**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.

**MAIN RESEARCH TOPICS:**

(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.

**SUMMARY OF MAIN RESEARCH
RESULTS:**

**(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

**(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 “,*

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

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

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

**(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 **H**_{0}. 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.

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.

Yu. P. Kalmykov and S. V. Titov,

Memory Function Approach to the Extended Rotational Diffusion Models”,

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