Fractional and bifractional Brownian motions can be defined on a metric space if the associated metric or distance function is conditionally negative definite (or of negative type). This paper ...introduces several forms of scalar or vector bifractional Brownian motions on various metric spaces and presents their properties. A metric space of particular interest is the arccos-quasi-quadratic metric space over a subset of
R
d
+
1
such as an ellipsoidal surface, an ellipsoid, or a simplex, whose metric is the composition of arccosine and quasi-quadratic functions. Such a metric is not only conditionally negative definite but also a measure definite kernel, and the metric space incorporates several important cases in a unified framework so that it enables us to study (bi, tri, quadri)fractional Brownian motions on various metric spaces in a unified manner. The vector fractional Brownian motion on the arccos-quasi-quadratic metric space enjoys an infinite series expansion in terms of spherical harmonics, and its covariance matrix function admits an ultraspherical polynomial expansion. We establish the property of strong local nondeterminism of fractional and bi(tri, quadri)fractional Brownian motions on the arccos-quasi-quadratic metric space.
Complex structures, consisting of a large number of interacting subsystems, have the ability to self-organize and evolve, when the scattering of energy coming from the outside ensures the maintenance ...of stationary ordered structures with an entropy less than the equilibrium entropy. One of the fundamental problems here is the role of quantum phenomena in the evolution of macroscopic objects. We provide experimental evidence for the active Brownian motion and evolution of structures driven by quantum effects for micron-sized grains levitating in superfluid helium. The active Brownian motion of grains was induced by quantum turbulence during the absorption of laser irradiation by grains. The intensity of Brownian motion associated with quantum vortices increased by 6-7 orders of magnitude compared to the values from the Einstein formula. We observed the grain structures in a state far from thermodynamic equilibrium and their evolution to more complex organized structures with lower entropy due to the quantum mechanism of exceedingly high entropy loss in superfluid helium.
In this paper we study the problem: How small are the increments of G-Brownian motion? We establish the Csörgő and Révész’s type theorem for the increments of G-Brownian motion.
Open quantum walks Sinayskiy, Ilya; Petruccione, Francesco
The European physical journal. ST, Special topics,
01/2019, Letnik:
227, Številka:
15
Journal Article
Recenzirano
Open quantum walks (OQWs) are a class of quantum walks, which are purely driven by the interaction with the dissipative environment. In this paper, we review theoretical advances on the foundations ...of discrete time OQWs, continuous time OQWs and a scaling limit of OQWs called open quantum Brownian motion. The main focus of the review is on the results and developments of discrete time OQW, covering general formalism, quantum trajectories for OQWs, central limit theorems, the microscopic derivation as well as possible generalisations and applications of OQWs.
The present study aims to investigate the impact of nanoparticle migration due to Brownian motion and thermophoresis on Ag-MgO/Water hybrid nanofluid natural convection. An enclosure with sinusoidal ...wavy walls is considered for this investigation; right and cold walls of this enclosure are in constant temperature while the upper and bottom walls are insulated. This simulation follows Buongiorno's mathematical model; Brownian and thermophoresis diffusion of Ag occurs in MgO-Water nanofluid while the diffusion of MgO happens in Ag-water nanofluid. The result indicates that Nu number increments up to 11% by increasing thermophoresis diffusion for both nanoparticles. Also, increasing Brownian diffusion of Ag augments nanoparticle concentration on hot wall, while the accumulation of nanoparticles unifies by incrementing Brownian diffusion of MgO.
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•Ag − MgO/water hybrid nanofluid utilizing Buongiorno's mathematical model.•Nanoparticle accumulates better on the cold wall due to thermophoresis migration.•Nu number increments up to 11% by increasing thermophoresis diffusion.•The accumulation of nanoparticles unifies by augmenting DBMgO.
Chemical methods developed over the past two decades enable preparation of colloidal nanocrystals with uniform size and shape. These Brownian objects readily order into superlattices. Recently, the ...range of accessible inorganic cores and tunable surface chemistries dramatically increased, expanding the set of nanocrystal arrangements experimentally attainable. In this review, we discuss efforts to create next-generation materials via bottom-up organization of nanocrystals with preprogrammed functionality and self-assembly instructions. This process is often driven by both interparticle interactions and the influence of the assembly environment. The introduction provides the reader with a practical overview of nanocrystal synthesis, self-assembly, and superlattice characterization. We then summarize the theory of nanocrystal interactions and examine fundamental principles governing nanocrystal self-assembly from hard and soft particle perspectives borrowed from the comparatively established fields of micrometer colloids and block copolymer assembly. We outline the extensive catalog of superlattices prepared to date using hydrocarbon-capped nanocrystals with spherical, polyhedral, rod, plate, and branched inorganic core shapes, as well as those obtained by mixing combinations thereof. We also provide an overview of structural defects in nanocrystal superlattices. We then explore the unique possibilities offered by leveraging nontraditional surface chemistries and assembly environments to control superlattice structure and produce nonbulk assemblies. We end with a discussion of the unique optical, magnetic, electronic, and catalytic properties of ordered nanocrystal superlattices, and the coming advances required to make use of this new class of solids.
We study optimal control problems for a class of second‐order stochastic differential equation driven by mixed‐fractional Brownian motion with non‐instantaneous impulses. By using stochastic analysis ...theory, strongly continuous cosine family, and a fixed point approach, we establish the existence of mild solutions for the stochastic system. Moreover, the optimal control results are derived without uniqueness of mild solutions of the stochastic system. Finally, the main results are validated with the aid of an example.
This paper applies Langevin idea to describe the Brownian motion of a particle characterized by an Ornstein–Uhlenbeck-type process. The original and clever method proposed by Langevin is based on ...Newton’s second law plus a fluctuating force whose solution for the mean square displacement consists in separating the fluctuating force from his equation to obtain a deterministic equation for the relevant physical variable. In this work the Langevin original idea is applied to calculate the mean square velocity for a field free particle case; then it is extended for a charged particle in a constant magnetic field. In a similar way, the strategy is also applied to calculate the mean square displacement for a Brownian harmonic oscillator in the overdamped regime, and also when a magnetic field is present. In particular, it is shown in the field free case that Langevin’s original strategy leads to the same results as those obtained using the statistical properties of a Gaussian white noise. All the theoretical results are compared with both the numerical simulation of the Langevin equation, and numerical solution of the corresponding deterministic differential equations.
•Langevin’s original approach.•Standard formulation of Brownian motion.•Brownian motion in a magnetic field.•Harmonic oscillator in a magnetic field.
Abstract
The Geometric Brownian Motion (GBM) model is used widely for model the dynamic of asset price movement. One of the company’s assets is a stock. The distribution of stock data that is ...normally distributed can be modeled using the Geometric Brownian Motion model. However, the distribution of stock data showed excess kurtosis and tail when using the Brown Geometry Motion model was less precise. One model for data showing excess kurtosis and tail was Variance Gamma (VG). In this research, the sample used was the stock data of PT Bank Danamon Indonesia Tbk for the period April 25
th
, 2018 to April 24
th
, 2020. The data sample was divided into two parts, namely training data and testing data. Based on the result of the stock description statistics, the value of skewness = -2.105417 and kurtosis = 22.16438 was obtained, while hypothesis testing concluded that the stock distribution did not spread normally. The resulting parameters for the VG model were σ = 0.08071, v = 8.00500 and θ = 0.01976. Based on the results of testing on the last 38 observations, the MAPE value was = 6.97560%. These results gave the conclusion that the VG model provided excellent forecasting results.
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