# Congruence of Linear Symplectic Forms by the Symplectic Group

Luchen Shi, Sunay Joshi, Ritwick Bhargava

(December 2022)

###### Abstract

This paper concerns the action of linear symplectomorphisms on linear
symplectic forms by conjugation in even dimensions. We prove that pfaffian and
$-\frac{1}{2}\operatorname{tr}(JA)$ (sum function) of $A$ are invariants on
the action. We use these invariants to provide a complete description of the
orbit space in dimension four. In addition, we investigate the geometric
shapes of the individual orbits in dimension four.

In symplectic geometry, our classification result in dimension four provides a
necessary condition for two symplectic forms on $\mathbb{R}^{4}$ to be
intertwined by symplectomorphisms of the standard symplectic form. This stands
in contrast to the lack of local invariants under diffeomorphisms.
Furthermore, we determine global invariants of a class of symplectic forms,
and we study an extension of a corollary of the Curry-Pelayo-Tang Stability
Theorem.

Lastly, we extend our results and investigate the action of linear
symplectomorphisms on linear symplectic forms in dimension $2n$. We determine
$n$ invariants of linear symplectic forms under this action, namely,
$s_{k}(A)$ we defined as $\sigma_{k}(A)$ which is the coefficient of term
$t^{k}$ in the polynomial expansion of pfaffian of $tJ+A$.

Keywords: Symplectic Forms, Symplectomorphisms, Invariants, Orbit Space,
Symplectic Geometry

##  1 Introduction

###  1.1 Motivation

Symplectic structures on $\mathbb{R}^{4}$ have been studied for decades.There
have been many important results on the diffeomorphism classes of symplectic
structures (the equivalence classes of symplectic structures under
diffeomorphisms): see references [2] [4] [5] [6] [8]. In this paper, we
restrict the use of diffeomorphisms to a narrower scope: the set of
symplectomorphisms of $(\mathbb{R}^{4},\omega_{\mathrm{std}})$. The
symplectomorphism group
$\operatorname{Symp}(\mathbb{R}^{4},\omega_{\mathrm{std}})$ consists of
diffeomorphisms of $\mathbb{R}^{4}$ which preserve the standard symplectic
form $\omega_{\mathrm{std}}$. The structure of
$\operatorname{Symp}(\mathbb{R}^{4},\omega_{\mathrm{std}})$ has been discussed
in references [9] [11] [12]. The elements of this group do not preserve
symplectic forms other than the scalar multiples of $\omega_{\mathrm{std}}$ in
general, and hence, we would like to know how a general symplectic form on
$\mathbb{R}^{4}$ can be acted by
$\operatorname{Symp}(\mathbb{R}^{4},\omega_{\mathrm{std}})$.

In this paper, we settle the linear version of this general question. The
answer to the linear problem yields a necessary condition for a symplectic
form on $\mathbb{R}^{4}$ to be transformed to another by
$\operatorname{Symp}(\mathbb{R}^{4},\omega_{\mathrm{std}})$, in the sense that
it provides both local invariants (the Pfaffian and the sum function) and
global invariants under symplectomorphisms. Note that this stands in contrast
to the lack of local invariants under diffeomorphisms. Furthermore, based on
our proof in four dimensions, we prove that the Pfaffian and sum function
remain invariant in every even dimension. These results may help in
understanding the structure of the symplectomorphism group
$\operatorname{Symp}(\mathbb{R}^{2n},\omega_{\mathrm{std}})$.

We also investigate the geometry of the orbits of the linear group action in
dimension four.

###  1.2 Goals

(Note: Throughout this paper, all vectors and matrices are written with
respect to the standard basis.)

Let

$S(2n,\mathbb{R})=\\{A\in M(2n,\mathbb{R})\mid A^{T}=-A,\det{A}\neq 0\\}$

denote the set of all linear symplectic forms on $\mathbb{R}^{2n}$, and let
the standard linear symplectic form on $\mathbb{R}^{2n}$ have matrix
representation

$J=\begin{bmatrix}J_{0}&O&\dots&O\\\ O&J_{0}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&O\\\ O&\dots&O&J_{0}\end{bmatrix},$

where $J_{0}=\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}$. Denote the set

$\operatorname{Sp}(2n)=\\{P\in M(2n,\mathbb{R})\mid P^{T}JP=J\\}$

as the group of linear symplectomorphisms on $\mathbb{R}^{2n}$.Consider the
group action $\rho:\operatorname{Sp}(2n)\times S(2n,\mathbb{R})\rightarrow
S(2n,\mathbb{R})$ given by $\rho(P,A)=P^{T}AP$ for $P\in\operatorname{Sp}(2n)$
and $A\in S(2n,\mathbb{R})$. In the case of $\mathbb{R}^{4}$, we have the
matrix representation $J$ as below according to the above definitions:

$J=\begin{bmatrix}0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\end{bmatrix},$

######  Question 1.

What is the orbit space $S(4,\mathbb{R})/\operatorname{Sp}(4)$?

######  Question 2.

Which quantities are invariant under the group action $\rho$ in
$\mathbb{R}^{2n}$?

######  Question 3.

What is the shape of each orbit, when considered as a subset of
$\mathbb{R}^{6}$?

######  Question 4.

How does the linear result apply to the nonlinear case?

######  Question 5.

What is the orbit space classification in higher dimensions?

The primary goal of this paper is to answer Question 1 and Question 2. At the
end of the paper, we partially address Questions 3, 4, and 5.

The structure of this paper is as follows: in Sections 3 and 4, we present
invariants of the group action $\rho$ in even dimensions as well as the Orbit
Space Classification of $S(4,\mathbb{R})/\operatorname{Sp}(4)$. We then
explore the geometric properties of the individual orbits of
$S(4,\mathbb{R})/\operatorname{Sp}(4)$ in Section 5. Finally, we employ the
language of symplectic forms to study the nonlinear case in $\mathbb{R}^{4}$
in Section 6 and to establish higher dimensional linear results in Section 7.

##  2 Main Results

###  2.1 Linear Case in $\mathbb{R}^{4}$

######  Definition 2.1 (Pfaffian of $4\times 4$ skew-symmetric matrix).

The Pfaffian of the $4\times 4$ skew-symmetric matrix

$A=\begin{bmatrix}0&a&b&c\\\ -a&0&d&e\\\ -b&-d&0&f\\\
-c&-e&-f&0\end{bmatrix},$

is the quantity $\operatorname{Pf}(A)=af-be+cd$.

It is well-known that for any $A\in S(4,\mathbb{R})$,
$\det(A)=\operatorname{Pf}(A)^{2}$ [1, Proposition 2.2].

######  Definition 2.2 (Sum function of $4\times 4$ skew-symmetric matrix).

The sum function of the $4\times 4$ skew-symmetric matrix

$A=\begin{bmatrix}0&a&b&c\\\ -a&0&d&e\\\ -b&-d&0&f\\\
-c&-e&-f&0\end{bmatrix},$

is the quantity $\operatorname{s}(A)=a+f$.

######  Theorem 2.1 (Invariants in $\mathbb{R}^{4}$).

The Pfaffian and sum function are invariant under the group action $\rho$.

######  Theorem 2.2 (Orbit Space Classification in $\mathbb{R}^{4}$).

The orbit space $S(4,\mathbb{R})/\operatorname{Sp}(4)$ consists of the
following families of distinct orbits:

  * •

For all $p>0$,

$\mathcal{J}_{p}^{+}=\\{\sqrt{p}J\\}$

  * •

For all $p>0$,

$\mathcal{J}_{p}^{-}=\\{-\sqrt{p}J\\}$

  * •

For all $p>0$, $q\in\mathbb{R}$,

$\mathcal{A}_{p,q}^{+}=\\{A\in S(4,\mathbb{R})\mid
A\not=\pm\sqrt{p}J,\operatorname{Pf}(A)=p,\operatorname{s}(A)=q\\}$

  * •

For all $p<0$, $q\in\mathbb{R}$,

$\mathcal{A}_{p,q}^{-}=\\{A\in
S(4,\mathbb{R})\mid\operatorname{Pf}(A)=p,\operatorname{s}(A)=q\\}$

We also present the shapes of two families of orbits of
$S(4,\mathbb{R})/\operatorname{Sp}(4)$.

######  Theorem 2.3 (Shapes of Orbits in $\mathbb{R}^{4}$).

Let $p\in\mathbb{R}^{*}$, $q\in\mathbb{R}$, and define
$\Delta=\frac{q^{2}}{4}-p$.

Then:

  * •

If $\Delta<0$, $\mathcal{A}_{p,q}$ is homeomorphic to
$(\mathbb{R}^{2}\backslash\\{0,0\\})\times\mathbb{R}^{2}$;

  * •

If $\Delta>0$, $\mathcal{A}_{p,q}$ is homeomorphic to
$\mathbb{S}^{2}\times\mathbb{R}^{2}$.

###  2.2 Nonlinear Case in $\mathbb{R}^{4}$ and $\mathbb{R}^{2n}$

######  Definition 2.3 (Linear symplectic form).

A linear symplectic form $\omega$ on $\mathbb{R}^{2n}$ is a map
$\omega:\mathbb{R}^{2n}\times\mathbb{R}^{2n}\rightarrow\mathbb{R}$ which
satisfies

  * •

$\omega$ is bilinear;

  * •

$\omega$ is skew-symmetric: if $v,w\in\mathbb{R}^{2n}$, then
$\omega(v,w)=-\omega(w,v)$; and

  * •

$\omega$ is non-degenerate: if $\omega(v,w)=0$ for all $w\in\mathbb{R}^{2n}$,
then $v=0$.

The pair $(\mathbb{R}^{2n},\omega)$ refers to the vector space
$\mathbb{R}^{2n}$ equipped with the linear symplectic form $\omega$. We call
$(\mathbb{R}^{2n},\omega)$ a symplectic vector space.

Any linear symplectic form $\omega$ can be represented by the action of a
characteristic matrix $\Omega$: if $\\{e_{1},\ldots,e_{2n}\\}$ is the standard
basis for $\mathbb{R}^{2n}$, and if $\Omega$ is the $2n\times 2n$ matrix such
that $\Omega_{i,j}=\omega(e_{i},e_{j})$ for all $1\leq i,j\leq 2n$, then
$\omega(v,w)=v^{T}\Omega w$ for all $v,w\in\mathbb{R}^{2n}$ [10, Proposition
2.8.14]. We shall refer to $\Omega$ as the matrix representation of $\omega$.

######  Definition 2.4 (Standard linear symplectic form).

Let $J$ denote the $2n\times 2n$ matrix with block matrix representation

$J=\begin{bmatrix}J_{0}&O&\dots&O\\\ O&J_{0}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&O\\\ O&\dots&O&J_{0}\end{bmatrix},$

where $J_{0}=\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}$. The standard linear
symplectic form on $\mathbb{R}^{2n}$ is the linear symplectic form
$\omega_{\mathrm{lstd}}$ defined as $\omega_{\mathrm{lstd}}(v,w)=v^{T}Jw$ for
all $v,w\in\mathbb{R}^{2n}$.

######  Definition 2.5 (Linear symplectomorphism).

A linear transformation $T:\mathbb{R}^{2n}\rightarrow\mathbb{R}^{2n}$ is a
linear symplectomorphism of $(\mathbb{R}^{2n},\omega_{\mathrm{lstd}})$ if
$\omega_{\mathrm{lstd}}(T(v),T(w))=\omega_{\mathrm{lstd}}(v,w)$ for all
$v,w\in\mathbb{R}^{2n}$.

Note that a linear transformation is a linear symplectomorphism if and only if
its matrix representation $P$ satisfies $P^{T}JP=J$ [10, Theorem 2.10.20].

We now turn to the definition of nonlinear symplectic forms [3, Definition
1.5].

######  Definition 2.6 ($2$-form).

A $2$-form on $\mathbb{R}^{2n}$ is an expression of the form

$\sum_{1\leq i<j\leq 2n}\omega_{i,j}\mathrm{d}x_{i}\wedge\mathrm{d}x_{j},$

where $\omega_{i,j}:\mathbb{R}^{2n}\rightarrow\mathbb{R}$ for all $1\leq
i<j\leq 2n$.

A $2$-form $\omega$ on $\mathbb{R}^{2n}$ is an alternating bilinear form
$\omega(x)$ at every point $x\in\mathbb{R}^{2n}$. Furthermore, the alternating
bilinear form $\omega(x)$ can be represented by the action of a characteristic
matrix $\Omega(x)$: if $\Omega(x)$ is the skew-symmetric matrix such that
$\Omega(x)_{i,j}=\omega_{i,j}$ for all $1\leq i<j\leq 2n$, then
$\omega(x)(v,w)=v^{T}\Omega(x)w$ for all $v,w\in\mathbb{R}^{2n}$. We shall
refer to $\Omega(x)$ as the matrix representation of $\omega(x)$.

######  Definition 2.7 (Symplectic form).

A symplectic form on $\mathbb{R}^{2n}$ is a $2$-form

$\omega=\sum_{1\leq i<j\leq
2n}\omega_{i,j}\mathrm{d}x_{i}\wedge\mathrm{d}x_{j}$

satisfying the following conditions:

  * •

$\omega$ is smooth: $\omega_{i,j}$ is smooth for all $1\leq i<j\leq 2n$;

  * •

$\omega$ is closed: for all $1\leq i<j<k\leq 2n$,

$\frac{\partial{\omega_{j,k}}}{\partial{x_{i}}}-\frac{\partial{\omega_{i,k}}}{\partial{x_{j}}}+\frac{\partial{\omega_{i,j}}}{\partial{x_{k}}}=0;$

and

  * •

$\omega$ is nondegenerate: $\det(\Omega(x))\neq 0$ for all
$x\in\mathbb{R}^{2n}$, where $\Omega$ is the matrix representation of
$\omega$.

As above, the symplectic vector space $(\mathbb{R}^{2n},\omega)$ represents
$\mathbb{R}^{2n}$ equipped with the symplectic form $\omega$.

Note that if $\omega$ is a symplectic form, then at each point
$x\in\mathbb{R}^{2n}$, the alternating bilinear form $\omega(x)$ is a linear
symplectic form.

######  Definition 2.8 (Standard symplectic form).

The standard symplectic form on $\mathbb{R}^{2n}$ is the symplectic form

$\omega_{\mathrm{std}}=\sum_{i=1}^{n}\mathrm{d}x_{2i-1}\wedge\mathrm{d}x_{2i}.$

Symplectomorphisms may be defined in terms of the pullback [10, Definition
7.3.1].

######  Definition 2.9 (Pullback of a $2$-form).

Let $\varphi:\mathbb{R}^{2n}\rightarrow\mathbb{R}^{2n}$ be a smooth map, and
let $\omega$ be a $2$-form. The pullback of $\omega$ by $\varphi$ is the
$2$-form defined as

$(\varphi^{*}\omega)(x)(v,w)=\omega(\varphi(x))(d\varphi_{x}(v),d\varphi_{x}(w))$

for all $v,w\in\mathbb{R}^{2n}$. (Here, $d\varphi_{x}$ denotes the
linearization of $\varphi$ at $x$.)

######  Definition 2.10 (Symplectomorphism).

A symplectomorphism $\varphi$ of $(\mathbb{R}^{2n},\omega_{\mathrm{std}})$ is
a diffeomorphism of $\mathbb{R}^{2n}$ satisfying
$\varphi^{*}\omega_{\mathrm{std}}=\omega_{\mathrm{std}}$.

We also introduce the notion of two symplectic forms being
$\omega_{\mathrm{std}}$-symplectomorphic.

######  Definition 2.11.

The symplectic forms $\omega_{1}$ and $\omega_{2}$ on $\mathbb{R}^{2n}$ are
$\omega_{\mathrm{std}}$-symplectomorphic if there exists
$\varphi\in\operatorname{Symp}(\mathbb{R}^{2n},\omega_{\mathrm{std}})$ such
that $\varphi^{*}\omega_{1}=\omega_{2}$.

With the above definitions in place, we may present our results for the
nonlinear case in $\mathbb{R}^{2n}$. Let $\mathbb{R}^{2n}$ have coordinates
$(x_{1},y_{1},x_{2},y_{2},...,x_{2n},y_{2n})$. First, we note that the
invariants of Theorem 2.1 provide a necessary condition for two nonlinear
symplectic forms on $\mathbb{R}^{2n}$ to be
$\omega_{\mathrm{std}}$-symplectomorphic.

######  Theorem 2.4.

Let $\omega_{1}$ and $\omega_{2}$ be symplectic forms on $\mathbb{R}^{2n}$.
Then $\omega_{1}$ and $\omega_{2}$ are
$\omega_{\mathrm{std}}$-symplectomorphic only if there exists
$\varphi\in\operatorname{Symp}(\mathbb{R}^{2n},\omega_{\mathrm{std}})$ such
that for every point $x\in\mathbb{R}^{2n}$, the Pfaffians and sum functions of
$\Omega_{1}(\varphi(x))$ and $\Omega_{2}(x)$ match. (Here,
$\Omega_{1}(\varphi(x))$ and $\Omega_{2}(x)$ are the matrix representations of
$\omega_{1}(\varphi(x))$ and $\omega_{2}(x)$, respectively.)

######  Theorem 2.5.

Consider symplectic forms

$\omega_{1}=\sum_{i=1}^{n}f_{i}(x_{i},y_{i})dx_{i}\wedge dy_{i}$

and

$\omega_{2}=\sum_{i=1}^{n}g_{i}(x_{i},y_{i})dx_{i}\wedge dy_{i}$

If $\phi\in\operatorname{Symp}(\mathbb{R}^{2n},\omega_{std})$, such that
$\phi^{*}\omega_{1}=\omega_{2}$, $\phi=(\phi_{1},\phi_{2},...,\phi_{n})$ Then
the set $\\{f_{i}(\phi^{2i-1},\phi^{2i})\\}$ and
$\\{g_{j}(\phi^{2j-1},\phi^{2j})\\}$ for some $i,j$ such that $1<i,j<n$ at
every point in $\mathbb{R}^{2n}$.

In particular, for the class of symplectic forms

$\omega_{1}=\sum_{i=1}^{n}f_{i}(x_{i},y_{i})dx_{i}\wedge dx_{i}$

where $f_{i},1\leq i\leq n$ are nowhere vanishing, we derive multiple global
invariants.

######  Theorem 2.6 (Global Invariants).

Consider the symplectic form

$\omega_{1}=\sum_{i=1}^{n}f_{i}(x_{i},y_{i})dx_{i}\wedge dx_{i}$

where $f_{i},1\leq i\leq n$ are nowhere vanishing. For $x\in\mathbb{R}^{2n}$,
define

$m_{\omega}(x)=\min\\{f_{i}(x_{i},y_{i}),1\leq i\leq n\\}$

and

$M_{\omega}(x)=\max\\{f_{i}(x_{i},y_{i}),1\leq i\leq n\\}.$

Then the following four quantities are global invariants of $\omega$:

  * •

$\inf\\{m_{\omega}(x)\mid x\in\mathbb{R}^{2n}\\}$,

  * •

$\sup\\{m_{\omega}(x)\mid x\in\mathbb{R}^{2n}\\}$,

  * •

$\inf\\{M_{\omega}(x)\mid x\in\mathbb{R}^{2n}\\}$,

  * •

$\sup\\{M_{\omega}(x)\mid x\in\mathbb{R}^{2n}\\}$.

In other words, if $\omega_{1}$ is $\omega_{\mathrm{std}}$-symplectomorphic to
the symplectic form

$\omega_{2}=\sum_{i=1}^{n}g_{i}(x_{i},y_{i})dx_{i}\wedge dy_{i}$

where $g_{i},1\leq i\leq n$ are nowhere vanishing, then the four global
invariants of $\omega_{1}$ and $\omega_{2}$ must match.

Finally, we consider the symplectic forms

$\omega_{t}=f_{1}(t,x_{1},y_{1})\mathrm{d}x_{1}\wedge\mathrm{d}y_{1}+f_{2}(t,x_{2},y_{2})\mathrm{d}x_{2}\wedge\mathrm{d}y_{2},t\in[0,1],$

where the functions $f_{i}$ are bounded away from zero with bounded time
derivative. The Curry-Pelayo-Tang Stability Theorem implies that there exists
a smooth path of $\varphi_{t}\in\operatorname{Diff}(\mathbb{R}^{4})$,
$t\in[0,1]$, such that $\varphi_{t}^{*}\omega_{t}=\omega_{0}$ [2, Example
4.1].

We present an infinite collection of smooth families of symplectic forms which
are diffeomorphic but not $\omega_{\mathrm{std}}$-symplectomorphic. For
instance, the family of symplectic forms

$\omega_{t}=(x_{1}^{2}+y_{1}^{2}+t+1)\mathrm{d}x_{1}\wedge\mathrm{d}y_{1}+(x_{2}^{2}+y_{2}^{2}+t+1)\mathrm{d}x_{2}\wedge\mathrm{d}y_{2},t\in[0,1]$

are diffeomorphic but not $\omega_{\mathrm{std}}$-symplectomorphic.

###  2.3 Higher Dimensional Linear Case

Lastly, we study a generalization of our original linear classification
problem to $\mathbb{R}^{2n}$.

Let

$S(2n,\mathbb{R})=\\{A\in M(2n,\mathbb{R})\mid A^{T}=-A,\det{A}\neq 0\\}$

denote the set of all linear symplectic forms on $\mathbb{R}^{2n}$, and let

$\operatorname{Sp}(2n)=\\{P\in M(2n,\mathbb{R})\mid P^{T}JP=J\\}$

denote the group of linear symplectomorphisms on $\mathbb{R}^{2n}$.

As above, we consider the group action $\rho:\operatorname{Sp}(2n)\times
S(2n,\mathbb{R})\rightarrow S(2n,\mathbb{R})$ given by $\rho(P,A)=P^{T}AP$ for
$P\in\operatorname{Sp}(2n)$ and for $A\in S(2n,\mathbb{R})$.

We present a general Equivalence Lemma for $\mathbb{R}^{2n}$. We also
demonstrate that the generalized Pfaffian and generalized sum function remain
invariants of the group action $\rho$ [1, Definition 2.1].

######  Definition 2.12 (Pfaffian of a $2n\times 2n$ matrix).

The Pfaffian of a $2n\times 2n$ matrix $A$ is the quantity

$\operatorname{Pf}(A)=\sum_{\sigma}\text{sgn}(\sigma)\prod_{i=1}^{n}A_{\sigma(2i-1),\sigma(2i)},$

where $\text{sgn}(\sigma)$ denotes the signature of $\sigma$ and the summation
is taken over all permutations $\sigma$ of $\\{1,2,\ldots,2n\\}$. (Here,
$A_{i,j}$ denotes the $(i,j)$-entry of $A$.)

Note that the identity $\det(A)=\operatorname{Pf}(A)^{2}$ holds in general [1,
Proposition 2.2].

######  Definition 2.13 (Sum function of $2n\times 2n$ matrix).

The sum function of a matrix $A\in S(2n,\mathbb{R})$ is the quantity

$\operatorname{s}(A)=\sum_{i=1}^{n}A_{2i-1,2i}.$

######  Theorem 2.7 (Invariants in $\mathbb{R}^{2n}$).

The Pfaffian of a matrix in $S(2n,\mathbb{R})$ is invariant under the group
action $\rho$. In addition, the sum function of a matrix in $S(2n,\mathbb{R})$
is invariant under the group action $\rho$.

######  Theorem 2.8 (Invariants in $\mathbb{R}^{2n}$).

Let $A\in S(2n,\mathbb{R})$, and let $0\leq k\leq n-1$. For a subset
$S\subset\\{1,2,...,n\\}$ with $|S|=k$, define the $(2n-2k)\times(2n-2k)$
matrix $A_{S}$ as the submatrix of $A$ formed by removing rows and columns
$2i-1$ and rows and columns $2i$ for all $i\in S$.Let $s_{k}(A)$ denote the
sum

$\sum_{\begin{subarray}{c}S\subseteq\\{1,2,\ldots,n\\}\\\
|S|=k\end{subarray}}\operatorname{Pf}(A_{S}).$

Then for any $P\in\operatorname{Sp}(2n)$, $s_{k}(A)=s_{k}(P^{T}AP)$. In other
words, $s_{k}$ is invariant under the group action $\rho$.

##  3 Proof of pfaffian and sum function in $\mathbb{R}^{4}$

In this section, we prove Theorem 2.1 and 2.7 together. We begin with the
invariance of the Pfaffian under $\rho$. We will need two well-known lemmas
[1, Proposition 2.3] [10, Corollary 7.3.5].

######  Lemma 3.1.

Let $A\in S(2n,\mathbb{R})$ and let $P\in M(2n,\mathbb{R})$. Then
$\operatorname{Pf}(P^{T}AP)=\operatorname{Pf}(A)\det(P)$.

######  Lemma 3.2.

If $P\in\operatorname{Sp}(2n)$, then $\det(P)=1$.

As a corollary of Lemmas 3.1 and 3.2, we have the invariance of the Pfaffian.

######  Lemma 3.3 (Pfaffian invariant in $\mathbb{R}^{4}$).

Let $A\in S(2n,\mathbb{R})$ and let $P\in\operatorname{Sp}(2n)$. Then
$\operatorname{Pf}(P^{T}AP)=\operatorname{Pf}(A)$. In other words, the
Pfaffian is invariant under the group action $\rho$.

###### Proof.

By Lemma 3.1, $\operatorname{Pf}(P^{T}AP)=\operatorname{Pf}(A)\det(P)$.
Furthermore, by Lemma 3.2, $\det{P}=1$. The result follows.

∎

Next, we shall prove the invariance of the sum function under $\rho$. Again,
we will need two lemmas.

######  Lemma 3.4.

If $P\in\operatorname{Sp}(2n)$, then $P^{T}\in\operatorname{Sp}(2n)$.

###### Proof.

By definition, if $P\in\operatorname{Sp}(2n)$, then $P^{T}JP=J$. Hence,

$JP^{T}JP=J^{2}=-I$ $\implies JP^{T}=-(JP)^{-1}=P^{-1}J$
$\implies(P^{T})^{T}JP^{T}=PJP^{T}=J,$

as desired.

∎

######  Lemma 3.5.

Let $P\in\operatorname{Sp}(2n)$. Let the column vectors of $P$ be
$c_{1},c_{2},\ldots,c_{2n}$, and let the row vectors of $P$ be
$r_{1},r_{2},\ldots,r_{2n}$. Then the sets $\\{c_{1},c_{2},\ldots,c_{2n}\\}$
and $\\{r_{1},r_{2},\ldots,r_{2n}\\}$ form symplectic bases for
$(\mathbb{R}^{2n},\omega_{\mathrm{lstd}})$.

###### Proof.

Collapse the column vectors $c_{1},c_{2},\ldots,c_{2n}$ of $P$ and write

$P=\begin{bmatrix}c_{1}&c_{2}&\dots&c_{2n}\end{bmatrix}.$

By definition, $P^{T}JP=J$. Therefore,

$\begin{bmatrix}c_{1}^{T}\\\ c_{2}^{T}\\\ \vdots\\\
c_{2n}^{T}\end{bmatrix}J\begin{bmatrix}c_{1}&c_{2}&\dots&c_{2n}\end{bmatrix}=\begin{bmatrix}c_{1}^{T}\\\
c_{2}^{T}\\\ \vdots\\\
c_{2n}^{T}\end{bmatrix}\begin{bmatrix}Jc_{1}&Jc_{2}&\dots&Jc_{2n}\end{bmatrix}$

$=\begin{bmatrix}c_{1}^{T}Jc_{1}&c_{1}^{T}Jc_{2}&\dots&c_{1}^{T}Jc_{2n}\\\
c_{2}^{T}Jc_{1}&c_{2}^{T}Jc_{2}&\dots&\vdots\\\
\vdots&\vdots&\ddots&c_{2n-1}^{T}Jc_{2n}\\\
c_{2n}^{T}Jc_{1}&\dots&c_{2n}^{T}Jc_{2n-1}&c_{2n}^{T}Jc_{2n}\end{bmatrix}$

$=\begin{bmatrix}\omega_{\mathrm{lstd}}(c_{1},c_{1})&\omega_{\mathrm{lstd}}(c_{1},c_{2})&\dots&\omega_{\mathrm{lstd}}(c_{1},c_{2n})\\\
\omega_{\mathrm{lstd}}(c_{2},c_{1})&\omega_{\mathrm{lstd}}(c_{2},c_{2})&\dots&\vdots\\\
\vdots&\vdots&\ddots&\omega_{\mathrm{lstd}}(c_{2n-1},c_{2n})\\\
\omega_{\mathrm{lstd}}(c_{2n},c_{1})&\dots&\omega_{\mathrm{lstd}}(c_{2n},c_{2n-1})&\omega_{\mathrm{lstd}}(c_{2n},c_{2n})\end{bmatrix}$

$=J=\begin{bmatrix}0&1&\dots&\dots&0\\\ -1&0&\ddots&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\vdots\\\ \vdots&\ddots&\ddots&0&1\\\
0&\dots&\dots&-1&0\end{bmatrix}.$

Equating corresponding entries, we find that $\\{c_{1},c_{2},\ldots,c_{2n}\\}$
forms a symplectic basis for $(\mathbb{R}^{2n},\omega_{\mathrm{lstd}})$. To
see that the same holds for the row vectors of $P$, note that by Lemma 7.6,
$P^{T}\in\operatorname{Sp}(2n)$. The result follows.

∎

We are now ready to prove the invariance of the sum function.

######  Lemma 3.6 (Sum function invariant in $\mathbb{R}^{2n}$).

Let $A\in S(2n,\mathbb{R})$ and let $P\in\operatorname{Sp}(2n)$. Then
$\operatorname{s}(P^{T}AP)=\operatorname{s}(A)$. In other words, the sum
function is invariant under the group action $\rho$.

###### Proof.

Let the matrix representation of $P$ be

$\begin{bmatrix}a_{1,1}&a_{1,2}&\dots&\dots&a_{1,2n}\\\
a_{2,1}&a_{2,2}&\ddots&\ddots&\vdots\\\ \vdots&\ddots&\ddots&\ddots&\vdots\\\
\vdots&\ddots&\ddots&a_{2n-1,2n-1}&a_{2n-1,2n}\\\
a_{2n,1}&\dots&\dots&a_{2n,2n-1}&a_{2n,2n}\end{bmatrix}.$

Let $c_{1},c_{2},\ldots,c_{2n}$ denote the column vectors of $P$, and let
$r_{1},r_{2},\ldots r_{2n}$ denote the row vectors of $P$. Let $\omega$ denote
the linear symplectic form with matrix representation $A$.

Then the matrix representation of $P^{T}AP$ is

$\begin{bmatrix}\omega(c_{1},c_{1})&\omega(c_{1},c_{2})&\dots&\omega(c_{1},c_{2n})\\\
\omega(c_{2},c_{1})&\omega(c_{2},c_{2})&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\omega(c_{2n-1},c_{2n})\\\
\omega(c_{2n},c_{1})&\dots&\omega(c_{2n},c_{2n-1})&\omega(c_{2n},c_{2n})\end{bmatrix}.$

Furthermore, since the matrix representation of $A$ can be rewritten as

$\begin{bmatrix}\omega(e_{1},e_{1})&\omega(e_{1},e_{2})&\dots&\omega(e_{1},e_{2n})\\\
\omega(e_{2},e_{1})&\omega(e_{2},e_{2})&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\omega(e_{2n-1},e_{2n})\\\
\omega(e_{2n},e_{1})&\dots&\omega(e_{2n},e_{2n-1})&\omega(e_{2n},e_{2n})\end{bmatrix},$

it suffices to show that
$\operatorname{s}(P^{T}AP)=\sum_{k=1}^{n}\omega(c_{2k-1},c_{2k})$ is equal to
$\operatorname{s}(A)=\sum_{k=1}^{n}\omega(e_{2k-1},e_{2k})$. To see this, note
that

$c_{k}=\sum_{j=1}^{2n}a_{j,k}e_{j}$

for all $1\leq k\leq 2n$. Hence, by the bilinearity of $\omega$,

$\sum_{k=1}^{n}\omega(c_{2k-1},c_{2k})=\sum_{1\leq i<j\leq
2n}\left(\sum_{k=1}^{n}(a_{i,2k-1}a_{j,2k}-a_{i,2k}a_{j,2k-1})\right)\omega(e_{i},e_{j})$

$=\sum_{1\leq i<j\leq
2n}\omega_{\mathrm{lstd}}(r_{i},r_{j})\omega(e_{i},e_{j}).$

Yet, by Lemma 7.7, $\\{r_{1},r_{2},\ldots,r_{2n}\\}$ is a symplectic basis for
$(\mathbb{R}^{2n},\omega_{\mathrm{lstd}})$. Therefore, the above sum reduces
to $\sum_{k=1}^{2n}\omega(e_{2k-1},e_{2k})=\operatorname{s}(A)$, as desired.

∎

Theorem 2.1 and 2.7 follows.

##  4 Proof of Orbit Space Classification in $\mathbb{R}^{4}$

###  4.1 Equivalence Lemma for $\mathbb{R}^{4}$

In order to establish the Orbit Space Classification of Theorem 2.2, we will
develop a method to classify equivalent skew-symmetric matrices based on their
so-called symplectic bases.

######  Definition 4.1 (Symplectic basis in $\mathbb{R}^{4}$).

Let $A\in S(4,\mathbb{R})$ and let $\mathcal{B}=\\{v_{1},v_{2},v_{3},v_{4}\\}$
be a basis for $\mathbb{R}^{4}$. The basis $\mathcal{B}$ is said to be a
symplectic basis for $A$ if the following equalities hold:

$v_{1}^{T}Av_{2}=1$ $v_{1}^{T}Av_{3}=0$ $v_{1}^{T}Av_{4}=0$
$v_{2}^{T}Av_{3}=0$ $v_{2}^{T}Av_{4}=0$ $v_{3}^{T}Av_{4}=1.$

The following theorem is well-known consequence of the Modified Gram-Schmidt
Algorithm [10, Theorem 2.10.4].

######  Theorem 4.1.

If $A\in S(4,\mathbb{R})$, then there exists a basis $\mathcal{B}$ for
$\mathbb{R}^{4}$ which is a symplectic basis for $A$.

######  Lemma 4.2.

Let $\\{v_{1},v_{2},v_{3},v_{4}\\}$ and $\\{w_{1},w_{2},w_{3},w_{4}\\}$ be two
bases for $\mathbb{R}^{4}$. Define the linear transformation
$P:\mathbb{R}^{4}\rightarrow\mathbb{R}^{4}$ such that $Pv_{i}=w_{i}$ for
$1\leq i\leq 4$. Then $P^{T}JP=J$ if and only if
$v_{i}^{T}Jv_{j}=w_{i}^{T}Jw_{j}$ for all $1\leq i,j\leq 4$.

###### Proof.

($\rightarrow$) If $P^{T}JP=J$, then

$w_{i}^{T}Jw_{j}=(Pv_{i})^{T}J(Pv_{j})=v_{i}^{T}(P^{T}JP)v_{j}=v_{i}^{T}Jv_{j},$

as desired.

($\leftarrow$) We may rewrite the given condition as

$v_{i}^{T}Jv_{j}=(Pv_{i})^{T}J(Pv_{j})=v_{i}^{T}(P^{T}JP)v_{j}$

for all $1\leq i,j\leq 4$. As $\\{v_{1},v_{2},v_{3},v_{4}\\}$ forms a basis
for $\mathbb{R}^{4}$, it follows that for all $x,y\in\mathbb{R}^{4}$,
$x^{T}Jy=x^{T}(P^{T}JP)y$. In particular, for the standard basis vectors
$x=e_{i}$ and $y=e_{j}$, we have
$J_{i,j}=e_{i}^{T}Je_{j}=e_{i}^{T}(P^{T}JP)e_{j}=(P^{T}JP)_{i,j}$. It follows
that all of the entries of $J$ and $P^{T}JP$ are equal, so that $J=P^{T}JP$.
The result follows.

∎

The following definition will allow us to present our primary method of matrix
classification, the Equivalence Lemma.

######  Definition 4.2 (Basis-values for $\mathbb{R}^{4}$).

Given a symplectic basis $\mathcal{B}=\\{v_{1},v_{2},v_{3},v_{4}\\}$ for the
matrix $A\in S(4,\mathbb{R})$, the set of basis-values of $\mathcal{B}$ is the
ordered $6$-tuple of real numbers

$(v_{1}^{T}Jv_{2},v_{1}^{T}Jv_{3},v_{1}^{T}Jv_{4},v_{2}^{T}Jv_{3},v_{2}^{T}Jv_{4},v_{3}^{T}Jv_{4}).$

At last, we may introduce the following corollary of Lemma 4.2.

######  Corollary 4.2.1 (Equivalence Lemma in $\mathbb{R}^{4}$).

Let $A,B\in S(4,\mathbb{R})$. Let
$\mathcal{B}_{1}=\\{v_{1},v_{2},v_{3},v_{4}\\}$ be a symplectic basis for $A$
and let $\mathcal{B}_{2}=\\{w_{1},w_{2},w_{3},w_{4}\\}$ be a symplectic basis
for $B$. If the set of basis-values of $\mathcal{B}_{1}$ is equal to the set
of basis-values of $\mathcal{B}_{2}$, then $A$ and $B$ are equivalent under
the group action $\rho$.

###### Proof.

Let $P:\mathbb{R}^{4}\rightarrow\mathbb{R}^{4}$ be the linear transformation
such that $Pv_{i}=w_{i}$ for $1\leq i\leq 4$. As the set of basis-values of
$\mathcal{B}_{1}$ is equal to the set of basis-values for $\mathcal{B}_{2}$,
by Lemma 4.2, it follows that $P^{T}JP=J$.

As $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are symplectic bases for $A$ and
$B$, respectively, $v_{i}^{T}Av_{j}=w_{i}^{T}Bw_{j}$ for all $1\leq i,j\leq
4$. Hence, $v_{i}^{T}Av_{j}=(Pv_{i})^{T}B(Pv_{j})^{T}$ for all $1\leq i,j\leq
4$, which implies that $v_{i}^{T}Av_{j}=v_{i}^{T}(P^{T}BP)v_{j}$ for all
$1\leq i,j\leq 4$.

As $\\{v_{1},v_{2},v_{3},v_{4}\\}$ is a basis for $\mathbb{R}^{4}$, it follows
that $x^{T}Ay=x^{T}(P^{T}BP)y$ for all $x,y\in\mathbb{R}^{4}$. In particular,
for the standard basis vectors $x=e_{i}$ and $y=e_{j}$, we have
$A_{i,j}=e_{i}^{T}Ae_{j}=e_{i}^{T}(P^{T}BP)e_{j}=(P^{T}BP)_{i,j}$. It follows
that all of the entries of $A$ and $P^{T}BP$ are equal, so that $A=P^{T}BP$.
The result follows.

∎

The Equivalence Lemma will be employed extensively in the following section. A
few remarks on its use are in order. Recall that the set of basis-values of a
basis $\mathcal{B}$ is an ordered tuple. Thus, in order to apply the
Equivalence Lemma to two sets of basis-values
$(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})$ and
$(b_{1},b_{2},b_{3},b_{4},b_{5},b_{6})$, we must ensure that $a_{i}=b_{i}$
holds for all $1\leq i\leq 6$.

###  4.2 Basis Classification

In this section, we complete the proof of the Orbit Space Classification of
Theorem 2.2. The proof is divided into two lemmas which establish the
equivalence of the matrices in each of the sets $\mathcal{A}_{p,q}^{+}$ and
$\mathcal{A}_{p,q}^{-}$.

######  Lemma 4.3.

Let $p>0$ and $q\in\mathbb{R}$. Then the elements of the set

$\mathcal{A}_{p,q}^{+}=\\{A\in S(4,\mathbb{R})\mid
A\not=\pm\sqrt{p}J,\operatorname{Pf}(A)=p,\operatorname{s}(A)=q\\}$

are equivalent under the group action $\rho$.

###### Proof.

Let $A\in\mathcal{A}_{p,q}^{+}$ be given as

$A=\begin{bmatrix}0&a&b&c\\\ -a&0&d&e\\\ -b&-d&0&f\\\ -c&-e&-f&0\\\
\end{bmatrix}.$

We shall split into cases in order to show that all matrices in
$\mathcal{A}_{p,q}^{+}$ are equivalent.

Case 1: $b\not=0$. In this case, the basis
$\mathcal{B}_{1}=\\{v_{1},v_{2},v_{3},v_{4}\\}$, where

$v_{1}=\begin{bmatrix}1\\\ 0\\\ 0\\\ 0\end{bmatrix}$

$v_{2}=\begin{bmatrix}0\\\ 0\\\ \frac{1}{b}\\\ 0\end{bmatrix}$

$v_{3}=\begin{bmatrix}-\frac{d}{b}\\\ 1\\\ -\frac{a}{b}\\\ 0\end{bmatrix}$

$v_{4}=\begin{bmatrix}-\frac{f}{af-be+cd}\\\ 0\\\ \frac{c}{af-be+cd}\\\
-\frac{b}{af-be+cd}\end{bmatrix},$

is a symplectic basis for $A$. Furthermore, the set of basis-values of
$\mathcal{B}_{1}$ is

$\left(0,1,0,0,-\frac{1}{af-be+cd},\frac{a+f}{af-be+cd}\right).$

As $af-be+cd=p$ and $a+f=q$ are fixed, the Equivalence Lemma implies that all
matrices in Case 1 are equivalent.

Case 2: $b=0$ and $c\neq 0$. In this case, the basis
$\mathcal{B}_{2}=\\{v_{1},v_{2},v_{3},v_{4}\\}$, where

$v_{1}=\begin{bmatrix}1\\\ 0\\\ 0\\\ 0\end{bmatrix}$

$v_{2}=\begin{bmatrix}0\\\ 0\\\ 0\\\ \frac{1}{c}\end{bmatrix}$

$v_{3}=\begin{bmatrix}-\frac{e}{c}\\\ 1\\\ 0\\\ -\frac{a}{c}\end{bmatrix}$

$v_{4}=\begin{bmatrix}-\frac{f}{af+cd}\\\ 0\\\ \frac{c}{af+cd}\\\
-\frac{b}{af+cd}\end{bmatrix},$

is a symplectic basis for $A$. Furthermore, the set of basis-values of
$\mathcal{B}_{2}$ is

$\left(0,1,0,0,-\frac{1}{af+cd},\frac{a+f}{af+cd}\right).$

As $af+cd=p$ and $a+f=q$, the Equivalence Lemma implies that all matrices in
Case 2 are equivalent. Furthermore, since the set of basis-values of the
matrices in Case 2 is equal to the set of basis-values of the matrices in Case
1, the Equivalence Lemma implies that all matrices in Cases 1 and 2 are
equivalent.

Case 3: $b=c=0$ and $d\neq 0$. In this case, we may apply the symplectic
permutation matrix $\tilde{P}\in\operatorname{Sp}(4)$ given by

$\tilde{P}=\begin{bmatrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
0&1&0&0\end{bmatrix}$

to yield the equivalent matrix $\tilde{P}^{T}A\tilde{P}$ in Case 2 with a
nonzero $(1,4)$-entry. Hence, the matrices of Case 3 are equivalent to the
matrices of Cases 1 and 2.

Case 4: $b=c=d=0$ and $e\neq 0$. As $af=p\neq 0$, it follows that $a\neq 0$
and $f\neq 0$. Hence, we may apply the matrix $P_{1}\in\operatorname{Sp}(4)$
given by

$P_{1}=\begin{bmatrix}0&\frac{e}{a}&0&0\\\ -\frac{a}{e}&0&0&\frac{a}{e}\\\
0&0&0&1\\\ 0&1&-1&0\\\ \end{bmatrix}$

to yield the equivalent matrix

$P_{1}^{T}AP_{1}=\begin{bmatrix}0&0&a&0\\\ 0&0&0&-f\\\ -a&0&0&a+f\\\
0&f&-a-f&0\\\ \end{bmatrix}$

with a nonzero $(1,3)$-entry. It follows that the matrices of Case 4 are
equivalent to those of Cases 1, 2, and 3.

Case 5: $b=c=d=e=0$. In this case, as $af=p$, the matrix $A$ is of the form

$A=\begin{bmatrix}0&a&0&0\\\ -a&0&0&0\\\ 0&0&0&\frac{p}{a}\\\
0&0&-\frac{p}{a}&0\\\ \end{bmatrix},$

with $a\neq\pm\sqrt{p}$. Hence, if $t=\frac{a}{\sqrt{p}}$, we may apply the
matrix $P_{2}\in\operatorname{Sp}(4)$ given by

$P_{2}=\begin{bmatrix}1&0&0&1\\\
0&\frac{1}{1-t^{2}}&\frac{t^{2}}{1-t^{2}}&0\\\
0&\frac{1}{1-t^{2}}&\frac{1}{1-t^{2}}&0\\\ t^{2}&1&1&1\\\ \end{bmatrix}$

to yield the equivalent matrix

$P_{2}^{T}AP_{2}=\sqrt{p}\cdot\begin{bmatrix}0&0&-t&0\\\ 0&0&0&\frac{1}{t}\\\
a&0&0&t+\frac{1}{t}\\\ 0&-\frac{1}{t}&-t-\frac{1}{t}&0\\\ \end{bmatrix},$

which has a nonzero $(1,3)$-entry. It follows that all matrices in Case 5 are
equivalent to those in Case 1. Therefore, all matrices in
$\mathcal{A}_{p,q}^{+}$ are equivalent, and we may conclude.

∎

######  Lemma 4.4.

Let $p<0$ and $q\in\mathbb{R}$. Then the elements of the set

$\mathcal{A}_{p,q}^{-}=\\{A\in
S(4,\mathbb{R})\mid\operatorname{Pf}(A)=p,\operatorname{s}(A)=q\\}$

are equivalent under the group action $\rho$.

###### Proof.

Let $A\in\mathcal{A}_{p,q}^{-}$ be given as

$A=\begin{bmatrix}0&a&b&c\\\ -a&0&d&e\\\ -b&-d&0&f\\\ -c&-e&-f&0\\\
\end{bmatrix}.$

By the analysis in the proof of Lemma 4.3, all matrices
$A\in\mathcal{A}_{p,q}^{-}$ with at least one of $b,c,d,e$ nonzero are
equivalent. It suffices to consider matrices $A$ with $b=c=d=e=0$.

In this case, as $af=p$, the matrix $A$ is of the form

$A=\begin{bmatrix}0&a&0&0\\\ -a&0&0&0\\\ 0&0&0&\frac{p}{a}\\\
0&0&-\frac{p}{a}&0\\\ \end{bmatrix}.$

Hence, if $t=\frac{a}{\sqrt{-p}}$ we may apply the matrix
$P\in\operatorname{Sp}(4)$ given by

$P=\begin{bmatrix}1&0&0&t^{2}+1\\\
0&\frac{1}{t^{2}+1}&-\frac{t^{2}}{(t^{2}+1)^{2}}&0\\\
0&1&\frac{1}{t^{2}+1}&0\\\ -\frac{t^{2}}{t^{2}+1}&1&\frac{1}{t^{2}+1}&1\\\
\end{bmatrix}$

to yield the equivalent matrix

$P^{T}AP=\sqrt{-p}\cdot\begin{bmatrix}0&0&-\frac{t}{t^{2}+1}&0\\\
0&0&0&-t-\frac{1}{t}\\\ \frac{t}{t^{2}+1}&0&0&t-\frac{1}{t}\\\
0&t+\frac{1}{t}&-t+\frac{1}{t}&0\\\ \end{bmatrix},$

which has a nonzero $(1,3)$-entry. It follows that all matrices in
$\mathcal{A}_{p,q}^{-}$ are equivalent, as desired.

∎

As a corollary of Theorem 2.1, the orbits $\mathcal{A}_{p,q}^{+}$ and
$\mathcal{A}_{p,q}^{-}$ are distinct for all $p,q$. Hence, by Lemmas 4.3 and
4.4, we have the Orbit Space Classification of Theorem 2.2.

##  5 Geometry of the Orbits in $\mathbb{R}^{4}$

In this section, we investigate the shapes of the individual orbits of
$S(4,\mathbb{R})/\operatorname{Sp}(4)$ when considered as subsets of
$\mathbb{R}^{6}$. We shall prove Theorem 2.3.

Let $p\in\mathbb{R}^{*}$, $q\in\mathbb{R}$. Then the orbits
$\mathcal{J}_{p}^{+}$ and $\mathcal{J}_{p}^{-}$ are singletons. The orbit
$\mathcal{A}_{p,q}$, on the other hand, is homeomorphic to the set

$\\{(a,b,c,d,e,f)\in\mathbb{R}^{6}\mid af-be+cd=p,a+f=q\\}.$

Let the discriminant $\Delta=\frac{q^{2}}{4}-p$, and consider the set

$D_{\Delta}=\\{(b,c,d,e)\in\mathbb{R}^{4}\mid be-cd\leq\Delta\\}.$

For each point $(a,b,c,d,e,f)\in\mathcal{A}_{p,q}$, $(b,c,d,e)$ is constrained
to $D_{\Delta}$. Furthermore, given $(b,c,d,e)\in D_{\Delta}$, there are at
most two corresponding values of $(a,f)$, namely

$(a,f)=\left(\frac{q}{2}+\sqrt{\Delta-(be-cd)},\frac{q}{2}-\sqrt{\Delta-(be-
cd)}\right)$

and

$(a,f)=\left(\frac{q}{2}-\sqrt{\Delta-(be-cd)},\frac{q}{2}+\sqrt{\Delta-(be-
cd)}\right).$

If $(b,c,d,e)$ lies on the boundary of $D_{\Delta}$, these two solutions are
identical. If $(b,c,d,e)$ lies in the interior of $D_{\Delta}$, these two
pairs are distinct. Hence, $\mathcal{A}_{p,q}$ is homeomorphic to the double
of $D_{\Delta}$: two copies of $D_{\Delta}$ glued along their boundaries.

In order to determine the shapes of the orbits $\mathcal{A}_{p,q}$ for
$\Delta>0$ and $\Delta<0$, we split into two cases.

Case 1: $\Delta>0$.

Let $D^{2}$ denote the closed unit disk. Consider the homeomorphism
$f:D^{2}\times\mathbb{R}^{2}\rightarrow D_{\Delta}$ defined as

$f(x,y,z,w)=(x\sqrt{z^{2}+w^{2}+\Delta}+z,w+y\sqrt{z^{2}+w^{2}+\Delta},w-y\sqrt{z^{2}+w^{2}+\Delta},x\sqrt{z^{2}+w^{2}+\Delta}-z).$

It follows that $D_{\Delta}$ is homeomorphic to $D^{2}\times\mathbb{R}^{2}$.
Furthermore, the double of $D^{2}$ is the sphere $\mathbb{S}^{2}$. Therefore,
in this case, $\mathcal{A}_{p,q}$ is homeomorphic to
$\mathbb{S}^{2}\times\mathbb{R}^{2}$, as desired.

Case 2: $\Delta<0$.

Consider the homeomorphism
$g:\mathbb{R}^{2}\times(D^{2}\backslash\\{0,0\\})\rightarrow D_{\Delta}$
defined as

$g(x,y,z,w)=\left(x+\frac{z\sqrt{x^{2}+y^{2}-\Delta}}{z^{2}+w^{2}},\frac{w\sqrt{x^{2}+y^{2}-\Delta}}{z^{2}+w^{2}}+y,\frac{w\sqrt{x^{2}+y^{2}-\Delta}}{z^{2}+w^{2}}-y,x-\frac{z\sqrt{x^{2}+y^{2}-\Delta}}{z^{2}+w^{2}}\right).$

It follows that $D_{\Delta}$ is homeomorphic to
$\mathbb{R}^{2}\times(D^{2}\backslash\\{0,0\\})$. The double of
$D^{2}\backslash\\{0,0\\}$ is homeomorphic to
$\mathbb{R}^{2}\backslash\\{0,0\\}$, and it follows that in this case,
$\mathcal{A}_{p,q}$ is homeomorphic to
$\mathbb{R}^{2}\times(\mathbb{R}^{2}\backslash\\{0,0\\})$. The result follows.

Cases 1, and 2 yield Theorem 2.3.

##  6 Nonlinear Case in $\mathbb{R}^{4}$

######  Proposition 6.1.

Given $\Omega_{1}$ and $\Omega_{2}$ matrix representations for symplectic form
$\omega_{1}$ and $\omega_{2}$ respectively, suppose that there is a
$\phi\in\operatorname{Symp}(\mathbb{R}^{2n},\omega_{\mathrm{std}})$ which
means that $\phi^{*}\omega_{1}=\omega_{2}$, then would the invariant
$s_{i},1\leq i\leq n$ match up for matrices $\Omega_{1}(\phi(x))$ and
$\Omega_{2}(x)$.

###### Proof.

By the equation $\phi^{*}\omega_{1}=\omega_{2}$, we get to know that
$P^{T}\Omega_{1}(\varphi(x))P=\Omega_{2}(x)$, where $P=d\varphi_{x}$ denotes
the linearization of $\varphi$ at $x$. Then we can conclude that
$\Omega_{1}(\varphi(x))$ and $\Omega_{2}(x)$ are equivalent under the group
action $rho$. Consequently by theorem 2.1, all the invariant will match up for
two matrices $\Omega_{1}(\phi(x))$ and $\Omega_{2}(x)$. ∎

In order to prove the global invariants of Theorem 2.6, we must restrict our
attention to the symplectic forms

$\omega=f_{1}(x_{1},y_{1})\mathrm{d}x_{1}\wedge\mathrm{d}y_{1}+f_{2}(x_{2},y_{2})\mathrm{d}x_{2}\wedge\mathrm{d}y_{2}$

and

$\omega^{\prime}=g_{1}(x_{1},y_{1})\mathrm{d}x_{1}\wedge\mathrm{d}y_{1}+g_{2}(x_{2},y_{2})\mathrm{d}x_{2}\wedge\mathrm{d}y_{2},$

where $f_{1},f_{2},g_{1}$, and $g_{2}$ are nowhere vanishing.

We will show that if $\omega$ and $\omega^{\prime}$ are
$\omega_{\mathrm{std}}$-symplectomorphic, then

$\inf\\{m_{\omega}(x)\mid
x\in\mathbb{R}^{4}\\}=\inf\\{m_{\omega^{\prime}}(x)\mid
x\in\mathbb{R}^{4}\\}.$

The three other global invariants are handled similarly.

We will need the following lemma, a corollary of Theorem 2.4.

######  Lemma 6.2.

Consider the symplectic forms

$\omega_{1}=f_{1}(x_{1},y_{1})\mathrm{d}x_{1}\wedge\mathrm{d}y_{1}+f_{2}(x_{2},y_{2})\mathrm{d}x_{2}\wedge\mathrm{d}y_{2}$

and

$\omega_{2}=g_{1}(x_{1},y_{1})\mathrm{d}x_{1}\wedge\mathrm{d}y_{1}+g_{2}(x_{2},y_{2})\mathrm{d}x_{2}\wedge\mathrm{d}y_{2},$

where $f_{1},f_{2},g_{1},$ and $g_{2}$ are nowhere vanishing. Let
$\varphi\in\operatorname{Symp}(\mathbb{R}^{4},\omega_{\mathrm{std}})$. If
$\varphi^{*}\omega_{1}=\omega_{2}$, then for each $x\in\mathbb{R}^{4}$,
either:

  * •

$f_{1}(\varphi^{1},\varphi^{2})=g_{1}(x_{1},y_{1})$ and
$f_{2}(\varphi^{3},\varphi^{4})=g_{2}(x_{2},y_{2})$, or

  * •

$f_{1}(\varphi^{1},\varphi^{2})=g_{2}(x_{2},y_{2})$ and
$f_{2}(\varphi^{3},\varphi^{4})=g_{1}(x_{1},y_{1})$.

(Here $\varphi^{i}:\mathbb{R}^{4}\rightarrow\mathbb{R}$ are the component
functions of $\varphi=(\varphi^{1},\varphi^{2},\varphi^{3},\varphi^{4})$.)

###### Proof.

By Theorem 2.4, the Pfaffian and sum functions of the matrices

$\Omega_{1}(\varphi(x))=\begin{bmatrix}0&f_{1}(\varphi^{1}(x),\varphi^{2}(x))&0&0\\\
-f_{1}(\varphi^{1}(x),\varphi^{2}(x))&0&0&0\\\
0&0&0&f_{2}(\varphi^{3}(x),\varphi^{4}(x))\\\
0&0&-f_{2}(\varphi^{3}(x),\varphi^{4}(x))&0\end{bmatrix}$

and

$\Omega_{2}(x)=\begin{bmatrix}0&g_{1}(x_{1},y_{1})&0&0\\\
-g_{1}(x_{1},y_{1})&0&0&0\\\ 0&0&0&g_{2}(x_{2},y_{2})\\\
0&0&-g_{2}(x_{2},y_{2})&0\end{bmatrix}$

must match at every point $x\in\mathbb{R}^{4}$.

Hence, for each $x\in\mathbb{R}^{4}$,

$f_{1}(\varphi^{1}(x),\varphi^{2}(x))f_{2}(\varphi^{3}(x),\varphi^{4}(x))=g_{1}(x_{1},y_{1})g_{2}(x_{2},y_{2})$

and

$f_{1}(\varphi^{1}(x),\varphi^{2}(x))+f_{2}(\varphi^{3}(x),\varphi^{4}(x))=g_{1}(x_{1},y_{1})+g_{2}(x_{2},y_{2}).$

The result follows.

∎

By Lemma 6.1,
$\\{f_{1}(\varphi^{1}(x),\varphi^{2}(x)),f_{2}(\varphi^{3}(x),\varphi^{4}(x))\\}=\\{g_{1}(x_{1},y_{1}),g_{2}(x_{2},y_{2})\\}$
for all $x\in\mathbb{R}^{4}$, and hence,
$m_{\omega}(\varphi(x))=m_{\omega^{\prime}}(x)$ for all $x\in\mathbb{R}^{4}$.
As $\varphi$ is a bijection, it follows that the sets $\\{m_{\omega}(x)\mid
x\in\mathbb{R}^{4}\\}$ and $\\{m_{\omega^{\prime}}(x)\mid
x\in\mathbb{R}^{4}\\}$ are equal. Therefore,

$\inf\\{m_{\omega}(x)\mid
x\in\mathbb{R}^{4}\\}=\inf\\{m_{\omega^{\prime}}(x)\mid
x\in\mathbb{R}^{4}\\},$

and Theorem 2.5 follows.

Lastly, we consider the following corollary of the Curry-Pelayo-Tang Stability
Theorem [2, Main Theorem].

######  Theorem 6.3 ([2, Example 4.1]).

Consider $\mathbb{R}^{4}$ with coordinates $(x_{1},y_{1},x_{2},y_{2})$. Let
$U$ be an open subset of $\mathbb{R}^{4}$, and let $f_{1},f_{2},f_{3},f_{4}\in
C^{\infty}(U)$. Note that

$\omega=f_{1}\mathrm{d}x_{1}\wedge\mathrm{d}y_{1}+f_{2}\mathrm{d}x_{2}\wedge\mathrm{d}y_{2}$

is a symplectic form if and only if each of the $f_{i}$ is nowhere vanishing
and depends only on the coordinates $x_{i},y_{i}$.

Define the symplectic forms

$\omega_{t}=f_{1}(t,x_{1},y_{1})\mathrm{d}x_{1}\wedge\mathrm{d}y_{1}+f_{2}(t,x_{2},y_{2})\mathrm{d}x_{2}\wedge\mathrm{d}y_{2},t\in[0,1],$

and suppose that the $f_{i}$ are bounded away from zero with bounded time
derivative. Then there exists a smooth path of
$\varphi_{t}\in\operatorname{Diff}(\mathbb{R}^{4})$, $t\in[0,1]$, such that
$\varphi_{t}^{*}\omega_{t}=\omega_{0}$.

As mentioned in Section 2.2, Theorem 6.2 motivates the following question.

######  Question 6.

Under the conditions of Theorem 6.1, does there exist a smooth path of
$\varphi_{t}\in\operatorname{Symp}(\mathbb{R}^{4},\omega_{\mathrm{std}})$,
$t\in[0,1]$, such that $\varphi_{t}^{*}\omega_{t}=\omega_{0}$?

As we shall see in the Example below, this is not necessarily the case. We use
Lemma 6.1 to provide an infinite collection of smooth families of symplectic
forms $\omega_{t}$, $t\in[0,1]$, which are diffeomorphic but not
$\omega_{\mathrm{std}}$-symplectomorphic.

######  Example.

Fix $t_{1},t_{2}\in[0,1]$, and assume without loss of generality that
$t_{1}>t_{2}$. Let $g:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a function
satisfying $\inf\\{g(x,y)\mid x,y\in\mathbb{R}\\}=c>0$. Let
$f_{1}(t,x_{1},y_{1})=g(x_{1},y_{1})+t$ and let
$f_{2}(t,x_{2},y_{2})=g(x_{2},y_{2})+t$. Consider the family of symplectic
forms

$\omega_{t}=f_{1}(t,x_{1},y_{1})\mathrm{d}x_{1}\wedge\mathrm{d}y_{1}+f_{2}(t,x_{2},y_{2})\mathrm{d}x_{2}\wedge\mathrm{d}y_{2},t\in[0,1].$

Then $\omega_{t_{1}}$ and $\omega_{t_{2}}$ are diffeomorphic but not
$\omega_{\mathrm{std}}$-symplectomorphic.

To see that $\omega_{t_{1}}$ and $\omega_{t_{2}}$ are diffeomorphic, apply
Theorem 6.2. To see that $\omega_{t_{1}}$ and $\omega_{t_{2}}$ are not
$\omega_{\mathrm{std}}$-symplectomorphic, note that by Lemma 6.1, if
$\varphi^{*}\omega_{t_{1}}=\omega_{t_{2}}$ for some
$\varphi\in\operatorname{Symp}(\mathbb{R}^{4},\omega_{\mathrm{std}})$, then
for each $x\in\mathbb{R}^{4}$, either
$f_{1}(t_{1},\varphi^{1}(x),\varphi^{2}(x))=f_{1}(t_{2},x_{1},y_{1})$ or
$f_{1}(t_{1},\varphi^{1}(x),\varphi^{2}(x))=f_{2}(t_{2},x_{2},y_{2})$. Select
$x=(u,v,u,v)$ such that $g(u,v)<c+(t_{1}-t_{2})$. In either of the two cases,
we find $g(\varphi^{1}(x),\varphi^{2}(x))+(t_{1}-t_{2})=g(u,v)$, which is a
contradiction to $g(u,v)<c+(t_{2}-t_{1})$. It follows that $\omega_{t_{1}}$
and $\omega_{t_{2}}$ are not $\omega_{\mathrm{std}}$-symplectomorphic, and
therefore, the symplectic forms $\omega_{t}$, $t\in[0,1]$ are diffeomorphic
but not $\omega_{\mathrm{std}}$-symplectomorphic.

In particular, consider $g(x,y)=x^{2}+y^{2}+1$, for which

$\omega_{t}=(x_{1}^{2}+y_{1}^{2}+t+1)\mathrm{d}x_{1}\wedge\mathrm{d}y_{1}+(x_{2}^{2}+y_{2}^{2}+t+1)\mathrm{d}x_{2}\wedge\mathrm{d}y_{2},t\in[0,1]$

From the above, it follows that the symplectic forms $\omega_{t}$, $t\in[0,1]$
are diffeomorphic but not $\omega_{\mathrm{std}}$-symplectomorphic.

##  7 Higher Dimensional Linear Case

###  7.1 Equivalence Lemma for $\mathbb{R}^{2n}$

In this section, we state the Equivalence Lemma in its full generality. First,
we must extend the notion of a symplectic basis as introduced in Section 4.1.

######  Definition 7.1 (Symplectic basis).

Let $\omega$ be a linear symplectic form on $\mathbb{R}^{2n}$, and let
$\mathcal{B}=\\{v_{1},v_{2},\ldots,v_{2n}\\}$ be a basis for
$(\mathbb{R}^{2n},\omega)$. Then $\mathcal{B}$ is said to be a symplectic
basis for $(\mathbb{R}^{2n},\omega)$ if the following equalities hold:

  * •

$\omega(v_{2i-1},v_{2i})=1$ for $1\leq i\leq n$; and,

  * •

$\omega(v_{i},v_{j})=0$ for $i<j$ and $(i,j)\neq(2k-1,2k)$ for all $1\leq
k\leq n$.

As in Theorem 4.1, the Modified Gram-Schmidt Algorithm may be used to prove
the the following result [10, Theorem 2.10.4].

######  Theorem 7.1.

Every symplectic vector space $(\mathbb{R}^{2n},\omega)$ has a symplectic
basis.

We continue with a generalization of Lemma 4.2.

######  Lemma 7.2.

Let $\\{v_{i}\\}_{i=1}^{2n}$ and $\\{w_{i}\\}_{i=1}^{2n}$ be two bases for
$\mathbb{R}^{2n}$. Define the linear transformation
$P:\mathbb{R}^{2n}\rightarrow\mathbb{R}^{2n}$ such that $Pv_{i}=w_{i}$ for all
$1\leq i\leq 2n$. Then $P^{T}JP=J$ if and only if
$\omega_{\mathrm{lstd}}(v_{i},v_{j})=\omega_{\mathrm{lstd}}(w_{i},w_{j})$ for
all $1\leq i\leq j\leq 2n$.

###### Proof.

($\rightarrow$) If $P^{T}JP=J$, then

$w_{i}^{T}Jw_{j}=(Pv_{i})^{T}J(Pv_{j})=v_{i}^{T}(P^{T}JP)v_{j}=v_{i}^{T}Jv_{j},$

as desired.

($\leftarrow$) Follows from Definition 2.5.

∎

######  Definition 7.2 (Basis-values).

Given a symplectic basis $\mathcal{B}=\\{v_{1},v_{2},\ldots,v_{2n}\\}$ for the
linear symplectic form $A$ on $\mathbb{R}^{2n}$, the set of basis-values of
$\mathcal{B}$ is the ordered ${n}\choose{2}$-tuple

$(\omega_{\mathrm{lstd}}(v_{1},v_{2}),\omega_{\mathrm{lstd}}(v_{1},v_{3}),\omega_{\mathrm{lstd}}(v_{1},v_{4}),\ldots,\omega_{\mathrm{lstd}}(v_{2n-1},v_{2n})).$

We are ready to state the Equivalence Lemma in $\mathbb{R}^{2n}$.

######  Corollary 7.2.1 (Equivalence Lemma in $\mathbb{R}^{2n}$).

Let $A,B\in S(2n,\mathbb{R})$. Let
$\mathcal{B}_{1}=\\{v_{1},v_{2},\ldots,v_{2n}\\}$ be a symplectic basis for
$A$ and let $\mathcal{B}_{2}=\\{w_{1},w_{2},\ldots,w_{2n}\\}$ be a symplectic
basis for $B$. If the set of basis-values of $\mathcal{B}_{1}$ is equal to the
set of basis-values of $\mathcal{B}_{2}$, then $A$ and $B$ are equivalent
under the group action $\rho$.

###### Proof.

Let $P:\mathbb{R}^{2n}\rightarrow\mathbb{R}^{2n}$ be the linear transformation
such that $Pv_{i}=w_{i}$ for $1\leq i\leq 2n$. As the set of basis-values of
$\mathcal{B}_{1}$ is equal to the set of basis-values for $\mathcal{B}_{2}$,
by Lemma 7.2, it follows that $P^{T}JP=J$.

As $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are symplectic bases for $A$ and
$B$, respectively, $v_{i}^{T}Av_{j}=w_{i}^{T}Bw_{j}$ for all $1\leq i,j\leq
2n$. Hence, $v_{i}^{T}Av_{j}=(Pv_{i})^{T}B(Pv_{j})^{T}$ for all $1\leq i,j\leq
2n$, which implies that $v_{i}^{T}Av_{j}=v_{i}^{T}(P^{T}BP)v_{j}$ for all
$1\leq i,j\leq 2n$.

As $\\{v_{1},v_{2},\ldots,v_{2n}\\}$ is a basis for $\mathbb{R}^{2n}$, it
follows that $x^{T}Ay=x^{T}(P^{T}BP)y$ for all $x,y\in\mathbb{R}^{2n}$. In
particular, for the standard basis vectors $x=e_{i}$ and $y=e_{j}$, we have
$A_{i,j}=e_{i}^{T}Ae_{j}=e_{i}^{T}(P^{T}BP)e_{j}=(P^{T}BP)_{i,j}$. It follows
that all of the entries of $A$ and $P^{T}BP$ are equal, so that $A=P^{T}BP$.
The result follows.

∎

###  7.2 Proof of Invariants $s_{k}$ in $\mathbb{R}^{2n}$

######  Definition 7.3.

In this section, we derive the invariants $s_{k}$ Theorem 2.8. The formal
definition of $s_{k}$ is stated as below:

Let $A\in S(2n,\mathbb{R})$, and let $0\leq k\leq n-1$. For a subset
$S\subset\\{1,2,...,n\\}$ with $|S|=k$, define the $(2n-2k)\times(2n-2k)$
matrix $A_{S}$ as the submatrix of $A$ formed by removing rows and columns
$2i-1$ and rows and columns $2i$ for all $i\in S$.Let $s_{k}(A)$ denote the
sum

$\sum_{\begin{subarray}{c}S\subseteq\\{1,2,\ldots,n\\}\\\
|S|=k\end{subarray}}\operatorname{Pf}(A_{S}).$

Then for any $P\in\operatorname{Sp}(2n)$, $s_{k}(A)=s_{k}(P^{T}AP)$. In other
words, $s_{k}$ is invariant under the group action $\rho$.

At the same time, we can notice that the pfaffian and sum function we
discussed in the section 3 of this paper are exactly $s_{0}$ and $s_{n-1}$
here respectively, which indicates that our result here is a generalized
version of the section 3 proof.

######  Definition 7.4 (Pfaffian of a $2n\times 2n$ matrix).

The Pfaffian of a $2n\times 2n$ matrix $A$ is the quantity

$\operatorname{Pf}(A)=\sum_{\sigma}\text{sgn}(\sigma)\prod_{i=1}^{n}A_{\sigma(2i-1),\sigma(2i)},$

where $\text{sgn}(\sigma)$ denotes the signature of $\sigma$ and the summation
is taken over all permutations $\sigma$ of $\\{1,2,\ldots,2n\\}$. (Here,
$A_{i,j}$ denotes the $(i,j)$ entry of $A$.)

According to the above permutation definition of pfaffian, we can expand
$\operatorname{Pf}{(tJ+A)}$ for $A\in S(2n,\mathbb{R})$ into a polynomial
$g(t)$ with $\deg(g)=n$, such that

$\operatorname{Pf}{(tJ+A)}=t^{n}+t^{n-1}\sigma_{n-1}(A)+t^{n-2}\sigma_{n-2}(A)+\cdot\cdot\cdot+\sigma_{0}(A)$
(1)

where $\sigma_{k}$ is the coefficient of $t^{k}$.

######  Definition 7.5.

Given a two form $\omega$ associated to any skew-symmetric $2n\times 2n$
matrice $A={a_{ij}}$, such that

$\omega=\sum_{i<j}a_{ij}e_{i}\wedge e_{j}.$

where $\\{e_{1},e_{2},...,e_{2n}\\}$ is the standard basis of
$\mathbb{R}^{2}n$. The pfaffian of matrix $A$ is then defined as

$\frac{1}{n!}\omega^{n}=\operatorname{Pf}(A)e_{1}\wedge
e_{2}\wedge\cdot\cdot\cdot\wedge e_{2n}$ (2)

Accordingly, we can use the above definition of pfaffian in terms of 2-forms
to prove the following result.

######  Proposition 7.3.

$\sigma_{k}$ is invariant under the group action $\rho$ which means that
$\sigma_{k}(P^{T}AP)=\sigma_{k}(A)$ for any $A\in S(2n,R)$ and
$P\in\operatorname{Sp}(2n)$.

###### Proof.

According to lemma 7.3 and 7.4, we have
$\operatorname{Pf}(tJ+A)=\operatorname{Pf}(P^{T}(tJ+A)P)$ for any
$P\in\operatorname{Sp}(2n)$. Consequently, $P^{T}JP=J$, and then

$\displaystyle\operatorname{Pf}(tJ+A)$
$\displaystyle=\operatorname{Pf}(P^{T}(tJ+A)P)$ (3)
$\displaystyle=\operatorname{Pf}(tJ+P^{T}AP)$ (4)
$\displaystyle=t^{n}+t^{n-1}\sigma_{n-1}(P^{T}AP)+t^{n-2}\sigma_{n-2}(P^{T}AP)+\cdot\cdot\cdot+\sigma_{0}(P^{T}AP)$
(5)

Comparing the coefficient of formula (1) to formula (4), we obtain that
$\sigma_{k}(A)=\sigma_{k}(P^{T}AP)$, for $k=0,...n-1$. ∎

Once we can prove that $\sigma_{k}=s_{k}$, we are done with the above
proposition on invariant. Let $\mathbf{N}=\\{1,2,...,2n\\}$, for any
$A=\\{a_{ij}\\}_{1\leq i,j\leq 2n}\in S(2n,\mathbb{R})$, we define the
associated 2-form $\omega$ such that

$\omega=\sum_{\begin{subarray}{c}i<j\\\
{i,j}\in\mathbf{N}\end{subarray}}a_{ij}dx_{i}\wedge dx_{j}.$

where the set $\\{dx_{1},dx_{2},...,dx_{2n}\\}$ is the standard basis for the
vector space $(T_{p}(\mathbb{R}^{2n}))^{*}$. Then the pfaffian of A is
determined by the following identity.

$\frac{1}{n!}\omega^{n}=\operatorname{Pf}(A)dx_{1}\wedge
dx_{2}\wedge\cdot\cdot\cdot\wedge dx_{2n}$ (6)

where $\omega^{n}=\omega\wedge\cdot\cdot\cdot\wedge\omega$.

######  Proposition 7.4.

Given coefficient $\sigma_{k}$ of term $t^{k}$ in the polynomial expansion of
$\operatorname{Pf}(tJ+A)$, standard symplectic form $\omega_{0}$ and 2-form
$\omega$ associated with $A$, we have

$\sigma_{k}dx_{1}\wedge dx_{2}\wedge\cdot\cdot\cdot\wedge
dx_{2n}=\frac{1}{k!(n-k)!}\omega_{0}^{k}\wedge\omega^{n-k}$ (7)

###### Proof.

For the standard symplectic form $\omega_{0}$ associated with the matrix
representation $J\in S(2n,\mathbb{R})$, we have

$\omega_{0}=\sum_{i=1}^{n}dx_{2i-1}\wedge dx_{2i}$

Now we consider the matrix $tJ+A$ which has the associated 2-form
$t\omega_{0}+\omega$. By formula (6), we then have

$\frac{1}{n!}(t\omega_{0}+\omega)^{n}=\operatorname{Pf}(tJ+A)dx_{1}\wedge
dx_{2}\wedge\cdot\cdot\cdot\wedge dx_{2n}$

Using the binomial theorem, we can then expand the
$\frac{1}{n!}(t\omega_{0}+\omega)^{n}$ and note that

$\frac{1}{n!}(t\omega_{0}+\omega)^{n}=\sum_{k=0}^{n}\frac{1}{k!(n-k)!}t^{k}\omega_{0}^{k}\wedge\omega^{n-k}$
(8)

Comparing the formula (8) with formula (5), we can then conclude that

$\sigma_{k}dx_{1}\wedge dx_{2}\wedge\cdot\cdot\cdot\wedge
dx_{2n}=\frac{1}{k!(n-k)!}\omega_{0}^{k}\wedge\omega^{n-k}$ (9)

∎

Therefore, the equivalence of $\sigma_{k}(A)=s_{k}(A)$ is given by the
following Proposition.

######  Proposition 7.5.

For $k=0,1$, we have

$\frac{1}{k!(n-k)!}\omega_{0}^{k}\wedge\omega^{n-k}=s_{k}dx_{1}\wedge
dx_{2}\wedge\cdot\cdot\cdot\wedge dx_{2n}$ (10)

###### Proof.

For $k=0$, formula (10) holds trivially which is just the formula (6).
For $k=1$, we have

$\frac{1}{(n-1)!}\omega_{0}^{k}\wedge\omega^{n-1}=\frac{1}{(n-1)!}\sum_{i=1}^{n}dx_{2i-1}\wedge
dx_{2i}\wedge\omega^{n-1}$ (11)

For each term $dx_{2i-1}\wedge dx_{2i}\wedge\omega^{n-1}$, we note that, only
those terms which do not contain $dx_{2i-1}$ or $dx_{2i}$ in $\omega^{n-1}$
term will contribute to the term $dx_{2i-1}\wedge
dx_{2i}\wedge\omega^{n-1}$.By this observation, if we construct a new 2-form
$\omega_{\hat{i}}$, such that

$\omega_{\hat{i}}=\sum_{\begin{subarray}{c}k<l\\\
{k,l}\in\mathbf{N}\setminus\\{2i-1,2i\\}\end{subarray}}a_{kl}dx_{k}\wedge
dx_{l}$ (12)

Consequently we would have

$dx_{2i-1}\wedge dx_{2i}\wedge\omega^{n-1}=dx_{2i-1}\wedge
dx_{2i}\wedge\omega_{\hat{i}}^{n-1}$

Consequently by formula (6), we have the following:

$\frac{\omega_{\hat{i}}^{n-1}}{(n-1)!}=\operatorname{Pf}(A_{i})dx_{1}\wedge
dx_{2}\wedge\cdot\cdot\wedge\widehat{dx_{2i-1}}\wedge\widehat{dx_{2i}}\cdot\cdot\cdot\wedge
dx_{2n}$ (13)

where the notation $\operatorname{Pf{i}}$ is the pfaffian of the submatrix
$A_{S}$ formed by formed by removing rows and columns $2i-1$ and rows and
columns $2i$ for all $i\in S$, where $S\subset\\{1,2,...,n\\}$ with $|S|=k$.
In the case of $k=1$, $\operatorname{Pf}(A_{i})$ refer to the pfaffian of some
submatrix $A_{S}$ formed by removing rows and columns $2i-1$ and rows and
columns $2i$ for one particular $i\in S$ as $|S|=1$. Therefore, the formula
$11$ gives

$\frac{1}{(n-1)!}\omega_{0}^{k}\wedge\omega^{n-1}=\sum_{i=1}^{n}\operatorname{Pf}(A_{i})\bigwedge_{i=1}^{2n}dx_{i}=s_{1}\bigwedge_{i=1}^{2n}dx_{i}$

∎

Inspired by the proof of invariant when $k=1$, we can derive the following
Proposition directly.

######  Proposition 7.6.

For $k=0,1...,n-1$, we have

$\frac{1}{k!(n-k)!}\omega_{0}^{k}\wedge\omega^{n-k}=s_{k}dx_{1}\wedge
dx_{2}\wedge\cdot\cdot\cdot\wedge dx_{2n}$ (14)

###### Proof.

Construct an increasing posive integer sequence
$\\{b_{i}\\},i\in\\{1,2,...,n\\}$, then we have

$\displaystyle\frac{1}{k!(n-k)!}\omega_{0}^{k}\wedge\omega^{n-k}$
$\displaystyle=\frac{1}{k!(n-k)!}(\sum_{i=1}^{n}dx_{2i-1}\wedge
dx_{2i})^{k}\wedge\omega^{n-k}$
$\displaystyle=\frac{1}{(n-k)!}\sum_{i=1}^{n}\bigwedge_{b_{i}}dx_{2b_{i}-1}\wedge
dx_{2b_{i}}\wedge\omega^{n-k}$
$\displaystyle=\bigwedge_{b_{i}}dx_{2b_{i}-1}\wedge
dx_{2b_{i}}\wedge\omega_{\hat{b_{i}}}^{n-k}.$

where $\omega_{\hat{b_{i}}}$ is constructed similar as in the case $k=1$, such
that

$\omega_{\hat{b_{i}}}=\sum_{k_{i}<l_{i}}\bigwedge_{b_{i}}dx_{2b_{i}-1}\wedge
dx_{2b_{i}}$

where $\\{k_{i}\\}$ and $\\{l_{i}\\}$ are two increasing positive sequences
such that $\\{k_{i},l_{i}\\}\in\mathbf{N}\setminus\\{2b_{i}-1,2b_{i}\\}$.
Using the similar techniques with $k=1$ case, we would have

$\frac{\omega_{\hat{a_{i}}}^{n-k}}{(n-k)!}=\operatorname{Pf}(A_{\\{a_{i}\\}})dx_{1}\wedge
dx_{2}\wedge\cdot\cdot\bigwedge_{b_{i}}\widehat{dx_{2b_{i}-1}\wedge
dx_{2b_{i}}}\cdot\cdot\cdot\wedge dx_{2n}$

Consequently we will get

$\frac{1}{k!(n-k)!}\omega_{0}^{k}\wedge\omega^{n-k}=\operatorname{Pf}(A_{a_{i}})\bigwedge_{i=1}^{2n}dx_{i}=s_{k}(A)\bigwedge_{i=1}^{2n}dx_{i}$

∎

Consequently, we can show that $\sigma_{k}=s_{k}$ which are the invariant.
According to the invariant proof of $\sigma_{k}$, we then get to know that the
quantity $s_{k}$ are also invariants. Hence the theorem 2.8 is justified.

##  8 Higher dimensional nonlinear case

###  8.1 The proof of invariants

######  Proposition 8.1.

Consider symplectic forms

$\omega_{1}=\sum_{i=1}^{n}f_{i}(x_{i},y_{i})dx_{i}\wedge dy_{i}$

and

$\omega_{2}=\sum_{i=1}^{n}g_{i}(x_{i},y_{i})dx_{i}\wedge dy_{i}$

If $\phi\in\operatorname{Symp}(\mathbb{R}^{2n},\omega_{std})$, such that
$\phi^{*}\omega_{1}=\omega_{2}$, $\phi=(\phi^{1},\phi^{2},...,\phi^{2n})$ Then
we would the following equality: Then the set
$\\{f_{i}(\phi^{2i-1},\phi^{2i})\\}$ and $\\{g_{j}(x_{i},y_{i})\\}$ for some
$i,j$ such that $1<i,j<n$ at every point in $\mathbb{R}^{2n}$.

###### Proof.

Given $\Omega_{1}(x)$ and $\Omega_{2}(x)$ are matrix representation of
$\omega_{1}$ and $\omega_{2}$ respectively, then we would have

$\operatorname{Pf}{(tJ+\Omega_{1}(\phi(x))}=\operatorname{Pf}{(tJ+\Omega_{2}(x))}$
(15)

Since both $\Omega_{1}(\phi(x))$ and $\Omega_{2}(x)$ are block diagonal
matrices and according to our proof of $n$ invariant in $\mathbb{R}^{2n}$, we
can conclude that the invariant $s_{i}(n)$= $(n-i)$th symmetric sum of zeros
of $P(t)$, formulated as below:

$P(t)=\prod_{i=1}^{n}(t+f_{i}(\phi^{2i-1},\phi^{2i}))=\prod_{i=1}^{n}(t+g_{i}(x_{i},y_{i}))$
(16)

for all $t\in\mathbb{R}^{2n}$. ∎

Hence the theorem 2.5 is justified by the above proposition. In the similar
way as $\mathbb{R}^{4}cases$, for $x\in\mathbb{R}^{2n}$ we can define

$m_{\omega}(x)=\min\\{f_{i}(x_{i},y_{i}),1\leq i\leq n\\}$

and

$M_{\omega}(x)=\max\\{f_{i}(x_{i},y_{i}),1\leq i\leq n\\}.$

In other words, if $\omega_{1}$ is $\omega_{\mathrm{std}}$-symplectomorphic to
the symplectic form

$\omega_{2}=\sum_{i=1}^{n}g_{i}(x_{i},y_{i})dx_{i}\wedge dy_{i}$

where $g_{i}$ are nowhere vanishing, then the four global invariants of
$\omega_{1}$ and $\omega_{2}$ must match.

  * •

$\inf\\{m_{\omega}(x)\mid x\in\mathbb{R}^{2n}\\}$,

  * •

$\sup\\{m_{\omega}(x)\mid x\in\mathbb{R}^{2n}\\}$,

  * •

$\inf\\{M_{\omega}(x)\mid x\in\mathbb{R}^{2n}\\}$,

  * •

$\sup\\{M_{\omega}(x)\mid x\in\mathbb{R}^{2n}\\}$.

Consequently, the theorem 2.6 is justified by the above natural constructions
as well.

##  9 Prospectives

In this section, we present conjectures and questions for further
investigation. We shall begin with propsectives regarding the linear case. Our
first conjecture stems from our observation that in $\mathbb{R}^{4}$, the
Pfaffian and sum function invariants determine the Orbit Space Classification
of $S(4,\mathbb{R})/\operatorname{Sp}(4)$.

######  Conjecture 1.

The $n$ invariants described in Conjecture 1 determine the orbit space
$S(2n,\mathbb{R})/\operatorname{Sp}(2n)$. In order words, if $A,B\in
S(2n,\mathbb{R})$ with $A\neq cJ$ for all $c\in\mathbb{R}$ and $B\neq cJ$ for
all $c\in\mathbb{R}$ and if $\operatorname{s_{k}}(A)=\operatorname{s_{k}}(B)$
for all $0\leq k\leq n-1$, then $A$ and $B$ are equivalent under the group
action $\rho$.

For example, Conjecture 2 implies that the orbit space
$S(6,\mathbb{R})/\operatorname{Sp}(4)$ consists of the following families of
distinct orbits:

  * •

For all $p>0$,

$\mathcal{J}_{p}^{+}=\\{\sqrt{p}J\\}$

  * •

For all $p>0$,

$\mathcal{J}_{p}^{-}=\\{-\sqrt{p}J\\}$

  * •

For all $p>0$, $q\in\mathbb{R}$, $r\in\mathbb{R}$,

$\mathcal{A}_{p,q,r}^{+}=\\{A\in S(6,\mathbb{R})\mid
A\not=\pm\sqrt{p}J,\operatorname{Pf}(A)=p,\operatorname{s_{1}}(A)=q,\operatorname{s_{2}}(A)=r\\}$

  * •

For all $p<0$, $q\in\mathbb{R}$, $r\in\mathbb{R}$,

$\mathcal{A}_{p,q,r}^{-}=\\{A\in
S(6,\mathbb{R})\mid\operatorname{Pf}(A)=p,\operatorname{s_{1}}(A)=q,\operatorname{s_{2}}(A)=r\\}$

We now turn to prospectives regarding the equivalence of nonlinear symplectic
forms. The first asks if Question 6 can be salvaged.

######  Question 7.

Let $f_{1},f_{2}$ be functions, and define the symplectic forms

$\omega_{t}=f_{1}(t,x_{1},y_{1})\mathrm{d}x_{1}\wedge\mathrm{d}y_{1}+f_{2}(t,x_{2},y_{2})\mathrm{d}x_{2}\wedge\mathrm{d}y_{2},t\in[0,1].$

What is a sufficient condition on $f_{1},f_{2}$ under which there exists a
smooth path of
$\varphi_{t}\in\operatorname{Symp}(\mathbb{R}^{4},\omega_{\mathrm{std}})$,
$t\in[0,1]$, such that $\varphi^{*}\omega_{t}=\omega_{0}$?

Finally, we present a question regarding nonlinear symplectic forms on
$\mathbb{R}^{2n}$.

######  Question 8.

How do the results of Section 6 generalize to $\mathbb{R}^{2n}$? What do the
proposed invariants of Conjectures 1 and 2 imply about nonlinear symplectic
forms on $\mathbb{R}^{2n}$?

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# Active stop and go motion: a strategy to improve spatial exploration?

Fernando Peruani<EMAIL_ADDRESS>Laboratoire de Physique Théorique et
Modélisation, UMR 8089, CY Cergy Paris Université, 95302 Cergy-Pontoise,
France  Debasish Chaudhuri<EMAIL_ADDRESS>Institute of Physics, Sachivalaya
Marg, Bhubaneswar 751005, India  Homi Bhaba National Institute, Anushaktigar,
Mumbai 400094, India

###### Abstract

We consider active Brownian particles switching between run and stop states.
Such intermittent dynamics are ubiquitous at all scales, from bacteria to
animals and in artificial active systems. We provide exact solutions for their
transport properties, e.g., velocity autocorrelations and diffusion
coefficients. The spread of particles depends on minute details, such as if
the memory of active orientation is retained across a stop event. We predict
dynamic schemes maximizing the effective diffusivity, potentially utilized by
active agents for better access to distributed food.

In most animal systems, active motion is not a continuous process but an
intermittent one. Individuals move actively for a while before they pause and
move again, restarting the cycle Kramer and McLaughlin (2001). Intermittent
active locomotion is often related to different ethological phases, such as
moving in search of food or stopping to rest Nathan  _et al._ (2008). Stop-
and-go (S-&-G) motion is also observed at the group level, with all group
members moving and stopping in a synchronized fashion Tunström  _et al._
(2013); Trillmich  _et al._ (2004); Ariel  _et al._ (2014); Gómez-Nava  _et
al._ (2022). Intermittent motility patterns are even observed at a microscopic
scale. The near-surface swimming bacteria such as enterohemorrhagic E. coli
perform S-&-G motion Perez Ipiña  _et al._ (2019). The locomotion of gliding
bacteria such as myxobacteria displays run and stop phases with “stick-and-
slip” statistics Gibiansky  _et al._ (2013). S-&-G motion is also observed in
active colloid systems subjected to an AC field Karani  _et al._ (2019);
Pradillo  _et al._ (2019). Despite the experimental relevance of S-&-G motion,
the large majority of active models have considered that particles are
permanently active, i.e., their self-propulsion is always switched on
Romanczuk  _et al._ (2012).

Here, we investigate the large-scale transport properties of active particles
that alternate between an active (moving) and a non-active (stop) state and
show that subtle details of the microscopic dynamics impact the diffusion
coefficient of these active particles. In particular, we analyze 4
representative models of experimental relevance using the same theoretical
framework. In model 1, the active heading direction of the particles
fluctuates as the particle moves and is reset when the particle stops. This
occurs, for instance, when bacteria in near-swimming perform transient
adhesion events to a surface Perez Ipiña  _et al._ (2019) or in Quincke
rollers driven by on-off-on electric field when the period is larger than the
Maxwell-Wagner relaxation time Karani  _et al._ (2019); Pradillo  _et al._
(2019). In contrast, in model 2, after a stop, the particle continues moving
in the direction was moving prior to the stop. This behavior is observed in
many insects and vertebrates that perform numerous short stops when moving
Kramer and McLaughlin (2001); Gómez-Nava  _et al._ (2022). These stops are
supposed to serve to scan for the presence of predators Kramer and McLaughlin
(2001); Gómez-Nava  _et al._ (2022). Model 3 assumes that the heading
direction and the particle polarization follow similar dynamics. This is
observed in cell migration, where fluctuations in cell polarization take place
even when the cell is not moving Schienbein and Gruler (1993); Vaidžiulytė
_et al._ (2022). Finally, in model 4, the active moving direction is kept
fixed while moving and evolves only during the stop phase. This behavior is
often called run-and-tumble motion and is characteristic of many bacterial
systems Figueroa-Morales  _et al._ (2020); Wadhwa and Berg (2022).

Model.– The state of the particle, in any dimension, is described by $\\{{\bf
x},{\bm{\sigma}},q\\}$, where the position ${\bf x}$ is determined by

$\displaystyle\dot{\bf x}=v[q]{\bm{\sigma}}\equiv{\bf u}(t)\,.$ (1)

The spin ${\bm{\sigma}}$ specifies the active heading direction of the
particle, and the behavioral state $q=\\{S,M\\}$ specifies the value of the
active speed: $v[q=S]=0$ and $v[q=M]=v_{M}$. The dynamics of $q$ follows:

$\displaystyle M\overset{\beta}{\underset{\gamma}{\rightleftharpoons}}S\,,$
(2)

where $\beta$ and $\gamma$ are constant rates. The dynamics of ${\bm{\sigma}}$
is different for model 1 to 4. The definition of these 4 models crucially
depends on the dimension of the system.

Models | Model 1 | Model 2 | Model 3 | Model 4
---|---|---|---|---
Rates: 1d |  $\alpha_{M}=\alpha$, $\alpha_{S}=0$, reset |  $\alpha_{M}=\alpha$, $\alpha_{S}=0$ | $\alpha_{M}=\alpha_{S}=\alpha$ |  $\alpha_{M}=0$, $\alpha_{S}=\alpha$
$D_{\rm eff}$ (1d) | $\frac{\langle v^{2}\rangle}{2\alpha+\beta}$ | $\frac{\langle v^{2}\rangle}{2\alpha}$ | $\frac{\langle v\rangle^{2}}{2\alpha}+\frac{\langle\delta v^{2}\rangle}{2\alpha+\beta+\gamma}$ | $\langle v^{2}\rangle\left(\frac{2\alpha+\gamma}{2\alpha\beta}\right)$
Rates: 2d |  $D_{M}=D_{r}$, $D_{S}=0$, reset |  $D_{M}=D_{r}$, $D_{S}=0$ | $D_{M}=D_{S}=D_{r}$ |  $D_{M}=0$, $D_{S}=D_{r}$
$D_{\rm eff}$ (2d) |  $\frac{1}{2}\frac{\langle v^{2}\rangle}{D_{r}+\beta}$ | $\frac{\langle v^{2}\rangle}{2D_{r}}$ | $\frac{\langle v\rangle^{2}}{2D_{r}}+\frac{1}{2}\frac{\langle\delta v^{2}\rangle}{D_{r}+\beta+\gamma}$ | $\langle v^{2}\rangle\left(\frac{D_{r}+\gamma}{2D_{r}\beta}\right)$
Examples | bacterial surface swimming Karani  _et al._ (2019); Perez Ipiña  _et al._ (2019) | animal movement Kramer and McLaughlin (2001); Gómez-Nava  _et al._ (2022) | cell migration Schienbein and Gruler (1993); Vaidžiulytė  _et al._ (2022) | bacterial run and tumble Figueroa-Morales  _et al._ (2020); Wadhwa and Berg (2022)
Table 1: Models and effective diffusion: Run-stop transitions are controlled
by rates $\beta,\,\gamma$ [Eq.(2)]. Orientational persistence in the run state
is controlled by rates $\alpha_{M}$ in 1D, $D_{M}$ in 2D, and that in the stop
state by $\alpha_{S}$ in 1D, $D_{S}$ in 2D. The first and third rows indicate
their specific choices. At reset, the stop state transitions to run with an
equal probability of all possible orientations. The calculated effective
diffusivities are listed in rows two and four. Examples corresponding to each
model are cited in the fifth row.

One dimensional systems.– In 1D, $\mathbf{x}\in\mathbb{R}$ and
${\bm{\sigma}}\in\\{+1,-1\\}$. The dynamics of the system can be described in
terms of the probability distribution $P_{q}^{{\bm{\sigma}}}(x,t)$, of finding
the particle at position $x$ at time $t$ in state $q$ and spin
${\bm{\sigma}}$. When the particle is in state $M$, the
${\bm{\sigma}}$-dynamics is given by
${\bm{\sigma}}\overset{\alpha_{M}}{\longrightarrow}-{\bm{\sigma}}$, while for
state $S$ it becomes
${\bm{\sigma}}\overset{\alpha_{S}}{\longrightarrow}-{\bm{\sigma}}$. The
probability distributions $P_{\text{M}}^{\pm}(x,t)$ and
$P_{\text{S}}^{\pm}(x,t)$ obey:

$\displaystyle\partial_{t}P_{\text{M}}^{+}$ $\displaystyle=$ $\displaystyle-
v_{M}\partial_{x}P_{\text{M}}^{+}-\alpha_{M}P_{\text{M}}^{+}+\alpha_{M}P_{\text{M}}^{-}-\beta
P_{\text{M}}^{+}+\gamma P_{\text{S}}^{+}\,,$
$\displaystyle\partial_{t}P_{\text{M}}^{-}$ $\displaystyle=$
$\displaystyle\,\,\,\,\,v_{M}\partial_{x}P_{\text{M}}^{-}-\alpha_{M}P_{\text{M}}^{-}+\alpha_{M}P_{\text{M}}^{+}-\beta
P_{\text{M}}^{-}+\gamma P_{\text{S}}^{-}\,,$
$\displaystyle\partial_{t}P_{\text{S}}^{+}$ $\displaystyle=$
$\displaystyle\beta P_{\text{M}}^{+}-\gamma
P_{\text{S}}^{+}-\alpha_{S}P_{\text{S}}^{+}+\alpha_{S}P_{\text{S}}^{-}\,,$
$\displaystyle\partial_{t}P_{\text{S}}^{-}$ $\displaystyle=$
$\displaystyle\beta P_{\text{M}}^{-}-\gamma
P_{\text{S}}^{-}-\alpha_{S}P_{\text{S}}^{-}+\alpha_{S}P_{\text{S}}^{+}\,.$ (3)

To obtain the effective diffusion coefficient $D_{\text{eff}}$ of these
particles, it is tempting to proceed as in the telegraph equation Zauderer
(2011) and perform an adiabatic elimination of the fast variables. To
illustrate this, let us define $\rho=P_{\text{M}}^{+}+P_{\text{M}}^{-}$,
$m=P_{\text{M}}^{+}-P_{\text{M}}^{-}$,
$\rho_{S}=P_{\text{S}}^{+}+P_{\text{S}}^{-}$, and
$m_{S}=P_{\text{S}}^{+}-P_{\text{S}}^{-}$. If we request
$\partial_{t}\rho_{S}=\partial_{t}m_{S}=0$, we get
$\rho_{S}=\frac{\beta}{\gamma}\rho$ and
$m_{S}=\frac{\beta}{\gamma+2\alpha_{S}}m$. Using these expressions and
$\partial_{t}m=0$, we arrive at
$\partial_{t}\rho=\frac{v_{M}^{2}}{2\alpha_{M}+\frac{\beta\alpha_{S}}{[\gamma/2+\alpha_{S}]}}\partial_{xx}\rho$.
This expression is, in general, wrong, though in the limit $\gamma=\beta=0$ we
recover the known result for continuously moving active particles:
$D_{\text{eff}}=\frac{v_{M}^{2}}{2\alpha_{M}}$. The reason for the failure of
the adiabatic elimination is that it cannot properly account for the time
scales involved.

In the following, we use an exact method that accounts for all involved time
scales to obtain $D_{\text{eff}}$. Since
$\langle\mathbf{x}\rangle(t)=\int_{0}^{t}dt^{\prime}\langle\mathbf{u}(t^{\prime})\rangle$
and
$\langle\mathbf{x}^{2}\rangle(t)=\int_{0}^{t}dt^{\prime}\int_{0}^{t}dt^{\prime\prime}\langle\mathbf{u}(t^{\prime})\mathbf{u}(t^{\prime\prime})\rangle$,
we compute $D_{\text{eff}}$ as
$D_{\text{eff}}=\lim_{t\to\infty}\frac{\langle\mathbf{x}^{2}\rangle}{2dt}$,
where $d$ is the dimension of the space. The challenge is then to calculate
$\langle\mathbf{u}(t)\cdot\mathbf{u}(t^{\prime})\rangle$. This quantity can be
computed as follows:

$\displaystyle\langle\mathbf{u}(t)\cdot\mathbf{u}(t^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\sum_{q_{1},q_{2}}\sum_{\sigma_{1},\sigma_{2}}\\!\\!v[q_{1}]v[q_{2}]\,{{\bm{\sigma}}}_{1}\cdot{{\bm{\sigma}}}_{2}\,$
(4) $\displaystyle\times$ $\displaystyle
P({\bm{\sigma}}_{2},q_{2},t|{\bm{\sigma}}_{1},q_{1},t^{\prime})P^{{\bm{\sigma}}_{1}*}_{q_{1}}\,,$

where $P({\bm{\sigma}}_{2},q_{2},t|{\bm{\sigma}}_{1},q_{1},t^{\prime})$ for
$t\geq t^{\prime}$ and $P^{{\bm{\sigma}}_{1}*}_{q_{1}}$ are, respectively, a
conditional and a steady state pdf. Note that neither
$P({\bm{\sigma}}_{2},q_{2},t|{\bm{\sigma}}_{1},q_{1},t^{\prime})$ nor
$P^{{\bm{\sigma}}_{1}*}_{q_{1}}$ depend on ${\bf x}$, and are the solutions of

$\displaystyle\partial_{t}\mathbf{P}=A\,\mathbf{P}\,$ (5)

where
$\mathbf{P}\equiv(P_{\text{M}}^{+},P_{\text{M}}^{-},P_{\text{S}}^{+},P_{\text{S}}^{-})$,
with

$\displaystyle A=\begin{pmatrix}-\alpha_{M}-\beta&\alpha_{M}&\gamma&0\\\
\alpha_{M}&-\alpha_{M}-\beta&0&\gamma\\\
\beta&0&-\alpha_{S}-\gamma&\alpha_{S}\\\
0&\beta&\alpha_{S}&-\alpha_{S}-\gamma\end{pmatrix}$ (6)

Note that Eq. (5) is nothing else than Eq. (Active stop and go motion: a
strategy to improve spatial exploration?) without the terms involving spatial
derivatives. From Eq. (5),
$P({\bm{\sigma}}_{2},q_{2},t|{\bm{\sigma}}_{1},q_{1},t^{\prime})$ for $t\geq
t^{\prime}$ is obtained using
$\mathbf{P}(t^{\prime})=(\delta_{+,{\bm{\sigma}}_{1}}\delta_{M,q_{1}},\delta_{-,{\bm{\sigma}}_{1}}\delta_{M,q_{1}},\delta_{+,{\bm{\sigma}}_{1}}\delta_{S,q_{1}},\delta_{-,{\bm{\sigma}}_{1}}\delta_{S,q_{1}})$
with $\delta_{ij}$ denoting the Kronecker delta function, while
$P^{{\bm{\sigma}}_{1}*}_{q_{1}}$ is the steady state of Eq. (5). We observe
that in Eq. (4), the only non-zero terms correspond to $q_{2}=q_{1}=M$ 111In
Supplementary Information, this Taylor-Kubo calculation of effective
diffusivity is illustrated further using the simple run-and-tumble model in
1D. .

In the following, we analyze the 4 models mentioned above. In model 1, we
consider a simple dynamics in which the particle, when actively moving,
stochastically switches from ${\bm{\sigma}}=+1$ to ${\bm{\sigma}}=-1$ and
vice-versa until eventually stops. After a stop, the particle moves with
probability $1/2$ in the direction ${\bm{\sigma}}=+1$ and with $1/2$ in
direction ${\bm{\sigma}}=-1$. This implies that the particle has no
recollection of its moving direction ${\bm{\sigma}}$ previous to stop: the
transition from $M$ to $S$ results in the resetting of ${\bm{\sigma}}$.
Mathematically, it is equivalent to assuming that
$P_{\text{S}}^{+}=P_{\text{S}}^{-}=P_{\text{S}}$, and thus we can describe the
dynamics using only 3 fields, and redefine
$\mathbf{P}=(P_{\text{M}}^{+},P_{\text{M}}^{-},P_{\text{S}})$ and reduce $A$
to:

$\displaystyle A=\begin{pmatrix}-\alpha_{M}-\beta&\alpha_{M}&\gamma/2\\\
\alpha_{M}&-\alpha_{M}-\beta&\gamma/2\\\ \beta&\beta&\gamma\end{pmatrix}$ (7)

After solving Eq. (5) using Eq. (7) and computing the velocity
autocorrelation, Eq. (4), we arrive at

$\displaystyle\langle{\bf u}(t)\cdot{\bf u}(0)\rangle=\langle v^{2}\rangle
e^{-(2\alpha+\beta)t}\,,$ (8)

where $\alpha_{M}=\alpha$, $\langle
v^{2}\rangle=v_{M}^{2}[P^{+*}_{M}+P^{+*}_{M}]$ with the steady-state
probabilities
$P^{+*}_{M}=P^{-*}_{M}=P^{\ast}=\frac{1}{2}\,\frac{\gamma}{\beta+\gamma}$ that
are common to the 4 models. Eq. (8) leads to an effective diffusion
coefficient:

$\displaystyle D_{\rm eff}^{(1)}=\frac{\langle v^{2}\rangle}{2\alpha+\beta}.$
(9)

The other three cases, models 2 to 4, cannot be reduced to less than 4 fields.
In model 2, the spin ${\bm{\sigma}}$ switches with rate $\alpha$ only in the
$M$ state, implying $\alpha_{S}=0$ and $\alpha_{M}=\alpha>0$. Consequently,
the transition from $M$ to $S$ does not erase the memory of the heading
direction before the stop. Following the procedure depicted above, we obtain:

$\displaystyle\langle{\bf u}(t)\cdot{\bf u}(0)\rangle=\frac{\langle
v^{2}\rangle}{2s}[(s+q)e^{-\lambda_{3}t}+(s-q)e^{-\lambda_{4}t}]$ (10)

where $q=2\alpha+\beta-\gamma$, $r=2\alpha+\beta+\gamma$,
$s=\sqrt{r^{2}-8\alpha\gamma}$, inverse correlation times
$\lambda_{3}=\frac{1}{2}(r+s)$, $\lambda_{4}=\frac{1}{2}(r-s)$. Integrating
Eq. (10, and taking the long-time limit, we obtain the effective diffusivity:

$\displaystyle D_{\rm eff}^{(2)}=\frac{\langle v^{2}\rangle}{2\alpha}.$ (11)

In model 3, on the other hand, ${\bm{\sigma}}$ switches with rate $\alpha$
both in the $M$ and $S$ states, and thus $\alpha_{M}=\alpha_{S}=\alpha$. This
implies that if the average time the particle remains in the stop state,
$\gamma^{-1}$, is larger than the characteristic flipping time $\alpha^{-1}$,
the transition to stop tends to erase the memory of the heading direction
prior to the stop. Otherwise, the particle is likely to recollect the previous
moving direction. For this model, the velocity autocorrelation reads:

$\displaystyle\langle{\bf u}(t)\cdot{\bf u}(0)\rangle=\frac{\langle
v^{2}\rangle}{\beta+\gamma}[\gamma e^{-\lambda_{2}t}+\beta
e^{-\lambda_{4}t}]\,,$ (12)

where $\lambda_{2}=2\alpha$ and $\lambda_{4}=(2\alpha+\beta+\gamma)$. From Eq.
(12), we arrive at the effective diffusion coefficient:

$\displaystyle D_{\rm eff}^{(3)}=\frac{\langle
v\rangle^{2}}{2\alpha}+\frac{\langle\delta
v^{2}\rangle}{2\alpha+\beta+\gamma}$ (13)

where $\langle v\rangle=v_{M}2P^{\ast}=v_{M}\,\gamma/(\beta+\gamma)$ and
$\langle\delta v^{2}\rangle=\langle v^{2}\rangle-\langle
v\rangle^{2}=v_{M}^{2}\,\beta\gamma/(\beta+\gamma)$. Note the increase in
diffusivity due to speed fluctuations, cf. Peruani and Morelli (2007); Shee
and Chaudhuri (2022).

Finally, in model 4, the spin ${\bm{\sigma}}$ remains fixed in the $M$ state
($\alpha_{M}=0$) and flips with rate $\alpha_{S}=\alpha$ only in the $S$
state. This implies that the memory of the heading direction can only be reset
during a stop. This occurs with high probability when the average stop time
$\gamma^{-1}$ is larger than $\alpha^{-1}$. Solving Eq. (5) and replacing it
into Eq. (4), we get the velocity correlation:

$\displaystyle\langle{\bf u}(t)\cdot{\bf u}(0)\rangle=\frac{\langle
v^{2}\rangle}{2s}[(s\\!+\\!f)e^{-\lambda_{3}t}\\!+\\!(s\\!-\\!f)e^{-\lambda_{4}t}]$
(14)

where $f=-2\alpha+\beta-\gamma$, $r=2\alpha+\beta+\gamma$,
$s=\sqrt{r^{2}-8\alpha\beta}$, $\lambda_{3}=\frac{1}{2}(r+s)$, and
$\lambda_{4}=\frac{1}{2}(r-s)$. Thus, the effective diffusivity reads:

$\displaystyle D_{\rm eff}^{(4)}=\langle
v^{2}\rangle\left(\frac{2\alpha+\gamma}{2\alpha\beta}\right)\,.$ (15)

The models and the main results are summarized in Table-1.

Two dimensional systems.– In the following, we generalize the 4 S-&-G models
to 2D. For all the models, the temporal evolution of the particle position is
given as before by Eq. (1), but where ${\bm{\sigma}}$ is not a discrete
variable adopting two ${+1,-1}$, but a unit vector
${\bm{\sigma}}(\theta)\equiv(\cos\theta,\sin\theta)$, whose orientation is
encoded by the angle $\theta$. On the other hand, the transition from the
states $M$ and $S$ is governed as before by Eq. (2). It is the dynamics of the
now continuum spin ${\bm{\sigma}}$ encoded by $\theta$ that needs to be
specified:

$\displaystyle\dot{\theta}=\sqrt{2\,D[q]\,}\eta(t)\,,$ (16)

where $\eta(t)$ is a white, $\delta$-correlated noise, and $D[q]$ is defined
as $D[q=M]=D_{M}$ and $D[q=S]=D_{S}$, with $D_{M}$ and $D_{S}$ angular
diffusion coefficients. Note that transitions $S\to M$ may also involve
$\theta(t)$ updates. This occurs, for instance, in model 1. The transition
$S\to M$ involves a resetting of ${\bm{\sigma}}$: the particle picks up a
random direction ${\bm{\sigma}}$, choosing $\theta$ with equal probability
from $[0,2\pi)$ when resuming active motion, and thus erasing the memory of
the heading direction prior to the stop.

In the following, we look for a description of the dynamics in terms of the
probability distribution functions of $M$ and $S$ states: $P_{M}(\theta,t)$
and $P_{S}(\theta,t)$, respectively. For model 1, where for simplicity we
assume $D_{S}=0$, the corresponding Fokker-Planck equations read:

$\displaystyle\partial_{t}P_{M}$ $\displaystyle=$ $\displaystyle
D_{M}\partial_{\theta}^{2}P_{M}-\beta
P_{M}+\frac{\gamma}{2\pi}\int\,d\theta^{\prime}\,P_{S}(\theta^{\prime},t)$
$\displaystyle\partial_{t}P_{S}$ $\displaystyle=$ $\displaystyle\beta
P_{M}-\gamma P_{S}.$ (17)

Using that $\int d\theta^{\prime}\,[P_{S}+P_{M}]=1$, Eq. (Active stop and go
motion: a strategy to improve spatial exploration?) can be recast as:

$\displaystyle\partial_{t}P_{M}\\!=\\!D_{M}\partial_{\theta}^{2}P_{M}\\!-\\!\beta
P_{M}\\!+\\!\frac{\gamma}{2\pi}\left[1\\!-\\!\\!\int
d\theta^{\prime}P_{M}(\theta^{\prime})\right]$ (18)

Inserting into Eq. (18) the ansatz
$P_{M}\\!=\\!\frac{a_{0}(t)}{2}\\!+\\!\sum_{n=1}^{N}\left[a_{n}(t)\cos(n\theta)\\!+\\!b_{n}(t)\sin(n\theta)\right]$,
we obtain equations and expressions for $a_{n}$ and $b_{n}$. Solving for
$P_{M}(\theta,t)$, we compute the velocity autocorrelation for model 1 using
Eq. (4):

$\displaystyle\langle{\bf u}(t)\cdot{\bf u}(0)\rangle={\langle
v^{2}\rangle}e^{-(D_{M}+\beta)t}\,,$ (19)

where $\langle v^{2}\rangle=v_{M}^{2}P_{M}^{\ast}$ with
$P_{M}^{\ast}=\gamma/(\beta+\gamma)$, and its associated diffusivity:

$\displaystyle D_{\rm eff}^{(1)}=\frac{1}{2}\frac{\langle
v^{2}\rangle}{D_{M}+\beta}$ (20)

For the remaining three models, the Fokker-Planck equations take the form:

$\displaystyle\partial_{t}P_{M}$ $\displaystyle=$ $\displaystyle
D_{M}\partial_{\theta}^{2}P_{M}-\beta P_{M}+\gamma P_{S}$
$\displaystyle\partial_{t}P_{S}$ $\displaystyle=$ $\displaystyle
D_{S}\partial_{\theta}^{2}P_{S}+\beta P_{M}-\gamma P_{S}.$ (21)

Eq. (Active stop and go motion: a strategy to improve spatial exploration?)
can be solved on a Fourier basis. One obtains the correlation function,

$\displaystyle\langle{\bf u}(t)\cdot{\bf u}(0)\rangle=\langle
v[q(t)]v[q(0)]\cos(\theta(t)-\theta(0))\rangle$ $\displaystyle=\frac{\langle
v^{2}\rangle}{\lambda_{+}^{(1)}-\lambda_{-}^{(1)}}[(\lambda_{+}^{(1)}-d_{1})e^{\lambda_{+}^{(1)}t}-(\lambda_{-}^{(1)}-d_{1})e^{\lambda_{-}^{(1)}t}]$
(22)

where $\langle v^{2}\rangle=v_{M}^{2}P_{M}^{\ast}$ with
$P_{M}^{\ast}=\gamma/(\beta+\gamma)$, $d_{1}=-(D_{S}+\gamma)$, and eigenvalues
corresponding to $n=1$ mode:

$\displaystyle\lambda_{\pm}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(D_{M}+D_{S}+\beta+\gamma)$
$\displaystyle\pm\frac{1}{2}[(D_{M}-D_{S}+\beta-\gamma)^{2}+4\beta\gamma]^{\frac{1}{2}}\,.$

The effective diffusion constant, obtained from Eq. (Active stop and go
motion: a strategy to improve spatial exploration?), takes the form:

$\displaystyle D_{\rm eff}=\frac{\langle
v^{2}\rangle}{2}\frac{-d_{1}}{\lambda_{+}^{(1)}\lambda_{-}^{(1)}}.$ (23)

We apply Eq. (23) to models 2, 3, and 4. In model 2, the spin ${\bm{\sigma}}$
only evolves in the $M$ states, implying that $D_{M}=D_{r}$ and $D_{S}=0$, and
thus:

$\displaystyle D_{\rm eff}^{(2)}=\frac{\langle v^{2}\rangle}{2D_{r}}.$ (24)

In model 3, the dynamics of ${\bm{\sigma}}$ is identical in both states, $M$
and $S$, i.e., $D_{M}=D_{S}=D_{r}$, and in consequence:

$\displaystyle D_{\rm eff}^{(3)}=\frac{\langle
v\rangle^{2}}{2D_{r}}+\frac{1}{2}\frac{\langle\delta
v^{2}\rangle}{D_{r}+\beta+\gamma}\,,$ (25)

where speed fluctuations $\langle\delta v^{2}\rangle=\langle
v^{2}\rangle-\langle v\rangle^{2}=v_{M}^{2}P_{M}^{\ast}P_{S}^{\ast}$ with
$P_{S}^{\ast}=\beta/(\beta+\gamma)$. Finally, in model 4, the memory of the
heading direction ${\bm{\sigma}}$ can be lost only in the $S$ state since
$D_{M}=0$ and $D_{S}=D_{r}$. This leads to a diffusivity:

$\displaystyle D_{\rm eff}^{(4)}=\frac{\langle
v^{2}\rangle}{2}\frac{D_{r}+\gamma}{D_{r}\beta}.$ (26)

Concluding remarks.– We studied 4 S-&-G models that, though differ only in
subtle details of their dynamics, exhibit distinct large-scale transport
properties, i.e., diffusivity. These models correspond to different
experimental situations. The resetting of the heading direction associated to
a stop in model 1 is found in bacterial surface exploration and Quincke
rollers under an AC field Perez Ipiña  _et al._ (2019); Karani  _et al._
(2019); Pradillo  _et al._ (2019). Active motion with pauses that keep the
memory of the active moving direction, i.e. model 2, occurs in several animal
systems, where stops are often used to scan potential predators Kramer and
McLaughlin (2001); Gómez-Nava  _et al._ (2022). In cell migration, changes in
the moving direction and cell polarization occur when the cell moves and
stops, as in model 3 Schienbein and Gruler (1993); Vaidžiulytė  _et al._
(2022). Finally, model 4 describes straight active runs follow by
reorientation events proportional to the stop duration, which is a
characteristic feature of bacterial run-and-tumble motion Figueroa-Morales
_et al._ (2020); Wadhwa and Berg (2022). In the following, we compare the
S-&-G strategy against the classical dynamics of active particles that move at
constant speed $v_{0}$ and change the heading direction at rate $\alpha$,
displaying diffusivity $D_{\rm eff}^{(0)}=v_{0}^{2}/2\alpha$.

We start by ensuring that both S-&-G and persistent active particles exhibit
the same mean speed $\langle v\rangle=v_{M}\frac{\gamma}{\beta+\gamma}=v_{0}$.
Under this condition, the diffusivity ratios are ${D_{\rm eff}^{(1)}}/{D_{\rm
eff}^{(0)}}=\frac{\gamma}{\gamma+\beta}[\frac{2\alpha}{2\alpha+\beta}]$,
${D_{\rm eff}^{(2)}}/{D_{\rm eff}^{(0)}}=\frac{\gamma}{\gamma+\beta}$,
$\frac{D_{\rm eff}^{(3)}}{D_{\rm
eff}^{(0)}}=\left[1+\frac{\beta}{\gamma}\frac{2\alpha}{2\alpha+\beta+\gamma}\right]$,
and ${D_{\rm eff}^{(4)}}/{D_{\rm
eff}^{(0)}}=\frac{\gamma}{\gamma+\beta}\frac{2\alpha+\gamma}{\beta}$, where
$D_{\rm eff}^{(i)}$ is the diffusivity of model $i$. This implies that if we
compare systems with the same average speed, the diffusivity of S-&-G
particles can be larger than that of persistently active particles, but the
energy dissipation rate of S-&-G particles is also larger. Let us recall that
assuming a viscous drag, the dissipation rate $q$ of an active particle moving
at speed $v$ is $q\propto\langle v^{2}\rangle$. Thus, the comparison of
dissipation rates can be expressed, for all models, as $q^{i}/q_{0}:=\langle
v^{2}\rangle/v_{0}^{2}=1+\beta/\gamma>1$ and $\langle\delta
v^{2}\rangle/v_{0}^{2}=\beta/\gamma$, implying that the excess of dissipation
can be attributed to the velocity fluctuations of S-&-G particles.

Now, instead of comparing systems with mean speed, we analyze systems that
display the same dissipation rate by requesting that $\langle
v^{2}\rangle=v_{M}^{2}\gamma/(\beta+\gamma)=v_{0}^{2}$. Under this constraint,
we find that ${D_{\rm eff}^{(1)}}/{D_{\rm
eff}^{(0)}}=[{2\alpha}/({2\alpha+\beta})]<1$, ${D_{\rm eff}^{(2)}}/{D_{\rm
eff}^{(0)}}=1$, $\frac{D_{\rm eff}^{(3)}}{D_{\rm
eff}^{(0)}}=\frac{\gamma}{\beta+\gamma}\left[1+\frac{\beta}{\gamma}\frac{2\alpha}{2\alpha+\beta+\gamma}\right]<1$,
and ${D_{\rm eff}^{(4)}}/{D_{\rm
eff}^{(0)}}=\left[({2\alpha+\gamma})/{\beta}\right]$. Note that assuming equal
energy expenditure, i.e., dissipation power, only model 4 – which corresponds
to the bacterial run-and-tumble motion – can exhibit a larger diffusivity than
persistently active particles. This suggests that by performing the run-and-
tumble motion, bacteria achieve longer spatial exploration at the same power
cost.

We note that the considered models assume that transitions between states obey
a Poissonian process. Our theory can easily be extended to incorporate more
complex transition times, for instance, via multiple internal states Perez
Ipiña  _et al._ (2019); Alirezaeizanjani  _et al._ (2020); Gupta  _et al._
(2019); Nava  _et al._ (2018).

Acknowledgments \- D.C. thanks C.Y. Cergy Paris Université for a Visiting
Professorship, SERB, India, for financial support through grant number
MTR/2019/000750 and International Center for Theoretical Sciences (ICTS-TIFR),
Bangalore for an Associateship. F.P. acknowledges financial support from C.Y.
Initiative of Excellence (grant Investissements d’Avenir ANR-16-IDEX-0008),
INEX 2021 Ambition Project CollInt and Labex MME-DII, projects 2021-258 and
2021-297.

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Supplemental Information

##  .1 Illustration of the Taylor-Kubo calculation using the 1D RTP model

Here we illustrate the method of calculating the velocity autocorrelation and
diffusivity that we used in the main text. We do this by using the example of
a simple RTP with right and left moving states $P_{\pm}(x,t)$ evolving as

$\displaystyle\partial_{t}P_{+}$ $\displaystyle=$ $\displaystyle-
v_{0}\partial_{x}P_{+}-\alpha P_{+}+\alpha P_{-}$
$\displaystyle\partial_{t}P_{-}$ $\displaystyle=$ $\displaystyle
v_{0}\partial_{x}P_{-}-\alpha P_{-}+\alpha P_{+}.$ (S1)

Note that the $\pm$ states are independent of the speed $v_{0}$. We utilize
this fact to write down the general solution
$(P_{+},P_{-})^{T}(t)=a_{1}e^{\lambda_{1}t}|\psi_{1}\rangle+a_{2}e^{\lambda_{2}t}|\psi_{2}\rangle$
in terms of the eigenvalues $\lambda_{1,2}$ and eigenvectors
$|\psi_{1,2}\rangle$ corresponding to the transition matrix
$((-\alpha,\alpha)^{T},(\alpha,-\alpha)^{T})$. The constants $a_{1,2}$ are
determined using the initial condition that at time $t=0$ the system is in
either $P_{+}(x,0)$ or $P_{-}(x,0)$ state. As a result, we obtain the four
transition probabilities $P(+,t|\pm,0)$, $P(-,t|\pm,0)$ and the two steady-
state solutions $P^{(s)}_{\pm}=1/2$ in the limit of $t\to\infty$. Thus, we can
calculate the steady-state auto-correlation function $\langle
v(t)v(0)\rangle=v_{0}^{2}[P(+,t|+,0)+P(-,t|-,0)-P(-,t|+,0)-P(+,t|-,0)]P_{\pm}^{(s)}$,
which leads to the mean-squared displacement $\langle
x^{2}(t)\rangle=\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}\langle
v(t_{1})v(t_{2})\rangle$. Using explicit calculations, one can easily check
that $\langle v(t_{1})v(t_{2})\rangle=v_{0}^{2}\exp(-2\alpha|t_{2}-t_{1}|)$,
and $\langle x^{2}(t)\rangle=\frac{v_{0}^{2}}{2\alpha^{2}}(e^{-2\alpha
t}-1+2\alpha t)$. This predicts a ballistic-diffusive crossover at
$t=1/2\alpha$ and an asymptotic effective diffusion constant $D_{\rm
eff}=\lim_{t\to\infty}\langle x^{2}(t)\rangle/2t=v_{0}^{2}/2\alpha$.