Generalized Hamiltonian for a two-mode fermionic model and asymptotic equilibria

In some recent papers, the so called \((H,\rho)\)-induced dynamics of a system \(\mathcal{S}\) whose time evolution is deduced adopting an operatorial approach, has been introduced. According to the formal mathematical apparatus of quantum mechanics, \(H\) denotes the Hamiltonian for \(\mathcal{S}\), while \(\rho\) is a certain rule applied periodically on \(\mathcal{S}\). In this approach the rule acts at specific times \(k\tau\), with \(k\) integer and \(\tau\) fixed, by modifying some of the parameters entering \(H\) according to the state variation of the system. As a result, a dynamics admitting an asymptotic equilibrium state can be obtained. Here, we consider the limit for \(\tau\rightarrow 0\), so that we introduce a generalized model leading to asymptotic equilibria. Moreover, in the case of a two-mode fermionic model, we are able to derive a relation linking the parameters involved in the Hamiltonian to the asymptotic equilibrium states.