Research Activities


            My scientific activity has a major theme the study of nonlinear and linear responses in dielectric and magnetic relaxation in gases liquids and solids and related physical phenomena.

Key words giving an overview of my contributions to research in this area are the following: Brownian motion (classical, quantum, fractional), Langevin and Fokker-Planck equations; random processes; Wigner representation of quantum mechanics, quantum master equations in phase space, dielectric relaxation and absorption in dielectric liquids, gases and liquid crystals; magnetic relaxation of ferrofluids, nanoparticles and spin systems; molecular spectroscopy; Josephson effect; Kerr effect; etc.




(A). Brownian motion (classical, quantum, fractional).

(B). Relaxation processes in nonlinear stochastic physical systems.

(C) Models of molecular reorientation in fluids. Application to dielectric relaxation and Kerr effect.

(D). Dielectric relaxation in nematic and ferroelectric liquid crystals.

(E). Magnetic relaxation of fine particles and spins.

(F) Molecular spectroscopy of gases.



(I) Applications of the Langevin and Fokker-Planck equation method to the study of

relaxation processes in various nonlinear physical systems.

1. In collaboration with others cited in the text I have developed an effective method for the calculation of nonlinear responses for various stochastic systems in the presence of strong external fields based on the direct averaging of nonlinear stochastic Langevin equations for the Brownian motion of particles over their realizations without recourse to the Fokker-Planck equation. The most important feature of the Langevin equation approach is that it allows us to calculate nonlinear responses in the same manner as linear responses. The reason is that the recurrence equations for the statistical averages as obtained from the Langevin equations have the same mathematical form both for linear and nonlinear responses.  In order to solve such recurrence equations for the statistical averages, we developed a powerful approach based on scalar and/or matrix continued fractions. This approach, first based on the Fokker-Planck equation by Risken [Fokker-Planck Equation, Springer, Berlin, on 1984], was extended using the Langevin equation by Coffey, Kalmykov and Waldron [The Langevin Equation, Word Scientific, Singapore, 2004]. The main feature of this method is that it provides us with exact solutions in the form of special functions as well as allowing us to solve divergent recurrence equations. The solution of recurrence equations in a computer is very simple since it requires at most 3 lines of computer algorithm and about fifteen lines of program. The continued fraction method has also allowed us to solve a variety of nonlinear Langevin equations and the corresponding Fokker-Planck equations. Finally, this approach can be applied to the solution of other equations of mathematical physics which can be reduced to the solution of recurrence equations such as the Schrödinger equation, the Master equation, etc. .

Various applications of the approach developed :

ª  Dielectric Relaxation in liquids and liquid crystals ;

ª  Kerr effect in liquids ;

ª  Supraparamagnetism;

ª  Josephson effect (linear and nonlinear impedances and current-voltage characteristics of Josephson junctions ;

ª    Ring laser gyroscopes;

ª    Brownian motion;

ª    Wave propagation in a random media ;

ª    Fractional diffusion;

ª    Quantum Brownian motion of particles  and spins  (phase space approach).


2. We discovered a depletion effect in a biased bistable potential, where the population in the shallower of the two potential wells may be substantially decreased by the application of the uniform bias force. This has a profound effect on the relaxation time because at a certain critical value of the bias force (which is much less than that required to destroy the bistable character of the potential) a switchover of the overall relaxation time from Arrhenius to non-Arrhenius behavior will take place. This is of importance in the stabilization of the magnetization in fine ferromagnetic particles, hence in the stabilization of information in magnetic recording systems.

3. We derived a general exact analytic equation for the nonlinear transient response relaxation time of a system whose dynamics is governed by a single-variable Fokker-Planck equation .

From Yu. P. Kalmykov et al., “Transient Nonlinear Dielectric Relaxation and Dynamic Kerr Effect from
Sudden Changes of a Strong dc Electric Field: Polar and Polarizable Molecules”,
Phys. Rev. E 60, 1475 (1999).


(II). Application of phase space methods to quantum Brownian motion of particles and spins

1. We have proposed an effective approach to the derivation and solution of the master equation for the Wigner quasiprobability distribution function W(x,p,t) in phase space (x,p) for the quantum Brownian motion of a particle in an anharmonic potential V(x,t). We have shown how Brinkman’s representation of the Fokker-Planck equation (via expansion of the momentum distribution in orthogonal polynomials) as a partial differential recurrence relation in configuration space and its associated solution methods could be also extended to the quantum regime. Furthermore, a heuristic method of determination of diffusion coefficients in the master equation is proposed. In addition, in the high damping (or noninertial) limit, using Brinkman’s method of derivation of the Smoluchowski equation governing the behaviour of the configuration space distribution function, we have derived a quantum Smoluchowski equation for the configuration space distribution function without using path integral methods. This approach has been successfully applied to various model anharmonic potentials (periodic, tilted cosine, double well, etc).

2. The Fourier series representation of the equilibrium quasiprobability density function or Wigner function of spin “orientations” for arbitrary spin Hamiltonians in a representation (phase) space of the polar angle (analogous to the Wigner function for translational motion) arising from the generalized coherent state representation of the density operator is evaluated explicitly for various nonaxially symmetric problems including a uniaxial paramagnet in a transverse external field, a biaxial, and a cubic system. By generalizing transition state theory (TST) to spins [i.e., calculating the escape rate using the equilibrium density function only] we have evaluated the reversal time of the magnetization for such systems. The quantum corrections to the TST escape rate equation for classical magnetic dipoles appear both in the prefactor and in the exponential part of the escape rate and exhibit a marked dependence on the spin number. Furthermore, the phase-space representation allows us to estimate the switching field curves and/or surfaces for spin systems because quantum effects in these fields can be estimated via Thiaville’s geometrical method [Phys. Rev. B 61, 12221 (2000)] for the study of the magnetization reversal of single-domain ferromagnetic particles.

3. Using a master equation for the quasiprobability distribution function of spin orientations in the configuration space of polar and azimuthal angles (analogous to the Wigner phase space distribution for translational motion) we have treated nonlinear longitudinal relaxation of spins in a uniform external dc magnetic field. We have demonstrated how the solution of the corresponding classical problem of the rotational Brownian motion of a magnetic moment in an external magnetic field can be carried over to the quantum regime yielding in closed form the dependence of the longitudinal spin relaxation on the spin size S as well as an expression for the integral relaxation time which in linear response reduces to that previously given by D. A. Garanin [Phys. Rev. E 55, 2569 (1997)] using the density matrix. The nonlinear relaxation is dominated by a single exponential having as time constant the integral relaxation time. Thus it is shown that a simple description in terms of a Bloch equation holds even for the nonlinear response of a giant spin.


(III) Development of models of molecular reorientation in fluids:

Application to the theory of dielectric relaxation and Kerr effect

1. We have studied the spectral properties of a system of polar molecules and analyzed the branches of the longitudinal and transverse excitations of the polarization. We have proved that the excitations of polarization of the system in the electrostatic field are stable .

2. Using the noninertial rotational Brownian motion model originally conceived of by Debye, we have obtained analytical expressions for the nonlinear response functions and relaxation times representing nonlinear dielectric relaxation and Kerr effect in systems of dipolar and polarisable molecules.

From Yu. P. Kalmykov, “Rotational Brownian motion and nonlinear dielectric relaxation of
asymmetric top molecules in strong electric fields “,
Phys. Rev. E. 2002, v. 65, 021101-11.


3. In the context of the extended rotational diffusion of linear molecules in a rectangular potential (confined rotator model), we have calculated the complex dielectric permeability and depolarized Rayleigh scattering spectra of various molecular liquids. A comparison with experimental data has shown that the model correctly describes the molecular absorption and scattering spectra in these liquids. We have also shown that the simultaneous description of experimental broad-band (0 – THz) dielectric relaxation and depolarized Rayleigh scattering data allows us to estimate values of the model parameters and, thus, gives the possibility of predicting the molecular absorption in liquids in a very broad  region of frequencies . This model has been verified and used by J. Janik et al., Mol. Cryst. Liq. Cryst. 98, 67, (1983) ; A. Kocot et al., Mol. Phys. 53, 67, (1984) ; T. Grochulski et al., Mol. Phys. 58, 67, (1986) ; M. Godlewska et al., Liquid Crystals. 1, 529, (1986) ; et  J. K. Vij et al., Mol. Phys. 72, 353, (1991) to study molecular motion in molecular liquids and liquid crystals.

4. We have calculated the complex dielectric susceptibility and depolarized Raman scattering spectra of molecular liquids composed of asymmetric top molecules by using extended rotational diffusion models.

(IV) Dielectric relaxation in nematic and ferroelectric liquid crystals

1. In the context of the model of the noninertial rotational Brownian motion of a polar molecule in a mean field potential, we have developed a theory of dielectric relaxation in nematic liquid crystals. We have derived simple analytical expressions for the longitudinal and transverse components of the dielectric permittivity tensor and for the relaxation times. The predictions of the theory have been verified experimentally by Urban et al. [Liq. Cryst. 25, 253 (1998); Z. Naturforsch. 53a, 134 (1998); Z. Naturforsch. 53a, 883 (1998)], M. Bates [Liq. Cryst. 32, 1365 (2005)], Merkel et al. [Phys. Rev. 73, 051702 (2006)], and others. We have also calculated the complex dielectric permittivity of a nematic liquid crystal with the aid of the extended rotational diffusion model of a linear molecule in a uniaxial (Maier-Saupe) potential and showed that this model allows one to calculate the spectra of the dielectric parameters of nematic liquid crystals in the 0-THz frequancy range .


Merkel et al., “Orientational order and dynamics of the dendritic liquid crystal organo-siloxane tetrapodes
determined using dielectric spectroscopy”,
Phys. Rev. E 73, 051702 (2006).

2. We have proposed a method of the calculation of dielectric parameters of ferroelectric liquid crystals in the Smectic A and Smectic C* phases (in bulk and SSFLC geometry).

From Yu. P. Panarin, Yu. P. Kalmykov, S. T. MacLughadha, H. Xu, and J. K. Vij ,
“Dielectric Response of Surface Stabilized Ferroelectric Liquid Crystal (SSFLC) Cells”,
Phys. Rev. E 50, 4763 (1994).

From Yu. P. Kalmykov, J. K. Vij , H. Xu, A. Rappaport, and M. D. Wand,
“The Dielectric Study of the Electroclinic Effect in the Smectic-A Phase”,
Phys. Rev. E 50, 2109 (1994).

(V) Superparamagnetism : Magnetization relaxation of single-domain particles

1. We have developed an analytic method of solution of Gilbert’s equation for the magnetization augmented by a random field which allows one to deduce (by direct averaging of that equation) over its realisations a system of recurrence equations for the relaxation functions characterizing the longitudinal and transverse magnetic relaxation of systems of superparamagnetic particles with various anisotropies (uniaxial, biaxial, cubic, etc.) subjected to a dc magnetic field H0. We have also proposed a method of solution of these recurrence equations using matrix continued fractions. This approach has allowed us to evaluate the longitudinal and transverse components of the complex magnetic susceptibility tensor as well as the magnetization relaxation times for various systems .

2. We have proposed a new method of measurements of the dissipation parameter of single-domain superparamagnetic particles .

From W. T. Coffey, D. S. F. Crothers, J. L. Dormann, Yu. P. Kalmykov, E. C. Kennedy, and W. Wernsdorfer
“Thermally Activated Relaxation Time of a Single Domain Ferromagnetic Particle Subjected to a Uniform Field at an Oblique Angle to the Easy Axis:
Comparison with Experimental Observations“,
Phys. Rev. Lett. 80, 5655 (1998).

3. We have solved the problem of the magnetization reversal in ferromagnetic nanoparticles in the presence of a constant magnetic field of arbitrary amplitude, oriented at an arbitrary angle with respect to the easy axis of the particle and have been able to calculate the relaxation time of the magnetization, switching field curves, signal-to-noise-ratio in stochastic resonance, and the complex magnetic susceptibility. We have discovered the depletion effect of a bias field, which is a general property of asymmetric bistable potentials and also applies in chemical physics. This is of importance in the stabilization of the magnetization in fine ferromagnetic particles, hence in the stabilization of information in magnetic recording systems .

(VI). Molecular spectroscopy of gases.

We have developed semiclassical versions of extended rotational diffusion models by using the memory function approach. In the context of this approach:

1. We have applied the semiclassical J-diffusion model to the calculation of the absorption of the molecular oxygen. This model takes into account the interference of the absorption lines and predicts correctly the collapse of the lines with increasing pressure so allowing one to describe the molecular absorption of oxygen at frequencies 50 - 70 GHz in a broad range of pressures (from 1 to 6000 kPa corresponding to altitude varying from 0 to 30 km in the Earth atmosphere) and temperatures.

From Yu. P. Kalmykov, S. V. Titov, and T. A. Novskova, “The Absorption Spectrum of Atmospheric Oxygen in the Frequency Range 50-70 GHz:
Collision Broadening in the Context of the J-Diffusion Model”
(in Russian), Radiotekh. Elektron. 43, 613 (1998) [English translation: J. Commun. Technol. Electron. 43, 565 (1998).] and
Yu. P. Kalmykov and S. V. Titov, “A Semiclassical Theory of Dielectric Relaxation and Absorption in Polar Fluids:
Memory Function Approach to the Extended Rotational Diffusion Models”,
in Relaxation Phenomena in Condensed Matter,
Ed. W. T. Coffey, A special volume of Advances in Chemical Physics, Wiley, New York, 1994, v.87, p.31-122.



2. We have proposed a model of molecular absorption in atmospheric water vapor. This model takes into account the finite duration of molecular collisions and allows one to describe the microwave / submillimeter absorption spectrum of water vapor.


Thus we have formulated a general approach allowing one to evaluate characteristic parameters of relaxation processes in gases, liquids, liquid and molecular crystals. By using it, we have solved various nonlinear and linear response problems. The essential results can be summarized as follows:

- development of efficient methods of solution of generalized nonlinear Langevin equations governing the dynamics of molecules (particles) in gases, liquids and solids (quantum and classical cases).

- development of methods of solution of the Master equations in phase space governing the quantum Brownian motion of particles and spins.

- elaboration of classical and semiclassical models of generalized rotation diffusion of the Brownian particle in a potential with account (or not) of the inertia of the particle;

- application of models so developed to the interpretation of experimental data on linear and nonlinear response characteristics (dielectric and magnetic permeability, relaxation functions, relaxation times, etc) of solids, liquids and gases as well as using them as a tool for the understanding of the physical mechanisms of relaxation in these media ;

- application of the methods developed to diverse physical phenomena such as electric birefringence of liquids, superparamagnetism, the Josephson effect, etc.

From Yu. P. Kalmykov et al., “Nonlinear Impedance of a Microwave-Driven Josephson Junction with Noise”, Phys. Rev. B 62, 3480 (2000).