# Metriplectic Structure of a Radiation-Matter-Interaction Toy Model
## 2 General Metriplectic Formalism
In order to complete the picture, the property \((H,A)=0\ \forall\ A\) is requested for the metric bracket and the total Hamiltonian \(H\), in order for dissipation not to \"delete\" the total energy, but just transform it. Regarding metriplectic systems, one further thing must be highlighted. Given the expression of the Poisson bracket \((\cdot,\cdot)\), many Casimir functions \(S\) may exist; choosing different forms of \(S\), the dynamics in Equations (1) and (2) will converge to different specific equilibria \(Z_{\rm eq}\). It must be underlined that this construction does not include \"all\" the dynamical systems referred to as \"metriplectic\" in literature: this is the construction of a _complete metriplectic system_ (CMS), while incomplete metriplectic systems (IMS) may be defined too, with the two brackets but the Hamiltonian as the only generator. IMS can be suitable tools to describe energetically open systems [4]. The development presented here makes the TPA process tractable in a very transparent way as a CMS, and points towards the systematic algebrization of non-linear optics. The system we introduce here in order to turn the TPA process into a CMS has three degrees of freedom: two real variables describe the electromagnetic radiation, one real variable describes the matter. Radiation is represented either via a complex phasor \(\psi\), or a couple of real variables, that in turn can be either the couple \((q,p)\) of the real and imaginary part of \(\psi\), or the couple \((\phi,\rho)\) of its phase and (square root) amplitude; the population of the excited level is given by some real positive variable \(n\). As mentioned before, the electromagnetic variable \(\psi\), or \((q,p)\) or \((\phi,\rho)\), are conceived as \"mechanical variables\", representing the \"exact\" state of the electromagnetic wave, while the excited population \(n\) is more a thermodynamic variable. During the _irreversible process_, the electromagnetic energy \(H_{0}(\psi)\) is converted into some energy \(U(n)\) associated with \(n\
eq 0\). In our \"metriplecticization\" scheme, one starts from the equations of motion of the state \(Z=(q,p,n)\) and observes that a suitable limit of them reduces the system to a Hamiltonian one. In this Hamiltonian limit, a Poisson bracket is defined, so that \(q\) and \(p\) are canonically conjugated \(\{q,p\}=1\), while \(n\) remains apparently outside the play as \(\{n,q\}=\{n,p\}=0\). As the population of the excited level is _in involution_ with \(q\) and \(p\), _any_ function \(S(n)\) will be a Casimir for \(\{\cdot,\cdot\}\). The program then is to find a suitable function \(H_{0}(q,p)\) that may play the role of Hamiltonian in the Hamiltonian limit. This represents the radiation energy, to be extended as \(H(q,p,n)=H_{0}(q,p)+U(n)\) to include the energy pertaining to the filling of the excited state, namely the internal energy of the environment \"atoms\". In order to complete the metriplectic framework, suitable forms for \(S(n)\) and for the metric bracket \((\cdot,\cdot)\) must be constructed, and this is essentially what is achieved in the present work. As anticipated, field variables \((q,p)\) may be replaced by some \((\phi,\rho)\) in which \(\dot{\rho}=0\) in the Hamiltonian limit and \(\phi\) evolves linearly with time. Then, the whole CMS can be re-expressed in the new variables \((\phi,\rho,n)\).