The fluctuations of thermodynamic systems in the presence of the fields are considered. The approach is of phenomenological nature and developed in a Gaussian approximation. The cases of a magnetizable continuum in a magnetoquasistatic field, as well as the so called discrete systems are used to exemplify the study. In the latter case one finds that the fluctuation estimators depend both on the intrinsic properties of the system and on the characteristics of the environment. Following earlier ideas of one of the authors we present a class of thermodynamic inequalities for the systems investigated in this paper. In the case of two variables these inequalities are nonquantum analogs of the well known quantum Heisenberg “uncertainty” relations. In this context, the fluctuation estimators support the idea that Boltzmann’s constant k has the signification of a generic indicator of stochasticity for thermodynamic systems.

A quaternion formulation is applied to the intrinsic transfer matrix for longitudinal elastic wave propagation through a multilayer medium in order to find the spectral response of a sonic crystal. Resonance conditions and the band structure of the crystal are obtained. The presence of a defect is also analysed. The analysis is carried out theoretically and through simulations. A coupled oscillators model is used to validate the obtained results from a phenomenological point of view. Experimental measurements are carried out for some periodic multilayer arrangements and they are correlated with theory. The obtained spectral response and band structure are essential in characterising the sonic crystal and also in optimising its structure in order to obtain specific passbands and stopbands. The adaptedness of the quaternion formulation to periodic structures and to the inclusion of defects is considered.

The higher-order moments of the fluctuations for thermodynamic systems in the presence of fields are investigated in the framework of a theoretical method. The method uses a generalized statistical ensemble consistent with an adequate expression for the internal energy. The applications refer to the case of a system in a magnetoquasistatic field. In the case of linear magnetic media, one finds that, for the description of the magnetic induction fluctuations, the Gaussian approximation is satisfactory. For nonlinear media, the corresponding fluctuations are non-Gaussian, having a non-null asymmetry. Furthermore, the respective fluctuations have characteristics of leptokurtic, mesokurtic and platykurtic type, depending on the value of the magnetic field strength as compared with a scaling factor of the magnetization curve.