Phase separation in binary systems with competing internal and external multiplicative noises

We have considered phase separation processes in binary stochastic systems with thermal diffusion and ballistic mixing representing irradiation influence. Introducing fluctuations of thermal flux and an external source of atom relocations due to ballistic diffusion into dynamics of a globally conserved field, we have shown that there are two competing mechanisms of phase transitions type of “order-disorder”: thermally assisted diffusion and irradiation induced atomic exchange. We have studied dynamics of the structure function at early stages of decomposition. In the framework of the mean field theory we have derived the effective Fokker-Planck equation to describe phase separation processes. It was shown that the ordering processes can be controlled by both regular and stochastic parts of external source influence. A reentrant behavior of a mean field order parameter versus the external noise intensity and fluctuations correlation radius is found.     

Control the relaxation properties of the diffuse bistable potential

The purpose of this work is to investigate procedures to control the population dynamics in a bistable potential. We use the solutions of the Fokker–Planck equation for the distribution function corresponding to a bistable potential under action of an external electric field. The control is performed with a piecewise protocol with time independent potentials. This protocol provides to be adequate for this problem, controlling the time necessary to balance of system probabilities. Some trajectories were controlled and the limits of procedure clarified. Nevertheless, with the application of the control protocol we clarify some simple phenomena of the system, associating the application of the external field with the system diffusion coefficient.     

From continuous time random walks to the fractional fokker-planck equation

We generalize the continuous time random walk (CTRW) to include the effect of space dependent jump probabilities. When the mean waiting time diverges we derive a fractional Fokker-Planck equation (FFPE). This equation describes anomalous diffusion in an external force field and close to thermal equilibrium. We discuss the domain of validity of the fractional kinetic equation. For the force free case we compare between the CTRW solution and that of the FFPE.     

Splitting Scheme Integration of the Two-and-One-Half-Dimensional Vlasov Equation

A numerical splitting scheme technique is applied to the integration of the Vlasov equation in two-space and three velocity dimensions. Calculations are performed for a cold, stationary ion, mobile electron plasma in the presence of a constant external magnetic field. Results for Landau damping and the two-stream instability are presented in both one and two velocity dimensions. The Landau damping results show a plateau structure in the electron velocity distribution function in both the parallel and perpendicular directions formed as a result of the damping of a high initial electric field propagating at 45° to the parallel velocity axis. The two-stream velocity distribution function shows pitch angle scattering and velocity diffusion for an electron beam initially propagating at 45° to the magnetic field direction through a low-temperature equal density background. The two-stream distribution function evolves to a stable quasi-isotropic plateau structure similar to that found in auroral sounding rocket data.     

Derivation of a hydrodynamic equation for Ginzburg-Landau models in an external field

The lattice approximation to a time-dependent Ginzburg-Landau equation is investigated in the presence of a small external field. The evolution law conserves the spin, but is not reversible. A nonlinear diffusion equation of divergence type is obtained in the hydrodynamic limit. The proof extends to certain stochastically perturbed Hamiltonian systems.     

Renormalized diffusion coefficients for electron-hole plasmas in crossed fields

The diffusion and drift of an excess plasma in a semiconductor is described with magnetohydrodynamic two-fluid equations including the fluctuating electric field produced by the equilibrium plasma in the sample. Using the weak coupling limit an equation of motion for the mean density of the excess plasma is established with renormalized drift and diffusion coefficients. With the aid of the fluctuation dissipation theorem these coefficients are expressed in terms of the dielectric function and discussed in detail for stable systems. The renormalized diffusion coefficient differs from the bare one by an additional term with thet ^–3/2-long time dependence. It is shown that this term in addition represents an anomalous diffusion rate proportionalB ^–1 which overweights the classical ambipolar diffusion for sufficiently strong fields, but decreases with increasing external electric field. The results are compared with experimental data.     

Coefficients of diffusion and conductivity of semiconductor carbon nanotubes in an external electric field

The coefficients of electron diffusion and conductivity have been calculated for single-layer semiconducting carbon nanotubes in an external electric field with the strength vector directed along the nanotube axis. The evolution of the electronic system of the nanotubes has been described using the Boltzmann kinetic equation in terms of the quasi-classical approximation of relaxation time. An analytical expression of the electron diffusion coefficient has been obtained, and its nonlinear field dependence has been revealed.     

Diffusion near nonequilibrium steady state

The diffusion equation describing the response of a system in a steady nonequilibrium state to a nonuniform external perturbation is derived from the Boltzmann equation. For transient processes, the diffusion equation governs the slow (final) stage of the process, irrespective to the form of the initial distribution. The initial velocity distribution becomes the local steady-state distribution during the first fast stage of the process. This fast initial relaxation affects only the initial condition for the diffusion equation. Some principal features of the nonequilibrium diffusion constant are discussed. As an illustration, the diffusion constant of hot electrons is calculated in the electron temperature approximation.     

Fractional generalized Langevin equation approach to single-file diffusion

Fractional generalized Langevin equation with external force is used to model single-file diffusion. It is found that for external force that varies with power law the solution for such a fractional Langevin equation gives the correct short and long time behavior for the mean square displacement of single-file diffusion when appropriate choice of parameters associated with fractional generalized Langevin equation are used. By considering some special cases of the fractional generalized Langevin equation, a new class of closed analytic expressions for the mean square displacement of single-file diffusion can be obtained. The effective Fokker-Planck equation associated with single-file diffusion is briefly considered.     

Diffusion dynamics in branched spherical structure

Diffusion on a spherical surface with trapping is a common phenomenon in cell biology and porous systems. In this paper, we study the diffusion dynamics in a branched spherical structure and explore the influence of the geometry of the structure on the diffusion process. The process is a spherical movement that occurs only for a fixed radius and is interspersed with a radial motion inward and outward the sphere. Two scenarios govern the transport process in the spherical cavity: free diffusion and diffusion under external velocity. The diffusion dynamics is described by using the concepts of probability density function (PDF) and mean square displacement (MSD) by Fokker-Planck equation in a spherical coordinate system. The effects of dead ends, sphere curvature, and velocity on PDF and MSD are analyzed numerically in detail. We find a transient non-Gaussian distribution and sub-diffusion regime governing the angular dynamics. The results show that the diffusion dynamics strengthens as the curvature of the spherical surface increases and an external force is exerted in the same direction of the motion.     

Electrical conductivity and diffusion coefficient of electrons in a graphene bilayer

The electron diffusion coefficient and the electrical conductivity of a graphene bilayer in an external electric field with a strength vector directed along the graphene sheet are calculated theoretically. The evolution of the electron system is simulated using the Boltzmann kinetic equation in the relaxation-time semi-classical approximation. Analytic expressions are obtained for the electron diffusion coefficient and the electrical conductivity, and the nonlinear dependences of these quantities on the electric field are established. The dependences of these quantities on the control electrostatic potential between graphene layers are analyzed.     

Stochastic diffusion in a periodic potential

This paper deals with two equations for classical stochastic diffusion in a potential. First, the full Fokker-Planck equation in phase-space for a Brownian particle in a periodic potential and linearly coupled to an external field is considered. The solution discussed previously by the author and co-worker is improved upon. An alternative approximation is introduced. Then, the simpler Smoluchowski equation, which is derivable from the Fokker-Planck equation under suitable conditions, is solved using Hill's determinant method. Finally a WKB-type method is proposed to solve the Smoluchowski equation for a general class of potentials.     

Nonexistence of Kirkwood's instability in a uniform fluid

Kirkwood's instability in the theory of fluid-solid transitions is proved to be impossible. Fluctuation of the one-particle distribution function in the first equation of the BGY hierarchy is investigated beyond Kunkin and Frisch's treatment. The second equation of the BGY hierarchy is utilized to eliminate the three-particle distribution function left in the Kunkin-Frisch result. The final expression for the first-order fluctuation of the one-particle distribution function under the presence of an external field is written in a form including only the pair correlation function and agrees identically with the one obtained from the direct expansion of the oneparticle distribution function in terms of the external field.     

The influence of a varying field on the non-Markovian migration of particles

The influence of an external varying field on the non-Markovian migration of particles described in the continuous-time random walk model (CTRWM) was analyzed theoretically. In terms of the Markovian representation for the CTRWM suggested earlier, a rigorous method for describing the influence of an external force was developed. This method reduced the problem to solving the non-Markovian stochastic Liouville equation (SLE) for the particle distribution function. An analysis of the derived SLE and its comparison with the earlier equations were performed. The method was used to study the characteristic features of the time dependence of the first and second moments of the distribution function for particles involved in subdiffusion motion in a uniform varying external field. Both oscillating and fluctuating fields were considered. In both cases, anomalously strong field effects on the second particle distribution moment (variance) were observed. This influence was especially strong for a fluctuating field, and in the limit of anomalously slow fluctuations at that.     

Statistical properties of random environment of 1<Emphasis Type="Italic">D</Emphasis> quantum <Emphasis Type="Italic">N</Emphasis>-particles system in external field

The investigation of 1D quantum N-particle system (PS) with relaxation in the random environment under the influence of external field is conducted within the framework of the stochastic differential equation of the Langevin—Schrödinger (L—Sch) type. Using L—Sch equation, the 2D second-order nonstationary partial differential equation is found, which describes the quantum distribution in the environment, depending on the energy of nonperturbed 1D quantum N-PS and on the external field parameters. It is shown that the average value of the interaction potential between 1D disordered quantum N-PS and the external field, has the ultraviolet divergence. This problem is solved by the renormalization of the equation for the function of quantum distribution. It is shown that it has a sense of dimensional renormalization which is characteristic for the quantum field theory. Critical properties of the environment are investigated in detail. The possibility of the first-order phase transition in the environment distribution depending on amplitude of an external field is been shown.     

Some New Features of Linear Theory for Describing Non-linear Diffusion

The non-linear flux equation, the non-linear Fokker-Planck equation (or Smoluchowski equation), and the non-linear Langiven equation are the basicequations for describing particle diffusion in non-ideal system subjected totime-dependent external fields. Nevertheless, the exact solution of thoseequations is still a challenge because of their inherent complexity of thenon-linear mathematics. Li et al. found that, based on the defined apparentvariables, the non-linear Fokker-Planck equation and the non-linear flux equation could be transformed to linear forms under the condition of strong friction limit or local equilibrium assumption. In this paper, some new features of the theory were found: (i) The linear flux equation for describing non-linear diffusion can be obtained from the irreversible thermodynamic theory; (ii) The linear non-steady state diffusion equation for describing non-linear diffusion of the non-steady state, which was described by the non-linear Fokker-Planck equation, can be derived more consistently from the microscopic molecular statistical theory; (iii) In the theory, thenon-linear Langiven equation also bears a linear form; (iv) For some special cases, e.g. diffusion in a periodic total potential system, the local equilibrium assumption or the strong friction limit is not required in establishing the linear theory for describing non-linear diffusion, so the linear theory may be important in the study of Brown motor.     

Generalized nonlinear master equations for local fluctuations: Dimensionality expansions and augmented mean field theory

By application of a projection operator technique we derive a formally exact generalization of the nonlinear mean field master equation introduced recently for the study of local fluctuations in a reacting medium. Our starting point is a phenomenological cell master equation. The results of our theory are applicable to the theory of a fluctuating hydrodynamic reacting system. The mean field equation is placed on a firm theoretical foundation by showing it to be the lowest order approximation in an expansion in the dimensionality of the physical space keeping the product of the number of nearest neighbors (an increasing function of dimensionality) and the typical diffusion coefficient constant. A more accurate nonlinear master equation that allows for the correlation and fluctuations in the environment of a given volume element is derived in the form of an augmented mean field equation.Work supported in part by a grant from the National Science Foundation.     

Macroscopic equations of motion and dynamic polarizability in the study of nuclear collective excitations

Time dependent Hartree-Fock equations are derived using a variational principle over a restricted part of the space of the Slater determinants, in the limit of small deformations (RPA). When an external oscillating field interacts with the nucleus, this method leads to an explicit expression of the nuclear response function (dynamic polarizability) as a function of the external frequency and of the deformation field, defining the nuclear deformation induced by the interaction. A linear differential equation for the deformation field is also obtained: in the limit ω → ∞ it has analytical solutions which satisfy the energy-weighted sum rule, evaluated in a HF ground state, in both isoscalar and isovector modes.     

Spin diffusion in multiple pulse spin-locking in solids containing paramagnetic impurities

Spin diffusion and spin-lattice relaxation in solids containing paramagnetic impurities under influence of a multiple-pulse spin-locking radio-frequency sequence are studied theoretically and experimentally. The diffusion equation obtained provides a clue for determination of the time dependent magnetization. The spin-lattice relaxation time is calculated as a function of the correlation time and multiple-pulse field parameters. From the experimental data the spin diffusion coefficient, the radius of the spin diffusion barrier, and the correlation time for very slow molecular motion in polycrystalline (C(2)F)(n) system are estimated and found to be D～7.1×10(-12)cm(2)/s, r(c)～4.8×10(-10)m, and τ(c)～10.2μs, respectively.     

The quantum acoustomagnetoelectric field in a quantum well with a parabolic potential

The acoustomagnetoelectric (AME) field in a quantum well with a parabolic potential (QWPP) has been studied in the presence of an external magnetic field. The analytic expression for the AME field in the QWPP is obtained by using the quantum kinetic equation for the distribution function of electrons interacting with external phonons. The dependence of the AME field on the temperature T of the system, the wavenumber q of the acoustic wave and external magnetic field B for the specific AlAs/GaAs/AlAs is achieved by using a numerical method. The problem is considered for both cases: The weak magnetic field region and the quantized magnetic field region. The results are compared with those for normal bulk semiconductor and superlattices to show the differences, and we use the quantum theory to calculate the AME field in the QWPP.