ANALYTICAL SOLUTION OF THE 2n-DIMENSIONAL FOKKER-PLANCK EQUATION WITH HARMONIC OSCILLATOR POTENTIAL

An analytical solution is obtained for the 2n-dimensiona Fokker-Planck equation (F-P equation for short) with the harmonic oscillator potential. A few steps are involved in the derivation. First,the Lagrangian subsidiary equation is solved; then with its integral constants as new variables of the F-P equation, the diffusion equation is obtained and solved; at last, expressed in the original phase space, the solution of the F-P equation .is finally obtained. The analysis for the solution is made. The solution is a Gaussian type function and a δ-function of time. If a particle moves in a well in ali directions, then as t→∞, the distribution function can reach a stationary nonzero distribution-Maxuwell-Boltzmann type distribution (M-B distribution for short).As an example, the 2-dimensional F-P equation is solved and discussed in detail.     

Long-time correlations in the stochastic regime

The phase space for Hamiltonians of two degrees of freedom is usually divided into stochastic and integrable components. Even when well into the stochastic regime, integrable orbits may surround ssmall stable regions or islands. The effect of these islands on the correlation function for the stochastic trajectories is examined. Depending on the value of the parameter describing the rotation number for the elliptic fixed point at the center of the island, the long-time correlation function may decay as t^?5 or exponentially, but more commonly it decays much more slowly (roughly as t^?1). As a consequence these small islands may have a profound effect on the properties of the stochastic orbits. In particular, there is evidence that the evolution of a distribution of particles is no longer governed by a diffusion equation.     

Evolution,migration and clustering of helium-vacancy complexes in RAFM steel- depth resolved positron annihilation Doppler broadening study

Indian Reduced Activation Ferritic Martensitic steel is implanted with 130 keV helium ions to a fluence of 5 × 10¹⁴ and 1 × 10¹⁶ ions/cm² and investigated using positron annihilation spectroscopy. The samples were characterised by defect sensitive S and W-parameters using depth resolved slow positron beam. A dose dependency is observed in the nucleation and growth of helium bubbles with annealing temperature. An experimental evidence for the migration of smaller helium-vacancy complexes is observed via the variation in thickness/width of irradiated layer with temperature. The S–W plot clearly shows the regions corresponding to defect annealing, bubble nucleation and growth.     

Radiation enhanced P-diffusion in germanium

Abstract

Cascade size may affect phase stability under irradiation because of two distinct contributions: the replacement to displacement cross-section ratio depends on the deposited energy density; ballistic jumps which tend to disorder ordered compounds occur by bursts, while thermal jumps which restore long range order occur one by one.

The latter effect cannot be handled by standard rate theory. A stochastic treatment of the problem, based on a Fokker Planck approximation of the relevant master equation is summarized. It is shown that the possible values of the long range order parameter under irradiation are not affected by the size of the cascade but that the respective stability of the former is cascade size sensitive. As a consequence, the stability diagram of phases under irradiation varies with the size of the cascades. A numerical example of this is given for the B2 structure.     

Laser-induced preferential domain nucleation in hafnium-doped congruent lithium niobate crystal

The focused visible laser-induced preferential domain nucleation effect is investigated in 3 mol% hafnium-doped congruent lithium niobate crystal. The local phase variation is in-situ monitored during laser-induced preferential domain nucleation. The variations of phase distributions are reconstructed by digital holographic interferometry. The nucleation field decreases exponentially with increasing irradiation intensity. The space charge field along the z direction is thought to be an important mechanism for the laser-induced preferential domain nucleation. Laser-induced hafnium-doped congruent lithium niobate crystals appear to be a promising candidate for further development of ferroelectric domain engineering.     

Observation of quantum particles on a large space-time scale

A quantum particle observed on a sufficiently large space-time scale can be described by means of classical particle trajectories. The joint distribution for large-scale multiple-time position and momentum measurements on a nonrelativistic quantum particle moving freely inR ^v is given by straight-line trajectories with probabilities determined by the initial momentum-space wavefunction. For large-scale toroidal and rectangular regions the trajectories are geodesics. In a uniform gravitational field the trajectories are parabolas. A quantum counting process on free particles is also considered and shown to converge in the large-space-time limit to a classical counting process for particles with straight-line trajectories. If the quantum particle interacts weakly with its environment, the classical particle trajectories may undergo random jumps. In the random potential model considered here, the quantum particle evolves according to a reversible unitary one-parameter group describing elastic scattering off static randomly distributed impurities (a quantum Lorentz gas). In the large-space-time weak-coupling limit a classical stochastic process is obtained with probability one and describes a classical particle moving with constant speed in straight lines between random jumps in direction. The process depends only on the ensemble value of the covariance of the random field and not on the sample field. The probability density in phase space associated with the classical stochastic process satisfies the linear Boltzmann equation for the classical Lorentz gas, which, in the limith0, goes over to the linear Landau equation. Our study of the quantum Lorentz gas is based on a perturbative expansion and, as in other studies of this system, the series can be controlled only for small values of the rescaled time and for Gaussian random fields. The discussion of classical particle trajectories for nonrelativistic particles on a macroscopic spacetime scale applies also to relativistic particles. The problem of the spatial localization of a relativistic particle is avoided by observing the particle on a sufficiently large space-time scale.     

Helium implanted CuHf as studied by TDPAC and positron lifetime measurements

¹⁸¹Ta time differential perturbed angular correlation (TDPAC) and positron lifetime measurements were carried out on homogeneously α-implanted CuHf samples. TDPAC measurements indicate the trapping of vacancy clusters and helium associated defect complexes by Hf atoms. The presence of helium-vacancy complexes and helium stabilised voids has been identified by positron lifetime measurements. Further the nucleation and growth stages of helium bubbles have been identified. TDPAC and positron lifetime measurements indicate that Hf atoms act as heterogeneous nucleating centers for helium bubbles. Hf atoms are found to suppress the bubble growth in CuHf as indicated by the results of positron lifetime measurements.     

Rare Events in Stochastic Partial Differential Equations on Large Spatial Domains

A methodology is proposed for studying rare events in stochastic partial differential equations in systems that are so large that standard large deviation theory does not apply. The idea is to deduce the behavior of the original model by breaking the system into appropriately scaled subsystems that are sufficiently small for large deviation theory to apply but sufficiently large to be asymptotically independent from one another. The methodology is illustrated in the context of a simple one-dimensional stochastic partial differential equation. The application reveals a connection between the dynamics of the partial differential equation and the classical Johnson–Mehl–Avrami–Kolmogorov nucleation and growth model. It also illustrates that rare events are much more likely and predictable in large systems than in small ones due to the extra entropy provided by space.     

Sharp Asymptotics for Stochastic Dynamics with Parallel Updating Rule

In this paper we study the metastability problem for a stochastic dynamics with a parallel updating rule; in particular we consider a finite volume Probabilistic Cellular Automaton (PCA) in a small external field at low temperature regime. We are interested in the nucleation of the system, i.e., the typical excursion from the metastable phase (the configuration with all minuses) to the stable phase (the configuration with all pluses), triggered by the formation of a critical droplet. The main result of the paper is the sharp estimate of the nucleation time: we show that the nucleation time divided by its average converges to an exponential random variable and that the rate of the exponential random variable is an exponential function of the inverse temperature β times a prefactor that does not scale with β. Our approach combines geometric and potential theoretic arguments.     

Large deviation estimates in the stochastic quantization ofϕ 2 4

We study in the small noise limit the behaviour of field trajectories for the process constructed by the authors in connection with the stochastic quantization of ₂ ⁴ . Due to the presence of infinite renormalization the usual large deviation techniques do not apply immediately and a new strategy has to be developed. We prove some estimates analogous to the Freidlin-Ventzel inequalities. From these it follows that the field trajectories suitably smeared in space over a scaler ₀ behave, when the noise is small, as the projection on the same scale of a field obeying a regularized stochastic equation with a large cut-off. However the estimates are not uniform in the cut-off and an interesting feature of the problem is that the scale over which the field is smeared determines whether the noise is sufficiently small for the estimates to apply.Laboratoire associé au CNRS UA 280     

Theory of switching of multiaxial ferroelectrics (Initial stage)

The classical theory of nucleation and growth is used to study the thermodynamics and kinetics of switching of multiaxial ferroelectrics. The initial stage of 180°-and 90°-domain switching is studied in the tetragonal, orthorhombic, and trigonal phases. The multidimensional kinetic theory of first-order phase transitions is applied to describe the initial stage of switching of ferroelectric crystals in the general case where three-dimensional growth (along the radius and height) of repolarized domains occurs. The energy of nucleus formation is calculated in the vicinity of the saddle point of an activation barrier in the space of sizes and shapes, and the dependence of the critical domain size on the switching field is found. The two-dimensional Fokker-Planck kinetic equation is reduced to a one-dimensional Zel’dovich equation, and a stationary solution to the Zel’dovich equation is obtained. The diffusion coefficients are derived in the size space for the normal and layer-by-layer mechanisms of domain growth. The main characteristic of the initial switching stage, namely, the steadystate flux of repolarized domains, is found as a function of the applied field.

     

Dynamics of cubic–quintic nonlinear Schr?dinger equation with different parameters

We study the dynamics of the cubic–quintic nonlinear Schr?dinger equation by the symplectic method. The behaviors of the equation are discussed with harmonically modulated initial conditions, and the contributions from the quintic term are discussed. We observe the elliptic orbit, homoclinic orbit crossing, quasirecurrence, and stochastic motion with different nonlinear parameters in this system. Numerical simulations show that the changing processes of the motion of the system and the trajectories in the phase space are various for different cubic nonlinear parameters with the increase of the quintic nonlinear parameter.     

The stochastic Burgers Equation

We study Burgers Equation perturbed by a white noise in space and time. We prove the existence of solutions by showing that the Cole-Hopf transformation is meaningful also in the stochastic case. The problem is thus reduced to the anaylsis of a linear equation with multiplicativehalf white noise. An explicit solution of the latter is constructed through a generalized Feynman-Kac formula. Typical properties of the trajectories are then discussed. A technical result, concerning the regularizing effect of the convolution with the heat kernel, is proved for stochastic integrals.     

Monocrystalline on seed diamond growth into HP/HT lpe autoautoclaves

Abstract

In the paper Authors analyse the possible ways of phase nucleation and the mechanisms of crystal growth, which suggest that, if we use as sp³ orbitals organized carbon source, diamond nucleation and growth by the way of spiondal decomposition and volumetic coalescence may take place. Such a process may be analysed as semihydrothermal-metalotheric coalescence may take place. Such a process may be analysed as semihydrothermal-metalothermal high pressure liquid phase epitaxy (MHPLPE) in separated autoautoclaves.     

Simulation of the early stage of binary alloy decomposition,based on the free energy density functional method

Based on the free energy density functional method, the early stage of decomposition of a onedimensional binary alloy corresponding to the approximation of regular solutions has been simulated. In the simulation, Gaussian composition fluctuations caused by the initial alloy state are taken into account. The calculation is performed using the block approach implying discretization of the extensive solution volume into independent fragments for each of which the decomposition process is calculated, and then a joint analysis of the formed second phase segregations is performed. It was possible to trace all stages of solid solution decomposition: nucleation, growth, and coalescence (initial stage). The time dependences of the main phase distribution characteristics are calculated: the average size and concentration of the second phase particles, their size distribution function, and the nucleation rate of the second phase particles (clusters). Cluster trajectories in the size–composition space are constructed for the cases of growth and dissolution.     

Universal Behaviour of (2+1)-Dimensional Stochastic Equations for Epitaxial Growth Processes

We investigate spatially discretized versions of a class of nonequilibrium continuum equations for epitaxial growth processes in (2+1)-dimensions using numerical integration. The epitaxial growth models include the most well-known Villain-Lai-Das Sarma (VLDS) equation and a stochastic differential equation recently proposed by Escudero (Phys. Rev. Lett. 101:196102, 2008). To suppress the instability in the VLDS equation, the nonlinear term is replaced by exponentially decreasing functions. The critical exponents in different regions are obtained. The roughness distributions at the steady states of the growth models show that the two equations are in good agreement with each other. Our results imply that the modified version of the VLDS equation with controlled instability and the equation proposed by Escudero belong to the same universality class. Anomalous scaling behaviour in these growth models are also discussed, and the nontrivial scaling properties are found very weak in (2+1)-dimensions.     

Stochastic quantization and path integral formulation of Fokker-Planck equation

The operator formalism (Fokker-Planck dynamics) associated to a general n-dimensional, non-linear drift, non-constant diffusion Fokker-Planck equation, is derived by a stochastic quantization from the corresponding path integral formulation in phase space.     

Quantum-networks: master equation and local measurements

Based on the SU(n)-algebra the Markoff master equation in discrete product space is reformulated to explicitly deal with composite systems. The resulting local (single node) and nonlocal (multi-node) state parameters allow a systematic approach to non-classical features of the state, like variance and covariance tensors. For local optical driving forces, inter-node interactions, and local damping channels the solution of the master equation is unraveled into stochastic quantum trajectories. Sampling leads to a joint distribution function in terms of those state parameters. Its linear moments define the ensemble-density matrix. The average variance and covariance are in terms of non-linear moments, which should be distinguished from their entirely statistical counterpairs. Non-classicality of the network dynamics is shown to reflect itself in the luminescence-photonstatistics.