Coupled physical and magnetodynamic rotational diffusion of a single-domain ferromagnetic nanoparticle suspended in a liquid.

A concise operator form of the Fokker-Planck equation agreeing with that proposed by Weizenecker [Phys. Med. Biol. 63, 035004 (2018)10.1088/1361-6560/aaa186] for the joint orientational distribution of the coupled physical and magnetodynamic rotational diffusion of a single-domain ferromagnetic nanoparticle suspended in a liquid is written from the postulated Langevin equations for the stochastic dynamics. Series expansion of its solution in a complete set yields, using the theory of angular momentum, differential-recurrence equations for statistical moments for coupled motion with uniaxial symmetry of the internal anisotropy-Zeeman energy of a nanoparticle. The numerical results via the matrix iteration method suggest that the susceptibility is adequately approximated by a single Lorentzian with peak frequency given by the inverse integral relaxation time and are discussed in relation to those of the well-known "egg model".

Anomalous diffusion of molecules with rotating polar groups: The joint role played by inertia and dipole coupling in microwave and far-infrared absorption.

Budó's generalization [A. Budó, J. Chem. Phys. 17, 686 (1949)10.1063/1.1747370] of the Debye rotational diffusion model of dielectric relaxation of polar molecules to an assembly with internal interacting polar groups is extended to inertial anomalous diffusion. Thus, the theory can be applied both in the GHz and the THz regions, accounting for anomalous behavior as well as the necessary return to optical transparency at very high frequencies. The linking of both dispersion regions in a single model including anomalous effects is accomplished via a fractional Fokker-Planck equation in phase space based on the continuous time random walk ansatz. The latter is written via the Langevin equations for the stochastic dynamics of pairs of interacting heavy polar groups embedded in the frame of reference of a particular molecule or molecular dimer rotating about a space-fixed axis. The fractional Fokker-Planck equation is then converted to a three-term matrix differential recurrence equation for the statistical moments. This is solved in the frequency domain for the linear dielectric response using matrix continued fractions. Thus, one has the complex susceptibility χ(ω) for extensive ranges of damping, group dipole moment ratio, and friction. The susceptibility, as inferred from the small oscillation limit, inherently comprises a low frequency (GHz) band with width depending on the anomalous parameter and a far-infrared (THz) or Poley peak of resonant character with a comblike structure of harmonic peaks. This behavior is due to the double transcendental nature of the after-effect function.

Generalization to anomalous diffusion of Budó's treatment of polar molecules containing interacting rotating groups.

A fractional Smoluchowski equation for the orientational distribution of dipoles incorporating interactions with continuous time random walk Ansatz for the collision term is obtained. This equation is written via the non-inertial Langevin equations for the evolution of the Eulerian angles and their associated Smoluchowski equation. These equations govern the normal rotational diffusion of an assembly of non-interacting dipolar molecules with similar internal interacting polar groups hindering their rotation owing to their mutual potential energy. The resulting fractional Smoluchowski equation is then solved in the frequency domain using scalar continued fractions yielding the linear dielectric response as a function of the fractional parameter for extensive ranges of the interaction parameter and friction ratios. The complex susceptibility comprises a multimode Cole-Cole-like low frequency band with width dependent on the fractional parameter and is analogous to the discrete set of Debye mechanisms of the normal diffusion. The results, in general, comprise an extension of Budó's treatment [A. Budó, J. Chem. Phys. 17, 686 (1949)] of the dynamics of complex molecules with internal hindered rotation to anomalous diffusion.

Anomalous diffusion of a dipole interacting with its surroundings.

A fractional Fokker-Planck equation based on the continuous time random walk Ansatz is written via the Langevin equations for the dynamics of a dipole interacting with its surroundings, as represented by a cage of dipolar molecules. This equation is solved in the frequency domain using matrix continued fractions, thus yielding the linear dielectric response for extensive ranges of damping, dipole moment ratio, and cage-dipole inertia ratio, and hence the complex susceptibility. The latter comprises a low frequency band with width depending on the anomalous parameter and a far infrared (THz) band with a comb-like structure of peaks. Several physical consequences of the model relevant to anomalous diffusion in the presence of interactions are discussed. The entire calculation may be regarded as an extension of the cage model interpretation of the dynamics of polar molecules to anomalous diffusion, taking into account inertial effects.

Cage model of polar fluids: Finite cage inertia generalization.

The itinerant oscillator model describing rotation of a dipole about a fixed axis inside a cage formed by its surrounding polar molecules is revisited in the context of modeling the dielectric relaxation of a polar fluid via the Langevin equation. The dynamical properties of the model are studied by averaging the Langevin equations describing the complex orientational dynamics of two bodies (molecule-cage) over their realizations in phase space so that the problem reduces to solving a system of three index linear differential-recurrence relations for the statistical moments. These are then solved in the frequency domain using matrix continued fractions. The linear dielectric response is then evaluated for extensive ranges of damping, dipole moment ratio, and cage-dipole inertia ratio and along with the usual inertia corrected microwave Debye absorption gives rise to significant far-infrared absorption with a comb-like structure of harmonic peaks. The model may be also regarded as an extension of Budó's [J. Chem. Phys. 17, 686 (1949)] treatment of molecules containing rotating polar groups to include inertial effects.