Higher – Order Relaxation Magnetic Phenomena and a Hierarchy of First-Order Relaxation Variables

In previous papers one of us (LR) has studied thermodynamic theories for magnetic relaxation phenomena due to several internal variables. In particular, she has obtained equations involving time derivatives of the magnetic field B up to the n-th order, and time derivatives of the magnetization M up to (n + 1)-th order. Such kind of equations generalize other kinds of physical phenomena, such as stresses τ as a function of small strains ε, and electrical polarizations P as a function of the electric field E. The aim of the present work is to provide a particular illustration of the theory, relating the mentioned n-th order relaxation equation to a hierarchy of first-order relaxation equations. Though this is only a simple situation, it may be helpful to relate the general equation to the microscopic structure of the system. Furthermore, we study in detail the form of the entropy and its consequences on the hierarchy of relaxation equations.