In this paper the concept random is investigated, for the notable reason that it is based on probability rather than determinism. Is the question relevant Can there be an alternative explanation Are experiments biased by the theory The random movement of particles in fluids is called Brownian motion. When particles in a fluid collider with fast-moving molecules the result is a random motion of these particles. The motion is not deterministic, i. e. the particles have what is called a random walk. At the fundamental level there are flows of packages, e. g. rivers or winds that carry the smaller particles from one place to another, i. e. aRb. The validity of the random theories cannot be proven in any experiments, since the circumstances are manipulated according to the theory. Within aRb it is Nature that decides, i. e. at the fundamental level, while at the surface level chaotic motion is admitted. Imagine sliding down a helter-skelter, wobbling from side to side, but still on your way down. The conclusion: 1. The Brownian motion, i. e. why particles move, depends on the flow of packages in nature.2. The question concerning Brownian motion is not relevant.3. The Kinetic Theory has to be discussed.
In this paper the concept random is investigated, for the notable reason that it is based on probability rather than determinism. Is the question relevant Can there be an alternative explanation Are experiments biased by the theory The random movement of particles in fluids is called Brownian motion. When particles in a fluid collider with fast-moving molecules the result is a random motion of these particles. The motion is not deterministic, i. e. the particles have what is called a random walk. At the fundamental level there are flows of packages, e. g. rivers or winds that carry the smaller particles from one place to another, i. e. aRb. The validity of the random theories cannot be proven in any experiments, since the circumstances are manipulated according to the theory. Within aRb it is Nature that decides, i. e. at the fundamental level, while at the surface level chaotic motion is admitted. Imagine sliding down a helter-skelter, wobbling from side to side, but still on your way down. The conclusion: 1. The Brownian motion, i. e. why particles move, depends on the flow of packages in nature.2. The question concerning Brownian motion is not relevant.3. The Kinetic Theory has to be discussed.
Introduction Despite the fact that classical mechanics employs deterministic dynamical laws, random dynamical processes often arise in classical physics, as well as in everyday life. A stochastic or random process is one in which states-of-affairs at successive times are not related to one another by deterministic laws, and instead probability theory is employed to describe whatever regularities exist. Tossing a coin or rolling a die several times in succession are examples of stochastic processes in which the previous history is of very little help in predicting what will happen in the future. The motion of a baseball is an example of a stochastic process which is to some degree predictable using classical equations of motion that relate its acceleration to the total force acting upon it. However, a lack of information about its initial state (e.g., whether it is spinning), its precise shape, and the condition and motion of the air through which it moves limits the precision with which one can predict its trajectory. The Brownian motion of a small particle suspended in a fluid and subject to random bombardment by the surrounding molecules of fluid is a well-studied example of a stochastic process in classical physics. Whereas the instantaneous velocity of the particle is hard to predict, there is a probabilistic correlation between successive positions, which can be predicted using stochastic dynamics and checked by experimental measurements. In particular, given the particle's position at a time t , it is possible to compute the probability that it will have moved a certain distance by the time t + Δ t .
The concept of a "random walk" has been widely used to describe many real-world systems from the Brownian motion of smoke particles to the fluctuation of prices on the stock market. The mathematics is usually explained though the analogy of a person dictating their movements by the toss of a coin: get heads, for example, and they move one step to the right; get tails and they shuffle one step to the left. After many coin tosses, the person's position is random – though, statistically, it is likely to be near the starting point.
The study of Schrödinger equation based on a hypothesis that every particle must move randomly in a quantum-sized volume has been done. In addition to random motion, every particle can do relative motion through the movement of its quantum-sized volume. On the other way these motions can coincide. In this proposed model, the random motion is one kind of intrinsic properties of the particle. The every change of both speed of randomly intrinsic motion and or the velocity of translational motion of a quantum-sized volume will represent a transition between two states, and the change of speed of randomly intrinsic motion will generate diffusion process or Brownian motion perspectives. Diffusion process can take place in backward and forward processes and will represent a dissipative system. To derive Schrödinger equation from our hypothesis we use time operator introduced by Nelson. From a fundamental analysis, we find out that, naturally, we should view the means of Newton’s Law F⃗ = ma⃗ as no an external force, but it is just to describe both the presence of intrinsic random motion and the change of the particle energy.
This problem in lieu of thesis is a discussion of two topics: Brownian movement and quantum computers. Brownian movement is a physical phenomenon in which the particle velocity is constantly undergoing random fluctuations. Chapters 2, 3 and 4, describe Brownian motion from three different perspectives. The next four chapters are devoted to the subject of quantum computers, which are the signal of a new era of technology and science combined together. In the first chapter I present to a reader the two topics of my problem in lieu of thesis. In the second chapter I explain the idea of Brownian motion, its interpretation as a stochastic process and I find its distribution function. The next chapter illustrates the probabilistic picture of Brownian motion, where the statistical averages over trajectories are related to the probability distribution function. Chapter 4 shows how to derive the Langevin equation, introduced in chapter 1, using a Hamiltonian picture of a bath with infinite number of harmonic oscillators. The chapter 5 explains how the idea of quantum computers was developed and how step-by-step all the puzzles for the field of quantum computers were created. The next chapter, chapter 6, discus the basic quantum unit of information namely, the so called qubit and its properties. Chapter 7 is devoted to quantum logic gates, which are important for conducting logic operation in quantum computers. This chapter explains how they were developed and how they are different from classical ones. Chapter 8 is about the quantum algorithm, Shor's algorithm. Quantum algorithm in quantum computers enables one to solve problems that are hard to solve on digital computers. The last chapter contains conclusions on Brownian movement and the field of quantum computers.
The aim of this chapter is to present and to discuss the general concept and mathematical structure of path integrals, introduced for the first time by N Wiener (1921, 1923, 1924, 1930), as a tool for solving problems in the theory of classical systems subject to random influences from the surrounding medium. The most famous and basic example of such a system is a particle performing the so-called Brownian motion. This phenomenon was discovered in 1828 by the British botanist R Brown, who investigated the pollen of different plants dispersed in water. Later, scientists realized that small fractions of any kind of substance exhibit the same behaviour, as a result of random fluctuations driven by the medium. The theory of Brownian motion emerged in the beginning of the last century as a result of an interplay between physics and mathematics and at present it has a wide range of applications in different areas, e.g., diffusion in stellar dynamics, colloid chemistry, polymer physics, quantum mechanics.
Introduction to statistical ensembles The concept of a random or stochastic process or function represents a generalization of the idea of a set of random variables x 1 , x 2 , …, when the set is no longer countable and the variables form a continuum. We therefore introduce a continuous parameter t , such as time, that labels the variates. We call x ( t ) a random process or a random function of t if x does not depend on t in a deterministic way. Random processes are encountered in many fields of science, whenever fluctuations are present. Examples of a real random process x ( t ) are the fluctuating voltage across an electrical resistor, and the coordinates of a particle under-going Brownian motion. We shall see shortly that the optical field generated by any realistic light source must also be treated as a random function of position and time. Of course the parameter t may also stand for some quantity other than time, but for simplicity we shall take it to represent time. In our applications x ( t ) will frequently represent a Cartesian component of the electric or magnetic field vector in a light beam. To begin with we shall take x ( t ) to be real, but complex random processes will also be encountered. The ensemble average As x does not depend on t deterministically, we can only describe its values statistically, by some probability distribution or probability density.