Anomalous diffusion, gigahertz, and terahertz spectroscopy of aliphatic ketones.

The experimental results of the complex dielectric permittivity of aliphatic ketones in dilute solutions of inert solvent cyclohexane in the gigahertz (GHz) and terahertz (THz) frequencies of the electromagnetic spectrum are examined in terms of the theory of inertial anomalous diffusion of polar molecules, considered as an assembly of molecules with interacting dipolar groups, in polar liquids. The theory is based on the generalization of the Debye rotational diffusion model of dielectric relaxation of polar molecules. The model comprises two interacting dipolar groups-one lighter and the other heavier; each has a finite moment of inertia and each experiences a finite friction with an extensive range of damping or drag coefficient and the dipole moment ratio of the two groups. The lighter group refers to the reference molecule, whereas the heavier group simulates the neighboring molecules, and the two groups interact with each other via the dipole-dipole interaction potential. The resulting approximate expression contains terms in both the Rocard form and the Сole-Сole form. The experimental data on aliphatic ketones are shown to fit extremely well with the theory, and parameters of the fit offer physical significance. An agreement of the plot of the experimental data on dielectric loss versus frequency to the formulas derived from the model offers a mathematical basis of the semiempirical equations used in the literature to fit the experimental data. Experimental results of the dielectric loss of neat polar liquid acetone in GHz and THz, as an example, are shown to fit the theory; however, the interaction potential parameter compared to its dilute solution counterpart is significantly increased, reflecting the increase in the dipole-dipole interaction energy.

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.

Golden rule kinetics of transfer reactions in condensed phase: the microscopic model of electron transfer reactions in disordered solid matrices.

The algorithm for a theoretical calculation of transfer reaction rates for light quantum particles (i.e., the electron and H-atom transfers) in non-polar solid matrices is formulated and justified. The mechanism postulated involves a local mode (an either intra- or inter-molecular one) serving as a mediator which accomplishes the energy exchange between the reacting high-frequency quantum mode and the phonon modes belonging to the environment. This approach uses as a background the Fermi golden rule beyond the usually applied spin-boson approximation. The dynamical treatment rests on the one-dimensional version of the standard quantum relaxation equation for the reduced density matrix, which describes the frequency fluctuation spectrum for the local mode under consideration. The temperature dependence of a reaction rate is controlled by the dimensionless parameter ξ0 = ℏω0/k(B)T where ω0 is the frequency of the local mode and T is the temperature. The realization of the computational scheme is different for the high/intermediate (ξ0 < 1 - 3) and for low (ξ0 ≫ 1) temperature ranges. For the first (quasi-classical) kinetic regime, the Redfield approximation to the solution of the relaxation equation proved to be sufficient and efficient in practical applications. The study of the essentially quantum-mechanical low-temperature kinetic regime in its asymptotic limit requires the implementation of the exact relaxation equation. The coherent mechanism providing a non-vanishing reaction rate has been revealed when T → 0. An accurate computational methodology for the cross-over kinetic regime needs a further elaboration. The original model of the hopping mechanism for electronic conduction in photosensitive organic materials is considered, based on the above techniques. The electron transfer (ET) in active centers of such systems proceeds via local intra- and intermolecular modes. The active modes, as a rule, operate beyond the kinetic regimes, which are usually postulated in the existing theories of the ET. Our alternative dynamic ET model for local modes immersed in the continuum harmonic medium is formulated for both classical and quantum regimes, and accounts explicitly for the mode∕medium interaction. The kinetics of the energy exchange between the local ET subsystem and the surrounding environment essentially determine the total ET rate. The efficient computer code for rate computations is elaborated on. The computations are available for a wide range of system parameters, such as the temperature, external field, local mode frequency, and characteristics of mode/medium interaction. The relation of the present approach to the Marcus ET theory and to the quantum-statistical reaction rate theory [V. G. Levich and R. R. Dogonadze, Dokl. Akad. Nauk SSSR, Ser. Fiz. Khim. 124, 213 (1959); J. Ulstrup, Charge Transfer in Condensed Media (Springer, Berlin, 1979); M. Bixon and J. Jortner, Adv. Chem. Phys. 106, 35 (1999)] underlying it is discussed and illustrated by the results of computations for practically important target systems.

Fractional diffusion in a periodic potential: Overdamped and inertia corrected solutions for the spectrum of the velocity correlation function.

Anomalous diffusion of a particle in a cosine periodic potential is treated using fractional diffusion equations in both phase and configuration space. Exact solutions of two distinct forms of the fractional Klein-Kramers (Fokker-Planck) equation for the distribution function in phase space are obtained via matrix continued fractions yielding the average velocity, the velocity autocorrelation function, its spectrum, etc. In the overdamped limit, the results yielded by both equations agree with those from a fractional probability density diffusion equation in configuration space. A simple analytic solution for the spectrum of the velocity correlation function is also given using the effective eigenvalue approximation. The results represent generalizations of the conventional solutions for the normal diffusion of a Brownian particle in a cosine potential to fractional dynamics (giving rise to anomalous diffusion).

[Biotope principles of sympatry and interspecies hybridization in mammals (by the example of the genus Spermophilus)].

Two hybridization zones were taken as an example (the russet ground squirrel (Spermophilus major) and the speckled ground squirrel (Spermophilus suslicus), the russet ground squirrel and the yellow ground squirrel (S. fulvus) to show that biotope characteristics determine segregation of sympatric species in contact populations. The heterogeneity of biotopes with regard to the requirements of sympatric species promotes their long-term and steady dispersal and, in rare cases, sporadic hybridization. A biotope with a homogenous environment enables wide interspecies hybridization and a rapid increase in the ratio of hybrids in the population.

[Breeding success and direction of animal crossing in the hybrid russet (Spermophilus major) and yellow (Spermophilus fulvus) ground squirrel population].

Reproductive relations between animals in the hybrid settlement of the russet and yellow ground squirrels are panmictic and are characterized by the absence of assortative crossings and by the presence of multimale pairings in females with evidently heterospermic broods. Maximum breeding success was observed for hybrid animals proving the fertility of hybrids. However, the hybrid breeding potential (young animal survival rate) is significantly lower as compared with russet and yellow ground squirrels, but the survival rate of the hybrid progeny in the hybrid population is not lower than in specific settlements.

Wigner function approach to the quantum Brownian motion of a particle in a potential.

Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.

Semiclassical master equation in Wigners phase space applied to Brownian motion in a periodic potential.

The quantum Brownian motion of a particle in a cosine periodic potential V(x)= -V{0}cos(x/x{0}) is treated using the master equation for the time evolution of the Wigner distribution function W(x,p,t) in phase space (x,p) . The dynamic structure factor, escape rate, and jump-length probabilities are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The escape rate so yielded is compared with that given analytically by the quantum-mechanical reaction rate solution of the Kramers turnover problem. The matrix continued fraction solution substantially agrees with the analytic solution.

Non-Markovian modification of the golden rule rate expression.

The reformulation of the standard golden rule approach considered in this paper for treating reactive tunneling reduces the computation of the reaction rate to a derivation of band shapes for energy levels of reactant and product states. This treatment is based on the assumption that the medium environment is actively involved as a partner in the energy exchange with the reactive subsystem but its reorganization effect is negligible. Starting from the quantum relaxation equation for the density matrix, the required band shapes are represented in terms of the spectral density function, exhibiting the continuum spectrum inherent to the interaction between the reactants and the medium in the total reactive system. The simplest Lorentzian spectral bands, obtained under Redfield approximation, proved to be unsatisfactory because they produced a divergent rate expression at low temperature. The problem is resolved by invoking a refined spectral band shape, which behaves as Lorentzian one at the band center but decays exponentially at its tails. The corresponding closed non-Markovian rate expression is derived and investigated taking as an example the photochemical H-transfer reaction between fluorene and acridine proceeding in the fluorene molecular crystal. The kinetics in this reactive system was thoroughly studied experimentally in a wide temperature range [B. Prass et al., Ber. Bunsenges. Phys. Chem. 102, 498 (1998)].

Thermally activated escape rate for a Brownian particle in a tilted periodic potential for all values of the dissipation.

The translational Brownian motion of a particle in a tilted washboard potential is considered. The dynamic structure factor and longest relaxation time are evaluated from the solution of the governing Langevin equation by using the matrix continued fraction method. The longest relaxation time is compared with the Kramers theory of the escape rate of a Brownian particle from a potential well as extended to the Kramers turnover region by Mel'nikov [Physics Reports 209, 1 (1991)]. It is shown that in the low temperature limit, the universal Mel'nikov expression for the escape rate provides a good estimate of the longest relaxation time for all values of dissipation including the very low damping (VLD), very high damping (VHD), and turnover regimes. For low barriers (where the Mel'nikov method is not applicable) and zero tilt, analytic equations for the relaxation times in the VLD and VHD limits are derived.

[A search for Y-chromosomal species-specific markers and their use for hybridization analysis in ground squirrels].

In four ground squirrel species from the Volga region-yellow (Spermophilus fulvus), russet (S. major), little (S. pygmaeus), and speckled (S. suslicus)--four hybridization variants (major/fulvus, major/pygmaeus, major/suslicus, and pygmaeus/suslicus) have been reliably described. Earlier we have shown that populations of S. major from the Volga region were characterized by wide introgression of mtDNA from S. fulvus and S. pygmaeus, which probably, resulted from ancient hybridization. In this study, the same populations were used to analyze the introgression of the Y chromosome, which (unlike mtDNA) is paternally inherited. Three genes, ZfY, SRY, and SmcY were tested as Y-chromosomal candidate markers. It was demonstrated that Y chromosome of ground squirrels lacked the ZfY gene, while its homologous structure, ZfY(X), was presumably linked to the X chromosome. The SRY region examined was rather conservative. In particular, the sequences determined in S. major and S. fulvus were identical, while three out of four substitutions found in S. pygmaeus were located in the coding region. The SmcY gene was found to be the most suitable marker, providing distinguishing of all of the four ground squirrel species by nine nucleotide substitutions. Introgression at the Y chromosome was observed only in two cases: in one S. major individual (out of 51 phenotypically pure animals) caught in the major/fulvus sympatry zone, and in four (one litter) out of fourteen S. fulvus individuals caught in close vicinity of the sympatry zone of these two species. Among 28 S. pymaeus and 9 S. suslicus individuals, no foreign SmcY genes were detected. Two colonies of the "hybrid accumulation" type were examined with eight major/suslicus hybrids analyzed in the first and seventeen major/fulvus hybrids in the second colony. The prevalence of the S. major paternal lineages was observed in both colonies (87.5 and 82.4%, respectively). The data obtained suggest that compared to wide mtDNA introgression, introgression of Y chromosome in the Volga region ground squirrels is statistically significantly less frequent event.

Thermally activated escape rate for a Brownian particle in a double-well potential for all values of the dissipation.

The translational Brownian motion in a (2-4) double-well potential is considered. The escape rate, the position correlation function and correlation time, and the generalized susceptibility are evaluated from the solution of the underlying Langevin equation by using the matrix-continued fraction method. The escape rate and the correlation time are compared with the Kramers theory of the escape rate of a Brownian particle from a potential well as extended by Mel'nikov and Meshkov [J. Chem. Phys. 85, 1018 (1986)]. It is shown that in the low-temperature limit, the universal Mel'nikov and Meshkov expression for the escape rate provides a good estimate of both escape rate and inverse position correlation time for all values of the dissipation including the very low damping (VLD), very high damping (VHD), and turnover regimes. Moreover, for low barriers, where the Mel'nikov and Meshkov method is not applicable, analytic equations for the correlation time in the VLD and VHD limits are derived.

Extended rotational diffusion and orientational relaxation of symmetric top molecules in a strong dc electric field: second-rank orientational correlation functions.

Second-rank orientational correlation functions (pertaining to Kerr effect relaxation and Raman scattering) are obtained using the extended rotational diffusion (J-diffusion) model of symmetric top polar molecules in a strong constant external field. It is shown that the shape of the molecule noticeably affects all second-rank correlation functions and relaxation times in the rare collision limit. In the opposite limit of frequent collisions, the quantities of interest are shown to be shape independent as a consequence of vanishingly small inertial effects. An interpolation formula for the orientation relaxation times in the intermediate regime between the rare and frequent collision limits is also given.

Anomalous diffusion and dielectric relaxation in an N-fold cosine potential.

The fractional Klein-Kramers (Fokker-Planck) equation describing the fractal time dynamics of an assembly of fixed axis dipoles rotating in an N-fold cosine potential representing the internal field due to neighboring molecules is solved using matrix continued fractions. The result can be considered as a generalization of the solution for the normal Brownian motion in a cosine periodic potential to fractional dynamics (giving rise to anomalous diffusion) and also represents a generalization of Fröhlich's model of relaxation over a potential barrier. The solution includes both inertial and strong internal field effects, which in combination produce a strong resonance peak (Poley absorption) at high frequencies due to librations of the dipoles in the potential, an anomalous relaxation band at low frequencies mainly arising from overbarrier relaxation, and a weaker relaxation band at midfrequencies due to the fast intrawell modes. The high-frequency behavior is controlled by the inertia of the dipole, so that the Gordon sum rule for dipolar absorption is satisfied, ensuring a return to optical transparency at very high frequencies. Application of the model to the broadband (0-THz) dielectric loss spectrum of a dilute solution of the probe dipolar molecule CH2Cl2 in glassy decalin is demonstrated.