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This paper presents a numerical solution to the problem of time-dependent blood flow via a w-shaped stenotic conduit, driven by pulsatile pressure gradient. The problem is formulated in cylindrical coordinates by employing the theoretical model of tangent hyperbolic fluid. The electro-osmotic effects are also taken into consideration. To simplify the non-dimensional governing equations of the flow problem, a mild stenosis assumption is utilized and the impact of the blood vessel wall is mitigated by employing a radial coordinate transformation. An explicit finite difference method is used to solve the resulting nonlinear system of differential equations, considering the auxiliary conditions specified at the boundary of the blood channel. After obtaining the numerical solution to the problem, an examination is carried out for various flow variables, such as axial velocity, temperature field, mass concentration, skin friction, Nusselt number, and Sherwood number. These results are presented graphically, and a concise explanation is provided using physical facts. An increase in flow rate and blood velocity leads to a rise in response, while an increase in stenosis height, Weissenberg number, and power-law exponent leads to a reverse response. Furthermore, the temperature field is significantly affected by the Brinkman number and the Prandtl number.

The tracking technique that is examined in this study considers the nanosensor’s velocity and distance as independent random variables with known probability density functions (PDFs). The nanosensor moves continuously in both directions from the starting point of the real line (the line’s origin). It oscillates while traveling through the origin (both left and right). We provide an analytical expression for the density of this distance using the Fourier-Laplace representation and a sequence of random points. We can take the tracking distance into account as a function of a discounted effort-reward parameter in order to account for this uncertainty. We provide an analytical demonstration of the effects this parameter has on reducing the expected value of the first collision time between a nanosensor and the particle and confirming the existence of this technique.

We give a connection between diffusion processes and classical mechanical systems in this paper. Precisely, we consider a system of plural massive particles interacting with an ideal gas, evolved according to classical mechanical principles, via interaction potentials. We prove the almost sure existence and uniqueness of the solution of the considered dynamics, prove the convergence of the solution under a certain scaling limit, and give the precise expression of the limiting process, a diffusion process.

We give a stochastic proof of the finite approximability of a class of Schrödinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman–Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.

The survival problem for a Brownian particle moving among random traps is considered in the case where the traps are correlated in a particular fashion being gathered in clusters. It is assumed that the clusters are statistically identical and independent of each other and are distributed in space according to a Poisson law. Mathematically, such a trapping medium is described via a Poisson cluster point process. We prove that the particle survival probability is increased at all times as compared to the case of noncorrelated (Poissonian) traps, which implies the slowdown of the trapping process. It is shown that this effect may be interpreted as the manifestation of the trap "attraction", thus supporting assertions on the qualitative influence of the trap "interaction" on the trapping rate claimed earlier in the physical literature. The long-time survival asymptotics (of Donsker–Varadhan type) is also derived. By way of appendix, FKG inequalities for certain functionals are proven and the limiting distribution for a Poisson cluster process, under clusters' scaling, is determined.

Recently the new technique to solve optimal stopping problems for Hunt processes is developed (see [S. Christensen, P. Salminen and B. Q. Ta, Optimal stopping of strong Markov processes, Stochastic Process. Appl. 123(3) (2013) 1138–1159]). The crucial feature of the approach is to utilize the representation of the r-excessive functions as expected suprema. However, it seems to be difficult when applying directly the approach to some concrete cases, e.g. one-sided problem for reflecting Brownian motion and two-sided problem for Brownian motion. In this paper, we review and exploit this approach to find explicit solutions of two problems above.

Computational studies have been widely applied for the thermal evaluation of the nanomaterial thermal feature in different industrial and scientific issues. The squeezed flow and heat transfer features for Al₂O₃-water nanofluid among analogous plates are investigated using the GOHAM and its validity is verified by comparison with existing numerical results. Novel aspects of Brownian motion and thermal force were accounted in the simulation of nanomaterial flow within parallel plate. Analytical investigation has been done for diverse governing factors namely: the squeeze, chemical reaction factors and Eckert number. The obtained outcomes show that $|C_f|$ has direct relationship with absolute values of squeeze factor. Nu increases for large Eckert number and absolute values of squeeze number.

From the stock markets of six countries with high GDP, we study the stock indices, S&P 500 (NYSE, USA), SSE Composite (SSE, China), Nikkei (TSE, Japan), DAX (FSE, Germany), FTSE 100 (LSE, Britain) and NIFTY (NSE, India). The daily mean growth of the stock values is exponential. The daily price fluctuations about the mean growth are Gaussian, but with a nonzero asymptotic convergence. The growth of the monthly average of stock values is statistically self-similar to their daily growth. The monthly fluctuations of the price follow a Wiener process, with a decline of the volatility. The mean growth of the daily volume of trade is exponential. These observations are globally applicable and underline regularities across global stock markets.

An accurate multiaffine analysis of 23 foreign currency exchange rates has been performed. The roughness exponent H₁ which characterizes the excursion of the exchange rate has been numerically measured. The degree of intermittency C₁ has been also estimated. In the (H₁,C₁) phase diagram, the currency exchange rates are dispersed in a wide region around the Brownian motion value (H₁=0.5,C₁=0) and have a significantly intermittent component (C₁≠0).

We investigate the classical Brownian motion of a particle in a two-dimensional noncommutative (NC) space. Using the standard NC algebra embodied by the symplectic Weyl–Moyal formalism we find that noncommutativity induces a nonvanishing correlation between both coordinates at different times. The effect stands out as a signature of spatial noncommutativity and thus could offer a way to experimentally detect the phenomena. We further discuss some limiting scenarios and the trade-off between the scale imposed by the NC structure and the parameters of the Brownian motion itself.

Studying the Brownian motion of a particle in a two-dimensional potential with two saddle-point passages connecting two wells, we compute the activation rate of the particle from one well into the other and illustrate a new technique for obtaining numerical solution to the Langevin equation for transition probability. By virtue of a Langevin equation with negative friction, this new method directly traces the active part of an activation event, without having to simulate the long period of small fluctuations in a well between two successful events, and computes the statistical weight for each successful activation. It makes feasible for us to numerically integrate the Langevin equation for transition probability even when the activation energy barrier (i.e. the potential difference between the saddle point and the well) is much greater than thermal energy k_BT where other methods fail to be tractable.

A path integral formulation is developed for the transient Brownian motion in response to a sudden change in temperature. Formulae are derived for the time-dependent probability distribution function and for the transient current, employing the negative friction Langevin equation. Numerical implementation of the theory for a double-well system gives a clear illustration of the transient of the transient behaviors of the system.

Recently there has been developed a new approach to the study of critical quantum systems in 1+1 dimensions which reduces them to problems in one-dimensional Brownian motion. This goes under the name of stochastic, or Schramm, Loewner Evolution (SLE). I review some of the recent progress in this area, from the point of view of many-body theory. Connections to random matrices also emerge.

This paper shows a novel calculation of the mean square displacement of a classical Brownian particle in a relativistic thermal bath. Also, the thermodynamic properties of a nondegenerate simple relativistic gas are reviewed in terms of a treatment performed in velocity space.

We derive the virial theorem for an overdamped system of rotating self-gravitating Brownian particles. We show that, in the two-dimensional case, it takes a closed form that can be used to obtain general results about the dynamics without being required to solve the Smoluchowski–Poisson system explicitly. In particular, we obtain the exact analytical expression of the mean square displacement 〈r²〉(t) of the interacting Brownian particles. We exhibit a critical temperature below which the system collapses, and above which it evaporates, and we determine how this temperature is affected by a solid rotation. We also develop an analogy between self-gravitating systems and two-dimensional point vortices. We derive a virial-like relation for point vortices at statistical equilibrium relating the angular velocity to the angular momentum and the temperature.

We review recent works on optomechanics of optically trapped microspheres and nanoparticles in vacuum, which provide an ideal system for studying macroscopic quantum mechanics and ultrasensitive force detection. An optically trapped particle in vacuum has an ultrahigh mechanical quality factor as it is well-isolated from the thermal environment. Its oscillation frequency can be tuned in real time by changing the power of the trapping laser. Furthermore, an optically trapped particle in vacuum may rotate freely, a unique property that does not exist in clamped mechanical oscillators. In this review, we will introduce the current status of optical trapping of dielectric particles in air and vacuum, Brownian motion of an optically trapped particle at room temperature, Feedback cooling and cavity cooling of the Brownian motion. We will also discuss about using optically trapped dielectric particles for studying macroscopic quantum mechanics and ultrasensitive force detection. Applications range from creating macroscopic Schrödinger's cat state, testing objective collapse models of quantum wavefunctions, measuring Casimir force, searching short-range non-Newtonian gravity, to detect gravitational waves.

The energetics of a Brownian heat engine and heat pump driven by position dependent temperature, known as the Büttiker–Landauer heat engine and heat pump, is investigated by numerical simulations of the inertial Langevin equation. We identify parameter values for optimal performance of the heat engine and heat pump. Our results qualitatively differ from approaches based on the overdamped model. The behavior of the heat engine and heat pump, in the linear response regime is examined under finite time conditions and we find that the efficiency is lower than that of an endoreversible engine working under the same condition. Finally, we investigate the role of different potential and temperature profiles to enhance the efficiency of the system. Our simulations show that optimizing the potential and temperature profile leads only to a marginal enhancement of the system performance due to the large entropy production via the Brownian particle's kinetic energy.

Here, we introduce a statistical approach derived from dynamics, for the study of the geophysical fluid dynamics phenomena characterized by a weak interaction among the variables of interest and the rest of the system. The approach is reminiscent of the one developed some years ago [M. Bianucci, R. Mannella, P. Grigolini and B. J. West, Phys. Rev. E51, 3002 (1995)] to derive statistical mechanics of macroscopic variables on interest starting from Hamiltonian microscopic dynamics. However, in the present work, we are interested to generalize this approach beyond the context of the foundation of thermodynamics, in fact, we take into account the cases where the system of interest could be non-Hamiltonian (dissipative) and also the interaction with the irrelevant part can be of a more general type than Hamiltonian. As such example, we will refer to a typical case from geophysical fluid dynamics: the complex ocean–atmosphere interaction that gives rise to the El Niño Southern Oscillation (ENSO). Here, changing all the scales, the role of the “microscopic” system is played by the atmosphere, while the ocean (or some ocean variables) plays the role of the intrinsically dissipative macroscopic system of interest. Thus, the chaotic and divergent features of the fast atmosphere dynamics remains in the decaying properties of the correlation functions and of the response function of the atmosphere variables, while the exponential separation of the perturbed (or close) single trajectories does not play a direct role. In the present paper, we face this problem in the frame of a not formal Langevin approach, limiting our discussion to physically based rather than mathematics arguments. Elsewhere, we obtain these results via a much more formal procedure, using the Zwanzing projection method and some elements from the Lie Algebra field.

The pausing-time distribution of thermal diffusion of hydrogen is analytically shown in the Gaussian density of state. The pausing-time distribution exhibits a log-normal distribution. It has been shown that the pausing-time distribution follows approximately power law, i.e., t^-1-α(t: pausing time). The diffusion coefficient of hydrogen is also obtained to be approximately τ^α-1 (τ: diffusion time). The value of α is the ratio of hydrogen temperature T_r to T_2σ, in which T_2σ, is a temperature corresponding to 2σ (σ: standard deviation). Finally, Brownian motion is shown to correspond to the case of σ = 0. The width of the energy distribution play an important role in hydrogen diffusion.

The diffusion coefficient of hydrogen is obtained for exponential energy distribution in hydrogenated amorphous silicon (a-Si:H). It is shown that the diffusion coefficient follows the form of τ^α-1 (τ: diffusion time) in the case of α < 1 and a larger τ, in which α is the ratio of hydrogen temperature to width for energy distribution function. In the case of α ≥ 1, as α reaches infinity at the limit, the hydrogen diffusion approaches Brownian motion.