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.
5 Key excerpts on "Brownian Motion"
eBook - ePubFinancial Statistics and Mathematical Finance
Methods, Models and Applications
- Ansgar Steland(Author)
- 2012(Publication Date)
- Wiley(Publisher)
...Chapter 5 Brownian Motion and Related Processes in Continuous Time This chapter is devoted to a basic introduction to stochastic processes in continuous time, particularly to Brownian Motion, its properties and some related processes. The study of Brownian Motion, discovered empirically by the botanist Robert Brown in 1827, is a highlight of modern science. He studied microscopic particles dispensed in a fluid and observed that they move in an irregular fashion. Albert Einstein developed in 1905 a physical theory and the related mathematical solution and calculus. He explained the irregular movement by the impact of the much smaller fluid molecules. Similar ideas were published by Marian Smoluchowski in 1906. However, in 1900 Louis Bachelier used the one-dimensional version t B t, t ≥ 0, of the Brownian Motion to model stock prices. The economic reasoning is quite similar. The stream of orders to buy or sell the stock result in small changes (ups and downs) of the stock prices. The rigorous mathematical foundation is due to Norbert Wiener, who established the existence of Brownian Motion that is therefore also called Wiener process. He constructed the Wiener measure that describes the distribution of Brownian Motion as a function of t. Brownian Motion has a couple of specific properties that appear puzzling at first glance. The trajectories are almost surely continuous, but nowhere differentiable. Further, the length of the curve t B t, t [ a, b ], a < b, is infinite for any interval. Here, we confine ourselves to a discussion of some useful general notions for stochastic processes in continuous time, including the extension of the definition of (semi-/super-) martingales to the continuous-time framework, a formal definition of Brownian Motion and its most important basic properties and rules of calculation...
eBook - ePubAerosol Technology
Properties, Behavior, and Measurement of Airborne Particles
- William C. Hinds, Yifang Zhu(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
...7 Brownian Motion and Diffusion In 1827, botanist Robert Brown first observed the continuous wiggling motion of pollen grains in water that we now call Brownian Motion. About 50 years later, a similar motion was observed for smoke particles in air, and the connection between this motion and that predicted for gas molecules by the kinetic theory of gases was first made. In the early 1900s, Einstein derived the relationships characterizing Brownian Motion, which were verified experimentally soon afterwards. Excluding convection, thermal diffusion is the primary transport and deposition mechanism for particles less than 0.1 μm in diameter. Thermal diffusion is responsible for the collection of these particles in situations where the transport distances are small, such as in a filter or in the airways of a human lung. Where the physical scale is large, convective or eddy diffusion of parcels of aerosol greatly exceeds the transport of aerosol particles by thermal diffusion. 7.1 Diffusion Coefficient Brownian Motion is the irregular wiggling motion of an aerosol particle in still air caused by random variations in the relentless bombardment of gas molecules against the particle. Diffusion of aerosol particles is the net transport of these particles in a concentration gradient. This transport is always from a region of higher concentration to a region of lower concentration. Both processes are characterized by the particle diffusion coefficient D. The larger the value of D, the more vigorous the Brownian Motion and the more rapid the mass transfer in a concentration gradient. The diffusion coefficient is the constant of proportionality that relates the flux J of aerosol particles (the net number of particles traveling through unit cross section each second) to the concentration gradient dn/dx. This relationship is called Fick's first law of diffusion. In the absence of any external forces, Fick's law is (7.1) Equation 7.1 for aerosol particles is the same as Eq...
eBook - ePub- A. Dixit(Author)
- 2013(Publication Date)
- Routledge(Publisher)
...1. Brownian Motion Brownian Motion is a continuous-time scalar stochastic process such that, given the initial value x 0 at time t = 0, the random variable x t for any t > 0 is normally distributed with mean (x 0 + μt) and variance (σ 2 t). The parameter μ measures the trend, and a the volatility, of the process. This process was first formulated to represent the motion of small particles suspended in a liquid. We shall sometimes refer to a 'particle' performing the Brownian Motion, x t as its 'position', and a graph of x t against t as its 'path'. We can think of Brownian Motion as the cumulation of independent identically normally distributed increments, the infinitesimal random increment dx over the infinitesimal time dt having mean μdt and variance σ 2 dt. Just as we would write a general normal (μ, σ) variable as μ + σw where w is a standard normal variable of zero mean and unit variance, we can write where w is a standardized Brownian Motion (Wiener process) whose increment dw has zero mean and variance dt. This is the usual shorthand notation for Brownian Motion. The (Itô) calculus of such infinitesimal random variables differs in some important ways from the usual non-random calculus. A fully rigorous treatment of Itô calculus is quite difficult. Therefore I shall develop a non-rigorous exposition that suffices for many economic applications. I shall approximate Brownian Motion by a discrete random walk. Then the normal distribution arises as the limit of a sum of independent binary variables Δ x over discrete time intervals Δ t, when these go to zero in a particular way. 1.1. Random walk representation Divide time into discrete periods of length Δ t, and let space consist of discrete points along a line, Δ h being the step-length or the distance between successive points...
eBook - ePubBedeviled
A Shadow History of Demons in Science
- Jimena Canales(Author)
- 2020(Publication Date)
- Princeton University Press(Publisher)
...But they had not killed him off entirely. The punctual effects of a single molecule moving in an unexpected direction could be more consequential than the entire regular movement of the rest of the crowd. RANDOM WALKS Brownian Motion was a crowded field by the time Einstein entered it. In his first paper, he repeated some of the same conclusions reached by previous scientists. “If it is really possible to observe the motion to be discussed here, along with the laws it is expected to obey, then classical thermodynamics can no longer be viewed as strictly valid even for microscopically distinguishable spaces,” he explained in 1905. 28 But in contrast to the Maxwellians and to Poincar é, Einstein tried to explain this anomaly without resorting to demons. In his view, underlying molecular forces were solely responsible for pushing these particles hither and yon. The year Einstein published his studies on Brownian Motion, the statistician Karl Pearson published a question in the journal Nature. His intention was to crowdsource a solution to a difficult mathematical problem: the problem of “the Random Walk.” If “a man starts from a point O and walks l yards in a straight line, he then turns through any angle whatever and walks another l yards in a second straight line,” and “he repeats the process n times,” where would he most likely end up? Where should a search team go to fetch such an aimless wanderer? When n was large in comparison to l, the answer was clear: “In open country the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point!” 29 When n was small in comparison to l, the inebriated man was much harder to find. Pearson’s example of the “Random Walk,” or the “Drunkard’s Walk,” was almost immediately identified with Brownian Motion and soon became famous. 30 The French scientist Jean Perrin joined these investigations, eventually earning a Nobel Prize for his contributions...
eBook - ePub- Rob Phillips, Jane Kondev, Julie Theriot, Hernan Garcia(Authors)
- 2012(Publication Date)
- Garland Science(Publisher)
...We are now ready to take the next step and consider the individual movements of discrete particles in water, ranging from molecules to organelles to viruses and the cells they attack. Over the next four chapters, we will develop this theme of biological motions, starting with the simplest case applying to nonliving and living systems alike, namely, Brownian or diffusive motion. What makes such processes especially intriguing is that despite the stochastic microscopic underpinnings, huge numbers of diffusing molecules over a large number of time steps can give the appearance of purposeful dynamics of particles down a concentration gradient. As discussed in Section 3.4.2 (p. 126), Brownian Motion is an inevitable outcome of the thermal jiggling of water molecules and does not indicate the activities of a living system. However, diffusive motion is always present at molecular length scales, and biological systems must tolerate, exploit, or inhibit Brownian Motion in order to perform directed dynamic processes. A familiar example of the physical limits put on organisms by the process of diffusion is something you experience with every breath you take. Human metabolism demands a constant high concentration of oxygen supplied to mitochondria throughout the body. Much smaller organisms that are oxygen-dependent can rely simply on diffusion of oxygen as a delivery mechanism, but this is only efficient over distances of the order of tens of microns. In order to grow to sizes exceeding 1 m, humans and other large animals have developed elaborate mechanisms to circulate oxygen and effectively enable its delivery to all tissues. In Chapter 7, we examined hemoglobin as a protein specialized for the sole purpose of carrying oxygen to parts of the body far from the lungs. Oxygen inhaled in air can diffuse through lung tissue over an effective distance of roughly 100 μm that is set not only by the free diffusion of oxygen, but also by its rate of consumption by cells in the tissue...
