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

TL;DR

This paper introduces a generalized Hamiltonian framework for a two-mode fermionic system, analyzing its asymptotic equilibrium states as the periodic rule's interval approaches zero, linking parameters to long-term behavior.

Contribution

It develops a generalized model for two-mode fermionic systems with a novel relation connecting Hamiltonian parameters to asymptotic equilibria.

Findings

Derived a relation between Hamiltonian parameters and equilibrium states.

Analyzed the limit as the rule interval approaches zero.

Established conditions for asymptotic equilibrium existence.

Abstract

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.