Brownian Simulations Research Articles

Diffusion in a dendritic spine: The role of geometry

Dendritic spines, the sites where excitatory synapses are made in most neurons, can dynamically regulate diffusing molecules by changing their shape. We present here a combination of theory, simulations, and experiments to quantify the diffusion time course in dendritic spines. We derive analytical formulas and compared them to Brownian simulations for the mean sojourn time a diffusing molecule stays inside a dendritic spine when either the molecule can reenter the spine head or not, once it is located in the spine neck. We show that the spine length is the fundamental regulatory geometrical parameter for the diffusion decay rate in the neck only. By changing the spine length, dendritic spines can be dynamically coupled or uncoupled to their parent dendrites, which regulates diffusion, and this property makes them unique structures, different from static dendrites.

Microstructure and Acoustical Macro-Behavior: Approach by Reconstruction of a Representative Elementary Cell

The fundamental issue of determining acoustic properties of porous media from their local geometry is examined in this Ph.D. dissertation thesis, thanks to a sample of open-cell aluminium foam analyzed by axial computed microtomography. Various geometric properties are measured to characterize the experimental sample at the cell size level. This is done in order to reconstruct a porous medium by means of idealized three- and two-dimensional unit cells. The frequency dependent thermal and velocity fields governing the propagation and dissipation of acoustic waves through rigid porous media are computed by Brownian motion simulation and the finite element method, respectively. Macroscopic behavior is derived by spatial averaging of the local fields. Our results are compared to experimental data obtained from impedance tube measurements. First, this approach leads to the identification of the macroscopic parameters involved in Pride and Lafarge semiphenomenological models. Second, it yields a direct access to thermal and viscous dynamic permeabilities. However, the bidimensional model underestimates the static viscous permeability as well as the viscous characteristic length; what thus require a three-dimensional implementation.

Physical properties of a very diffuse HI structure at high Galactic latitude

The main goal of this analysis is to present a new method to estimate the physical properties of diffuse cloud of atomic hydrogen observed at high Galactic latitude. This method, based on a comparison of the observations with fractional Brownian motion simulations, uses the statistical properties of the integrated emission, centroid velocity and line width to constrain the physical properties of the 3D density and velocity fields, as well as the average temperature of HI. We applied this method to interpret 21 cm observations obtained with the Green Bank Telescope of a very diffuse HI cloud at high Galactic latitude located in Firback North 1. We first show that the observations cannot be reproduced solely by highly-turbulent CNM type gas and that there is a significant contribution of thermal broadening to the line width observed. To reproduce the profiles one needs to invoke two components with different average temperature and filling factor. We established that, in this very diffuse part of the ISM, 2/3 of the column density is made of WNM and 1/3 of thermally unstable gas (T ~2600 K). The WNM gas is mildly supersonic ( ~1) and the unstable phase is definitely sub-sonic ( ~0.3). The density contrast (i.e., the standard deviation relative to the mean of density distribution) of both components is close to 0.8. The filling factor of the WNM is 10 times higher that of the unstable gas, which has a density structure closer to what would be expected for CNM gas. This field contains a signature of CNM type gas at a very low level (N_H ~ 3 x 10^19) which could have been formed by a convergent flow of WNM gas.

Primitive Chain Network Simulations of Damping Functions for Shear, Uniaxial, Biaxial and Planar Deformations

Damping functions of entangled polymers for shear, uniaxial, biaxial, and planar deformations, as well as normal stress ratios for shear deformations, were obtained from Brownian simulations making use of the primitive chain network model. To investigate the effect of the force balance over the entanglements and of the convective constraint release mechanism, comparisons with predictions of earlier theories and with experimental data in the literature were performed. It was found that the obtained damping functions are close to the three-chain theory [Marrucci, Greco, and Ianniruberto, Macromol Symp, 158, 57 (2000)] suggesting that the force balance is a dominant correction over the basic Doi-Edwards theory as compared with the effect of convective constraint release. Furthermore, the predicted normal stress ratio in shear, i.e., a quantity very sensitive to the different assumptions, is in good agreement with experiments, suggesting that the combination of force balance, convective constraint release, and other relaxation modes, as accounted for through the primitive chain network model, is quite acceptable.

Primitive chain network simulations for branched polymers

We present simulations of branched polymer dynamics based on a sliplink network model, which also accounts for topological change around branch points, i.e., for branch-point diffusion. It is well-known that, with the exception of stars, branched polymers may show a peculiar rheological behavior due to the exceptionally slow relaxation of the backbone chains bridging branch points. Though Brownian simulations based on sliplinks are powerful tools to study the motion of polymers and to predict rheological properties, none of the existing methods can simulate the relaxation of the bridge chains. The reason for that is lack of a rule for network topology rearrangement around branch points, so that entanglements between bridge chains cannot be renewed. Therefore, we introduce in this paper one possible branch-point mobility rule into our primitive chain network model. For star polymers, diffusion coefficients were calculated and compared with experiments. For both star and H-shaped polymers, diffusion was simulated both with and without the new rule, and the effect on linear viscoelasticity was also determined in one case.

Diffusion-influenced excited-state reversible transfer reactions, A*+B⇌C*+D, with two different lifetimes: Theories and simulations

We report accurate Brownian simulation results for the kinetics of the pseudo-first-order diffusion-influenced excited-state reversible transfer reaction A(*) + Bright harpoon over left harpoonC(*) + D with two different lifetimes using two different propagation algorithms. The results are used to test approximate solutions for this many-particle problem. Available theories fail when one of the two reactions or (decay) rate constants is large. To remedy this situation, we develop two uniform approximations, which are based on introducing a generalized Smoluchowski term into the relaxation-time approximation. The best of these is the extended unified theory of reversible target reactions, which reduces correctly in all limits and exhibits superior agreement with simulations.

Viscous Permeability of Random Fiber Structures: Comparison of Electrical and Diffusional Estimates with Experimental and Analytical Results

An investigation of the permeability for viscous flow through random fiber structures used as preforms in composite fabrication processes is presented. A method derived from the electrical conduction principles is employed to predict the viscous permeability of the fiber structures directly from their formation factor, specific surface area, and porosity. The recent Brownian diffusion random-walk simulation results on the molecular survival time are used to derive an upper bound and an approximation to the viscous permeability of these structures. The results are compared to experimental data and theoretical models of the literature. It is found that the conduction-based method provides very good permeability estimates in most cases, resulting in an overall ratio of the theoretical prediction to experimental measurement in the proximity of 1 for the over 500 experimental points utilized. This is one of the most comprehensive experimental validations of the conduction-based method to date and its first validation for anisotropic particle beds. A tortuosity analysis indicates a stronger pore-geometry effect on viscous flow relative to any type of diffusional flow, enhancing the understanding of the differences between pressure-driven and diffusion-driven composite manufacturing techniques.

Memoryless control of boundary concentrations of diffusing particles

Flux between regions of different concentration occurs in nearly every device involving diffusion, whether an electrochemical cell, a bipolar transistor, or a protein channel in a biological membrane. Diffusion theory has calculated that flux since the time of Fick (1855), and the flux has been known to arise from the stochastic behavior of Brownian trajectories since the time of Einstein (1905), yet the mathematical description of the behavior of trajectories corresponding to different types of boundaries is not complete. We consider the trajectories of noninteracting particles diffusing in a finite region connecting two baths of fixed concentrations. Inside the region, the trajectories of diffusing particles are governed by the Langevin equation. To maintain average concentrations at the boundaries of the region at their values in the baths, a control mechanism is needed to set the boundary dynamics of the trajectories. Different control mechanisms are used in Langevin and Brownian simulations of such systems. We analyze models of controllers and derive equations for the time evolution and spatial distribution of particles inside the domain. Our analysis shows a distinct difference between the time evolution and the steady state concentrations. While the time evolution of the density is governed by an integral operator, the spatial distribution is governed by the familiar Fokker-Planck operator. The boundary conditions for the time dependent density depend on the model of the controller; however, this dependence disappears in the steady state, if the controller is of a renewal type. Renewal-type controllers, however, produce spurious boundary layers that can be catastrophic in simulations of charged particles, because even a tiny net charge can have global effects. The design of a nonrenewal controller that maintains concentrations of noninteracting particles without creating spurious boundary layers at the interface requires the solution of the time-dependent Fokker-Planck equation with absorption of outgoing trajectories and a source of ingoing trajectories on the boundary (the so called albedo problem).

Instability of a fluid–fluid interface in driven colloidal mixtures

A Rayleigh–Taylor-like interface instability is studied in a compressible Brownian Yukawafluid mixture on the ‘molecular’ scales of length and time of the individual particles. As amodel, a two-dimensional phase-separated symmetric binary mixture of colloidal particlesof type A and B with a fluid–fluid interface separating an A-rich phase from a B-rich phaseis investigated, by means of Brownian computer simulations, when brought intonon-equilibrium via a constant external driving field which acts differently on the differentparticles and perpendicular to the interface. Two different scenarios are observed whichoccur either for high or for low interfacial free energies as compared to the drivingforce. In the first scenario for high interfacial tension, the critical wavelengthλc of the unstable interface modes is in good agreement with the classical Rayleigh–Taylor formulaprovided that dynamically rescaled values for the interfacial tension are used. The wavelengthλc increases with time, representing an effect of self-healing of the interface due to a localdensity increase near the interface. The Rayleigh–Taylor formula is confirmed even ifλc is of the order of a molecular correlation length. In the second scenario for very largedriving forces as compared to the interfacial line tensions, on the other hand, the particlespenetrate the interface easily due to the driving field and form microscopic lanes with awidth different from the predictions of the classical Rayleigh–Taylor formula. The resultsare of relevance for phase-separating colloidal mixtures in a gravitational or electric field.

Computation of the dynamic thermal properties of a three-dimensional unit cell of porous media by Brownian motion simulation

Acoustic dissipation in porous media is mainly due to viscous and thermal mechanisms that occur in the pores of the microstructure. The purpose of this study is the determination of the macroscopic dynamic acoustic bulk modulus and thermal permeability of real foams from a local scale approach. To achieve this goal, two distinct steps are followed. First, the local geometry of a real foam is obtained using computed microtomography (μCT), then a periodic and regularly paving space tetrakaidecahedron cell is identified from the microstructure. Second, the heat equation is solved for the geometrical model. The paper provides a three-dimensional application of the efficient simulation technique of Brownian motion proposed by Torquato et al. for steady state diffusion-controlled problems [Appl. Phys. Lett. 55, 1847–1849 (1989)] and adapted by Lafarge [Poromechanics II, 708 (2002)] in a bi-dimensional case. The influence of the model’s microstructural details (anisotropy, and struts junction and cross-section) on the macroscopic properties are studied. The predictions of the macroscopic properties using this local scale approach are then compared to experimental measurements.

Nonlinear sliding friction of adsorbed overlayers on disordered substrates

We study the response of an adsorbed monolayer on a disordered substrate under a driving force using Brownian molecular-dynamics simulation. We find that the sharp longitudinal and transverse depinning transitions with hysteresis still persist in the presence of weak disorder. However, the transitions are smeared out in the strong disorder limit. The theoretical results here provide a natural explanation for the recent data for the depinning transition of Kr films on gold substrate.

체커보드 형상을 가진 3차원 복합소재의 연결도와 전도율

We consider the problem of whether the three-dimensional checkerboard has the connectivity. For this purpose, we first consider the problem of determining the effective conductivity of a checkerboard-shaped composite material by the Brownian motion simulation method. Specifically, we use the efficient first-passage-time technique. Simulation results show that the effective conductivity of the three-dimensional checkerboard increases faster than the two-dimensional counterpart as the contrast between the phase conductivities increases. This implies that the three-dimensional checkerboard's connectivity is stronger than the two-dimensional checkerboard's and thus each phase material of the three-dimensional checkerboard is more likely to be connected than not to be connected.<br/>

Brownian Ionic Channel Simulation

The simulation of ion transport in biological ion channels presents numerous interesting challenges. Since there are accurate structures for only a handful of channel proteins it is difficult to predict solely from the conformation how internal and external structure of the molecule will affect it's ionic transport characteristics. This problem is compounded by the fact that the electrostatics and dynamic behaviour of these molecules is, only now, beginning to be understood. We present an ion transport simulation methodology based on a self-consistent Brownian/Poisson technique, that is capable of resolving ion transport on ps-μs timescales including the effect of long range electric fields in complex dielectric structures. The results of the self-consistent Brownian simulation are compared against the commercial drift diffusion simulation package Taurus and experimental measurements of the ion conduction for the Kcsa bacterial ion channel protein.

Unified theory of reversible target reactions

We discuss two fundamental reversible diffusion influenced reactions: (i) A+B⇌C and (ii) A+B⇌C+D. In the pseudo-unimolecular case, we prove that reaction (i) is a special case of (ii), which thus constitutes a unified reversible problem of multiparticle kinetics. For static A and C (the “target” limit), we suggest to treat this problem as follows. First we generalize the Smoluchowski theory to reaction (ii). In Laplace space, we combine this with a power-law theory, determining the combination coefficient so that the unified theory reduces correctly in all known limits. We also show how to rewrite it in the time domain, with the generalized Smoluchowski theory as the leading term. Comparison with Brownian simulations shows near perfect agreement for both versions of our theory under all conditions.

Accurate solution for the many body ABCD problem

An accurate semi-analytic solution for reversible diffusion-influenced kinetics is presented in the pseudo-unimolecular target limit. It is shown to exhibit excellent agreement with Brownian simulations of the A+B⇌C+D reaction.

On the Use of Fractional Brownian Motion Simulations to Determine the Three‐dimensional Statistical Properties of Interstellar Gas

Based on fractional Brownian motion (fBm) simulations of three-dimensional gas density and velocity fields, we present a study of the statistical properties of spectroimagery observations (channel maps, integrated emission, and line centroid velocity) in the case of an optically thin medium at various temperatures. The power spectral index ?W of the integrated emission is confirmed to be that of the three-dimensional density field (?n) provided that the medium's depth is at least of the order of the largest transverse scale in the image, and the power spectrum of the centroid velocity map is found to have the same index ?C as that of the velocity field (?v). Further tests with non-fBm density and velocity fields show that this last result holds and is not modified by the effects of density-velocity correlations. A comparison is made with the theoretical predictions of Lazarian & Pogosyan.

What drives the translocation of stiff chains?

We study the dynamics of the passage of a stiff chain through a pore into a cell containing particles that bind reversibly to it. Using Brownian molecular dynamics simulations we investigate the mean first-passage time as a function of the length of the chain inside for different concentrations of binding particles. As a consequence of the interactions with these particles, the chain experiences a net force along its length whose calculated value from the simulations accounts for the velocity at which it enters the cell. This force can in turn be obtained from the solution of a generalized diffusion equation incorporating an effective Langmuir adsorption free energy for the chain plus binding particles. These results suggest a role of binding particles in the translocation process that is in general quite different from that of a Brownian ratchet. Furthermore, nonequilibrium effects contribute significantly to the dynamics; e.g., the chain often enters the cell faster than particle binding can be saturated, resulting in a force several times smaller than the equilibrium value.

An efficient brownian motion simulation method for the conductivity of a digitized composite medium

We use the first-passage-time formulation by Torquato, Kim and Cule [J. Appl. Phys., Vol. 85. pp. 1560-1571 (1999)], which makes use of the first-passage region in association with the diffusion tracer’s Brownian movement, and develop a new efficient Brownian motion simulation method to compute the effective conductivity of digitized composite media. By using the new method, one can remarkably enhance the speed of the Brownian walkers sampling the medium and thus reduce the computation time. In the new method, we specifically choose the first-passage regions such that they coincide with two, four, or eight digitizing units according to the dimensionality of the composite medium and the local configurations around the Brownian walkers. We first obtain explicit solutions for the relevant first-passage-time equations in two- and three-dimensions. We then apply the new method to solve the illustrative benchmark problem of estimating the effective conductivities of the checkerboard-shaped composite media, for both periodic and random configurations. Simulation results show that the new method can reduce the computation time about by an order of magnitude.

Synaptic fusion pore structure and AMPA receptor activation according to Brownian simulation of glutamate diffusion.

The rising phase of fast, AMPA-mediated Excitatory Post Synaptic Currents (EPSCs) has a primary role in the computational ability of neurons. The structure and radial expansion velocity of the fusion pore between the vesicle and the presynaptic membrane could be important factors in determining the time course of the EPSC. We have used a Brownian simulation model for glutamate neurotransmitter diffusion to test two hypotheses on the fusion pore structure, namely, the proteinaceous pore and the purely lipidic pore. Three more hypotheses on the radial expansion velocity were also tested. The rising phases of the EPSC, computed under various conditions, were compared with experimental data from the literature. Our present results show that a proteinaceous fusion pore should produce a more marked foot at the beginning of the rising phase of the EPSC. They also confirm the hypothesis that the structure of the fusion pore and its radial expansion velocity play significant roles in shaping the fast EPSC time course.

Addressing the bias in Monte Carlo pricing of multi-asset options with multiple barriers through discrete sampling

An efficient conditioning technique, the so-called Brownian Bridge simulation, has previously been applied to eliminate pricing bias that arises in applications of the standard discrete-time Monte Carlo method to evaluate options written on the continuous-time extrema of an underlying asset. It is based on the simple and easy to implement analytic formulas for the distribution of one-dimensional Brownian Bridge extremes. This paper extends the technique to the valuation of multi-asset options with knock-out barriers imposed for all or some of the underlying assets. We derive formula for the unbiased option price estimator based on the joint distribution of the multi-dimensional Brownian Bridge dependent extrema. As analytic formulas are not available for the joint distribution in general, we develop upper and lower biased option price estimators based on the distribution of independent extrema and the Fr\'echet lower and upper bounds for the unknown distribution. All estimators are simple and easy to implement. They can always be used to bind the true value by a confidence interval. Numerical tests indicate that our biased estimators converge rapidly to the true option value as the number of time steps for the asset path simulation increases in comparison to the estimator based on the standard discrete-time method. The convergence rate depends on the correlation and barrier structures of the underlying assets.