Physics
Brownian Motion
Brownian motion refers to the random movement of particles suspended in a fluid, caused by collisions with the fluid molecules. This phenomenon was first observed by botanist Robert Brown in 1827 and later explained by Albert Einstein in 1905 as evidence for the existence of atoms and molecules. Brownian motion is a key concept in understanding the behavior of particles at the microscopic level.
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10 Key excerpts on "Brownian Motion"
- Peter F. Green(Author)
- 2005(Publication Date)
- CRC Press(Publisher)
51 2 Brownian Motion 2.1 Introduction Under equilibrium conditions, the dynamics of a dilute concentration of par-ticles of microscopic dimensions immersed in a liquid at a temperature T are random, and if the average velocity of a particle is measured over a sufficiently long time interval, for example in the x-direction, it would be zero, = 0. This is a consequence of the fact that the particle can move in any direction with equal probability. Whereas the average velocity is zero, the velocity of the particle at a given instant is typically not zero because fluctuations of the velocity occur and are specified by . These fluctuations increase with temperature, and more massive objects experience smaller fluc-tuations. The random, statistically fluctuating, and incessant motions of the particles in the liquid typify the phenomenon of Brownian Motion. What are believed to be the first well-documented observations of this phenomenon were made in 1828, by an English Botanist, Robert Brown, after whom the effect is named. Brown made careful observations of the motions of pollen grains in water using an optical microscope. He reported that the motions of the pollen grains were incessant and that their behavior could not be reconciled with currents in the fluid or with evaporation. We now know that the dynamics of these particles manifest the random inces-sant bombardment by the molecules in the liquid. If measurements of the displacements of a tiny particle in a liquid during fixed time intervals were to be performed, a distribution function that char-acterizes its dynamics could be constructed. Specifically, two parameters would be of interest: 1) the magnitude and direction of the displacement of a particle, ∆ x , during each fixed interval ∆ t; and 2) the number, n , of occur-rences of such displacements.
- Hui Wang(Author)
- 2012(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 2 Brownian Motion Brownian Motion was discovered in 1827 by the English botanist Robert Brown when he was studying the movement of microscopic pollen grains suspended in a drop of water. The rigorous mathematical foundation of Brownian Motion was established by Norbert Wiener around 1923. For this reason, Brownian Motion is also called the Wiener process. In mathematical finance, Brownian Motion has been used extensively in the modeling of security prices. The celebrated Black–Scholes option pricing formula was derived upon the assumption that the underlying stock price is a geometric Brownian Motion. The purpose of this chapter is to introduce Brownian Motion and its basic properties. We suggest that the reader go over Appendix A before reading this chapter since the multivariate normal distributions are indis-pensable for the study of Brownian Motion. 2.1 Brownian Motion A continuous time stochastic process W = { W t : t ≥ 0 } is a collection of random variables indexed by “time” t . For each fixed ω ∈ Ω , the mapping t mapsto → W t ( ω ) is called a sample path . We say W is a standard Brownian Motion if the following conditions hold: 1. Every sample path of the process W is continuous. 2. W 0 = 0. 3. The process has independent increments, that is, for any sequence 32 CHAPTER 2. Brownian Motion 0 = t 0 < t 1 < · · · < t n , the increments W t 1 -W t 0 , W t 2 -W t 1 , · · · , W t n -W t n -1 are independent random variables. 4. For any s ≥ 0 and t > 0, the increment W s + t -W s is normally dis-tributed with mean 0 and variance t . It is immediate from the definition that W t = W t -W 0 is normally dis-tributed with mean 0 and variance t . Figure 2.1 shows some representative sample paths of a standard Brow-nian motion. They all exhibit a certain kind of ruggedness. Actually, it can be shown that with probability one, the Brownian Motion sample paths are nowhere differentiable and nowhere monotonic [18].
eBook - PDFEnergy Systems
Adaptive Complexity
- T. Ohta(Author)
- 2013(Publication Date)
- Elsevier Science(Publisher)
Then, its thermal motion may be observed, under a microscope, to display a zig-zagged path (so-called random walk) due to random thermal collisions with liquid molecules. The collisions give rise to unbalanced forces, called Brownian or Langevin forces, acting on the particle. This random behavior of the particle has been called Brownian Motion. The Brownian force causes an abrupt change of the particle velocity in both its magnitude and direction, but the magnitude decays, on average, to the equilibrium value of the Maxwell velocity distribution in a relaxation time, given by r B = mI γ where γ — βπηα (1.1) 0306-2619/00/$ - see front matter © 2000 Elsevier Science Ltd. All rights reserved. PII: S0306-2619(00)00007-6 62 M. Takeo I Applied Energy 67 (2000) 61-89 for which m and a are the particle's mass and radius, respectively, and η the shear viscosity of the liquid, which induces the friction coefficient γ. The relaxation time is of the order of 10~ 9 s for a 0.1 μιη sphere in water at room temperature. This is short compared with the lifetime of an excited atomic state. Thus, if optical scattering is used to investigate the random walk of a particle in water, the detailed behavior during the relaxation process cannot be observed. Because of the Brownian Motion, however, the particle macroscopically diffuses, as an averaged behavior in the liquid, with a much smaller velocity, i.e. the diffusion velocity. This diffusion process is, in general, affected by the presence of other par-ticles, if the suspension is concentrated. The statistics of Brownian Motion in a turbulent fluid flow are quite different from the steady case [1,2], and we will not discuss the turbulent case here. In recent years, there has been growing interest in the study of the dynamic properties of interacting Brownian particles.
eBook - PDF- K. Patterson(Author)
- 2010(Publication Date)
- Palgrave Macmillan(Publisher)
160 6 Brownian Motion: Basic Concepts 160 Introduction Brownian Motion is a key concept in economics in two respects. It underlies an important part of stochastic finance, which includes the pricing of risky assets, such as stock prices, bonds and exchange rates. For example a central model for the price of a risky asset is that of geometric Brownian Motion (see Chapter 7). It also plays a key role in econometrics, especially in the distribution theory underlying test statistics for a unit root. For example, the limiting distribution of the familiar Dickey-Fuller pseudo-t test for a unit root is a functional of Brownian Motion. In both cases, but in different contexts, it is possible to relate the importance of Brownian Motion to the limit of a random walk process in which the steps of the random walk become smaller and smaller. The process can be viewed in the limit as occurring in continuous time; the stochastic difference equation becomes a stochastic differential equation and the random walk generates a sample path of Brownian Motion. Solving the stochastic differential equation requires use of the Itô calculus. This chapter outlines the key concept of Brownian Motion, whereas the next chapter is a non-technical introduction to Itô calcu- lus. Together these chapters provide some familiarity with the language that is used in more advanced texts and especially in the asymptotic distribution theory for unit root tests. A number of references to more advanced works are provided at the end of the chapter for the interested reader. This chapter is organised as follows. Section 6.1 introduces the definition of Brownian Motion and Section 6.2, picking up on Chapter 5, links this to the limit of a random walk; Section 6.3 outlines the function space Brownian Motion: Basic Concepts 161 on which BM is defined and Section 6.4 summarises some key proper- ties of Brownian Motion.
- Carlo Di Castro, Roberto Raimondi(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
11 Brownian Motion and transport in disordered systems 11.1 Einstein’s theory of Brownian Motion According to Einstein (1905) a colloid, a fine suspension of particles or grains in a liquid, is not qualitatively different from a solution. A solution is made of molecules of a substance mixed with those of another substance, in a fine suspension; instead, the grains can be made of many molecules and may have dimensions ranging from the microscopic to macroscopic size. As a consequence of this analogy, a colloid must show the phenomenon of osmotic pressure, i.e. if we indicate with n the number of grains per unit volume, they act as a gas of diluted particles and the pressure exerted by the grains of the solute follows the law P = nk B T . (11.1) Still using the analogy with solutions, one may expect the grains of the solute be in thermal motion as the molecules of the solvent. But given the big difference in mass between solvent and solute, the motion of the grains of the latter is much slower than that of the former. The molecules of the solvent, being thermally agitated, move randomly in all directions and in so doing collide with the grains. The exchanged momentum in each collision is randomly oriented. As a result the grains, after a transient time, will move with a pattern typical of Brownian Motion. Einstein showed that such a motion can be described in terms of a diffusive process. Let us assume that the colloidal solution is in the field of gravity and indicate with M the mass of the grains. In statistical equilibrium the force of gravity acting on the grains must compensate the difference in osmotic pressure at two different levels, in complete analogy with the barometric formula valid for a perfect gas (see Eq. (N.1) of the answer to Problem 4.1). Let us take the force of gravity directed towards the negative x -axis and let n(x ) be the density of grains at level x . The total mass of the grains contained in a volume element dx d y dz is Mn(x )dx d y dz .
eBook - PDFMolecular Machines
A Materials Science Approach
- Giovanni Zocchi(Author)
- 2018(Publication Date)
- Princeton University Press(Publisher)
1 Brownian Motion The phenomenon of Brownian Motion connects equilibrium and non- equilibrium statistical mechanics. It connects diffusion—a nonequilibrium phenomenon—with thermal fluctuations—an equilibrium concept. More precisely, diffusion with a net flow of particles, driven by a concentration gradient, pertains to a nonequilibrium system, since there is a net cur- rent. Without a concentration gradient, the system is macroscopically in equilibrium, but each individual particle undergoes self-diffusion just the same. In this sense, Brownian Motion is at the border of equilibrium and nonequilibrium statistical mechanics. Understanding Brownian Motion led Einstein, in one of his famous 1905 papers, to a form of the fluctuation– dissipation theorem. Here we give an introduction to the main ideas. 1.1 Random Walk The simplest model of Brownian Motion is a random walk on a lattice. It is the following process: a particle starts at a lattice site, and makes random steps to the neighboring sites. For example, we may visualize a square lattice in 2-D (figure 1.1). With equal probability, the particle steps up, down, right, or left. After a time interval τ 0 the process is repeated. Thus time is connected to the number of steps N by t = Nτ 0 . (1.1) The ith step is specified by a vector ⃗ r i (there are only 4 such vectors for a square lattice in 2-D); a particular realization of the random walk of N steps is the set of vectors random walk: {⃗ r i ∶ i = 1, 2, … , N }, |⃗ r i | = 𝓁 ∀ i, (1.2) 1 2 CHAPTER 1 r i → FIGURE 1.1. Square lattice and the vector ⃗ r i representing the ith step of a random walk. where 𝓁 is the step size (the lattice spacing). The end-to-end distance of the walk (the displace- ment of the particle after N steps) is ⃗ R = N ∑ i=1 ⃗ r i . (1.3) The ⃗ r i ’s are random variables, and so is ⃗ R. Now we ask what the typical displacement of the particle is, that is, we want ensemble averages.
eBook - PDFNonequilibrium Statistical Physics
A Modern Perspective
- Roberto Livi, Paolo Politi(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
On the contrary, the experimental observation of the erratic motion of the pollen particle indicated that the distance of the particle from its original position grew over sufficiently long time intervals as the square root of time, thus showing the diffusive nature of its motion. Repeating many times the same experiment, where the pollen particle, the solvent, and the temperature of the solvent are the same, the particle in each realization follows different paths, but one can perform a statistical average over these realizations that enforces the conclusion that the particle exhibits a diffusive motion. The universal character of this phenomenon was confirmed by the experimental observations that a diffusive behavior was found also when the type of Brownian particle, the solvent, and the temperature were changed, yielding different values of the proportionality constant between time and the average squared distance of the particle from its initial position. A convincing explanation of Brownian Motion had to wait for the fundamental contribution of Albert Einstein, which appeared in 1905, the same year as his contributions on the theories of special relativity and the photoelectric effect. Einstein’s phenomenological theory of Brownian Motion, relying on simple physical principles, inspired the French scientist Paul Langevin, who proposed a mechanistic approach. The basic idea was to write a Newton-like ordinary differential equation where, for the first time, a force was attributed a stochastic nature. In fact, the microscopic forces exerted by the solvent particles through elastic collisions with the Brownian particle are represented as uncorrelated fluctuations in space and time, whose square amplitude is assumed to be proportional to the thermal energy; according to kinetic theory, this amounts to the solvent temperature T, provided the Brownian particle is at thermodynamic equilibrium with the solvent.
eBook - PDFEinstein's Miraculous Year
Five Papers That Changed the Face of Physics
- Albert Einstein, John Stachel(Authors)
- 2021(Publication Date)
- Princeton University Press(Publisher)
Part Two Einstein on Brownian Motion Einstein as a student at the ETH, or shortly thereafter. (Courtesy of Hebrew University of Jerusalem) Einstein's study of Brownian Motion constitutes one of the high points in the long tradition of research on the kinetic theory of heat and of his own contributions to this field. Some of the consequences of his work were of great signif-icance for the development of physics in the twentieth cen-tury. Einstein's derivation of the laws governing Brownian Motion, and their subsequent experimental verification by Perrin and others, contributed significantly to the acknowl-edgment of the physical reality of atoms by the then still-numerous skeptics. His papers on Brownian Motion helped to establish the study of fluctuation phenomena as a new branch of physics. The methods he created in the course of his research prepared the way for statistical thermodynam-ics, later developed by Szilard and others, and for a general theory of stochastic processes. Since at least the middle of the nineteenth century, a growing number of physicists and chemists had accepted the atomic hypothesis. The assumption that matter consists of atoms and molecules suggested a number of relations be-tween phenomena, both physical and chemical, that are un-expected from a purely macroscopic point of view. Various methods for the determination of molecular dimensions gave values that were often in surprisingly good agreement. The physical reality of atoms was not, however, universally ac-cepted by the end of the century. There were still some fer-vent opponents of the atomic hypothesis, such as Wilhelm Ostwald and Georg Helm, who called themselves energeti-cists to indicate that they regarded the concept of energy as the most fundamental ontological concept of science. Oth-ers, such as Ernst Mach, while adopting a hostile position 73
eBook - PDFMiddle World
The Restless Heart of Matter and Life
- M. Haw(Author)
- 2016(Publication Date)
- Palgrave Macmillan(Publisher)
He went beyond Gouy's ideas by adding concrete maths, equations enabling num- erical predictions that could be tested in experiments. How, Einstein asked, can we describe the motion of a molecule in a liquid? The liquid is so crowded that no molecule can simply move in a straight line: it will continually collide with its neigh- bours and cannon off in some other direction. So let's assume, Einstein went on, that the path of the molecule is like a randomly 3 Some claim that Einstein didn't even know there was such a phe- nomenon as 'Brownian Motion' when he published his first paper: his predictions had simply come straight out of the maths. It's notori- ously hard to trace the footsteps of Einstein - the sources of his ideas. He was sparing when it came to citing others in his own papers. Letters indicate that he knew something about Brownian Motion, though had perhaps not read much of the literature, before 1905. A THEORY OF THE BROWNIAN WORLD 97 varying quantity: in a long enough time interval the molecule suffers so many collisions that, from the beginning of one time interval to the beginning of the next, the molecule's speed and direction are 'randomized'. Given the molecule's speed and direction at one time instant, because of the randomizing effect of so many millions of colli- sions, you might just as well roll dice to decide which way the molecule will be going at the next time instant. The molecule car- ries out a 'random walk', hopping now this way, now that - driven by the randomizing effect of many collisions. Einstein constructed a mathematical description of this molec- ular random-walk motion 4 . Then he went further. Now take a large particle, proposed Einstein, one much bigger than the molecules of the liquid. Place it in the liquid - immerse it in the sea of randomly jiggling liquid molecules. How does the large particle move? The particle's motion, Einstein pointed out, depends on its energy.
- A. O. Caldeira(Author)
- 2014(Publication Date)
- Cambridge University Press(Publisher)
However, as we have seen before, in SQUID rings things can be very different on reaching the appro- priate domain of the circuit parameters when quantum mechanics comes into play. 4.1 Classical Brownian Motion 89 In this circumstance we have to deal with dissipation, fluctuations, and quantum effects on the same footing, and it is our main goal in this book to tie all these effects together. We shall return to this point in the next section. Although we could jump directly to the development of our strategy to tackle the above-mentioned problem, we will, for the sake of completeness, introduce some key concepts in the classical theory of Brownian Motion. 4.1.1 Stochastic processes The Langevin equation involves the concept of fluctuating forces, which intro- duces a probabilistic character into the dynamics of the variable q (t ). One way to understand this process is by creating a statistical ensemble of equally prepared Brownian particles. After a time t each particle will occupy a different position due to the fact that it is being subject to a random force. Therefore, we can define the probability P (q , t )dq to find the Brownian particle within the interval [q , q + dq ] at time t , which allows us to compute the average value of any function g(q ). Let us then start by introducing some useful concepts of stochastic processes (Reichl, 2009). Let y (t ) be a variable which can assume any value in the interval −∞ < y (t ) < ∞. The probability density that the variable y (t ) has the value y 1 at time t 1 is P 1 ( y 1 , t 1 ). This concept can be generalized to the case of n events through the probability that y (t ) has the value y 1 at time t 1 , y 2 at t 2 , . . ., y n at t n , which is described by the joint probability density P n ( y n , t n ; y n−1 , t n−1 ; . . .; y 1 , t 1 ). (4.3) The functions P n are normalized as R n P n ( y n , t n ; y n−1 , t n−1 ; . . .; y 1 , t 1 ) dy 1 dy 2 .
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