Spatial Diffusion Coefficient Research Articles

Temperature and spatial diffusion of atoms cooled in a 3D lin ⊥lin bright optical lattice

We present a detailed experimental study of a three-dimensional lin$\perp$lin bright optical lattice. Measurements of the atomic temperature and spatial diffusion coefficients are reported for different angles between the lattice beams, i.e. for different lattice constants. The experimental findings are interpretated with the help of numerical simulations. In particular we show, both experimentally and theoretically, that the temperature is independent of the lattice constant.

Transport of a single ion in an optical lattice: spatial diffusion and energy

This paper presents a numerical study of the dynamics of a single ion subjected to a polarization gradient field, a weak confining electric potential and a periodic driving force. Comparison with laboratory experiments requires that an amplitude be chosen for the periodic force; a single value gives satisfactory agreement for both the spatial diffusion coefficient and the potential energy in the confining potential. The solution of the Fokker–Planck equation for the system in the diffusion limit allows one to explain the apparent rise of the spatial diffusion coefficient when the periodic force amplitude is increased.

Velocity Correlation and the Spatial Diffusion Coefficients of Cosmic Rays: Compound Diffusion

We consider the transport of charged particles in a stochastic magnetic field using a method based on the velocity correlation function ⟨vi(0)vj(t)⟩ developed by R. Kubo. This can be used under very general conditions to evaluate the corresponding spatial diffusion coefficients, if the fluctuations are statistically homogeneous in space and time. Although Kubo's theory is quite general, it is not obvious how it can be applied to describe compound diffusion when particles are strictly tied to the magnetic field lines and perpendicular transport results solely from the random walk of the field lines. This motion is non-Markovian and leads to a slower ⟨Δx2⟩ ∝ t1/2 diffusion in contrast to the ⟨Δx2⟩ ∝ t dependence of the standard diffusion. We demonstrate how compound diffusion fits into Kubo's formalism. As intuitively as can be anticipated, the non-Markovian nature of the motion results in a long-term anticorrelation in ⟨vj(0)vi(t)⟩, which causes the ordinary spatial diffusion coefficient to vanish identically. The ⟨Δx2⟩ ∝ t1/2 dependence of the compound diffusion can also be recovered from the Laplace transform of the velocity correlation function. Some implications of the long-term anticorrelation are discussed.

Diffusion of energetic particles in focusing fields

We address the modification of the effective spatial diffusive coefficient of energetic charged particles in magnetic field configurations where the divergence of the field lines and the consequent weakening of the field strength lead to adiabatic focusing. Particles propagating along the magnetic field lines and undergoing pitch angle scattering and adiabatic focusing are considered. In the presence of significant focusing the conditions for the applicability of a diffusive description are not strictly valid, thus focusing modifies the effective parallel diffusion coefficient κ‖. We calculate the appropriate spatial diffusion coefficients from a method based on the use of adjoint Green functions. The correspondence between this method and the method based on the velocity correlation function < νi(0)νj(t) > developed by Kubo [1957] is discussed. We target the modulation of galactic and anomalous cosmic rays, which can be best approximated by the assumption of a constant, or slowly varying, spatial gradients, for which the method of adjoint Green functions is most suitable. We show that this assumption leads to an effective κ‖ identical to that derived by Bieber and Burger [1990] from a Born approximation. The derivation also results in a nonsteady analytical solution to the Fokker‐Planck equation, which describes a distribution of constant spatial gradient moving at a constant speed. We also discuss the case of hemispherical scattering, when scattering is effective within each of the μ < 0 and μ > 0 hemispheres (where μ is the cosine of pitch angle) but is restricted between the two hemispheres. A refinement to the hemispherical equation of Isenberg [1997] and Schwadron [1998] is suggested.

Trapping and cooling cesium atoms in a speckle field

We present the results of two experiments where cold cesium atoms are trapped in a speckle field. In the first experiment, a YAG laser creates the speckle pattern and induces a far-detuned dipole potential which is a nearly-conservative potential. Localization of atoms near the intensity maxima of the speckle field is observed. In a second experiment we use two counterpropagating laser beams tuned close to a resonance line of cesium and in the lin lin configuration, one of them being modulated by a holographic diffuser that creates the speckle field. Three-dimensional cooling is observed. Variations of the temperature and of the spatial diffusion coefficient with the size of a speckle grain are presented.

Atom cooling and trapping by disorder

We demonstrate the possibility of three-dimensional cooling of neutral atoms by illuminating them with two counterpropagating laser beams of mutually orthogonal linear polarization, where one of the lasers is a speckle field, i.e. a highly disordered but stationary coherent light field. This configuration gives rise to atom cooling in the transverse plane via a Sisyphus cooling mechanism similar to the one known in standard two-dimensional optical lattices formed by several plane laser waves. However, striking differences occur in the spatial diffusion coefficients as well as in local properties of the trapped atoms.

Nonisothermal activation: nonlinear transport theory

We present the statistical mechanical foundation of nonisothermal stochastic processes, thereby generalizing Kramers' Fokker–Planck model for thermal activation and providing a microscopic context for Rolf Landauer's original ideas on state-dependent diffusion. By applying projection operator methods suitable for nonlinear mesoscopic systems coupled to a heat bath, we develop the theory of classical Brownian motion (in position and momentum) including the local temperature as a dynamical variable. The ensuing stochastic process involves a microcanonical effective mean force different from the free energy gradient, while the equilibrium potential is given by the availability. The effective spatial diffusion coefficient in the Smoluchowski limit is calculated. The microcanonical analysis corresponds to the case of small thermal conductance.

Quasi‐linear Theory of Cosmic Ray Transport and Acceleration: The Role of Oblique Magnetohydrodynamic Waves and Transit‐Time Damping

We calculate quasi-linear transport and acceleration parameters for cosmic ray particles interacting resonantly with undamped fast-mode waves propagating in a low-β plasma. For super-Alfvenic particles and a vanishing cross-helicity state of the fast-mode waves, we demonstrate that the rate of adiabatic deceleration vanishes, and that the momentum and spatial diffusion coefficients can be calculated from the Fokker-Planck coefficient Dμμ. Adopting isotropic fast-mode turbulence with a Kolmogorov-like turbulence spectrum, we demonstrate that Dμμ is the sum of contributions from transit-time damping and gyroresonant interactions. Gyroresonance refers to | n | ≠ 0 resonant particle-wave interactions. Transit-time damping refers to the n = 0 interaction of particles with the compressive magnetic field component of the fast-mode waves. We show that transit-time damping provides the dominant contribution to pitch-angle scattering in the interval ≤ | μ | ≤ 1, where is the ratio of Alfven to particle speed. In the interval | μ | < , transit-time damping does not occur, and gyroresonance provides a small but finite contribution to particle scattering. As a consequence, the momentum diffusion coefficient is mainly determined by the transit-time damping contribution. On the other hand, since the spatial diffusion coefficient and the related mean free path are given by the average over μ of the inverse of Dμμ, these spatial transport parameters are determined by the contribution from the interval | μ | < . We also calculate the cosmic ray transport parameters for plasma turbulence consisting of a mixture of isotropic fast-mode waves and slab Alfven waves. Here, the momentum diffusion coefficient is determined by the transit-time damping of the fast-mode waves, and is a factor ln -1 larger than in the case of pure slab Alfven wave turbulence. The mean free path and the spatial diffusion coefficient are modified significantly from the pure fast-mode case, since the crucial scattering at | μ | < is now provided by gyroresonances with slab Alfven waves. The mean free path is a constant at nonrelativistic energies, and may account for the legendary λfit-λQLT discrepancy of solar energetic particles.

Recent Developments In Quasilinear Cosmic Ray Particle Acceleration Theories

Several recent results of quasilinear cosmic ray transport theory are reviewed. It is shown that modelling the plasma turbulence beyond the magnetostatic approximation is most important for a correct description of cosmic ray dynamics. In particular, the effects of finite plasma wave speeds play a crucial role both for the behaviour of the particles' spatial diffusion coefficient and the diffusive acceleration by super-Alfvenic shock waves.

Towards a microscopic theory of nonisothermal stochastic processes

Using a nonlinear projection operator technique, the Markovian stochastic dynamics of a classical Brownian particle is derived in the case that the effective temperature is not constant. After recapitulating the general formalism and its application to isothermal Brownian motion, an isolated ‘particle plus environment’ system is analyzed in the microcanomical ensemble. The ensuing generalized Kramers equation is shown to involve a microcanonical potential of mean force different from the conditional free energy, and the conditional entropy emerges as the relevant equilibrium potential. The Smoluchowski limit is presented, and the effective spatial diffusion coefficient is calculated. Upon coupling the system to a heat bath, the microcanonical theory is seen to correspond to the case of small heat conductance. The final result for the stochastic process in the position, momentum, and energy variables is shown to be consistent with both isothermal Kramers theory and nonlinear equilibrium thermodynamics.

Single beam laser induced fluorescence technique for plasma transport measurements

A technique for measuring ion transport using laser-induced fluorescence has been developed and tested in an argon plasma. It uses only one broadband beam thus being simpler than some previous techniques because no detection beam is required. First, a 5 μs laser pulse centered on 611 nm stimulates a transition from the metastable state in Ar(II) 3d 2G9/2 to 4p 2F7/20. A 4p 2F7/20 to 4s2D5/2transition rapidly results with emission at 461 nm. Upon cessation of the laser pulse, the 461 nm light in the detection volume does not return to its background level immediately because the 3d 2G9/2 level is partially depleted. The time history of the 461 nm signal in returning to steady-state background intensity provides a means of determining ion transport because the recovery signal is due to processes including ion excitation, diffusion, convection, and thermal motion. Measurements of the ion velocity distribution yield the contributions of thermal and convective effects to ion transport. By varying the laser beam diameter and the detection volume the plasma ion spatial diffusion coefficient D, and the time, τp it takes for processes other than transport to bring the 461 nm emission back to the steady-state background level are determined. For example, in one set of plasma conditions D=0.58±0.16 m2/s and τp=59±7 μs were found.

Existence of solutions to coagulation-fragmentation systems with diffusion

Abstract We show existence of solutions to an infinite system of parabolic equations obtained by adding spatial diffusion to the classical coagulation-fragmentation equations. In the case where the spatial diffusion coefficients depend on cluster size, local existence is shown. A global solution is obtained in the case where all the diffusivities are equal. As done in previous studies on the coagulation-fragmentation system, the method of proof relies on the truncation of the infinite system; essential in the derivation of the necessary estimates is the fact that the total concentration of particles satisfies a parabolic equation, to which the maximum principle can be applied.

Cosmic ray momentum diffusion in the presence of nonlinear Alfvén waves

Abstract. The relation between the spatial diffusion coefficient along the magnetic field, kII, and the momentum diffusion coefficient, Dp, for relativistic cosmic ray particles is modelled using Monte Carlo simulations. Wave fields with vanishing wave helicity and cross-helicity, constructed by superposing 'Alfvén-like' waves are considered. As the result, particle trajectories in high amplitude wave fields and then - by averaging over these trajectories - the values of transport coefficients are derived. The modelling is performed at various wave amplitudes, from δ B/B0 = 0.15 to 2.0, and for a number of wave field types. At our small amplitudes approximately the quasi-linear theory (QLT) estimates for kII and Dp are reproduced. However, with growing wave amplitude the simulated results show a small divergence from the QLT ones, with kII decreasing slower than theoretical prediction and the opposite being true for Dp. The wave field form gives only a slight influence on the wave-particle interactions at large wave amplitudes δ B/B0 ~ 1. The parameter characterizing the relative efficiency of the second-order to the first-order acceleration at shock waves, Dp κII is given in the QLT approximation by the Skilling formula V2A p2 / 9. In simulations together with increasing δ B it increases above this scale in all the cases under our study. Consequences of the present results for the second-order Fermi acceleration at shock waves are briefly addressed.

Brownian dynamics simulation of the lateral distribution of charged membrane components.

Brownian dynamics simulations were performed to study the contribution of electric interactions between charged membrane components to their lateral distribution in a two-dimensional viscous liquid (bilayer lipid membrane). The electrostatic interaction potential was derived from an analytical solution of the linearized Poisson-Boltzmann equation for point charges in an electrolyte solution--membrane--electrolyte solution system. Equilibrium as well as dynamic quantities were investigated. The lateral organization of membrane particles, modelled by mobile cylinders in a homogeneous membrane separating two electrolyte solutions was described by spatial distribution functions, diffusion coefficients and cluster statistics. Disorder, local order and crystal-like arrangements were observed as a function of the particle charge, the closest possible distances between the charges and the particle density. The simulations revealed that the system is very sensitive to the position of the charges with respect to the electrolyte solution-membrane interface. Electrostatic interactions of charges placed directly on the membrane surface were almost negligible, whereas deeper charges demonstrated pronounced interaction. Biologically relevant parameters corresponded at most to local and transient ordering. It was found that lateral electric forces can give rise to a preferred formation of clusters with an even number of constituents provided that the closest possible charge-charge distances are small. It is concluded that lateral electrostatic interactions can account for local particle aggregations, but their impact on the global arrangement and movement of membrane components is limited.

Three-dimensional spatial diffusion in optical molasses

We have studied the expansion of a small cloud of85Rb atoms in three-dimensional optical molasses (lin ⊥ lin and σ+ − σ− configurations) and observed diffusive motion. We determined the spatial-diffusion coefficients for various laser intensities and detunings, and compared them (in the case of lin ⊥ lin molasses) to values calculated from friction and momentum-diffusion coefficients of a one-dimensional (1D) theory of laser cooling. The predicted variations of the spatial-diffusion coefficient with laser intensity and detuning are in good qualitative agreement with the experimental data. We found that the minimal value observed experimentally, ≈ 6 × 10−4 cm2/s, lies within a factor of 3 of the 1D theoretical minimum, ≈, 26ħ/M, whereM is the atomic mass.

Experimental studies of the radial diffusion of energetic electrons in lower-hybrid wave heated plasmas in the Alcator-C tokamak

An upper limit for the cross-field spatial diffusion coefficient of energetic electrons in moderate density tokamak discharges has been measured. The energetic electrons were generated by square wave-modulated lower-hybrid (LH) wave injection. The decay of the fast electron population for electron energies εd≥100 keV following LH radio-frequency (RF) power shutoff was inferred from the time dependence of the radial profiles of the hard x-ray bremsstrahlung emission. The experimental results were fitted to a diffusive model that includes collisional losses and electric field acceleration. The fast electron diffusion coefficient consistent with the data is Df≤0.2 m2/s, at least in the inner half of the plasma. The corresponding global confinement time of the energetic electrons is estimated to be at least a factor of 2 greater than the energy confinement time of the bulk plasma in discharges of density n̄e=8×1019 m −3.

Relative Diffusion and Two-Point Microscale Correlation in Phase Space of Electrons in Localized Langmuir Fields11The project partially supported by National Natural Science Foundation of China through Grant No. 1880109.

We discuss both relative spatial diffusion coefficient and relative velocity diffusion coefficient for electrons interacting with coherent, localized Langmuir wave packets. We find that and have nearly the same positions in which they get a nonzero minimum for small relative position and relative velocity and the diffusion coefficients are greatly reduced dose to minima. We also find that the position in which has a minimum shifts when and vary, while the position for does not move. So we show that the coherent, localized wave packets can drive the formation of strong two-point microscale correlation in phase space only when r and u are small. We also discuss the lifetime of clump.

First-order Fermi acceleration in spherically symmetric flows - Solutions including quadratic losses

view Abstract Citations (22) References (44) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS First-Order Fermi Acceleration in Spherically Symmetric Flows: Solutions Including Quadratic Losses Becker, Peter A. Abstract We obtain a new, exact solution for the Green's function describing the transport of relativistic particles in steady, spherically symmetric background flows, including the effects of first-order Fermi acceleration, spatial diffusion, bulk advection, and losses proportional to the square of the particle momentum. The flow velocity of the background (scattering) plasma and the spatial diffusion coefficient are assumed to vary as v(r) is proportional to r^-α^ and k(p,r) is proportional to rbeta^K(p), respectively, where r is the radius and K(p) is an arbitrary function of the particle momentum p. The momentum loss rate is assumed to vary quadratically with the particle momentum as _loss_ = Ap^2^v/r, where A = constant. Examples of quadratic loss mechanisms include synchrotron and inverse Compton; in the synchrotron case, the implied magnetic field variation is B^2^ is proportional to v/r, which may be satisfied in galaxy cluster cooling flows. Losses exactly balance first-order Fermi acceleration at the critical momentum p_c_ = (2 - α)/(3A), which is a single constant for the entire flow if A = constant. Finite losses cause a compression of the dynamic range of the momentum variable relative to the lossless case (A = 0), since particles injected with momentum p_0_ are confined to the interval (p_0_, p_c_). The sign of v is unrestricted, and therefore our model can be used to study the transport of relativistic particles in both winds and accretion flows. Previous studies of particle transport in radio sources have considered the effects of spatial diffusion and synchrotron losses on relativistic electrons propagating through static background plasmas. The inclusion of bulk motions of the plasma affects both the energization of the electrons (via Fermi acceleration) and the spatial transport (via advection). In accretion flows, advection tends to drag the electrons inward, reducing the number escaping from the flow, while Fermi acceleration tends to increase the energy of the electrons that do escape. We find that the spectrum of electrons at large radii (and hence the associated synchrotron emission) is significantly harder in the presence of bulk motions, and that extended, power-law radio spectra are a natural consequence of either monoenergetic or power-law electron injection. In particular, we conclude that Fermi acceleration may help to explain the production of core-halo radio emission in cooling flows. Therefore core halos may provide independent evidence for dynamical cooling processes in clusters of galaxies. It has been previously pointed out that the flux of relativistic particles emerging from an accretion flow vanishes for certain values of the parameters (α,β,γ ) when K(p) is proportional to pgamma. This region of the parameter space is consequently unobservable if the relativistic particles are photons. However, if the relativistic particles are electrons, we show that observable power-law radio emission can be produced even when the emergent flux of electrons vanishes. Publication: The Astrophysical Journal Pub Date: September 1992 DOI: 10.1086/171769 Bibcode: 1992ApJ...397...88B Keywords: Cooling Flows (Astrophysics); Green'S Functions; Particle Acceleration; Radio Sources (Astronomy); Relativistic Particles; Electrons; Galactic Clusters; Plasma Physics; Radio Emission; Relativity; Synchrotron Radiation; Astrophysics; ACCELERATION OF PARTICLES; GALAXIES: COOLING FLOWS; METHODS: ANALYTICAL; PLASMAS; RADIO LINES: MOLECULAR: INTERSTELLAR; RELATIVITY full text sources ADS |

Diffusive acceleration of electrons in SN 1987A

We suggest that synchrotron radiation emitted by electrons undergoing diffusive acceleration at the supernova blast wave is the cause of the reappearance of radio emission from SN 1987A in 1990 July. Making reasonable assumptions concerning adiabatic losses and the magnetic field in the preexplosion stellar wind, we find the light curve and spectrum which would be produced by a constant rate of particle injection occurring in a localized region of the shock front; i.e., in a clump or knot. Observations indicate that two such clumps have been encountered to date. From the observed delay in the switch-on of the emission between 843 MHz and 4.8 GHz we find a spatial diffusion coefficient for electrons of κ<SUB>normal</SUB>_ = 2 x 10^20^ m^2^ s^-1^. This value agrees with that found for the second clump and is substantially greater than the Bohm value. Estimating the magnetic field to be 10^-7 T and assuming the diffusion coefficient is constant leads to the conclusion that the shock encountered the first clump on about day 900 and the second clump on day 1190. From the observed spectrum we find the compression ratio of the shock responsible for electron acceleration to be ~2.7. This implies strong modification of the shock front which may be due to the acceleration of cosmic rays.

Structure of the Halley bow shock

The similarities between cosmic ray mediated shocks and the bow shock of comet Halley are discussed briefly and a two-fluid model is presented. Since the shock is weak, we use a perturbation scheme to study the shock structure. We show that the shock is smoothed by the pick-up ions and that the length scale of the cometary ion foot is determined by the diffusive length scale of the cometary ions. Our results suggest that the spatial diffusion coefficient of the cometary ions is close to its “Bohm” limit in the vicinity of the shock. In addition, we suggest that small irregularities on the smoothed shock may be explained by means of a cometary ion driven instability.