Deep inelastic collisions viewed as brownian motion

The theory of Brownian motion as described by nonlinear Langevin equations and the corresponding Fokker–Planck equations is discussed. The general problems of the theory of nonlinear Brownian motion considered are: Brownian motion in a medium with nonlinear friction; the critical analysis of three forms of the relevant Langevin and Fokker–Planck equations (Ito's form, Stratonovich's form, and the kinetic form); the Smoluchowski equations and master equations for different cases; two methods of transition from master equation to Fokker–Planck equation; master equations for one-step processes; traditional and nontraditional definition of transition probabilities; evolution of free energy and entropy in Brownian motion; Lyapunov functionals. The following particular examples are considered: Brownian motion in self-oscillatory systems; H-theorem for the van der Pol oscillator; S-theorem; oscillator with inertial nonlinearity; bifurcation of energy of the limiting cycle; oscillator with multistable stationary states; oscillators in discrete time; bifurcations of energy of the limiting cycle and the period of oscillations; criterion of instability upon transition to discrete time, based on the H-theorem; Brownian motion of quantum atoms oscillators in the equilibrium electromagnetic field; Brownian motion in chemically reacting systems; partially ionised plasmas; the Malthus–Verhulst process.

In both the kinetic theory and in the theory of Brownian motion we dealt with the transition from the kinetic description in six-dimensional phase space to a more coarse description in the space of three dimensions. In Ch. 11, for example, it was the transition from the Boltzmann equation to the comparatively simple equations of gas dynamics. Then, in the theory of Brownian motion, we went over from Fokker — Planck equation for the distribution function f(r,v,t) to the simpler Einstein — Smoluchowski equation in the spatial distribution f(r,t). As indicated in Sect. 15.10, this transition corresponds to the simplified hydrodynamic description of Brownian motion

Random walk is the basic concept in mathematics. Brownian motion is one of the most popular application. The core of the Brownian motion is the continuous path for small particles. After being developed by Wiener, it is used in many other studies, including biology, chemistry, and economics. Brownian motion is contemporarily used for predicting possible path. For animals scientists can know the probable range of activities for animals in order to keep people from bothering them and keep them out of danger. For chemical reactions, it is easy to predict every tiny particle about their movement so that people can directly understand the core of the reaction and one can predict the speed and the phenomenon of the reaction. For economics, economists can use the Brownian motion to calculate the probability for stock to up or down. This paper aims to present the theory of Brownian motion and illustrate the applications. This allows people to better choose the time to buy and sell in order to get the highest profit, and makes the economic market more controllable.

Stochastic curtailment has been considered for the interim monitoring of group sequential trials (Davis and Hardy, 1994). Statistical boundaries in Davis and Hardy (1994) were derived using theory of Brownian motion. In some clinical trials, the conditions of forming a Brownian motion may not be satisfied. In this paper, we extend the computations of Brownian motion based boundaries, expected stopping times, and type I and type II error rates to fractional Brownian motion (FBM). FBM includes Brownian motion as a special case. Designs under FBM are compared to those under Brownian motion and to those of O’Brien–Fleming type tests. One- and two-sided boundaries for efficacy and futility monitoring are also discussed. Results show that boundary values decrease and error rates deviate from design levels when the Hurst parameter increases from 0.1 to 0.9, these changes should be considered when designing a study under FBM.

On the theory of brownian motion

In his theory of Brownian motion, Einstein has made a monumental contribution to thermodynamics. Specifically, he has accounted in his theory for the time‐rate of change of the particle momentum, associated with thermal motion, to study the diffusion of dilute particles in the liquid, but excluding its surface. This paper shows first that Einstein’s unique approach to Brownian motion is justified, not only for the problem treated by the distinguished author, but for thermodynamic systems generally, including interfacial systems. The qualitative procedure is based on the principle of detailed balancing, which matured about two decades after Einstein’s original work on Brownian motion. Quantitative justification for the inclusion of the particle thermal momentum in thermodynamics will further be shown for a specific general system on the basis of the first two laws and the quantum theory. The paper then reviews a generalization of Einstein’s transport theory that accounts for both the particle kinetic energy and momentum and is further applicable to interfacial systems. To be briefly described are numerous, new, interdisciplinary consequences of the generalized theory, which have been corroborated by extensive experiments.

The relative merits of 'Boltzmann entropy' and 'Gibbs entropy' are discussed. It is found that, for describing the irreversible features of Brownian motion the 'Gibbs entropy' is the superior concept. In terms of this the Brownian motion gives an excellent illustration of the second law of thermodynamics. The historical origins of the 'Gibbs entropy' are examined, and it is concluded that Ludwig Boltzmann must be given a substantial part of the credit for it. On the other hand, the equation S=k log W, which has been attributed posthumously to Boltzmann, does less than full justice to his concept of entropy, and has led to serious misunderstandings in the theory of Brownian motion.

Brownian motion theory is always challenging how to describe diffusion phenomena with the main issue is how to extend the classical theory of Brownian motion to the special relativity framework. In this study, we formulated dynamics and distribution of a Brownian particle in relativistic framework by using Langevin and Fokker-Planck equation. By representing Brownian particle dynamics by Langevin equation, the velocity curves were dependent on the presence of viscous friction coefficient (heat bath), and were used generalized in special relativity theory, A relativistic Langevin equation reduces to the classical theory at low velocities. Likewise, the distribution of Brownian particles is represented as a stationary solution of the relativistic Fokker-Planck equation. From numerical results, we found that the probability density in the relativistic Fokker-Planck equation for was reduced to the standard Fokker-Planck equation in Netownian classical theory. For the calculation result showed that the Hanggi-Klimontovich approach has a consistent result to the relativistic Maxwell distribution. This work could open a promising interpretation to formulate the diffusion phenomena into general relativity theory.

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

The Schrödinger and Dirac Free Particle Equations without Quantum Mechanics

The dependence of quadrupole relaxation in NMR upon molecular shape is examined by regarding the molecule as a rigid ellipsoidal body which shows anisotropic rotational Brownian motions. Dependence of relaxation times upon the angle between the direction of the maximum field gradient and that of the molecular axis is considered also; the complex angular dependence of the relaxation times is derived, which suggests that careful examinations must be taken when we use the information on molecular motion from the other measurements. For example, the effective correlation time obtained from the quadrupole relaxation cannot be used directly to compute the proton relaxation times in an accurate precision. The use of the ordinary theory of anisotropic Brownian motion in molecular system is examined, where the constituents of the ``liquid'' have comparable dimension with that of the solute molecule and they are by no means ``homogeneous.'' The improved theory is applied to the quadrupole relaxation in the system of axial ratio=2, which predicts that the spectral density is about 2∼8 times larger than the one given by the Debye—Bloembergen—Perrin—Favro theory.

Suppose that S is a subordinator with a nonzero drift and W is an independent 1-dimensional Brownian motion. We study the subordinate Brownian motion X defined by X t = W(S t ). We give sharp bounds for the Green function of the process X killed upon exiting a bounded open interval and prove a boundary Harnack principle. In the case when S is a stable subordinator with a positive drift, we prove sharp bounds for the Green function of X in (0, ∞ ), and sharp bounds for the Poisson kernel of X in a bounded open interval.

The scattering of beam electrons in a beam formed, magnetized plasma has been measured experimentally. Evidence for an effective inverse mean free path which depends upon the intensity and width of the beam excited fluctuations is presented. The model used to interpret the experiment is based upon an elementary application of the Fokker-Planck equation and the theory of Brownian motion. It includes the effects of velocity space diffusion. Mean free paths between 10 and 103 cm are found to be in agreement with the predictions of our model. The ordinary collisional attenuation of the beam under the same operating conditions is several orders of magnitude smaller.

History of Noise Research