Brownian Motion Dynamics Research Articles

Dynamics of Brownian motion and nonlinear mixed convective conditions on thermophoretic slip flow embedded with non-Fourier flux

In this paper, the importance of mass and heat transfer characteristics of mixed convective conditions on nonlinear free convection flow embedded with non-Fourier flux and thermal radiation is numerically investigated. A partial Casson slip and mixed impact of thermophoresis and Brownian motion is considered at the elastic surface. To change the highly nonlinear partial differential equations into ordinary differential system, we used the similarity transformations and then the shooting method is employed to solve resulting equations. The significance of various physical parameters on the velocity, concentration and temperature profiles are investigated for both negative value of ferromagnetic interaction ([Formula: see text]) and positive value of ferromagnetic interaction ([Formula: see text]). The governing physical aspects of coefficient of skin friction, the Sherwood and Nusselt numbers are obtained for various embedded parameters. Our findings concluded that the ferromagnetic interaction is positive and then more mass transfer rate is identified compared to the negative value of ferromagnetic interaction, but whereas in friction and heat transfer rate, it was completely mixed sense (i.e., different effects have different results). Brownian motion has less Nusselt number values in positive ferromagnetism compared to negative ferromagnetism and quite opposite behavior is observed in the presence of thermophoresis, this happened due to nonlinear flux and buoyancy.

Dynamics of Brownian motion and flux conditions on naturally unsteady thermophoretic flow with variable fluid properties

AbstractThis work examines the boundary flow difficulties of the past and the heat transfer properties of Blasius and Sakiadis flows under prescribed concentration flux and prescribed heat flux. The nanofluid is also taken into account in this model, along with impacts from Brownian motion and thermophoresis. The modified system governing partial differential equations is numerically solved by using the R‐K method along with the shooting technique. Various values of physical quantities like thermophoresis parameter, Brownian motion parameter, Eckert number, thermal radiation parameter, heat source parameter, and magnetic field parameter along with the were calculated using the temperature, concentration, and velocity profiles. Finally, we demonstrated how the Brownian motion, radiation, and thermophoresis parameters can significantly increase the temperature distributions. The concentration distributions were decelerated with an increase in Brownian motion parameters for both Blasius and Sakiadis cases.

Simultaneous Random Number Generation and Optical Tweezers Calibration Employing a Learning Algorithm Based on the Brownian Dynamics of a Trapped Colloidal Particle

True random number generators are in high demand for secure cryptographic algorithms. Unlike algorithmically generated pseudo-random numbers they are unclonable and non-deterministic. A particle following Brownian dynamics as a result of the stochastic Ornstein-Uhlenbeck process is a source of true randomness because the collisions with the ambient molecules are probabilistic in nature. In this paper, we trap colloidal particles in water using optical tweezers and record its confined Brownian motion in real-time. Using a segment of the initial incoming data we train our learning algorithm to measure the values of the trap stiffness and diffusion coefficient and later use those parameters to extract the “white” noise term in the Langevin equation. The random noise is temporally delta correlated, with a flat spectrum. We use these properties in an inverse problem of trap-calibration to extract trap stiffnesses, compare it with standard equipartition of energy technique, and show it to scale linearly with the power of the trapping laser. Interestingly, we get the best random number sequence for the best calibration. We test the random number sequence, which we have obtained, using standard tests of randomness and observe the randomness to improve with increasing sampling frequencies. This method can be extended to the trap-calibration for colloidal particles confined in complex fluids, or active particles in simple or complex environments so as to provide a new and accurate analytical methodology for studying Brownian motion dynamics using the newly-emerged but robust machine learning platform.

Statistical physics of flux-carrying Brownian particles

Chern–Simons gauge field theory has provided a natural framework to gain deep insight about many novel phenomena in two-dimensional condensed matter. We investigate the nonequilibrium thermodynamics properties of a (two-dimensional) dissipative harmonic particle when the Abelian topological gauge action and the (linear) Brownian motion dynamics are treated on an equal footing. We find out that the particle exhibits remarkable magneticlike features in the quantum domain that are beyond the celebrated Landau diamagnetism: this could be viewed as the non-relativistic Brownian counterpart of the composite excitation of a charge and magneticlike flux. Interestingly, it is shown that the properties of such flux-carrying Brownian particle are in good agreement with the classical statistical mechanics at sufficient high temperatures, as well as are widely consistent with the Third Law of thermodynamics in the studied dissipative scenarios. Our findings also suggest that its ground state may be far from trivial, i.e. it is a seemingly degenerate state.

Thermal Brownian Motion of Skyrmion for True Random Number Generation

The true random number generators (TRNGs) have received extensive attention because of their wide applications in information transmission and encryption. The true random numbers generated by TRNG are typically applied to the encryption algorithm or security protocol of the information security core. Recently, TRNGs have also been employed in emerging stochastic computing paradigm for reducing power consumption. Roughly speaking, TRNG can be divided into circuits-based, e.g., oscillator sampling or directly noise amplifying; and quantum physics-based, e.g., photoelectric effect. The former generally requires a large area and has a large power consumption, whereas the latter is intrinsic random but is more difficult to implement and usually requires additional post-processing circuitry. Very recently, magnetic skyrmion has become a promising candidate for implementing TRNG because of their nanometer size, high stability, and intrinsic thermal Brownian motion dynamics. In this work, we propose a TRNG based on continuous skyrmion thermal Brownian motion in a confined geometry at room temperature. True random bitstream can be easily obtained by periodically detecting the relative position of the skyrmion without the need for additional current pulses. More importantly, we implement a probability-adjustable TRNG, in which a desired ratio of 0 and 1 can be acquired by adding an anisotropy gradient through voltage-controlled magnetic anisotropy (VCMA) effect. The behaviors of the skyrmion-based TRNG are verified by using micromagnetic simulations. The National Institute of Standards and Technology (NIST) test results demonstrate that our proposed random number generator is TRNG with good randomness. Our research provides a new perspective for efficient TRNG realization.

Generalization of Langevin Dynamics from Spatio-Temporal Dressed Dynamics Perspective.

Understanding multiscale dynamics characteristics has been the holy grail in a broad range of scientific disciplines from physics, chemistry, to biology, and beyond. The seminal Langevin equation successfully unravels remarkable details of Brownian motion dynamics involving stochastic collisions from the fluidic medium. However, extraordinary phenomena beyond the scope of Langevin dynamics were observed to exhibit a series of multiscale dynamic features in recent years. Here, an explicit spatio-temporal coupled kernel is developed to provide the microscopic account for delicate dynamic coupling between particle and medium in the dressed dynamics perspective. This methodology is applied to investigate an aqueous solvation shell model with an explicit spatial boundary to illustrate the significance of spatial and temporal coupling, leading to a general temporal profile of dressed dynamics over 12 orders of magnitude in time In particular, the time-resolved viscosity is formulated to address the remarkable enhancement of viscosity measured by the liquid cell electron microscopy. The understanding of dressed dynamics processes will be greatly enriched by further studies on the detailed dynamics that manifests the microscopic inhomogeneity of medium.

Rainbows and Transforms: Semi-Analytic Formulae

In this work we show how the techniques introduced by Hurd and Zhou can be used to derive a pricing framework for rainbow options by using the joint characteristic function of the logarithm of the underlying assets. Semi-analytic formulae will be achieved by splitting the symbol, applying different dampings to its constituent components, and finally calculating the Fourier transforms of said components allowing the pricing of call and put options on the maximum and minimum of multiple assets; we shall also revisit Margrabe’s classical formula. Our work extends a number of classical results that focus on Geometric Brownian motion dynamics, and provides an alternative pricing methodology to that proposed in some of the most recent literature.

Expected utility of the drawdown-based regime-switching risk model with state-dependent termination

In this paper, we model an entity’s surplus process X using the drawdown-based regime-switching (DBRS) dynamics proposed in Landriault et al. (2015a). We introduce the state-dependent termination time to the model, and provide rationale for its introduction in insurance contexts. By examining some related potential measures, we first derive an explicit expression for the expected terminal utility of the entity in the DBRS model with Brownian motion dynamics. The analysis is later generalized to time-homogeneous Markov framework, where the spectrally negative Lévy model is also discussed as a special example. Our results show that, even considering the impact of the termination risk, the DBRS strategy can still outperform its counterparts in either single regime strategy. This study shows that the DBRS model is not myopic, as it not only helps to recover from significant losses, but also may improve the insurer’s overall welfare.

Diffusion dynamics and concentration of toxic materials from quantum dots-based nanotechnologies: an agent-based modeling simulation framework

Due to their favorable electrical and optical properties, quantum dots (QDs) nanostructures have found numerous applications including nanomedicine and photovoltaic cells. However, increased future production, use, and disposal of engineered QD products also raise concerns about their potential environmental impacts. The objective of this work is to establish a modeling framework for predicting the diffusion dynamics and concentration of toxic materials released from Trioctylphosphine oxide-capped CdSe. To this end, an agent-based model simulation with reaction kinetics and Brownian motion dynamics was developed. Reaction kinetics is used to model the stability of surface capping agent particularly due to oxidation process. The diffusion of toxic Cd2+ ions in aquatic environment was simulated using an adapted Brownian motion algorithm. A calibrated parameter to reflect sensitivity to reaction rate is proposed. The model output demonstrates the stochastic spatial distribution of toxic Cd2+ ions under different values of proxy environmental factor parameters. With the only chemistry considered was oxidation, the simulation was able to replicate Cd2+ ion release from Thiol-capped QDs in aerated water. The agent-based method is the first to be developed in the QDs application domain. It adds both simplicity of the solubility and rate of release of Cd2+ ions and complexity of tracking of individual atoms of Cd at the same time.

Real-time visualization of 3-D dynamic microscopic objects using optical diffraction tomography

3-D refractive index (RI) distribution is an intrinsic bio-marker for the chemical and structural information about biological cells. Here we develop an optical diffraction tomography technique for the real-time reconstruction of 3-D RI distribution, employing sparse angle illumination and a graphic processing unit (GPU) implementation. The execution time for the tomographic reconstruction is 0.21 s for 96(3) voxels, which is 17 times faster than that of a conventional approach. We demonstrated the real-time visualization capability with imaging the dynamics of Brownian motion of an anisotropic colloidal dimer and the dynamic shape change in a red blood cell upon shear flow.

Design and Pricing of Derivative Contracts in a Spectrum Market

We propose a secondary spectrum market that allows wireless providers to purchase spectrum access licenses of short duration in the form of spot contracts and derivative contracts on spectrum. A spot contract provides immediate access to one or more wireless channels and cannot be further traded. On the other hand, derivative contracts on spectrum typically involve purchase of spectrum licenses in the future for predefined terms, and they can play an important role in risk management objectives of wireless providers. In this article, we utilize a model for the spot price of spectrum licenses in which the price increases with increasing congestion in spectrum usage caused by the primary demand for spectrum. The spot price process is modeled as driven by a fractional Brownian motion process owing to the self-similarity properties of the wireless traffic. Utilizing fractional stochastic calculus, we obtain the value of derivative contracts in the proposed spectrum market. The value of a derivative contract is expressed as the expected discounted payoff of the derivative contract under risk-neutral Brownian motion dynamics for the spot price. Finally, we design a variety of derivative contracts considering the risk profile of both the buyers and sellers of spectrum. Through a detailed numerical study, we examine the value of the derivative contracts for changes in spot price volatility and the parameters that define the contracts.

Bragg scattering and Brownian motion dynamics in optically induced crystals of submicron particles

A set of four confocal laser beams of 1064-nm wavelength is used to prepare optically induced crystals of submicron particles in aqueous solution. Thousands of polystyrene spheres of about 200 nm in diameter are trapped in three dimensions. Bragg scattering patterns obtained with a probe beam of 532-nm wavelength are in agreement with the calculated lattice structure and its polarization dependence. The decay and rise of the Bragg scattering intensity upon switching the lattice off and on reveal the Brownian motion dynamics of the particles in the periodic optical trapping potential. Experimental results agree well with results from trajectory simulations based on the Langevin equation. The results exhibit the interplay between Brownian motion and deterministic forces in an inhomogeneous (near-)periodic optical trapping potential.

The immersed molecular finite element method

In this paper, we present the framework of the immersed molecular finite element method (IMFEM). The framework incorporates Brownian motion dynamics, a molecular phenomenon predominantly present at the nanoscale, into the previously developed immersed finite element method, thus referred to as the immersed molecular finite element method. The thermal fluctuations are embedded in the fluid equations via fluctuating stress terms that are constructed at each timestep in accordance to the fluctuation dissipation theorem. The coupling between the immersed objects and the surrounding fluctuating fluid is accomplished via the hydrodynamic forces that are naturally introduced in the fluid–structure interaction term. A three-dimensional implementation of the framework is presented and verified with an example problem where a comparison is made with the analytic solution. Incorporating a molecular-type force field superimposed with a Coulomb potential between the immersed objects allows us to use the IMFEM to comprehensively model self-assembly of nanoscale structures. Our preliminary investigations suggest that this integrated simulation package will assist in achieving a fundamental understanding of nanodiamond self-assembly process, which represents an important component in the advancement of therapeutic and diagnostic applications.

Modeling the complex dynamics of enzyme-pathway coevolution

Metabolic pathways must have coevolved with the corresponding enzyme gene sequences. However, the evolutionary dynamics ensuing from the interplay between metabolic networks and genomes is still poorly understood. Here, we present a computational model that generates putative evolutionary walks on the metabolic network using a parallel evolution of metabolic reactions and their catalyzing enzymes. Starting from an initial set of compounds and enzymes, we expand the metabolic network iteratively by adding new enzymes with a probability that depends on their sequence-based similarity to already present enzymes. Thus, we obtain simulated time courses of chemical evolution in which we can monitor the appearance of new metabolites, enzyme sequences, or even entire organisms. We observe that new enzymes do not appear gradually but rather in clusters which correspond to enzyme classes. A comparison with Brownian motion dynamics indicates that our system displays biased random walks similar to diffusion on the metabolic network with long-range correlations. This suggests that a quantitative molecular principle may underlie the appearance of punctuated equilibrium dynamics, whereby enzymes occur in bursts rather than by phyletic gradualism. Moreover, the simulated time courses lead to a putative time-order of enzyme and organism appearance. Among the patterns we detect in these evolutionary trends is a significant correlation between the time of appearance and their enzyme repertoire size. Hence, our approach to metabolic evolution may help understand the rise in complexity at the biochemical and genomic levels.

Measurement of the Instantaneous Velocity of a Brownian Particle

Brownian motion of particles affects many branches of science. We report on the Brownian motion of micrometer-sized beads of glass held in air by an optical tweezer, over a wide range of pressures, and we measured the instantaneous velocity of a Brownian particle. Our results provide direct verification of the energy equipartition theorem for a Brownian particle. For short times, the ballistic regime of Brownian motion was observed, in contrast to the usual diffusive regime. We discuss the applications of these methods toward cooling the center-of-mass motion of a bead in vacuum to the quantum ground motional state.

Entropic particle transport in periodic channels

The dynamics of Brownian motion has widespread applications extending from transport in designed micro-channels up to its prominent role for inducing transport in molecular motors and Brownian motors. Here, Brownian transport is studied in micro-sized, two-dimensional periodic channels, exhibiting periodically varying cross-sections. The particles in addition are subjected to an external force acting alongside the direction of the longitudinal channel axis. For a fixed channel geometry, the dynamics of the two-dimensional problem is characterized by a single dimensionless parameter which is proportional to the ratio of the applied force and the temperature of the particle environment. In such structures entropic effects may play a dominant role. Under certain conditions the two-dimensional dynamics can be approximated by an effective one-dimensional motion of the particle in the longitudinal direction. The Langevin equation describing this reduced, one-dimensional process is of the type of the Fick–Jacobs equation. It contains an entropic potential determined by the varying extension of the eliminated channel direction, and a correction to the diffusion constant that introduces a space dependent diffusion. Different forms of this correction term have been suggested before, which we here compare for a particular class of models. We analyze the regime of validity of the Fick–Jacobs equation, both by means of analytical estimates and the comparisons with numerical results for the full two-dimensional stochastic dynamics. For the nonlinear mobility we find a temperature dependence which is opposite to that known for particle transport in periodic potentials. The influence of entropic effects is discussed for both, the nonlinear mobility and the effective diffusion constant.

Structural Default Models Applied to South African Banks

We modify the structural default model of Merton to make it more readily applicable to banking firms in South Africa. In essence the modification assumes that both assets and liabilities follow geometric Brownian motion dynamics and then treats the equity of the bank as the value of swap options rather than a call option as is done in the traditional Merton model. We fit this new model to the data of three leading South African banks and find that it fits quite well and leads to useful and realistic failure probability and buffer capital results.

Quantum dynamics without the wavefunction

When suitably generalized and interpreted, the path integral offers an alternative to the more familiar quantal formalism based on state vectors, self-adjoint operators and external observers. Mathematically one generalizes the path-integral-as-propagator to a quantal measure μ on the space Ω of all ‘conceivable worlds’, and this generalized measure expresses the dynamics or law of motion of the theory, much as Wiener measure expresses the dynamics of Brownian motion. Within such ‘histories-based’ schemes new and more ‘realistic’ possibilities open up for resolving the philosophical problems of the state-vector formalism. In particular, one can dispense with the need for external agents by locating the predictive content of μ in its sets of measure zero: such sets are to be ‘precluded’. But unrestricted application of this rule engenders contradictions. One possible response would remove the contradictions by circumscribing the application of the preclusion concept. Another response, more in the tradition of ‘quantum logic’, would accommodate the contradictions by dualizing Ω to a space of ‘co-events’ and effectively identifying reality with an element of this dual space.

Numerical solution of the space fractional Fokker–Planck equation

The traditional second-order Fokker–Planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of Brownian motion. Densities of α-stable type have been used to describe the probability distribution of these motions. The resulting governing equation of these motions is similar to the traditional Fokker–Planck equation except that the order α of the highest derivative is fractional. In this paper, a space fractional Fokker–Planck equation (SFFPE) with instantaneous source is considered. A numerical scheme for solving SFFPE is presented. Using the Riemann–Liouville and Grünwald–Letnikov definitions of fractional derivatives, the SFFPE is transformed into a system of ordinary differential equations (ODE). Then the ODE system is solved by a method of lines. Numerical results for SFFPE with a constant diffusion coefficient are evaluated for comparison with the known analytical solution. The numerical approximation of SFFPE with a time-dependent diffusion coefficient is also used to simulate Lévy motion with α-stable densities. We will show that the numerical method of SFFPE is able to more accurately model these heavy-tailed motions.

Doubling an investment

We study the issue of optimal long-term portfolio management in continuous time multi-asset financial markets. Rather than following the abstract notion of ‘utility’ and its implied paradigm of ‘maximization of expected utility’ we suggest a different approach: The investor sets a goal—such as reaching a desired fortune level, or doubling the initial investment—and then operates to minimize the expected time-to-goal, i.e., achieving the goal as quick as possible. We assume the ‘standard model’ of multi-asset financial markets where assets are governed by correlated Geometric Brownian motion dynamics, and study optimality under the criteria of ‘minimization of the expected time-to-goal’. We explicitly compute: (i) the optimal holding strategies; (ii) the dynamics and behavior of the optimal investment portfolios; and, (iii) the statistics—mean, variance, and Laplace transform—of the time-to-goal (under the optimal investment strategy). Also, an investment paradox arising in this context—in which some portfolios have exponential mean growth but have a positive probability of never doubling their initial value—is discussed and explained.