between the stochastic acceleration
due to electromagnetic turbulence and collisional thermalization, ion
distribution function shows a distinct super-Gaussian shape, and the ion
kinetics are manifested in neutron yields and spectra. Our results have well
explained the recent unmagnetized experimental observations, and the findings
of magnetized scenario can be verified by current astrophysical experiments.

††preprint: APS/123-QED

The electromagnetic instabilities, which usually arise from the
interpenetrating plasma systems of the ejections from supernova explosions and
the surrounding interstellar medium, have become one of long-standing research
focuses in laboratory astrophysics communities. It is believed to be the
mechanism driving collisionless shocks and accelerating particles in several
astrophysical events, such as supernova remnants Miceli  _et al._ (2019) and
gamma-ray bursts Pruet  _et al._ (2001). In such so-called collisionless
systems, the mean-free-path of Coulomb collision $\lambda_{mfp}$ is larger
than the scale of interest $L$, allowing the plasma flows to interpenetrate
each other. It is noteworthy that particle motions can be modified by self-
generated electromagnetic fields, originating from the Biermann battery effect
Widrow (2002) or Weibel-type instabilities Weibel (1959); Fried (1959), which
can also affect the nuclear reaction process of charged particles.

In fact, although the growth of plasma instabilities is limited by collisional
dissipations in experiments Ryutov  _et al._ (2014), the electromagnetic field
structures are successfully observed in laser-driven counter-streaming plasmas
Kugland  _et al._ (2012); Ross  _et al._ (2013); Fox  _et al._ (2013);
Huntington  _et al._ (2015); Park  _et al._ (2015); Huntington  _et al._
(2017); Ross  _et al._ (2017); Fiuza  _et al._ (2020); Higginson  _et al._
(2019). Figure 1(a) shows the schematic diagram, in which counter-streaming
plasmas are produced with lasers irradiating two separated targets, and
nuclear reactions occur when two plasma jets collide. Such a counter-streaming
plasma system provides an important platform to investigate the particle
temperature coupling Ross  _et al._ (2013), collisionless shock Park  _et al._
(2015); Fiuza  _et al._ (2020), electron acceleration Fiuza  _et al._ (2020),
and neutron generation Ross  _et al._ (2017); Higginson  _et al._ (2019). For
the plasma flows with relatively high density and velocity, both thermonuclear
($\lambda_{mfp}\leq L$) and beam-beam ($\lambda_{mfp}\geq L$) reactions are
usually included. The thermonuclear neutrons are usually produced by thermal
collisions when the plasmas are in equilibrium, i.e., $\lambda_{mfp}\leq L$,
such as DD fusion reactions in interpenetrating plasmas of CD/CH case. While
for CD/CD scenario, neutron generations are dominated by the counter-streaming
CD/CD flows, namely beam-beam reaction, which is a non-equilibrium process and
can occur when $\lambda_{mfp}\geq L$ Higginson  _et al._ (2019). Recent
studies showed that the discrepancies in experimental observations and
simulation results for neutron yields Ross  _et al._ (2017) and velocities
Higginson  _et al._ (2019) are considered to be likely attributed to
electromagnetic effects, which are produced by plasma instabilities Kugland
_et al._ (2012); Ross  _et al._ (2013); Fox  _et al._ (2013); Huntington  _et
al._ (2015, 2017); Park  _et al._ (2015); Fiuza  _et al._ (2020). To the best
of our knowledge, distinguishing the effects of electromagnetic instabilities
on ion kinetics in counter-streaming plasmas is challenging for experiments.
Besides, laboratory-scale particle-in-cell (PIC) simulations including
electromagnetic turbulence, collisional thermalization, and neutron diagnosis
are not yet available in the publications due to the huge computational
burden. Therefore, the influences of electromagnetic instability and
turbulence on ion kinetics and neutron generation remain largely ambiguous.

In this Letter, we utilize a recently developed high-order implicit PIC code
to quantitatively investigate how electromagnetic turbulence affects ion
kinetics under realistic laboratory conditions. Supported by neutron
diagnostics, including neutron yields and spectra, we demonstrate that the
electromagnetic turbulence is driven by both ion two-stream instability (ITSI)
and ion filamentation instability (IFI). Furthermore, in magnetized scenarios,
the growth rates of these instabilities increase with the strengthening of the
applied magnetic field, leading to a significant enhancement of turbulence
fields and the emergence of super-Gaussian shapes in the ion distribution
functions. This strongly be associated with turbulence field amplification
Bohdan  _et al._ (2021); Bell (2004); Peterson  _et al._ (2021, 2022) and
particle acceleration Comisso and Sironi (2018); Zhdankin  _et al._ (2017);
Petrosian (2012); Zhdankin  _et al._ (2019). Additionally, following the
definitions of unstable modes provided by Bret and Lazar $et\ al$. Bret  _et
al._ (2005); Lazar  _et al._ (2009), we systematically derive and discuss
various plasma instabilities, including ITSI, IFI, ion Weibel instability
(IWI), and drift-kink instability (DKI).

A full linear kinetic model allows us to self-consistently account for ITSI,
IFI, IWI, and DKI in the unmagnetized counter-streaming systems, and the
detailed derivations of dispersion functions are presented in Sec. I of the
Supplemental Material Sup . It is found that the ion Weibel mode is stable,
and when considering the timescale of full development of instabilities (i.e.,
several hundred picoseconds), the DKI can be negligible due to its
characteristic growing-up time of $\sim$7 ns. We find that the electrostatic
ITSI is an oblique instability mode Bret  _et al._ (2010); Kato and Takabe
(2010); Skoutnev  _et al._ (2019) with growth rate of
$\Gamma_{\mathrm{ITSI}}\simeq 2.2\times 10^{-3}\omega_{pe}$, and its
characteristic growing-up time is $\sim$1 ps, which can provide a possible
seed for the development of turbulence fields; the IFI with growth rate of
$\Gamma_{\mathrm{IFI}}\simeq 4.3\times 10^{-3}\omega_{pi}$ and growing-up
timescale of $\sim$100 ps is another factor to drive electromagnetic
turbulence, and its growth rate depends on the anisotropy of ion beams, given
by Sup

$A_{i}=\frac{2v_{0}^{2}+v_{ti,\parallel}^{2}}{v_{ti,\perp}^{2}}-1,$ (1)

indicating that $A_{i}$ not only depends on the thermal anisotropy but also on
ratio of flow velocity to thermal velocity, where $v_{ti,\parallel}$ and
$v_{ti,\perp}$ are respectively the parallel and perpendicular thermal
velocities of ions with respect to flow velocity $v_{0}$, $\omega_{pe}$ and
$\omega_{pi}$ are the plasma frequencies of electron and ion, respectively. In
such counter-streaming systems, $A_{i}$ mainly depends on
$2v_{0}^{2}/v_{ti,\perp}^{2}$ for most recent experiments Kugland  _et al._
(2012); Ross  _et al._ (2013); Fox  _et al._ (2013); Huntington  _et al._
(2015); Park  _et al._ (2015); Huntington  _et al._ (2017); Ross  _et al._
(2017); Fiuza  _et al._ (2020); Higginson  _et al._ (2019) due to $v_{0}\gg
v_{ti}$ at the linear stage, which have been validated by our numerical
solutions Sup . Note that $A_{i}$ is only determined by thermal anisotropy
when $v_{0}\rightarrow 0$, reducing to classical Weibel mode Weibel (1959).

Figure 1:  (a) Schematic diagram of two counter-streaming plasmas generated by
laser-ablation targets, there are three possible unstable modes: the
filamentation mode (labeled with “FI”) with $\mathbf{k}\perp\mathbf{v}_{0}$,
Weibel mode (labeled with “WI”) with $\mathbf{k}\parallel\mathbf{v}_{0}$, and
two-stream mode (labeled with “TS”) with $\mathbf{k}\parallel\mathbf{v}_{0}$,
where $\mathbf{v}_{0}$ is the flow velocity. Maps of filamentous currents from
PIC simulations in $z$-$x$ (b) and $y$-$x$ (c) planes.

Compared to the linear regime, the nonlinear processes usually play a more
important role, and in fact, the nonlinear electromagnetic turbulence and its
influence on ion kinetics, which depend on both the plasma flow and ambient
medium conditions, are far from completely being understood. Fortunately, $ab\
initio$ PIC simulations have significantly improved our ability to study the
nonlinear processes associated with turbulence fields and ion kinetics. Here,
a series of large-scale two-dimensional (2D) PIC kinetic simulations are
performed by employing the LAPINS code Wu  _et al._ (2019, 2021), which is
based on a high-order implicit algorithm, eliminating the numerical cooling
found in the standard implicit PIC methods by the use of pseudo-electric-field
Wu  _et al._ (2019). Recently, a pairwise nuclear reaction module was
developed in the LAPINS code along with the binary collision algorithm Wu  _et
al._ (2021), it thus possesses the capability to self-consistently study
kinetic instabilities, collisional thermalizations, and nuclear reactions Wu
_et al._ (2019); Liu  _et al._ (2022). It should be remarked that the scaling
transformation PIC method Huntington  _et al._ (2015, 2017); Fiuza  _et al._
(2020), in which artificially scaling up beam velocity and reducing ion-to-
electron mass ratio, is no longer applicable because the collisional
thermalization and nuclear reaction cannot be achieved. Moreover, the effect
of nonphysical mass ratio on ion filamentation dynamics should not be
negligible at the nonlinear stage Ruyer  _et al._ (2016).

Figure 2:  Top: unmagnetized case with $B_{0}=0$ T. Bottom: magnetized case
with $B_{0}=50$ T. Snapshots of magnetic field (a),(b) and electric field
(c),(d) at $t=800$ ps. (e) and (f) are 2D Fourier transforms of magnetic field
shown in (a) and (b), respectively. (g),(h) Distribution functions of the
deuteron beams at $t=6$ ns, where the blue dots and solid (dashed) lines
respectively represent the simulation data and Gaussian (super-Gaussian)
fitting curves with the form of $\exp\left[-(v_{y}/v_{th})^{\alpha}\right]$
for high-energy ions. The inset of (g) for the case without considering
electromagnetic effects, i.e., the electromagnetic fields are not applied to
the particles. We also perform another magnetized case where $B_{0}=10$ T, and
the simulation results about the distributions of electromagnetic fields,
Fourier transform, and velocity distribution function can be seen in the
Supplemental Material Sup

In our simulations, the uniform $\mathrm{CD}/\mathrm{CH}$ and
$\mathrm{CD}/\mathrm{CD}$ plasma flows
($n_{\mathrm{C}}:n_{\mathrm{D}}=n_{\mathrm{C}}:n_{\mathrm{H}}=1:2$), with a
realistic ion mass ratio (e.g., $m_{\mathrm{D}}/m_{e}=3672$ for $\mathrm{D}$
and $m_{\mathrm{H}}/m_{e}=1836$ for $\mathrm{H}$), propagate along the $\pm y$
directions. The beams follow the typical experimental parameters of NIF and
OMEGA with initial velocity of $v_{0}=2000\ \text{km/s}$, density of
$n_{e}=4\times 10^{19}\ \text{cm}^{-3}$, and temperature of $T=100\
\mathrm{eV}$ (Kugland  _et al._ , 2012; Ross  _et al._ , 2013; Fox  _et al._ ,
2013; Huntington  _et al._ , 2015; Park  _et al._ , 2015; Huntington  _et al._
, 2017; Ross  _et al._ , 2017; Fiuza  _et al._ , 2020; Higginson  _et al._ ,
2019). The plasma flows are magnetized with a uniform magnetic field
$\mathbf{B}_{\mathrm{ext}}=B_{0}\mathbf{\hat{e}}_{x}$, and the corresponding
initial magnetization is
$\sigma_{0}=B_{0}^{2}/\left(\mu_{0}\sum_{i}{n_{i}}m_{i}v_{i}^{2}\right)\simeq
10^{-3}$ Demidem  _et al._ (2023), which is available for current laboratories
Fujioka  _et al._ (2013); Law  _et al._ (2016), where $\mu_{0}$ is vacuum
permeability, $n_{i}$ and $v_{i}$ are number density and velocity of ion
beams, respectively. The simulation window of sizes $0.8\ \mathrm{mm}\
(y)\times 0.4\ \mathrm{mm}\ (x)$ and $0.4\ \mathrm{mm}\ (z)\times 0.4\
\mathrm{mm}\ (x)$ for different configurations with periodic boundary
conditions for both particles and fields are adopted. The timestep of 3.3 fs
and grid size of $2.0\ \mu$m ($\simeq 0.02c/\omega_{pi}\simeq 2c/\omega_{pe}$)
with 528 particles per cell are used, where $c$ is light speed in vacuum.

To get a deeper insight of electromagnetic turbulence at the nonlinear stage,
we plot the magnetic field $B_{z}$ and electric field $E_{y}$ from the
unmagnetized case in Figs. 2(a) and 2(c), which show typical turbulence
distributions at a nonlinear moment $t=800$ ps. The average turbulent magnetic
field induced by the IFI is on the same order of magnitude as that predicted
by magnetic trapping theory Sup ; Davidson  _et al._ (1972)

$B_{\mathrm{sat}}\simeq\frac{\Gamma_{\mathrm{IFI}}^{2}m_{i}c}{q_{i}v_{0}k_{\mathrm{sat}}}\simeq
40\ \mathrm{T},$ (2)

where $k_{\mathrm{sat}}$ is the fastest-growing wave number. This nonlinear
electromagnetic turbulence is closely related to the plasma instabilities in
the counter-streaming system, including both the ITSI and IFI.

It should be noted that for the simulations of the $z$-$x$ plane, the
longitudinal ITSI can be excluded, only the transverse IFI develops, and the
saturated magnetic field can be well predicted by Eq. (2). The simulation
results obtained from this configuration are also presented in Supplemental
Material Sup .

PIC Simulation results indicate that the turbulence emerges spontaneously from
the nonlinear evolution of instabilities. In the magnetized scenario, the
electromagnetic turbulence with vortex structures is observed in Figs. 2(b)
and 2(d), and turbulence field amplification shows a significant
transformation of kinetic energy into waves. The average turbulent magnetic
fields can reach $\sim$90 T for $B_{0}=10$ T and $\sim$200 T for $B_{0}=50$ T,
and corresponding magnetization
$\sigma_{1}=B_{1}^{2}/\left(\mu_{0}\sum_{i}{n_{i}m_{i}v_{i}^{2}}\right)$ for
self-generated field $B_{1}$ are $\sim$0.0401 and $\sim$0.113, respectively,
which are more than 1 order of magnitude higher than that in the unmagnetized
case ($\sigma_{1}\simeq 0.0028$). Moreover, we also derive the dispersion
relations of the IFI and ITSI for the magnetized counter-streaming plasmas in
Sec. III of Supplemental Material Sup , which indicates that the linear growth
rates of both are enhanced with the increase of applied magnetic field
$\mathbf{B_{\mathrm{ext}}}$. Consequently, the saturated turbulent magnetic
fields are amplified due to $B_{\mathrm{sat}}\propto\Gamma^{2}$. This
electromagnetic turbulence is also analyzed in the spectral space, and in
Figs. 2(e) and 2(f), we show the amplitude of Fourier transforms of Figs. 2(a)
and 2(b). The turbulent magnetic field has a significant anisotropy
distribution with a wave number peaked at
$k_{m}=\left(k_{x}^{2}+k_{y}^{2}\right)^{1/2}\simeq 7\omega_{pi}/c$ in the
unmagnetized case. While for the magnetized scenario, the turbulence field
oscillations with a large wavelength ($\propto 1/k_{m}$) are observed, and
corresponding wave number drops to $k_{m}\simeq 3\omega_{pi}/c$ [see Fig.
2(f)], and it tend to exhibit isotropic turbulence Kolmogorov (1941).

Figure 3:  Temporal evolution of work done by turbulent electric field for
sampled deuterons in the unmagnetized case with $B_{0}=0$ T (a) and magnetized
case with $B_{0}=50$ T (b), where the colored curves represent different
tracked deuterons. Temporal evolution of the deuteron energy spectra (colored
by time) for three scenarios of excluding electromagnetic field effects (w/o
EB) (c), $B_{0}=0$ T (d), and $B_{0}=50$ T (e). (f) Neutron yields as the
functions of time in the $\mathrm{CD}/\mathrm{CH}$ case. (g) Average
reactivity of DD thermonuclear fusion as the function of temperature Hively
(1977); Atzeni and Meyer–ter–Vehn , where the marks of blue circle, magenta
pentagram, and red diamond, with $6.31\times 10^{-20}$, $2.20\times 10^{-19}$,
and $1.88\times 10^{-18}\ \mathrm{cm^{3}s^{-1}}$, correspond to three
temperatures in (c)-(e), respectively.

To illustrate the influences of such electromagnetic turbulence on ion
kinetics, we analyze the velocity distributions and energy spectra of
deuterons. As shown in Fig. 2(g), the ion distribution function features a
non-Maxwellian profile, and for the high-velocity tail, the falloff is steeper
than a usual Maxwellian shape, which is fitted by a super-Gaussian function of
$\exp\left[-(v_{y}/v_{th})^{\alpha}\right]$ with $\alpha\simeq 2.6$, where
$v_{th}$ is a constant. Moreover, driven by strong turbulence fields in the
magnetized case, the distribution function is severely deformed with a
exponential factor of $\alpha\simeq 2.8$, as illustrated in Fig. 2(h). This
distinct super-Gaussian shape is a result of the competition between
stochastic acceleration due to electromagnetic turbulence and collisional
thermalization. Note that similar super-Gaussian distributions of electrons
are observed due to inverse bremsstrahlung absorption in laser-plasma
interactions Matte  _et al._ (1988); Milder  _et al._ (2021). As expected,
when the electromagnetic effects are not taken into account, a standard
Maxwellian distribution with $\alpha\simeq 2$ appears in the inset of Fig.
2(g). Fig. 3(a) plots the work done by turbulent electric field on deuterons,
which shows that the deuterons experience stochastic acceleration or
deceleration in turbulence fields, and some of them obtain energies of about
25 keV, leading to a broadened energy spectrum in the unmagnetized case,
compared with the case where the electromagnetic effects are ignored, as shown
in Figs. 3(c) and 3(d). By fitting the high-energy tail of ion spectra, the
ion temperatures in these two cases are $T\simeq 4.5$ keV and $6.6$ keV,
respectively. Furthermore, in the magnetized case with $B_{0}=50\ \mathrm{T}$,
the deuteron beams can be further heated with a temperature of $T\simeq 16.5$
keV and cutoff energy of $E_{c}\simeq 0.29$ MeV due to the work done by a
stronger turbulent electric field [see Figs. 3(b) and 3(e)].

Figure 4:  Temporal evolution of neutron yields (a) and energy spectra at
$t=6$ ns (b) in different $\mathrm{CD}/\mathrm{CD}$ cases of without
considering electromagnetic effects (w/o EB), $B_{0}=0$ T, $B_{0}=10$ T, and
$B_{0}=50$ T. The blue line in (a) represents the yield difference between the
cases of w/o EB and $B_{0}=0\ \mathrm{T}$ as the function of time. Neutron
velocity shifts as the function of emitted angle $\theta$ in the cases of w/o
EB (c) and $B_{0}=0\ \mathrm{T}$ (d), where the red dashed lines represent
$\Delta V_{n}=m_{r}v_{r}^{2}/(2v_{n0}m_{\star})$, $v_{n0}$ is the velocity of
a neutron created from a cold and stationary deuterium plasma,
$m_{r}/m_{\star}\simeq 0.748$ and $v_{r}=2v_{0}$.

Compared with the low-energy deuterons, the high-energy ones corresponding to
larger nuclear reaction cross sections have a greater contribution to DD
reactions $\mathrm{D}(\mathrm{d},\mathrm{n})^{3}\mathrm{He}\ \left(Q=3.269\
\mathrm{MeV}\right)$ Higginson  _et al._ (2019); Atzeni and Meyer–ter–Vehn .
In the unmagnetized $\mathrm{CD}/\mathrm{CH}$ case, the turbulence fields
result in an about fivefold enhancement of neutron yield [see Fig. 3(f)],
which provides an explanation for a confusing point that was not fully
clarified in the work of Ross $et\ al.$ Ross  _et al._ (2017), where the
experimentally observed neutron yield is larger than that obtained from the
simulation without considering electromagnetic effects by a factor of
$\sim$6\. When considering pure thermonuclear reactivities Hively (1977);
Atzeni and Meyer–ter–Vehn , we note that the neutron yield is expected to
increase by approximately 3.5 times due to the electromagnetic effects [see
Fig. 3(g)], which indicates that the neutron yield originates not only from
thermonuclear reactions but also from the beam-beam reactions triggered by
$\sim$2.4% reversed deuterons [see Fig. 2(g)] due to the electromagnetic
turbulence. For the magnetized case where $B_{0}=50\ \mathrm{T}$, $N_{n}\simeq
2.1\times 10^{7}$, which is about 1 order of magnitude higher than that in the
unmagnetized case [see purple line in Fig. 3(f)].

For $\mathrm{CD}/\mathrm{CD}$ case, the thermonuclear reaction process is very
similar to that in $\mathrm{CD}/\mathrm{CH}$ case. However, the difference of
neutron yield caused by electromagnetic turbulence fields becomes more complex
because the beam-beam reactions from different flows are dominant. When
$t\gtrsim 0.15$ ns, the yield difference $\delta N_{n}$ between the cases of
w/o EB and $B_{0}=0\ \mathrm{T}$ is rapidly enlarged with $\left|\delta
N_{n}\right|\sim 10^{6}$, as shown in Fig. 4(a), which corresponds to the
linear period of the IFI growth. This is explained as follows, as illustrated
in Figs. 1(b) and 1(c), we witness the well-defined ion filamentous currents
at the linear stage, showing a self-organized misalignment distribution
between two ion beams, which can diminish the beam-beam reactivity Sup , and a
significant drop can be observed in the magnetized cases due to the enhanced
instabilities [see red and purple lines in Fig. 4(a)]. When $t\gtrsim 4$ ns,
$\delta N_{n}$ starts to increase, indicating that the electromagnetic
turbulence plays an increasingly important role in ion energization at the
nonlinear stage.

Additionally, a striking distinction in the neutron spectra is clearly visible
in Fig. 4(b), in which the full width at half maximum (FWHM) of energy
spectrum from the case of $B_{0}=0\ \mathrm{T}$ is $\Delta E_{n}\simeq 0.142\
\mathrm{MeV}$, which is $\sim$34% larger than that in the case excluding
electromagnetic effects, where $\Delta E_{n}\simeq 0.106\ \mathrm{MeV}$. This
is because the turbulence fields create greater center-of-mass velocity
$v_{cm}$ and relative velocity $v_{r}$, which broadens the neutron velocity
shifts $v_{n}-v_{n0}\equiv\Delta V_{n}$, formulated by Higginson  _et al._
(2019)

$\Delta V_{n}\simeq
v_{cm}\cos\theta-\frac{v_{cm}^{2}}{2v_{n0}}\sin^{2}\theta+\frac{1}{2v_{n0}}\frac{m_{r}}{m_{\star}}v_{r}^{2},$
(3)

as shown in the corresponding cases [see Figs. 4(c) and 4(d)]. Note that
$\Delta E_{n}$ can be up to about 0.24 MeV due to a larger $\Delta V_{n}$ (not
shown) for the magnetized case with $B_{0}=50$ T, leading to further
broadening of neutron spectrum due to $\Delta E_{n}\propto\Delta V_{n}^{2}$.

In conclusion, driven by plasma instabilities, significant amplification of
electromagnetic turbulence and the emergence of a non-Maxwellian ion
distribution are observed in counter-streaming plasmas. Through theoretical
analysis and numerical simulations, we have demonstrated that as the applied
magnetic field perpendicular to the flows increases, both the growth rates and
turbulence fields intensify. The ion distribution function exhibits a distinct
super-Gaussian shape due to the competition between stochastic acceleration of
turbulence and collisional thermalization. This ion kinetics is manifested in
the neutron yields and spectra. Notably, the thermonuclear neutron yield is
enhanced by more than 1 order of magnitude in the CD/CH case with $B_{0}=50\
\mathrm{T}$, and the neutron spectra exhibit significant broadening in the
magnetized CD/CD scenarios. Our results provide a good explanation for recent
experimental observations in unmagnetized plasmas. We anticipate that the
findings on magnetized plasmas will be validated through current astrophysical
experiments utilizing available laser facilities. These insights offer
profound implications for the amplification of turbulence fields and the
generation of energetic particles in laboratory astrophysics Bohdan  _et al._
(2021); Bell (2004); Peterson  _et al._ (2021, 2022); Comisso and Sironi
(2018); Zhdankin  _et al._ (2017); Petrosian (2012); Zhdankin  _et al._
(2019).

J. Zhang, D. Wu, and D. W. Yuan initiated this work. P. Liu and D. Wu
performed the simulations and data analysis. P. Liu and D. Wu drafted the
manuscript. All the authors contributed to physical analysis.

We would like to thank the anonymous referees for their constructive comments
and suggestions. This work was supported by the Strategic Priority Research
Program of Chinese Academy of Sciences (Grant No. XDA250050500), the National
Natural Science Foundation of China Grants (No. 12075204, No. 11875235, No.
61627901 and No. 11873061), the Shanghai Municipal Science and Technology Key
Project (No. 22JC1401500), the Chinese Academy of Sciences Youth
Interdiscipline Team (JCTD-2022-05), and the National Supercomputing Tianjin
Center Fusion Support Program. D. Wu thanks the sponsorship from Yangyang
Development Fund.

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