Normal Diffusive Behavior Research Articles

Continuous-time random walk: exact solutions for the probability density function and first two moments

We consider the decoupled continuous-time random walk model with a finite characteristic waiting time and approximate jump length variance. We take the waiting time probability density function (PDF) given by a combination of the exponential and the Mittag-Leffler function. Using this waiting time PDF, we investigate the diffusion behavior for all times. We obtain exact solutions for the first two moments and the PDF for the force-free and linear force cases. Due to the finite characteristic waiting time and jump length variance, the model presents, for the force-free case, normal diffusive behavior in the long-time limit. Further, the model can describe anomalous behavior at intermediate times.

Crossover from Normal to Anomalous Diffusion in Systems of Field-Aligned Dipolar Particles

Using molecular dynamics simulations we investigate the translational dynamics of particles with dipolar interactions in homogenous external fields. For a broad range of concentrations, we find that the anisotropic, yet normal diffusive behavior characterizing weakly coupled systems becomes anomalous both parallel and perpendicular to the field at sufficiently high dipolar coupling and field strength. After the ballistic regime, chain formation first yields cagelike motion in all directions, followed by transient, mixed diffusive-superdiffusive behavior resulting from cooperative motion of the chains. The enhanced dynamics disappears only at higher densities close to crystallization.

Heavy-tailed targets and (ab)normal asymptotics in diffusive motion

We show that, under suitable confinement conditions, the ordinary Fokker–Planck equation may generate non-Gaussian heavy-tailed probability density functions (pdfs) (like, for example, Cauchy or more general Lévy stable distributions) in its long-time asymptotics. In fact, all heavy-tailed pdfs known in the literature can be obtained this way. For the underlying diffusion-type processes, our main focus is on their transient regimes and specifically the crossover features, when an initially infinite number of pdf moments decreases to a few or none at all. The time dependence of the variance (if in existence), ∼tγ with 0<γ<2, may in principle be interpreted as a signature of subdiffusive, normal diffusive or superdiffusive behavior under confining conditions; the exponent γ is generically well defined in substantial periods of time. However, there is no indication of any universal time rate hierarchy, due to a proper choice of the driver and/or external potential.

Self-gravitating Brownian particles in two dimensions: the case of N = 2 particles

We study the motion of N=2 overdamped Brownian particles in gravitational interaction in a space of dimension d=2. This is equivalent to the simplified motion of two biological entities interacting via chemotaxis when time delay and degradation of the chemical are ignored. This problem also bears some similarities with the stochastic motion of two point vortices in viscous hydrodynamics [Agullo & Verga, Phys. Rev. E, 63, 056304 (2001)]. We analytically obtain the density probability of finding the particles at a distance r from each other at time t. We also determine the probability that the particles have coalesced and formed a Dirac peak at time t (i.e. the probability that the reduced particle has reached r=0 at time t). Finally, we investigate the variance of the distribution <r^2> and discuss the proper form of the virial theorem for this system. The reduced particle has a normal diffusion behaviour for small times with a gravity-modified diffusion coefficient <r^2>=r_0^2+(4k_B/\xi\mu)(T-T_*)t, where k_BT_{*}=Gm_1m_2/2 is a critical temperature, and an anomalous diffusion for large times <r^2>~t^(1-T_*/T). As a by-product, our solution also describes the growth of the Dirac peak (condensate) that forms in the post-collapse regime of the Smoluchowski-Poisson system (or Keller-Segel model) for T<T_c=GMm/(4k_B). We find that the saturation of the mass of the condensate to the total mass is algebraic in an infinite domain and exponential in a bounded domain.

Continuous-time random walk: Crossover from anomalous regime to normal regime

We consider decoupled continuous time random walk with finite characteristic waiting time and jump length variance. We take approximate jump length probability distribution and waiting time probability distribution given by a product of power-law and exponential function. Using this waiting time probability distribution we study diffusion behaviors for all the time. Due to the finite characteristic waiting time and jump length variance the model presents normal diffusive behavior in the long-time limit. However, the model can describe anomalous behavior at the short and intermediate times. In particular, the model can describe subdiffusive, normal, and superdiffusive behaviors at the short times. Moreover, exact solution for probability distribution of the system is also investigated.

Discriminating between normal and anomalous random walks

Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, {x2(t) proportional t, while anomalous behavior is expected to show a different time dependence, x2(t) proportional t{delta} with delta<1 for subdiffusive and delta>1 for superdiffusive motions. Here we explore in details the fact that this kind of qualification, if applied straightforwardly, may be misleading: there are anomalous transport motions revealing perfectly "normal" diffusive character (x2(t) proportional t) yet being non-Markov and non-Gaussian in nature. We use recently developed framework of Monte Carlo simulations which incorporates anomalous diffusion statistics in time and space and creates trajectories of such an extended random walk. For special choice of stability indices describing statistics of waiting times and jump lengths, the ensemble analysis of anomalous diffusion is shown to hide temporal memory effects which can be properly detected only by examination of formal criteria of Markovianity (fulfillment of the Chapman-Kolmogorov equation).

Effect of aspect ratio on transverse diffusive broadening: A lattice Boltzmann study

We study scaling laws characterizing the interdiffusive zone between two miscible fluids flowing side by side in a Y-shape laminar micromixer using the lattice Boltzmann method. The lattice Boltzmann method solves the coupled three-dimensional (3D) hydrodynamics and mass transfer equations and incorporates intrinsic features of 3D flows related to this problem. We observe the different power-law regimes occurring at the center of the channel and close to the top/bottom wall. The extent of the interdiffusive zone scales as the square root of the axial distance at the center of the channel. At the top/bottom wall, we find an exponent 1/3 at early stages of mixing as observed in the experiments of Ismagilov [Appl. Phys. Lett. 76, 2376 (2000)]. At a larger distance from the entrance, the scaling exponent close to the walls changes to 1/2 [J.-B. Salmon and A. Adjari, J. Appl. Phys. 101, 074902 (2007)]. Here, we focus on the effect of the finite aspect ratio on diffusive broadening. Interestingly, we find the same scaling laws regardless of the channel's aspect ratio. However, the point at which the exponent 1/3 characterizing the broadening at the top/bottom wall reverts to the normal diffusive behavior downstream strongly depends on the aspect ratio. We propose an interpretation of this observation in terms of the shear rate at the side walls. A criterion for the range of aspect ratios with non-negligible effect on diffusive broadening is also provided.

Diffusion of Nanoparticles in Semidilute and Entangled Polymer Solutions

We studied the diffusion of gold nanoparticles in semidilute and entangled solutions of polystyrene (PS) in toluene using fluctuation correlation spectroscopy (FCS). The polymer concentration was varied from approximately 6c* to 20c*, where c* is the overlap concentration. In our experiments, the particle radius (R approximately 2.5 nm) was much smaller compared to the radius of gyration (Rg approximately 18 nm) of the chain but comparable to the average mesh size (xi) of the fluctuating polymer network. The diffusion coefficient (D) of the particles decreased monotonically with polymer concentration and it can be fitted with a stretched exponential function, D=D0 exp(-microcnu), with the value of the scaling parameter, nu approximately 0.9. At high concentration of the polymer, a clear subdiffusive motion of the particles was observed. The results were compared with the diffusion of free dyes (coumarin 480), which showed normal diffusive behavior for all concentrations.

Probing the Dynamic Organization of Transcription Compartments and Gene Loci within the Nucleus of Living Cells

The three-dimensional organization of nuclear compartments within living cells determines genome function and yet their underlying self-organizing principles are unclear. We visualize in real-time transcriptionally active compartments (TCs) by the transient enrichment of fluorescently-labeled uridine 5′-triphosphate molecules within living cells. These TCs partially colocalize with active RNA-Pol II in the cell nucleus. Fluorescence anisotropy maps of chromatin compaction evidences a more open chromatin structure at the TCs. Using live-cell timelapse imaging, heterogeneity in the dynamic behavior of TCs has been revealed which falls into three distinct classes: subdiffusive, super-diffusive, and normal diffusive behavior. In contrast, the mobility of a candidate gene locus, either in the repressed or activated state, undergoes a differential restricted motion that is coupled to TC movement. Further TC dynamics is directly affected by small molecule chromatin structure modulators and adenosine triphosphate depletion. This heterogeneous behavior in TC dynamics within living cells could provide an interesting paradigm to explore the spatiotemporal dimension to gene transcription control.

The Grain-Boundary Diffusion of Zn in α-Fe

Using semi-infinite amounts of pure Zn and pure Fe as the diffusing media and maintaining constant boundary conditions, the diffusion of Zn into αFe has been experimentally studied at temperatures of 400, 500, and 725 °C. Long diffusion times (4 days) were used to allow the various intermetallic phases to become well developed. It was determined that diffusion of Zn into Fe is not “anomalously rapid,” but follows normal diffusion behavior for polycrystalline metals in that both lattice diffusion and grain-boundary diffusion contribute to mass transport for the temperatures investigated. The grain-boundary diffusion coefficients determined are consistent with those for many other metal systems.

Thermal diffusion behavior of implanted germanium atoms in silicon dioxide film measured by high-resolution RBS

The depth profile is a very important information for nanoparticle formation in very thin SiO 2 films by ion implantation prior to TEM observation, because of the thermal diffusion of implanted atoms. Here, we have investigated depth profiles of Ge atoms in SiO 2 thin films by high-resolution Rutherford backscattering spectrometry (HR-RBS). Samples were prepared by Ge negative ion implantation into 25-nm-thick SiO 2 films on Si substrates at 10 keV with a fluence of 1 × 10 15 ions/cm 2, and subsequent annealing at various temperatures. As implanted, and after annealing at less than 700 °C, the depth profiles were Gaussian and corresponded well to the calculated profiles. Jointed-half-Gaussian curves were fitted to the profiles. The Ge diffusion coefficients were estimated by applying a linear diffusion equation to the fitting curves, and they indicated a rapid diffusion to the surface side rather than the interface side. After annealing at 900 °C, there appeared two clear Ge peaks in the HR-RBS spectra for SiO 2, with strong asymmetric shape at the SiO 2/Si interface. Cross-sectional TEM observations revealed that a Ge concentration of about 1 at.% yields Ge nanoparticles of about 2 nm in diameter. In this low energy implantation, the vacancies due to the implantation were found near the surface, indicating anisotropic diffusion. At high temperature of 900 °C, more normal Ge diffusion behavior should occur, such as strong anisotropic diffusion in the thin SiO 2 film with an interface of crystalline Si. The HR-RBS was useful to reveal nanoscale profiles, and these results will assist in planning Ge nanoparticle fabrication in thin SiO 2 films on Si.

Diffusive Dynamics of Vesicles Tethered to a Fluid Supported Bilayer by Single-Particle Tracking

We recently introduced a method to tether intact phospholipid vesicles onto a fluid supported lipid bilayer using DNA hybridization (Yoshina-Ishii, C.; Miller, G. P.; Kraft, M. L; Kool, E. T.; Boxer, S. G. J. Am. Chem. Soc. 2005, 127, 1356-1357). Once tethered, the vesicles can diffuse in two dimensions parallel to the supported membrane surface. The average diffusion coefficient, D, is typically 0.2 microm(2)/s; this is 3-5 times smaller than for individual lipid or DNA-lipid conjugate diffusion in supported bilayers. In this article, we investigate the origin of this difference in the diffusive dynamics of tethered vesicles by single-particle tracking under collision-free conditions. D is insensitive to tethered vesicle size from 30 to 200 nm, as well as a 3-fold change in the viscosity of the bulk medium. The addition of macromolecules such as poly(ethylene glycol) reversibly stops the motion of tethered vesicles without causing the exchange of lipids between the tethered vesicle and supported bilayer. This is explained as a depletion effect at the interface between tethered vesicles and the supported bilayer. Ca ions lead to transient vesicle-vesicle interactions when tethered vesicles contain negatively charged lipids, and vesicle diffusion is greatly reduced upon Ca ion addition when negatively charged lipids are present both in the supported bilayer and tethered vesicles. Both effects are interesting in their own right, and they also suggest that tethered vesicle-supported bilayer interactions are possible; this may be the origin of the reduction in D for tethered vesicles. In addition, the effects of surface defects that reversibly trap diffusing vesicles are modeled by Monte Carlo simulations. This shows that a significant reduction in D can be observed while maintaining normal diffusion behavior on the time scale of our experiments.

Transport in polygonal billiards

We present in this work a numerical study of the dynamics of ensembles of point particles within a polygonal billiard chain. This billiard is a system with no exponential instability. Our numerical results suggest that some members of the family exhibit normal diffusive behavior while others present anomalous diffusion. Our conclusions are drawn from the numerical evaluation of the mean square displacement, the velocity autocorrelation function and its spectral analysis. Furthermore we analyze the properties of the incoherent scattering function. The super Burnett coefficient seems to be ill defined in all systems. The multifractal analysis of the spectrum of the velocity autocorrelation functions is also reported. Finally, we study the heat conduction in our polygonal chain.

Thermal diffusion of supersonic solitons in an anharmonic chain of atoms.

We study the nonequilibrium diffusion dynamics of supersonic lattice solitons in a classical chain of atoms with nearest-neighbor interactions coupled to a heat bath. As a specific example we choose an interaction with cubic anharmonicity. The coupling between the system and a thermal bath with a given temperature is made by adding noise, delta correlated in time and space, and damping to the set of discrete equations of motion. Working in the continuum limit and changing to the sound velocity frame we derive a Korteweg-de Vries equation with noise and damping. We apply a collective coordinate approach which yields two stochastic ODEs which are solved approximately by a perturbation analysis. This finally yields analytical expressions for the variances of the soliton position and velocity. We perform Langevin dynamics simulations for the original discrete system which confirm the predictions of our analytical calculations, namely, noise-induced superdiffusive behavior which scales with the temperature and depends strongly on the initial soliton velocity. A normal diffusion behavior is observed for solitons with very low energy, where the noise-induced phonons also make a significant contribution to the soliton diffusion.

Anomalous tag diffusion in the asymmetric exclusion model with particles of arbitrary sizes.

Anomalous behavior of correlation functions of tagged particles are studied in generalizations of the one-dimensional asymmetric exclusion problem. In these generalized models the range of the hard-core interactions are changed and the restriction of relative ordering of the particles is partially broken. The models probing these effects are those of biased diffusion of particles having size S=0,1,2, em leader, or an effective negative "size" S=-1,-2, em leader, in units of lattice space. Our numerical simulations show that irrespective of the range of the hard-core potential, as long some relative ordering of particles are kept, we find suitable sliding-tag correlation functions whose fluctuations growth with time anomalously slow (t1/3), when compared with the normal diffusive behavior (t1/2). These results indicate that the critical behavior of these stochastic models are in the Kardar-Parisi-Zhang (KPZ) universality class. Moreover a previous Bethe-ansatz calculation of the dynamical critical exponent z, for size S> or =0 particles is extended to the case S<0 and the KPZ result z=3/2 is predicted for all values of S in Z.

Statics and dynamics of ethane molecules in AlPO(4)-5: a molecular dynamics simulation study.

From an experimental perspective, there has been disagreement among researchers on whether ethane would display single-file or normal diffusive behavior in the channels of AlPO(4)-5. Pulsed field gradient nuclear magnetic resonance measurements implied single-file diffusion, while quasielastic neutron scattering showed normal diffusion. In this paper we present the results of extensive classical molecular dynamics simulations of the diffusion of ethane molecules adsorbed in AlPO(4)-5. Our aim is to provide microscopic details of the static and dynamic properties of the adsorbed molecules in order to verify whether the conditions for the single-file regime can be achieved in a nondefective AlPO(4)-5 crystal structure.

Spectral properties and anomalous transport in a polygonal billiard.

We analyze a class of polygonal billiards, whose behavior is conjectured to exhibit a variety of interesting dynamical features. Correlation functions are numerically investigated, and in a subclass of billiard tables they give indications about a singular continuous spectral measure. By lifting billiard dynamics we are also able to study transport properties: the (normal or anomalous) diffusive behavior is theoretically connected to a scaling index of the spectral measure; the proposed identity is shown to agree with numerical simulations. (c) 2000 American Institute of Physics.

Diffusion in Metals, Quasicrystals, and Intermetallic Compounds

ABSTRACTIn this paper recent developments of diffusion in three related areas will be reviewed: The first part is devoted to self- and solute diffusion in metals with particular emphasis on Al. Contrary to most other metallic solvents, diffusion of transition elements in Al is anomalous in several respects: diffusion is very slow, activation enthalpies, pre-exponential factors and activation volumes are unusually high. By contrast, non-transition elements in Al show more or less normal solute diffusion behaviour. The anomalous behaviour is attributed to a strong repulsive interaction between transition metal solutes and vacancies. Ab-initio calculations could help to understand this well-documented diffusion problem in detail.In the second part very recent diffusion studies on single crystals of the Al-base quasicrystalline intermetallic compound Al-Pd-Mn will be discussed. Diffusion of Zn, Ge, Mn, Fe, Co, Pd and Au has been studied by various groups. At least in the high-temperature regime diffusion in the quasicrystal – despite some differences in detail – shows striking similarities to diffusion in Al, for which a vacancy-type mechanism is generally accepted. The activation volumes of +0.67 and +0.74 atomic volumes measured for Mn- and Zn-diffusion in Al-Pd-Mn strongly favour a vacancy mechanism as well. For the low temperature regime of Pd and Au diffusion the possibility of a phason-related mechanism is discussed.The third part deals with recent investigations of iron-aluminides and iron-silicides for which Fe self-diffusion and diffusion of selected foreign elements (Ge in Fe–Si, In and Zn in Fe–Al) has been investigated. There is no doubt that vacancies mediate the diffusion process. Within this general mechanism a number of factors like the crystal structure, the state of order, the composition, and the type of bonding have strong influence on diffusion. Such factors are discussed also in connection with results from positron annihilation and M6ssbauer spectroscopy.

Intrinsic self-diffusion and substitutional Al diffusion in α-Ti

Self-diffusion and Al impurity diffusion were studied in the α (h.c.p.) phase of Ti. We used four Ti materials with different impurity contents, including ultrapure Ti with extremely small concentrations of interstitial impurities (Fe, Co and Ni). The self-diffusion measurements are performed with the radiotracer 44Ti and the ion beam sputtering technique. For Al diffusion measurements in-depth profiling by secondary ion mass spectrometry is applied. The measurements are made both perpendicular (⊥) and parallel (‖) to the c axis using single crystals and coarse-grained polycrystals. The measurements on ultrapure α-Ti yield the Arrhenius parameters D 0⊥ = 1.35 × 10 −3m 2/s and Q ⊥ = 303 ± 2kJ/mol for self-diffusion and D 0⊥ = 6.6 × 10 −3m 2/s and Q ⊥ = 329 ± 2kJ/mol for Al diffusion. The anisotropy factor D |/ D ⊥ ≈ 0.5 for self-diffusion and ≈ 0.65 for Al diffusion. These results are treated as intrinsic diffusion properties of α-Ti. It is demonstrated that they are well consistent with the normal diffusion behaviour in other h.c.p. metals. We conclude that both self-diffusion and substitutional solute diffusion in α-Ti are intrinsically normal and dominated by the vacancy mechanism. Diffusion in less pure α-Ti occurs faster and with a smaller activation energy. This effect is explained by the enhancement of atomic mobility in the matrix material owing to the interstitially dissolved fast-diffusing impurities.

Influence of dislocation loops created by amorphizing implants on point defect and boron diffusion in silicon

An experiment has been performed in which wafers with boron-doped, buried marker layers were implanted with 100 keV, 2×1015 cm−2 Si. This is an amorphizing implant. A second implant of B was introduced prior to any post-implant annealing such that the implanted B was completely contained within the preamorphized region. After the implants, samples were annealed at various temperatures for various times and secondary-ion mass spectroscopy was used to obtain the dopant profiles. It was found that the buried marker layer exhibited normal transient-enhanced diffusion behavior. However, the B in the preamorphized region did not experience any significant amount of motion. This suggests that the solid phase regrowth of the amorphous layer did not cause a redistribution of the dopant atoms within that layer, and also the plane of dislocation loops that form at the amorphous/crystalline interface is an effective barrier against the interstitial damage diffusing upwards from the nonamorphized tail of the amorphizing Si implant. The same behavior was observed when As or P implants are used instead of the B implant. This type of behavior has been simulated using a model considering the growth of stacking faults bounded by dislocation loops.