# Metriplectic Structure of a Radiation-Matter-Interaction Toy Model
## 2 General Metriplectic Formalism  
Before describing how the metriplectic formalism is applied to the TPA, it is useful to sketch briefly the construction of a metriplectic system. Typically, one starts from a Hamiltonian system described by a set of variables \(X\), the dynamics of which are generated by some Hamiltonian \(H_{0}(X)\) and some Poisson bracket \(\{\cdot,\cdot\}\) so that \(\dot{X}=\{X,H_{0}(X)\}\). Then, some quantity \(S\) is defined, with the property of being _in involution_ with _any_ possible function of \(X\), \(\{S,A\}=0\ \forall\ A(X)\), i.e., to be _a Casimir of_\(\{\cdot,\cdot\}\). This quantity \(S\) may either depend on the original variables \(X\) only (as for the kinetic theories or for the rigid body mentioned before), or on some \"environmental\" variable \(Y\) too (as in the case of a particle motion with friction, or those of non-ideal hydrodynamics or magneto-hydrodynamics; this will be the case here too). The Casimir \(S\) becomes the generator of a new non-Hamiltonian component of the motion through the introduction of a new bracket, \((\cdot,\cdot)\), with properties of symmetry and semi-definiteness [3]. The extended system, based on the old Hamiltonian one, now has new dynamics in which the variables \(X\) evolve according to  
\[\dot{X}=\{X,H(X,Y)\}+\chi(X,S(X,Y)), \tag{1}\]  
while the motion of the environmental variables, if any, is typically influenced by \(S\) and the metric bracket only:  
\[\dot{Y}=\chi(Y,S(X,Y)). \tag{2}\]  
Metriplectic systems describe the evolution of dissipative dynamics to _asymptotically stable equilibria_, so in general, with \(Z=(X,Y)\), there will exist some \(Z_{\text{eq}}\) to which \(Z(t)\) converges for \(t\to+\infty\). The state \(Z_{\text{eq}}\) is the equilibrium towards which the system \"thermalizes\". In Equation (1), the Hamiltonian \(H(X,Y)\) may be different from the original \(H_{0}(X)\), as it may include a term depending on \(Y\) in order to close the system energetically, and may take into account of the irreversible consumption of \(H_{0}(X)\) (dissipation); the difference \(U=H-H_{0}\) is the _internal energy_ of the environment. In Equations (1) and (2), the factor \(\chi\) is a coefficient representing a coupling condition between \(X\) and \(Y\), and characterizing the asymptotically stable equilibrium (this \(\chi\) is related to the thermodynamic temperature, when dissipation is due to some thermal bath); the strength of the dissipative interaction, defining the non-dissipative (Hamiltonian) regime in some suitable limit of its, is some \(\alpha\) included in the definition of \((\cdot,\cdot)\), so that \(\alpha\to 0\) turns off dissipation. The mathematical expressions (1), (2) of \(\dot{X}\) and \(\dot{Y}\) depend on this \(\alpha\), so one may well say:  
\[\lim_{\alpha\to 0}\dot{X}(\alpha)=\{X,H_{0}(X,Y)\},\ \ \lim_{\alpha\to 0}\dot{Y}(\alpha)=0. \tag{3}\]From Equations (1) and (2) it appears that the limit for \(\chi\to 0\) also gives the ODEs in Equation (3); however, this does not switch off dissipation, but simply describes a condition in which it is uneffective as in the condition of zero absolute temperature, see Sections 4 and 6. As far as the bracket \((\cdot,\cdot)\) and the Casimir \(S\) are concerned, the semi-definiteness of the first one, \((A,B)\leq 0\ \forall\ A,B\), and a suitable choice of the sign of \(\chi\), i.e., \(\chi<0\), implies that \(S\) will grow monotonically along the system motion \(\hat{S}\geq 0\), until some asymptotically stable equilibrium \(Z_{\rm eq}=(X_{\rm eq},Y_{\rm eq})\) is reached, so that \(\hat{S}(Z_{\rm eq})=0\)[3]. In other words, the Casimir \(S\) turns out to be _a Lyapunov function around_\(Z_{\rm eq}\), and it can be understood as a form of _entropy_[12] (of course, all the reasoning just presented is rephrased \"without \(Y\)\" for those metriplectic systems in which no \"environment\" needs to be defined, as in the case of kinetic theories).