Debye-Langevin equation

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

In Eq. (4), the atomic polarizability is generally so small comparing to the electronic polarizability. That it can be neglected [62], that is, ao = ae. Therefore, the total polarizability, a, of the molecule may be ascribed by the sum of two contributions of polarizability (so-called Debye-Langevin equation), such as... 

Debye in 1912 extended Langevin s procedure to dielectrics, by attributing to the molecules a permanent (natural) electric dipole moment defined as in equation (34), The classical permanent magnetic dipole moment of a molecule is defined as follows (as an axial vector) ... 

In fact, generalized Langevin equations (Section 8.2.6) need to be used in such applications to account for the fact that the system is usually faster than the bath. Indeed, the spectral density, Eq. (8.65) should reflect the spectral characteristics of the bath, including its Debye frequency. 

Classic Brownian motion has been widely applied in the past to the interpretation of experiments sensitive to rotational dynamics. ESR and NMR measurements of T and Tj for small paramagnetic probes have been interpreted on the basis of a simple Debye model, in which the rotating solute is considered a rigid Brownian rotator, sueh that the time scale of the rotational motion is much slower than that of the angular momentum relaxation and of any other degree of freedom in the liquid system. It is usually accepted that a fairly accurate description of the molecular dynamics is given by a Smoluchowski equation (or the equivalent Langevin equation), that can be solved analytically in the absence of external mean potentials. 

Equation (5.4) is the Langevin equation of the problem. This equation is now analogous to the equation of motion of a polar molecule under the influence of an electric field [52]. The quantity that directly corresponds to the dipole moment of the polar molecule p. in the Debye theory is the magnetic moment vM of an individual ferrofluid particle, not the magnetization M which is the magnetic moment per unit volume. Equation... 

If we consider only the zero frequency and a particle j with a permanent dipole moment is reduced to the Debye-Langevin equation ... 

Tsekov and Ruckenstein considered the dynamics of a mechanical subsystem interacting with crystalline and amorphous solids [39, 40]. Newton s equations of motion were transformed into a set of generalized Langevin equations governing the stochastic evolution of the atomic co-ordinates of the subsystem. They found an explicit expression for the memory function accounting for both the static subsystem—solid interaction and the dynamics of the thermal vibrations of the solid atoms. In the particular case of a subsystem consisting of a single particle, an expression for the fiiction tensor was derived in terms of the static interaction potential and Debye cut-off fi equency of the solid. 

Experimental determination of microscopic characteristics is based on the Debye-Langevin (eq. (4.2.28)) and Lorentz-Lorentz (eq. (4.2.30)) equations. They connect macroscopic molecular characteristics, measured directly in physical experiment, with microscopic ones. Measuring dielectric permeability s in an electrostatic field, the molar polarization 11 can be found ... 

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