# On the Generating Functions of Irreversible port-Hamiltonian Systems

Jonas Kirchhoff, Bernhard Maschke

### Abstract

We study the geometric structure of the drift dynamics of Irreversible port-
Hamiltonian systems. This drift dynamics is defined with respect to a product
of quasi-Poisson brackets, reflecting the interconnection structure and the
constitutive relations of the irreversible phenomena occuring in the system.
We characterize this product of quasi-Poisson brackets using a covariant
4-tensor and an associated function. We derive various conditions for which
this 4-tensor and the associated function may be reduced to a product of
quasi-Poisson brackets.

### Keywords

port-Hamiltonian Systems, Nonlinear Systems, Irreversible Thermodynamics,
Energy and Entropy based Modelling, Geometrical Methods.

Corresponding author: Jonas Kirchhoff
Jonas Kirchhoff
Institut für Mathematik, Technische Universität Ilmenau, Weimarer Straße 25,
98693 Ilmenau, Germany
E-mail<EMAIL_ADDRESS>

Bernhard Maschke
Univ. Lyon, Université Claude Bernard Lyon 1, CNRS, LAGEPP UMR 5007, France
E-mail<EMAIL_ADDRESS>

Jonas Kirchhoff thanks the Technische Universität Ilmenau and the Freistaat
Thüringen for their financial support as part of the Thüringer
Graduiertenförderung.

##  1 Introduction

Irreversible port-Hamiltonian Systems [RMS13a] differ intrinsically from other
dynamical models of irreversible thermodynamic systems, in the sense that they
are defined with respect to a product of quasi-Poisson brackets and not a
symmetric non-negative bracket such as in e.g. [BE91, Mor86]. However the
properties of this product of brackets are poorly understood. In this paper,
we shall associate a four-tensor and a corresponding set of functions with
this product and analyze their relation.

##  2 Irreversible port-Hamiltonian Systems

Irreversible port-Hamiltonian Systems are an extension of port-Hamiltonian
Systems which has been defined to represent not only the energy conservation
but also the irreversible entropy creation [RMS13b, RMS13a] and is defined as
follows.

######  Definition 2.1.

[RMS13a] An Irreversible port-Hamiltonian System (IPHS) is a nonlinear control
system

$\displaystyle\frac{dx}{dt}=\gamma\left(x,\mathrm{d}H(x)\right)\left\\{S,H\right\\}_{J}J\mathrm{d}H(x)+W\left(x,\mathrm{d}H(x)\right)$
(2.1) $\displaystyle+g\left(x,\mathrm{d}H(x)\right)u,$

where $x\left(t\right)\in\mathbb{R}^{n}$ is the state vector,
$u\left(t\right)\in\mathbb{R}^{m}$ is the control input, and $dH(x)$ denotes
the differential of the function $H$ at $x$ in standard coordinates, defined
by

(i) two (smooth) real functions called _Hamiltonian function_ $H(x)\in
C^{\infty}(\mathbb{R}^{n})$ and _entropy function_ $S(x)\in
C^{\infty}(\mathbb{R}^{n})$,

(ii) the _structure matrix_ $J\in\mathbb{R}^{n\times n}$ which is constant and
skew-symmetric, defining the Poisson bracket

$\left\\{S,H\right\\}_{J}:=\mathrm{d}S^{\top}\left(x\right)J\mathrm{d}H$

(iii) a real function $\gamma(x,\mathrm{d}H(x))=\hat{\gamma}(x)\in
C^{\infty}(\mathbb{R}^{n})$, strictly positive function of the states and co-
states

(iv) the vector field $W(x,\mathrm{d}H(x))\in\mathbb{R}^{n}$ and matrix field
$g(x,\mathrm{d}H(x))\in\mathbb{R}^{n\times m}$ defining the input map.

Note that the drift dynamics, which may be rewritten for any function
$f\in\mathcal{C}^{\infty}(\mathbb{R}^{n})$

$\frac{df}{dt}=\gamma\left(x,\mathrm{d}H\right)\left\\{S,H\right\\}_{J}\left\\{f,H\right\\}_{J}$
(2.2)

is defined in such a way that the two axioms of Irreversible Thermodynamics
are satisfied for the isolated thermodynamic system

  * •

energy conservation: $\frac{dH}{dt}=0$

  * •

irreversible entropy production:

$\frac{dS}{dt}=\gamma\left(x,\mathrm{d}H\right)\left\\{S,H\right\\}_{J}^{2}\geq
0$

Note that taking account of the input leads to the energy balance equation

$\frac{dH}{dt}=\mathrm{d}H^{\top}\left(W+gu\right)$

and the entropy balance equation

$\frac{dS}{dt}=\underbrace{\gamma\left(x,\mathrm{d}H\right)\left\\{S,H\right\\}_{J}^{2}}_{=\sigma_{int}\geq
0}+\mathrm{d}H^{\top}\left(W+gu\right)$ (2.3)

The reader may find various examples of irreversible systems fitting into this
frame and ranging from heat exchangers to chemical reaction dynamics and the
gas-piston system [RMS13a, RMS13b]. In these physical systems, the bracket
$\left\\{S,H\right\\}_{J}$ appears to be the driving force of the irreversible
phenomenon such as the temperature difference for the heat conduction or the
chemical affinity for the chemical reaction dynamics. The function
$\gamma(x,\mathrm{d}H(x))=\hat{\gamma}(x)$, corresponds to the constitutive
relation of the phenomenon such as Fourier’s law or the chemical reaction
kinetics.

Examining the drift dynamics, it may be observed that it is defined by two
functions, the energy and the entropy function and is generated by a product
of Poisson brackets. This departs radically from all alternative suggestions
of structured description of irreversible thermodynamical systems ranging from
the single generator function bracket approach in [BE91, BE94, Mor86] to the
two generator functions approach in [Kau84, GO97a, GO97b] where the
irreversible phenomena (and the irreversible entropy creation) are described
by a _symmetric positive bracket_ associated with a (pseudo-)gradient
dynamics.

In this paper, we shall discuss the definition of the drift dynamics and ask
the question whether the functions
$E_{J}:=(f,S,H)\mapsto\gamma\\{S,H\\}_{J}\\{f,H\\}_{J}$ appearing in the
definition of the dynamics (2.2) is the only expression with minimal
homogeneity that depend only on gradients of $f,S$ and $H$ so that $H$ is
preserved and $S$ increases (nontrivially) monotonically along trajectories of
the system.

##  3 Conservative-irreversible functions

Recall first the classical definition of a derivation.

######  Definition 3.1 (Derivation, see [Coh85, p. 39]).

A function
$\delta:\mathcal{C}^{\infty}(\mathbb{R}^{n})\to\mathcal{C}^{\infty}(\mathbb{R}^{n})$
is a derivation if, and only if, $\delta$ is linear and fulfills the Leibniz
rule,

$\displaystyle\forall
f,g\in\mathcal{C}^{\infty}(\mathbb{R}^{n}):\delta(fg)=f\delta(g)+\delta(f)g.$

And define _conservative-irreversible functions_ , as follows.

######  Definition 3.2.

A function

$\displaystyle
E:\left(\mathcal{C}^{\infty}(\mathbb{R}^{n})\right)^{3}\to\mathcal{C}^{\infty}(\mathbb{R}^{n})$

is conservative-irreversible if, and only if, there is a function

$\displaystyle
e:\left(\mathcal{C}^{\infty}(\mathbb{R}^{n})\right)^{4}\to\mathcal{C}^{\infty}(\mathbb{R}^{n})$

with the properties

  1. (i)

$e$ is a derivation in each of the four arguments (a _four-derivation_),

  2. (ii)

$e$ is symmetric in the third and fourth argument,

  3. (iii)

$e(h,\cdot,h,h)\equiv 0$ for all $h\in\mathcal{C}^{\infty}(\mathbb{R}^{n})$,

  4. (iv)

$e(\cdot,\cdot,h,h)$ is pointwise symmetric and positive semidefinite for all
$h\in\mathcal{C}^{\infty}(\mathbb{R}^{n})$,

so that

$\displaystyle\forall
f,s,h\in\mathcal{C}^{\infty}(\mathbb{R}^{n}):E(f,s,h)=e(f,s,h,h).$ (3.1)

######  Example 3.3.

Let $J\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n\times n})$ be
pointwise skew-symmetric. Consider the function

$\displaystyle e_{J}:\left(\mathcal{C}^{\infty}(\mathbb{R}^{n})\right)^{3}$
$\displaystyle\to\mathcal{C}^{\infty}(\mathbb{R}^{n}),$
$\displaystyle(f,s,h,g)\mapsto\\{s,g\\}_{J}\\{f,h\\}_{J}.$

This function has the properties (i)-(iv) and induces the conservative-
irreversible function

$\displaystyle E_{J}:\left(\mathcal{C}^{\infty}(\mathbb{R}^{n})\right)^{3}$
$\displaystyle\to\mathcal{C}^{\infty}(\mathbb{R}^{n}),$
$\displaystyle(f,s,h)\mapsto\\{s,h\\}_{J}\\{f,h\\}_{J}.$

$\diamond$

######  Remark 3.4.

The symmetry condition (ii) states that we choose a symmetric representative
of all four-derivations that generate a given conservative-irreversible
function $E$. Indeed, if $e$ and $e^{\prime}$ are any four-derivations with
the properties (iii) and (iv) which fulfill (3.1), then, for all
$\lambda\in\mathbb{R}$, the four-derivation

$\displaystyle(f,s,h,g)\mapsto\lambda
e(f,s,h,g)+(1-\lambda)e^{\prime}(f,s,g,h)$

has the properties (iii) and (iv) and fulfills (3.1), too. Especially, the
symmetrisation of $e$,

$\displaystyle(f,s,h,g)\mapsto\frac{1}{2}e(f,s,h,g)+\frac{1}{2}e(f,s,g,h),$

is a generating four-derivation for $E$.

Algebraically, the set of conservative-irreversible functions has naturally
the structure of a module over the smooth, pointwise nonnegative functions, as
can easily been seen.

######  Lemma 3.5.

Equipped with pointwise addition and multiplication, the set of nonnegative
smooth functions

$\displaystyle\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}_{\geq
0}):=\left\\{f\in\mathcal{C}^{\infty}(\mathbb{R}^{n})\,\big{|}\,\forall
x\in\mathbb{R}^{n}:f(x)\geq 0\right\\}.$

is a commutative ring with 1.

Proof. Since $\mathbb{R}$ is an ordered field and since the product of smooth
real valued functions is smooth (see [AMR88, p. 86]),
$\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}_{\geq 0})$ is indeed a
commutative ring with $1$. $\square$

Denote the set of all conservative-irreversible functions with

$\displaystyle\mathfrak{CI}(\mathbb{R}^{n}):=\left\\{E:\left(\mathcal{C}^{\infty}(\mathbb{R}^{n})\right)^{3}\to\mathcal{C}^{\infty}(\mathbb{R}^{n})\,\big{|}\,E\leavevmode\nobreak\
\text{cons.\leavevmode\nobreak\ irrev.}\right\\}.$

and equip it with the pointwise addition

$\displaystyle(E+G)(f,s,h)(x):=E(f,s,h)(x)+G(f,s,h)(x),$

and the multiplication

$\displaystyle(\lambda\cdot E)(f,s,h)(x):=\lambda(x)E(f,s,h)(x)$

for all $E\in\mathfrak{CI}(\mathbb{R}^{n})$ and
$\lambda\in\mathbb{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}_{\geq 0})$.

######  Lemma 3.6.

$(\mathfrak{CI}(\mathbb{R}^{n}),+,\cdot)$ is a module over
$\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}_{\geq 0})$.

Proof. Let $E,G\in\mathfrak{CI}(\mathbb{R}^{n})$ and
$\lambda\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}_{\geq 0})$ be
arbitrary. We show that $\lambda E+G\in\mathfrak{CI}(\mathbb{R}^{n})$. Since
$E$ and $G$ are conservative-irreversible functions, there are functions $e$
and $g$ as in Definition 3.2 so that

$\displaystyle E=e(f,s,h,h),\quad G(f,s,h)=g(f,s,h,h)$

for all $f,s,h\in\mathcal{C}^{\infty}(\mathbb{R}^{n})$. Then, $\lambda E+G$
has the representation $\lambda e+f$. It remains to verify that $\lambda e+f$
has the properties (i)–(iv) in Definition 3.2. (i) holds, since derivations on
$\mathcal{C}^{\infty}(\mathbb{R}^{n})$ are a module over
$\mathcal{C}^{\infty}(\mathbb{R}^{n})$. Let
$f,s,h,i\in\mathcal{C}^{\infty}(\mathbb{R}^{n})$. Then

$\displaystyle(\lambda f+g)(f,s,h,i)$ $\displaystyle=\lambda
f(f,s,h,i)+g(f,s,h,i)$ $\displaystyle=\lambda f(f,s,i,h)+g(f,s,i,h)$
$\displaystyle=(\lambda f+g)(f,s,i,h)$

and hence (ii) holds. The remaining properties can be shown analogously from
the pointwise definition of the addition and scalar multiplication on
$\mathfrak{CI}(\mathbb{R}^{n})$. This shows that the operations $+$ and
$\cdot$ are well-defined. It is easy to verify that
$(\mathfrak{CI}(\mathbb{R}^{n}),+,\cdot)$ is indeed a module over
$\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}_{\geq 0})$; we omit the
details. $\square$

Besides being a module, $\mathfrak{CI}(\mathbb{R}^{n})$ is a convex cone.

######  Lemma 3.7.

$\mathfrak{CI}(\mathbb{R}^{n})$ is a convex cone in the real vector space of
functions from $\left(\mathcal{C}^{\infty}(\mathbb{R}^{n})\right)^{3}$ to
$\mathcal{C}^{\infty}(\mathbb{R}^{n})$.

Proof. Since constant functions are smooth, this is a direct consequence of
Lemma 3.6. $\square$

We want to characterise the functions $e$ that induce conservative-
irreversible functions as in (3.1).

######  Proposition 3.8.

A four-linear function

$\displaystyle
e:\left(\mathcal{C}^{\infty}(\mathbb{R}^{n})\right)^{4}\to\mathcal{C}^{\infty}(\mathbb{R}^{n})$

has the properties (i)–(iv) from Definition 3.2 if, and only if, there is some
$\varepsilon\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n\times n\times
n\times n})$ with the properties

  1. (a)

$\varepsilon_{i,j,k,\ell}=\varepsilon_{i,j,\ell,k}$,

  2. (b)

$\varepsilon_{i,j,k,\ell}+\varepsilon_{k,j,\ell,i}+\varepsilon_{\ell,j,i,k}=0$,

  3. (c)

the function

$\displaystyle
h\mapsto\left[\sum_{k,\ell=1}^{n}\varepsilon_{i,j,k,\ell}\mathrm{d}_{k}h\mathrm{d}_{\ell}h\right]_{i,j=1}^{n}$

is pointwise symmetric and positive semidefinite,

for all $i,j,k,\ell\in\underline{n}$ so that

$\displaystyle
e(f,s,h,q)=\sum_{i,j,k,\ell=1}^{n}\varepsilon_{i,j,k,\ell}\mathrm{d}_{i}f\mathrm{d}_{j}s\mathrm{d}_{k}h\mathrm{d}_{\ell}q.$
(3.2)

Proof. We split the proof into steps.

Step 1: We show that $e$ has a tensor representation $\varepsilon$. By taking
the basis of the tangent space associated to the standard coordinates, we see
that each four-derivation

$\displaystyle\eta:\left(\mathcal{C}^{\infty}(\mathbb{R}^{n})\right)^{4}\to\mathcal{C}^{\infty}(\mathbb{R}^{n})$

has a tensor representation

$\displaystyle\eta(f,s,h,q)=\sum_{i,j,k,\ell=1}^{n}\eta_{i,j,k,\ell}\mathrm{d}_{i}f\mathrm{d}_{j}s\mathrm{d}_{k}h\mathrm{d}_{\ell}q$

for all $f,s,h,q\in\mathcal{C}^{\infty}(\mathbb{R}^{n})$. Thus, $e$ has a
representation
$\varepsilon\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n\times n\times
n\times n})$ with (3.2). It remains to prove that $e$ has the properties
(ii)–(iv) from Definition 3.2 if, and only if, $\varepsilon$ has the
properties (a)–(c).

Step 2: We show that $e$ fullfills the symmetry condition (ii) if, and only
if, its tensor representation $\varepsilon$ fulfills (a). Let
$i,j,k,\ell\in\underline{n}$ and assume that $e$ fulfills (ii). By evaluating
$e$ at the coordinate functions, we find

$\displaystyle\varepsilon_{i,j,k,\ell}=e(x_{i},x_{j},x_{k},x_{\ell})=e(x_{i},x_{j},x_{\ell},x_{k})=\varepsilon_{i,j,\ell,k}.$

Thus, $\varepsilon$ fulfills (a). The converse implication can be equally easy
proven.

Step 3: We show that $e$ fulfills (iii) if, and only if,

$\displaystyle\varepsilon_{i,j,i,\ell}+\varepsilon_{i,j,\ell,i}+\varepsilon_{\ell,j,i,i}=0$
(3.3)

for all $i,j,\ell\in\underline{n}$ and

$\displaystyle\varepsilon_{i,j,k,\ell}+\varepsilon_{k,j,i,\ell}+\varepsilon_{k,j,\ell,i}+\varepsilon_{\ell,j,k,i}+\varepsilon_{i,j,\ell,k}+\varepsilon_{\ell,j,i,k}=0$
(3.4)

for all $j\in\underline{n}$ and pairwise different $jik,\ell\in\underline{n}$.

“$\implies$” We show first necessity. By evaluating $e$ at the coordinate
functions, we get

$\displaystyle\forall
i,\ell\in\underline{n}:0=e(x_{\ell},x_{i},x_{\ell},x_{\ell})=\varepsilon_{\ell,i,\ell,\ell}.$

Let now $b\in\mathbb{R}\setminus\\{0\\}$. We have

$\displaystyle 0$
$\displaystyle=e(x_{i}+bx_{j},x_{k},x_{i}+bx_{j},x_{i}+bx_{j})$
$\displaystyle=b\varepsilon_{i,k,i,j}+b\varepsilon_{i,k,j,i}+b\varepsilon_{j,k,i,i}+b^{2}\varepsilon_{j,k,j,i}$
$\displaystyle\quad+b^{2}\varepsilon_{j,k,i,j}+b^{2}\varepsilon_{i,k,j,j}$

for all $i\neq j\in\underline{n}$ and $k\in\underline{n}$. If we divide by $b$
and let $b$ tend to zero, we get

$\displaystyle\forall
i,j,k\in\underline{n}:0=\varepsilon_{i,i,j}+\varepsilon_{i,j,i}+\varepsilon_{j,i,i}$

and hence (3.3) is indeed necessary. The property (3.4) can now be concluded
by calculating
$e(x_{i}+x_{k}+x_{\ell},x_{j},x_{i}+x_{k}+x_{\ell},x_{i}+x_{k}+x_{\ell})$ for
$j\in\underline{n}$ and pairwise different $i,j,k\in\underline{n}$; we omit
the details.

“$\impliedby$” It is now straightforward to show sufficiency. First, we note
that (3.3) for $i=\ell$ yields $\varepsilon_{i,j,i,i}=0$ for all
$i,j\in\underline{n}$. Let $s,h\in\mathcal{C}^{\infty}(\mathbb{R}^{n})$ be
arbitrary. Then we have

$\displaystyle e(h,s,h,h)$
$\displaystyle=\sum_{i,j,k,\ell=1}^{n}\varepsilon_{i,j,k,\ell}\mathrm{d}_{i}h\mathrm{d}_{j}s\mathrm{d}_{k}h\mathrm{d}_{\ell}h$
$\displaystyle=\sum_{j=1}^{n}\mathrm{d}_{j}s\Bigg{(}\sum_{i=1}^{n}\varepsilon_{i,j,i,i}(\mathrm{d}_{i}h)^{3}$
$\displaystyle\quad+\sum_{i\neq
k=1}^{n}(\varepsilon_{i,j,i,k}+\varepsilon_{i,j,k,i}+\varepsilon_{k,j,i,i})(\mathrm{d}_{i}h)^{2}\mathrm{d}_{j}h$
$\displaystyle\quad+\sum_{i=1}^{n}\sum_{k=i+1}^{n}\sum_{\ell=k+1}^{n}\varepsilon_{i,k,\ell}\mathrm{d}_{i}h\mathrm{d}_{k}h\mathrm{d}_{\ell}h\Bigg{)}$
$\displaystyle=\sum_{j=1}^{n}\mathrm{d}_{j}s\sum_{i=1}^{n}\sum_{k=i+1}^{n}\sum_{\ell=k+1}^{n}(\varepsilon_{i,j,k,\ell}+\varepsilon_{i,j,\ell,k}$
$\displaystyle\quad+\varepsilon_{k,j,i,\ell}+\varepsilon_{k,j,\ell,i}+\varepsilon_{\ell,j,i,k}+\varepsilon_{\ell,j,k,i})\mathrm{d}_{i}h\mathrm{d}_{j}h\mathrm{d}_{k}h$
$\displaystyle=0.$

This shows the assertion.

Step 4: We show that, given the symmetry condition (a), (3.3) and (3.4) are
equivalent to (b). When we plug the symmetry condition (a) in, then we see
that (3.4) is equivalent to

$\displaystyle\varepsilon_{i,j,k,\ell}+\varepsilon_{k,j,\ell,i}+\varepsilon_{\ell,j,i,k}=0$

for all $j\in\underline{n}$ and pairwise different $i,k,\ell\in\underline{n}$.
Combining this with (3.3), we see that (3.3) and (3.4) are, in view of (a),
indeed equivalent to (b).

Step 5: From the relation (3.2), it is immediately clear that (iv) and (c) are
equivalent. This shows that the proposition holds indeed true. $\square$

In our following considerations, we identify therefore the four-derivations
$e$ from definition 3.2 with their tensorial representations
$\varepsilon\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n\times n\times
n\times n})$.

##  4 Relation to port-Hamiltonian systems

We have introduced conservative-irreversible functions. In Example 3.3, we
have seen that IPHS can be geometrically formulated as being generated by
conservative-irreversible functions according to (2.2). In this section, we
shall be interested in the converse question: which are the conservative-
irreversible functions which may be related to the expression in (2.2)
generated by a quasi-Poisson bracket, i.e. Poisson bracket not satisfying the
Jacobi identity?

Before investigating this problem, let us consider the two-dimensional case
and give an example of a conservative-irreversible function from which one may
derive such a bracket.

######  Example 4.1.

Let $n=2$ and define $\varepsilon\in\mathbb{R}^{2\times 2\times 2\times 2}$ by

$\displaystyle\varepsilon_{i,j,k,\ell}:=\begin{cases}2,&(i,j,k,\ell)\in\\{(1,1,2,2),(2,2,1,1)\\},\\\
-1,&(i,j,k,\ell)\in\left\\{\begin{array}[]{l}(1,2,1,2),(1,2,2,1),\\\
(2,1,1,2),(2,1,2,1)\end{array}\right\\},\\\ 0,&\text{else}.\end{cases}$

$\varepsilon$ is symmetric in the first two arguments and in the last two
arguments. It fullfills, by construction, the condition (b) in Proposition
3.8. When checking condition (c) in Proposition 3.8, we have to consider, for
all $h\in\mathcal{C}^{\infty}(\mathbb{R}^{2})$,

$\displaystyle\begin{bmatrix}\sum_{k,\ell=1}^{2}\varepsilon_{1,1,k,\ell}\mathrm{d}_{k}h\mathrm{d}_{\ell}h&\sum_{k,\ell=1}^{2}\varepsilon_{1,2,k,\ell}\mathrm{d}_{k}h\mathrm{d}_{\ell}h\\\
\sum_{k,\ell=1}^{2}\varepsilon_{2,1,k,\ell}\mathrm{d}_{k}h\mathrm{d}_{\ell}h&\sum_{k,\ell=1}^{2}\varepsilon_{2,2,k,\ell}\mathrm{d}_{k}h\mathrm{d}_{\ell}h\end{bmatrix}$
$\displaystyle=\begin{bmatrix}2\mathrm{d}_{2}h^{2}&-2\mathrm{d}_{1}h\mathrm{d}_{2}h\\\
-2\mathrm{d}_{1}h\mathrm{d}_{2}h&2\mathrm{d}_{1}h^{2}\end{bmatrix}=:M(h).$

We can calculate the characteristic polynomial of $M(h)$ as

$\displaystyle p_{M(x)}(\lambda)$
$\displaystyle=(\lambda-2\mathrm{d}_{2}h^{2})(\lambda-2\mathrm{d}_{1}h^{2})-4\mathrm{d}_{1}h^{2}\mathrm{d}_{2}h^{2}$
$\displaystyle=\lambda^{2}-2\lambda(\mathrm{d}_{2}h^{2}+\mathrm{d}_{1}h^{2})$

and hence we conclude that $M(h)$ has eigenvalues $0$ and
$2\mathrm{d}_{1}h^{2}+2\mathrm{d}_{2}h^{2}$. This shows that $M(h)$ is
pointwise symmetric and positive semidefinite. Therefore, we conclude with
Proposition 3.8 that $(\varepsilon_{i,j,k,\ell})$ defines a conservative-
irreversible function.
We relate this tensor to a quasi-Poisson bracket. The vector space of skew-
symmetric matrices is one-dimensional and spanned by

$\displaystyle J:=\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}.$

A tensorial representation
$e\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n\times n\times n\times
n})$ of the conservative-irreversible function

$\displaystyle(f,s,h)\mapsto\\{f,h\\}_{J}\\{s,h\\}_{J}$

can thus be calculated as

$\displaystyle\forall
i,j,k,\ell\in\underline{2}:e_{i,j,k,\ell}=J_{i,k}J_{j,\ell}$

and hence we have

$\displaystyle
e_{i,j,k,\ell}=\begin{cases}1,&(i,j,k,\ell)\in\\{(1,1,2,2),(2,2,1,1)\\},\\\
-1,&(i,j,k,\ell)\in\\{(1,2,2,1),(2,1,1,2)\\},\\\ 0,&\text{else}.\end{cases}$

When we symmetrise $2e$ in the last two entries, i.e. consider the four-tensor

$\displaystyle[e_{i,j,k,\ell}+e_{i,j,\ell,k}]_{i,j,k,\ell=1}^{2},$

then we see that $\varepsilon$ is precisely this symmetrisation. Thus, the
conservative-irreversible function represented by $\varepsilon$ is

$\displaystyle(f,s,h)\mapsto 2\\{s,h\\}_{J}\\{f,h\\}_{J}.$

$\diamond$

This example leads to study the set of conservative-irreversible functions
that can be split in the form

$\displaystyle(f,s,h)\mapsto\\{s,h\\}_{1}\\{f,h\\}_{2}$ (4.1)

for biderivations $\\{\cdot,\cdot\\}_{1},\\{\cdot,\cdot\\}_{2}$. This yields
the question: Which conservative-irreversible functions allow a splitting
(4.1)?

######  Theorem 4.2.

Let
$E:\left(\mathcal{C}^{\infty}(\mathbb{R}^{n})\right)^{3}\to\mathcal{C}^{\infty}(\mathbb{R}^{n})$
be a conservative-irreversible function. Then $E$ has the representation

$\displaystyle E(f,s,h)=\\{s,h\\}_{1}\\{f,h\\}_{2}$ (4.2)

for some biderivations $\\{\cdot,\cdot\\}_{1}$, $\\{\cdot,\cdot\\}_{2}$ if,
and only if, there is a quasi-Poisson bracket $\\{\cdot,\cdot\\}_{J}$ and
$\gamma\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}_{\geq 0})$ so that

$\displaystyle E(f,s,h)=\gamma\\{s,h\\}_{J}\\{f,h\\}_{J}.$ (4.3)

Proof. Sufficiency of (4.3) follows from Example 3.3; we show necessity. Let
$E$ have the representation (4.2) and the tensorial representation
$e\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n\times n\times n\times
n})$. The biderivations $\\{\cdot,\cdot\\}_{1},\\{\cdot,\cdot\\}_{2}$ have the
representation

$\displaystyle\forall
f,g\in\mathcal{C}^{\infty}(\mathbb{R}^{n}):\Bigg{\\{}\begin{array}[]{ll}\\{f,g\\}_{1}&=\mathrm{d}f^{\top}A\mathrm{d}g,\\\
\\{f,g\\}_{2}&=\mathrm{d}f^{\top}B\mathrm{d}g\end{array}$

for some $A,B\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n\times n})$.
Then, $e$ can be calculated as

$\displaystyle\forall
i,j,k,\ell\in\underline{n}:e_{i,j,k,\ell}=A_{i,k}B_{j,\ell}.$ (4.4)

Since the properties (a)–(c) in Proposition 3.8 are in a pointwise manner, we
can without loss of generality assume that $A$ and $B$ (and therefore $e$) are
constant. Further, if either of the brackets is trivial, then the proposition
holds obviously true, so that we can restrict ourself without loss of
generality to nontrivial biderivations.

We show: Given matrices $A,B\in\mathbb{R}^{n\times
n}\in\mathbb{R}^{n}\setminus\\{0\\}$, the four-tensor
$e:=[A_{i,k}B_{j,\ell}]_{i,j,k,\ell=1}^{n}\in\mathbb{R}^{n\times n\times
n\times n}$ fulfills (b) and (c) – equivalently (iii) and (iv) – if, and only
if, $A$ is skew-symmetric and $A$ and $B$ are linearly dependent. We split the
proof into steps.

Step 1: In the proof of Proposition 3.8, we have shown that the property (iii)
of $e$ is equivalent to (3.3) and (3.4). We prove that

$\displaystyle\forall
i,j,k\in\underline{n}:e_{i,j,i,k}+e_{i,j,k,i}+e_{k,j,i,i}=0$ (7)

and

$\displaystyle\forall j\in\underline{n}\leavevmode\nobreak\ \forall
i,k,\ell\in\underline{n}\leavevmode\nobreak\
\text{pairwise\leavevmode\nobreak\ different}:e_{i,j,k,\ell}+e_{k,j,i,\ell}$
(8)
$\displaystyle+e_{i,j,\ell,k}+e_{\ell,j,i,k}+e_{k,j,\ell,i}+e_{\ell,j,k,i}=0$

hold true if, and only if, $A$ is skew-symmetric.

“$\implies$” If we plug the definition of $e_{i,j,k,\ell}$ into (7), then we
have

$\displaystyle\forall
i,j,k\in\underline{n}:A_{i,i}B_{j,k}+B_{j,i}(A_{i,k}+A_{k,i})=0.$

Especially, for $i=k$, we get

$\displaystyle\forall i,j\in\underline{n}:A_{i,i}B_{j,i}=0,$

so that we conclude

$\displaystyle\forall i\in\underline{n}:A_{i,i}=0\vee B_{\cdot,i}=0.$

If $B_{\cdot,i}=0$ for some fixed $i\in\underline{n}$, then we get

$\displaystyle\forall j,k\in\underline{n}:A_{i,i}B_{j,k}=0$

and hence, since we assume $B\neq 0$, $A_{i,i}=0$ in any case. Thus, we derive

$\displaystyle\forall i,j,k\in\underline{n}:B_{j,i}(A_{i,k}+A_{k,i})=0$

from which we derive that $B_{\cdot,i}=0$ or $A_{i,\cdot}=-A_{\cdot,i}$ for
all $i\in\underline{n}$. If we plug this find into (ii), then we get, for all
$j\in\underline{n}$ and all pairwise different $i,k,\ell\in\underline{n}$,

$\displaystyle 0$
$\displaystyle=(A_{i,k}+A_{k,i})B_{j,\ell}+(A_{i,\ell}+A_{\ell,i})B_{j,k}+(A_{k,\ell}+A_{\ell,k})B_{j,i}.$

If $A$ is not skew-symmetric, then we have some $i\neq\ell\in\underline{n}$ so
that $A_{i,\ell}\neq-A_{\ell,i}$ and thus $B_{j,i}=B_{j,\ell}=0$ for all
$j\in\underline{n}$. Hence we conclude

$\displaystyle\forall j\in\underline{n}\leavevmode\nobreak\ \forall
k\in\underline{n}\setminus\\{i,j\\}:0=(A_{i,\ell}+A_{\ell,i})B_{j,k}.$

Therefore

$\displaystyle\forall j\in\underline{n}\leavevmode\nobreak\ \forall
k\in\underline{n}\setminus\\{i,j\\}:B_{j,k}=0,$

so that we conclude with our previous findings that $B=0$, contrary to our
assumption. Hence $A$ is indeed skew-symmetric.

“$\impliedby$” This follows from a straightforward calculation using the
definition of $e$. We have, for all $i,j,k,\ell\in\underline{n}$

$\displaystyle
e_{i,j,k,\ell}+e_{k,j,i,\ell}+e_{i,j,\ell,k}+e_{\ell,j,i,k}+e_{k,j,\ell,i}+e_{\ell,j,k,i}$
$\displaystyle\quad=(A_{i,k}+A_{k,i})B_{j,\ell}+(A_{i,\ell}+A_{\ell,i})B_{j,k}+(A_{k,\ell}+A_{\ell,k})B_{j,i}$
$\displaystyle\quad=0$

by skew-symmetry of $A$ and hence (8) holds true. Putting $i=k$, we readily
conclude that (7) holds equally true.

This proves that $e$ has the property (iii) if, and only if, $A$ is skew-
symmetric.

Step 2: It is easy to verify that the symmetry of $e(\cdot,\cdot,h,h)$ in
property (iii) is equivalent to

$\displaystyle\forall
y\in\mathbb{R}^{n}:Ayy^{\top}B^{\top}=Byy^{\top}A^{\top}.$ (4.5)

We show: If $A$ and $B$ fulfill (4.5), then $A_{\cdot,i}$ and $B_{\cdot,i}$
are linearly dependent for all $i\in\underline{n}$. Especially, if $A$ and $B$
fulfill (4.5), then, for each $i\in\underline{n}$, the matrix

$\displaystyle Ae_{i}e_{i}^{\top}B^{\top}=A_{\cdot,i}B_{\cdot,i}^{\top},$

where $e_{i}$ denotes, as usual, the $i$-th standard unit vector, is
symmetric. Therefore, we conclude

$\displaystyle\forall\lambda\in\mathbb{R}\leavevmode\nobreak\ \forall
j,k\in\underline{n}:B_{j,i}(A_{k,i}-\lambda B_{k,i})=B_{i,k}(A_{j,i}-\lambda
B_{j,i});$

the same holds true with $A$ and $B$ interchanged. If $B_{\cdot,i}\neq 0$,
then there is some $k\in\underline{n}$ with $B_{k,i}\neq 0$ and some
$\mu_{i}\in\mathbb{R}$ so that $A_{k,i}=\mu_{i}B_{k,i}$ and thus we have

$\displaystyle\forall
j\in\underline{n}:0=B_{j,i}(A_{k,i}-\mu_{i}B_{k,i})=B_{k,i}(A_{j,i}-\mu_{i}B_{j,i}),$

so that we conclude $A_{\cdot,i}=\mu_{i}B_{\cdot,i}$. Interchanging $A$ and
$B$, we conclude that if $A_{\cdot,i}\neq 0$, then there is some
$\lambda_{i}\in\mathbb{R}$ so that $B_{\cdot,i}=\lambda_{i}A_{i}$. If, on the
other hand, $B_{\cdot,i}=0$ and $A_{\cdot,i}=0$, then $A_{\cdot,i}$ and
$B_{\cdot,i}$ are linearly dependent, anyway. Therefore,

$\displaystyle\forall i\in\underline{n}\leavevmode\nobreak\
\exists(\lambda_{i},\mu_{i})\in\mathbb{R}^{2}\setminus\\{0\\}:\lambda_{i}A_{\cdot,i}+\mu_{i}B_{\cdot,i}=0.$

Step 3: In this step, we exploit the skew-symmetry of $A$, which we verified
in Step 1. We show that if $A$ and $B$ fullfill (4.5) and $A\neq 0$ is skew-
symmetric, then $Ayy^{\top}B^{\top}$ is symmetric for all $y\in\mathbb{R}^{n}$
if, and only if, $A$ and $B$ are linerly dependent. Let $i,j\in\underline{n}$
be so that $A_{i,j}\neq 0$ (especially $i\neq j$); after renormalisation we
can w.l.o.g. assume that $A_{i,j}=1$. Then $A_{i,j}=-A_{j,i}$ so that we can
w.l.o.g. further assume that $i<j$ and thus

$\displaystyle
A(e_{i}+e_{j})=\left(\begin{array}[]{ccccccccccc}\star&\cdots&\star&1&\star&\cdots&\star&-1&\star&\cdots&\star\end{array}\right)^{\top}$

By Step 2 there are $\lambda_{i},\lambda_{j}\in\mathbb{R}$ so that
$B_{\cdot,i}=\lambda_{i}A_{\cdot,i}$ and $B_{\cdot,j}=\lambda_{j}A_{\cdot,j}$.
Plugging this in, we get

$\displaystyle\big{(}A(e_{i}+e_{j})(e_{i}+e_{j})^{\top}B^{\top}\big{)}_{i,j}=\lambda_{i}$

and

$\displaystyle\big{(}A(\lambda e_{i}+\mu e_{j})(\lambda e_{i}+\mu
e_{j})^{\top}B^{\top}\big{)}_{j,i}=\lambda_{j}.$

By (4.5), we conclude $\lambda_{i}=\lambda_{j}$. Define the graph
$G=(\underline{n},E)$ with

$\displaystyle(\iota,\kappa)\in E:\iff\big{(}\exists
i\in\underline{n}:A_{\iota,i},A_{\kappa,i}\neq 0\big{)}\vee
A_{\iota,\kappa}\neq 0$

and the equivalence relation $\sim$ that is canonically associated to $G$ and
induced by the connected components of $G$. Then we conclude for all
$i,j\in\underline{n}$ with $i\in[j]_{/\sim}$ that $\lambda_{i}=\lambda_{j}$.
It remains to consider the case $A_{\cdot,i}=0$. In this case, we have

$\displaystyle
A(e_{i}+e_{j})(e_{i}+e_{j})^{\top}B^{\top}=Ae_{i}e_{i}^{\top}B^{\top}+A_{\cdot,j}B_{\cdot,i}^{\top}.$

Thus, $A_{\cdot,j}B_{\cdot,i}^{\top}$ is symmetric. Analogously to Step 2, we
conclude that for all $j\in\underline{n}$ with $A_{\cdot,j}\neq 0$, there is
some $\lambda_{i,j}\in\mathbb{R}$ so that
$B_{\cdot,i}=\lambda_{i,j}A_{\cdot,j}$. Since $A\neq 0$, and since $A$ is
skew-symmetric, $A$ contains at least two (non-zero) linearly independent
columns. Thus, we have

$\displaystyle B_{\cdot,i}\in\bigcap_{i\in\underline{n},A_{\cdot,i}\neq
0}\mathrm{span}\\{A_{\cdot,i}\\}=\\{0\\}.$

Hence, after possibly performing some simultaneous permutations of the rows
and likewise the columns of $A$ and $B$ – which are orthogonal transformations
that do not change the symmetry properties of $A$ and $B$ – we can split

$\displaystyle A=\begin{bmatrix}A^{1}\\\ &A^{2}\\\ &&\ddots\\\
&&&A^{k}\end{bmatrix},\qquad B=\begin{bmatrix}B^{1}\\\ &B^{2}\\\ &&\ddots\\\
&&&B^{k}\end{bmatrix}$

for some $k\in\underline{n}$, where the $A^{i}\in\mathbb{R}^{n_{i}\times
n_{i}}$ and $B^{i}\in\mathbb{R}^{n_{i}\times n_{i}}$ have the form

$\displaystyle A^{i}$ $\displaystyle=\begin{bmatrix}&&&&&&A^{i}_{1}\\\
&&&&&\udots\\\ &&&&A^{i}_{\ell_{i}}\\\ &&&0\\\ &&-(A^{i}_{\ell_{i}})^{\top}\\\
&\udots\\\ -(A^{i}_{1})^{\top}\end{bmatrix},$ $\displaystyle B^{i}$
$\displaystyle=\begin{bmatrix}&&&&&&\lambda_{i,1}A^{i}_{1}\\\ &&&&&\udots\\\
&&&&\lambda_{i,\ell_{i}}A^{i}_{\ell_{i}}\\\ &&&0\\\
&&-\lambda_{i,\ell_{i}}(A^{i}_{\ell_{i}})^{\top}\\\ &\udots\\\
-\lambda_{i,1}(A^{i}_{1})^{\top}\end{bmatrix}$

We show first that $\lambda_{i,j}=\lambda_{i,k}$ for all $i\in\underline{n}$
and $j,k\in\underline{\ell_{i}}$. Let $y\in\mathbb{R}^{n_{i}}$ be arbitrary.
Then we have

$\displaystyle A^{i}yy^{\top}B^{i}$
$\displaystyle=\begin{pmatrix}A^{i}_{1}y_{n}\\\ \vdots\\\
A^{i}_{\ell_{i}}y_{\ell_{i}+2}\\\ 0\\\ -A^{i}_{\ell_{i}}y_{\ell_{i}}\\\
\vdots\\\
-A^{i}_{1}y_{1}\end{pmatrix}\begin{pmatrix}\lambda_{i,j}y_{n-j+1}^{\top}(A^{i}_{1})^{\top}\end{pmatrix}_{j=1}^{n}$

which, in turn, equals the rather lengthy matrix depictured in Figure 1.

$\displaystyle\begin{bmatrix}\lambda_{i,1}A^{i}_{1}y_{n}y_{n}^{\top}(A^{i}_{1})^{\top}&\cdots&\lambda_{i,\ell_{1}}A^{i}_{1}y_{n}y_{\ell_{i}+2}^{\top}(A^{i}_{\ell_{i}})^{\top}&0&-\lambda_{i,\ell_{1}}A^{i}_{1}y_{n}y_{\ell_{i}}^{\top}(A^{i}_{\ell_{i}})^{\top}&\cdots&-\lambda_{i,1}A^{i}_{1}y_{n}y_{1}^{\top}(A^{i}_{1})^{\top}\\\
\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
\lambda_{i,1}A^{i}_{\ell_{i}}y_{\ell_{i}+2}y_{n}^{\top}(A^{i}_{1})^{\top}&\cdots&\lambda_{i,\ell_{1}}A^{i}_{\ell_{i}}y_{\ell_{i}+2}y_{\ell_{i}+2}^{\top}(A^{i}_{\ell_{i}})^{\top}&0&-\lambda_{i,\ell_{1}}A^{i}_{\ell_{i}}y_{\ell_{i}+2}y_{\ell_{i}}^{\top}(A^{i}_{\ell_{i}})^{\top}&\cdots&-\lambda_{i,1}A^{i}_{\ell_{i}}y_{\ell_{i}+2}y_{1}^{\top}(A^{i}_{1})^{\top}\\\
0&\cdots&0&0&0&\cdots&0\\\
-\lambda_{i,1}A^{i}_{\ell_{i}}y_{\ell_{i}}y_{n}^{\top}(A^{i}_{1})^{\top}&\cdots&-\lambda_{i,\ell_{1}}A^{i}_{\ell_{i}}y_{\ell_{i}}y_{\ell_{i}+2}^{\top}(A^{i}_{\ell_{i}})^{\top}&0&\lambda_{i,\ell_{1}}A^{i}_{\ell_{i}}y_{\ell_{i}}y_{\ell_{i}}^{\top}(A^{i}_{\ell_{i}})^{\top}&\cdots&\lambda_{i,1}A^{i}_{\ell_{i}}y_{\ell_{i}}y_{1}^{\top}(A^{i}_{1})^{\top}\\\
\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
-\lambda_{i,1}A^{i}_{1}y_{1}y_{n}^{\top}(A^{i}_{1})^{\top}&\cdots&-\lambda_{i,\ell_{1}}A^{i}_{1}y_{1}y_{\ell_{i}+2}^{\top}(A^{i}_{\ell_{i}})^{\top}&0&\lambda_{i,\ell_{1}}A^{i}_{1}y_{1}y_{\ell_{i}}^{\top}(A^{i}_{\ell_{i}})^{\top}&\cdots&\lambda_{i,1}A^{i}_{1}y_{1}y_{1}^{\top}(A^{i}_{1})^{\top}\end{bmatrix}.$
(4.6) Figure 1: Structure of $A^{i}yy^{\top}B^{i}$.

Since this matrix is symmetric by assumption, we conclude that indeed for all
$j,k\in\underline{\ell_{i}}$ either $\lambda_{i,j}=\lambda_{i,k}$ or
$A^{i}_{j}y_{n-j+1}y_{n-k+1}^{\top}(A^{i}_{k})^{\top}=0$ for all
$y_{n-j+1},y_{n-k+1}$; but if the latter is the case, then $A^{i}_{j}=0$ or
$A^{i}_{k}=0$ and we can w.l.o.g. assume that $\lambda_{i,j}=\lambda_{i,k}$
anyway. This shows that it remains to consider the case

$\displaystyle A=\begin{bmatrix}A^{1}\\\ &\ddots\\\ &&A^{k}\end{bmatrix},\quad
B=\begin{bmatrix}\lambda_{1}A^{1}\\\ &\ddots\\\
&&\lambda_{k}A^{k}\end{bmatrix}$

for some $k\in\mathbb{N}^{*}$ and
$\lambda_{1},\ldots,\lambda_{k}\in\mathbb{R}$. If $k=1$, then $A$ and $B$ are
linearly dependent. Consider the case $k=2$. Then we have for all
$y=(y_{1},y_{2})\in\mathbb{R}^{n}$

$\displaystyle
Ayy^{\top}B^{\top}=\begin{bmatrix}\lambda_{1}A_{1}y_{1}y_{1}^{\top}A_{1}^{\top}&\lambda_{2}A_{1}y_{1}y_{2}^{\top}A_{2}^{\top}\\\\[6.0pt]
\lambda_{1}A_{2}y_{2}y_{1}^{\top}A_{1}^{\top}&\lambda_{2}A_{2}y_{2}y_{2}^{\top}A_{2}^{\top}\end{bmatrix}$

and hence we conclude
$(\lambda_{1}-\lambda_{2})A_{2}y_{2}y_{1}^{\top}A_{1}^{\top}=0$. This yields
$A_{1}=0$ or $A_{2}=0$ or $\lambda_{1}=\lambda_{2}$. But if $A_{1}=0$ or
$A_{2}=0$, then we can choose $\lambda_{1}=\lambda_{2}$ anyways, so that we
conclude that w.l.o.g. $\lambda_{1}=\lambda_{2}$. By induction over $k$, we
conclude that indeed $\lambda_{1}=\cdots=\lambda_{k}$. This concludes the
proof of the theorem. $\square$

We have seen that the conservative-irreversible functions induced by quasi-
IPHS are all conservative-irreversible functions that allow a splitting (4.2).
Immediately, this raises the question: Are these functions
$E\in\mathfrak{CI}(\mathbb{R}^{n})$ all so that their associated four-
derivation $e$ induces the family
$(e(\cdot,s,h,\cdot))_{s,h\in\mathcal{C}^{\infty}(\mathbb{R}^{n})}$ of quasi-
Poisson brackets? Unfortunately, we have not the room to answer this question.
We can, however, characterize the functions
$E\in\mathfrak{CI}(\mathbb{R}^{n})$ with this property.

######  Proposition 4.3.

Let
$E:\left(\mathcal{C}^{\infty}(\mathbb{R}^{n})\right)^{3}\to\mathcal{C}^{\infty}(\mathbb{R}^{n})$
be a conservative-irreversible function with associated four-derivation $e$.
All biderivations $e(\cdot,s,h,\cdot)$ are quasi-Poisson if, and only if,

$\displaystyle\forall
i,j,k,\ell\in\underline{n}:e_{i,j,k,\ell}=-e_{\ell,j,k,i}.$ (4.7)

Proof. In view of Proposition 3.8, we identify $e$ with its representation
$e\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n\times n\times n\times
n})$. Then, it is evident that, for all
$s,h\in\mathcal{C}^{\infty}(\mathbb{R}^{n})$, the matrix representation of
$e(\cdot,s,h,\cdot)$ is

$\displaystyle[e(\cdot,s,h,\cdot)]=\sum_{j,k=1}^{n}e_{i,j,k,\ell}\mathrm{d}_{j}s\mathrm{d}_{k}h.$

Thus, $e(\cdot,s,h,\cdot)$ is a quasi-Poisson bracket if, and only if,

$\displaystyle\sum_{j,k=1}^{n}e_{i,j,k,\ell}\mathrm{d}_{j}s\mathrm{d}_{k}h=-\sum_{j,k=1}^{n}e_{\ell,j,k,i}\mathrm{d}_{j}s\mathrm{d}_{k}h.$

We show that latter and (4.7) are equivalent. It is clear that (4.7) is
sufficient. We show necessity. By plugging in the coordinate functions
$x_{\iota}$ and $x_{\kappa}$, we see that

$\displaystyle e_{i,\iota,\kappa,\ell}$
$\displaystyle=\sum_{j,k=1}^{n}e_{i,j,k,\ell}\mathrm{d}_{j}x_{\iota}\mathrm{d}_{k}x_{\kappa}$
$\displaystyle=-\sum_{j,k=1}^{n}e_{\ell,j,k,i}\mathrm{d}_{j}x_{\iota}\mathrm{d}_{k}x_{\kappa}$
$\displaystyle=-e_{\ell,\iota,\kappa,i}.$

This shows that (4.7) is indeed necessary. $\square$

##  5 Conclusion and outlook

We have suggested a geometric characterization of the quasi-Poisson brackets
of Irreversible port-Hamiltonian Systems in terms of covariant 4-tensors and
an associated functions, called _conservative-irreversible functions_. We have
shown that all conservative-irreversible functions that can be split into two
biderivations are induced by an quasi-Poisson bracket as defined for
Irreversible port-Hamiltonian Systems. Lastly, we have characterized the
conservative-irreversible functions that induce quasi-pseudo-Hamiltonian
systems.

Unfortunately, it remains unanswered whether the conservative-irreversible
functions are precisely the functions that allow a splitting (4.1). In future
works, we want to answer this question. Next, we have not taken account that
the function $\gamma$ defining the dissipative constitutive relations for
Irreversible port-Hamiltonian Systems, may depend explicitly on the co-energy
variable. Note that, as long as this function is positive, the entropy
creation term remain positive. Finally future work shall make use of these
conservative-irreversible functions in order to define with an input- and an
output-map compatible with irreversibel thermodynamics. Finally, having
characterized, possibly more general systems retaining both the energy and
entropy balance equations, might lead to novel desired closed-loop systems
useful for the control design.

Acknowledgements We thank our colleague Manuel Schaller (Ilmenau) for his
valuable critique.

## References

  * [AMR88] R. Abraham, J. E. Marsden, and T. Ratiu.  Manifolds, Tensor Analysis, and Applications.  Springer, 2nd edition, 1988.
  * [BE91] A. N. Beris and B. J. Edwards.  Noncanonical Poisson bracket for nonlinear elasticity with extensions to viscoelasticity.  Journal of Physics A: Mathematical and General, 24(11):2461–2480, 1991.
  * [BE94] A. N. Beris and B. J. Edwards.  Thermodynamics of Flowing Systems with Internal Microstructure.  Oxford University Press, 1994.
  * [Coh85] P. M. Cohn.  Free rings and their relations.  Academic Press, 1985.
  * [GO97a] M. Grmela and H. C. Öttinger.  Dynamics and thermodynamics of complex fluids. I. Development of a general formalism.  Physical Review E, 56:6620–6632, Dec 1997.
  * [GO97b] M. Grmela and H. C. Öttinger.  Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism.  Physical Review E, 56:6633–6655, Dec 1997.
  * [Kau84] A. N. Kaufman.  Dissipative Hamiltonian systems: A unifying principle.  Physics Letters A, 100(8):419–422, 1984.
  * [Mor86] P. J. Morrison.  A paradigm for jointed Hamiltonian and dissipative systems.  Phys. D, 18(1-3):410–419, 1986.
  * [RMS13a] H. Ramirez, B. Maschke, and D. Sbarbaro.  Irreversible port-Hamiltonian systems: A general formulation of irreversible processes with application to the cstr.  Chemical Engineering Science, 89:223–234, 2013.
  * [RMS13b] H. Ramirez, B. Maschke, and D. Sbarbaro.  Modelling and control of multi-energy systems: An irreversible port-Hamiltonian approach.  European Journal of Control, 19(6):513–520, 2013.


* Ledoit and Wolf, (2012) Ledoit, O. and Wolf, M. (2012).  Nonlinear shrinkage estimation of large-dimensional covariance matrices.  Annals of Statistics, 46(6):1024–1060.
  * Ledoit et al., (2018) Ledoit, O., Wolf, M., and Zhao, Z. (2018).  Efficient sorting: A more powerful test for cross-sectional anomalies.  Journal of Financial Econometrics.
  * Lee, (2016) Lee, J. H. (2016).  Predictive quantile regression with persistent covariates: Ivx-qr approach.  Journal of Econometrics, 192(1):105–118.
  * Li et al., (2021) Li, Z., Wang, Q., and Li, R. (2021).  Central limit theorem for linear spectral statistics of large dimensional kendalls rank correlation matrices and its applications.  The Annals of Statistics, 49(3):1569–1593.
  * Lin et al., (2022) Lin, W., Olmo, J., and Taamouti, A. (2022).  Portfolio selection under systemic risk.  Journal of Money, Credit and Banking.
  * Liu and Zhang, (2021) Liu, L. and Zhang, D. (2021).  Robust estimation of high-dimensional vector autoregressive models.  arXiv preprint arXiv:2109.10354.
  * Markowitz, (1952) Markowitz, H. (1952).  Portfolio selection.  Journal of Finance, 7:77–91.
  * Markowitz, (1956) Markowitz, H. (1956).  The optimization of a quadratic function subject to linear constraints.  Naval research logistics Quarterly, 3(1-2):111–133.
  * McGee and Olmo, (2020) McGee, R. and Olmo, J. (2020).  Optimal characteristic portfolios.  Available at SSRN.
  * Negahban et al., (2017) Negahban, S., Oh, S., and Shah, D. (2017).  Rank centrality: Ranking from pairwise comparisons.  Operations Research, 65(1):266–287.
  * Newey and Powell, (1987) Newey, W. K. and Powell, J. L. (1987).  Asymmetric least squares estimation and testing.  Econometrica: Journal of the Econometric Society, pages 819–847.
  * Olmo, (2021) Olmo, J. (2021).  Optimal portfolio allocation and asset centrality revisited.  Quantitative Finance, 21(9):1475–1490.
  * Peralta and Zareei, (2016) Peralta, G. and Zareei, A. (2016).  A network approach to portfolio selection.  Journal of Empirical Finance, 38:157–180.
  * Roverato and Whittaker, (1998) Roverato, A. and Whittaker, J. (1998).  The isserlis matrix and its application to non-decomposable graphical gaussian models.  Biometrika, 85(3):711–725.
  * Sadler, (2022) Sadler, E. (2022).  Ordinal centrality.  Journal of Political Economy, 130(4):000–000.
  * Salgueiro et al., (2005) Salgueiro, M. F., Smith, P. W., and McDonald, J. W. (2005).  Power of edge exclusion tests in graphical gaussian models.  Biometrika, 92(1):173–182.
  * Schenone, (2018) Schenone, P. (2018).  Causality: a decision theoretic foundation.  arXiv preprint arXiv:1812.07414.
  * Sharpe, (1964) Sharpe, W. F. (1964).  Capital asset prices: A theory of market equilibrium under conditions of risk.  The Journal of Finance, 19(3):425–442.
  * Van Mieghem, (2014) Van Mieghem, P. (2014).  Graph eigenvectors, fundamental weights and centrality metrics for nodes in networks.  arXiv preprint arXiv:1401.4580.
  * Vargas Jr et al., (2017) Vargas Jr, R., Waldron, A., Sharma, A., Flórez, R., and Narayan, D. A. (2017).  A graph theoretic analysis of leverage centrality.  AKCE International Journal of Graphs and Combinatorics, 14(3):295–306.
  * White et al., (2015) White, H., Kim, T.-H., and Manganelli, S. (2015).  Var for var: Measuring tail dependence using multivariate regression quantiles.  Journal of Econometrics, 187(1):169–188.
  * Yu, (1991) Yu, K.-B. (1991).  Recursive updating the eigenvalue decomposition of a covariance matrix.  IEEE Transactions on Signal Processing, 39(5):1136–1145.

SUPPLEMENTARY APPENDIX

Statistical Estimation for Covariance Structures with Tail Estimates

using Nodewise Quantile Predictive Regression Models

##### I. Discussion of Computational Aspects

In this Section, we discuss some relevant computational aspects to our
framework. In terms of the implementation of the algorithm we use the
Statistical Package R. To reduce the execution time we utilize parallel
programming techniques as well as related R packages for this purpose, such as
the Rcpp library and the Rccp Armadillo linear algebra library. For example,
we use parallel computing techniques with R using the Iridis4 High Performance
Computing Facility of the University of Southampton. An innovation of our
proposed methodology which reduces significantly computational time is the
algorithm we use for the estimation of the risk matrix. The R installation on
the Iridis4 cluster allows the use of 16 cores per submitted job. Moreover, by
submitting a number of jobs at the same time we were able to utilize in the
order of 64 cores in parallel for our computation. We define with
$\boldsymbol{\Gamma}$ the $\mathsf{VaR-\Delta CoVaR}$ risk matrix. Then, the
elements of the risk matrix are estimated using the procedure
below131313Notice that the novelty of our Algorithm 2 is the parallel
estimation of the elements of the matrix by avoiding restimating the VaR in
each iteration we estimate the corresponding CoVaR. Researchers investigating
estimation problems for high dimensional covariance and precision matrices
have proposed various methodologies for estimating $\mathbf{\Sigma}^{-1}$
column-by-column implementable with parallel computing such as the nodewise
Lasso and the constrained $\ell_{1}-$minimization for inverse matrix
estimation (CLIME) (shu2019estimationmatrix).. Notice that we correct for
possible estimation bias due to the existence of the generated regressor using
a bootstrap procedure which is implemented internally in R when estimating
each pair of $\mathsf{VaR}$ and $\mathsf{CoVaR}$’s using the R package
quantreg.

    VaR_DCoVar_forecast_function <- function( Nr_C = Nr_C, nhist = nhist,
                              returns = returns_hist, macro = macro_hist, tau = tau )
    {#begin of function

      # Initialize inputs
      Nr_C  <- Nr_C
      nhist <- nhist
      tau   <- tau

      returns <- returns_hist
      macro   <- macro_hist

      nr <- NROW(returns)
      p <- ncol(macro)

      # Step 1: Estimate the VaR-DCoVaR matrix (no change of signs in this step)
      VaR.DCoVar.forecast <- Risk_Matrix_forecast_function( Nr_C = Nr_C, nhist = nhist, returns=returns_hist, macro=macro_hist, tau = tau )

      # Step 2: Take the negative VaRs and DCoVaRs
      VaR.DCoVar.forecast.positive <- VaR.DCoVar.forecast
      for (i in 1: Nr_C )
      {
        for (j in 1: Nr_C )
        {
          VaR.DCoVar.forecast.positive[i,j] <- - ( VaR.DCoVar.forecast[i,j] )
        }
      }

      # Second I construct the non-symmetric version of the proposed Gamma risk matrix
      # Take the estimated VaRs into a vector
      VaR.forecasts <- matrix( 0, nrow = Nr_C, ncol = 1 )
      for (i in 1:Nr_C )
      {
        VaR.forecasts[i,1] <- abs( VaR.DCoVar.forecast.positive[i,i] )
      }
      VaR.DCoVar.forecast.positive.new <- diag( Nr_C )

      # Step 3: Construct our proposed risk matrix
      for (i in 1:Nr_C)
      {
        for (j in 1:Nr_C)
        {# begin for-loop
          if (j!=i)
          {# begin if
            VaR.DCoVar.forecast.positive.new[j,i] <- ( ( VaR.forecasts[j,1]*VaR.forecasts[i,1] )^0.5 ) * VaR.DCoVar.forecast.positive[j,i]
          }# end if
        }# end for-loop
      }

      # Next I replace the diagonal with the positive estimated VaRs
      for (i in 1:Nr_C )
      {
        VaR.DCoVar.forecast.positive.new[i,i] <- VaR.forecasts[i,1]
      }

      # Step 4: Take the symmetric part of the Gamma matrix
      risk.matrix <- 0.5 * ( VaR.DCoVar.forecast.positive.new + t(VaR.DCoVar.forecast.positive.new) )

      return( risk.matrix )

    }#end of function

##### II. Discussion of Asymptotic Behaviour of Eigenvalues of Large
Covariance Matrices

Related studies include Fan and Johnstone,  (2017) who consider estimating the
unknown covariance structure with linear random effect models and construct an
asymptotic test of the global sphericity null hypothesis, such that
$H_{0}:\Sigma_{r}=\sigma_{t}^{2}$, for every $r\in\left\\{1,...,k\right\\}$.

######  Lemma 3.

Assume that assumptions holds. Then, there exists $n_{0}\geq 1$ such that, for
all $n>n_{0}$, $A_{n}$ is diagonalizable in the form

$\displaystyle A_{n}=P_{n}D_{n}P_{n}^{-1},\ \ \text{with}\ \ \
D_{n}=\mathsf{diag}\big{(}\lambda_{n,1},...,\lambda_{n,p}\big{)}$ (8.16)

containing order distinct eigenvalues
$1>|\lambda_{n,1}|\geq...\geq|\lambda_{n,p}|>0$. In addition, it holds that
$\left\lVert P_{n}\right\rVert\leq\mathcal{C}$ and $\left\lVert
P_{n}^{-1}\right\rVert\leq\mathcal{C}$.

######  Theorem 5.

Assume that conditions hold and that
$\mathbb{E}\big{[}\left|\epsilon_{1}\right|^{2+\nu}\big{]}=\eta_{\nu}<+\infty$
for some $\nu>0$. Then, if the eigenvalues of $A$ are real, we have the
asymptotic normality such that

$\displaystyle\sqrt{n}V_{n}^{-1/2}P_{n}^{\top}\left(\widehat{\theta}_{n}-\theta_{n}\right)\overset{d}{\to}\mathcal{N}\big{(}0,H_{0}^{-1}\big{)}$
(8.17)

where $\widehat{\theta}_{n}$ is the OLS estimator in the nearly unstable
$AR(p)$ process, $V_{n}$ corresponds to the matrix of rates and $H_{0}$ is a
standardized positive precision matrix. Furthermore, if some eigenvalues of
$A$ are complex, we have the asymptotic normality such that

$\displaystyle\sqrt{n\kappa_{n}}\langle
L_{n},\widehat{\theta}_{n}-\theta_{n}\rangle\overset{d}{\to}\mathcal{N}\big{(}0,h_{0}^{2}\big{)},$
(8.18)

where

$\displaystyle
L_{n}=\left(1,\frac{1}{\lambda_{1}\rho_{n}},...,\frac{1}{(\lambda_{1}\rho_{n})^{p-1}}\right)^{\top}\
\ \ \text{and}\ \ \ h_{0}^{2}=\frac{2c}{\pi^{2}_{11}}>0.$ (8.19)

###### Proof.

Suppose that the eigenvalues of $A$ are real, and consider the filtration
$\mathcal{F}_{n,k}=\sigma\big{(}\Phi_{n,0},\varepsilon_{1},...,\varepsilon_{k}\big{)}$,
$\forall n\geq 1,\forall 1\leq k\leq n$.

For all $a\in\mathbb{R}^{p}\backslash\left\\{0\right\\}$, let also

$\displaystyle
m_{n,k}^{(a)}=a^{\top}V_{n}^{1/2}P_{n}^{-1}\Phi_{n,k-1}\varepsilon_{k}\ \ \
\text{and}\ \ \ M_{n}^{(a)}=\sum_{k=1}^{n}m_{n,k}^{(a)}$ (8.20)

Notice that the sequence $\left(m_{n,k}^{(a)}\right)$ is a scalar martingale
difference array with respect to $\mathcal{F}_{n,k}$ at fixed $n$ and for
$1\leq k\leq n$. Moreover, the predictable quadratic variation of
$M_{n}^{(a)}$ is given by

$\displaystyle\langle M^{(a)}\rangle_{n}$
$\displaystyle=\sum_{k=1}^{n}\mathbb{E}\left[\left(m_{n,k}^{(a)}\right)^{2}\big{|}\mathcal{F}_{n,k-1}\right]$
$\displaystyle=\sigma^{2}a^{\top}V_{n}^{1/2}P_{n}^{-1}\left[\sum_{k=1}^{n}\Phi_{n,k-1}\Phi_{n,k-1}^{\top}\right]\left(P_{n}^{-1}\right)^{\top}V_{n}^{1/2}a$

since $\left(\varepsilon_{k}\right)$ is a white noise and $\Phi_{n,k-1}$ is
$\mathcal{F}_{n,k-1}-$measurable, which results to the following convergence

$\displaystyle\frac{1}{n}\langle
M^{(a)}\rangle_{n}\overset{p}{\to}\sigma^{2}a^{\top}H\alpha>0.$ (8.21)

where $H$ is the covariance matrix which is positive definite.

To apply the central limit theorem for arrays of martingales, it remains to
show that the Linderberg’s condition is satisfied such that

$\displaystyle\forall\ \varepsilon>0,\
\frac{1}{n}\sum_{k=1}^{n}\mathbb{E}\left[\left(m_{n,k}^{(a)}\right)^{2}\boldsymbol{1}\left\\{\left|m_{n,k}^{(a)}\right|>\epsilon\sqrt{n}\right\\}\big{|}\mathcal{F}_{n,k-1}\right]\overset{p}{\to}0.$
(8.22)

In order to prove the above result we first see that for any $1\leq k\leq n$,

$\displaystyle\mathbb{E}\left[\left(m_{n,k}^{(a)}\right)^{2}\boldsymbol{1}\left\\{\left|m_{n,k}^{(a)}\right|>\epsilon\sqrt{n}\right\\}\big{|}\mathcal{F}_{n,k-1}\right]$
$\displaystyle=T_{n,k-1}^{(a)}\xi_{n,k}$ $\displaystyle\leq
T_{n,k-1}^{(a)}\underset{1\leq k\leq n}{\mathsf{sup}}\xi_{n,k}$

where
$T_{n,k-1}^{(a)}\overset{\Delta}{=}a^{\top}V_{n}^{1/2}P_{n}^{-1}\Phi_{n,k-1}\Phi_{n,k-1}^{\top}\left(P_{n}^{-1}\right)^{\top}V_{n}^{1/2}a>0$
and using the Holder’s and Markov’s inequalities we get that

$\displaystyle\xi_{n,k}$
$\displaystyle\overset{\Delta}{=}\mathbb{E}\left[\varepsilon_{k}^{2}\boldsymbol{1}\left\\{\left|m_{n,k}^{(a)}\right|>\epsilon\sqrt{n}\right\\}\big{|}\mathcal{F}_{n,k-1}\right]$
$\displaystyle\leq\eta_{\nu}^{\frac{2}{(2+\nu)}}\mathbb{P}\left(\left(m_{n,k}^{(a)}\right)^{2}>\epsilon^{2}n\big{|}\mathcal{F}_{n,k-1}\right)^{\frac{\nu}{(2+\nu)}}\leq\mathcal{C}\left(\frac{T_{n,k-1}^{(a)}}{n}\right)^{\frac{2}{(2+\nu)}}$

Therefore, it is sufficient to establish that

$\displaystyle\forall\ a\in\mathbb{R}^{p}\backslash\left\\{0\right\\},\ \ \
\frac{M_{n}^{(a)}}{\sqrt{n}}\overset{d}{\to}\mathcal{N}\big{(}0,\sigma^{2}a^{\top}Ha\big{)}.$
(8.23)

Since $a$ is arbitrary, we can use the Cramer-Wold device to get the
convergence of the $p-$dimensional vector such that

$\displaystyle\frac{M_{n}^{(a)}}{\sqrt{n}}\overset{d}{\to}\mathcal{N}\big{(}0,\sigma^{2}H\big{)}.$
(8.24)

with

$\displaystyle
M_{n}=V_{n}^{1/2}P_{n}^{-1}\sum_{k=1}^{n}\Phi_{n,k-1}\varepsilon_{k}.$ (8.25)

∎

######  Remark 11.

Notice that the above results are given in the excellent study of Badreau and
Proïa,  (2023) and are particularly useful for obtaining relevant asymptotic
theory for the eigenvalues of the proposed risk matrix in this article.
Additional useful properties to be considered in this direction include the
separation property of tail or extreme events. In particular the separation
property shows that extremes induced from different factors belong to
independent blocks. Further aspects relevant to the separation property are
more formally in the study of Krizmanić and Katsouris (2023+): "Weak
Convergence of Self-Normalized Partial Sum Processes in the $M_{1}$ topology".