Brownian motion in the context of Discrete dynamical system

In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model recognizes seventeen distinct particles—twelve fermions and five bosons. As a consequence of flavor and color combinations and antimatter, the fermions and bosons are known to have 48 and 13 variations, respectively. These 61 elementary particles include electrons and other leptons, quarks, and the fundamental bosons. Subatomic particles such as protons or neutrons, which contain two or more elementary particles, are known as composite particles.

Ordinary matter is composed of atoms, themselves once thought to be indivisible elementary particles. The name atom comes from the Ancient Greek word ἄτομος (atomos) which means indivisible or uncuttable. Despite the theories about atoms that had existed for thousands of years, the factual existence of atoms remained controversial until 1905. In that year, Albert Einstein published his paper on Brownian motion, putting to rest theories that had regarded molecules as mathematical illusions. Einstein subsequently identified matter as ultimately composed of various concentrations of energy.

In the physics of aerosols, deposition is the process by which aerosol particles collect or deposit themselves on solid surfaces, decreasing the concentration of the particles in the air. It can be divided into two sub-processes: dry and wet deposition. The rate of deposition, or the deposition velocity, is slowest for particles of an intermediate size. Mechanisms for deposition are most effective for either very small or very large particles. Very large particles will settle out quickly through sedimentation (settling) or impaction processes, while Brownian diffusion has the greatest influence on small particles. This is because very small particles coagulate in few hours until they achieve a diameter of 0.5 micrometres. At this size they no longer coagulate. This has a great influence in the amount of PM-2.5 present in the air.

Deposition velocity is defined from F = vc, where F is flux density, v is deposition velocity and c is concentration. In gravitational deposition, this velocity is the settling velocity due to the gravity-induced drag.

"Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen" (English: "On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat") is the 1905 journal article, by Albert Einstein, that proved the reality of atoms, the modern understanding of which had been proposed in 1808 by John Dalton. It is one of the four groundbreaking papers Einstein published in 1905, in Annalen der Physik, in his miracle year.

In 1827, botanist Robert Brown used a microscope to look at dust grains floating in water. He found that the floating grains were moving about erratically; a phenomenon that became known as "Brownian motion". This was thought to be caused by water molecules knocking the grains about. In 1905, Albert Einstein proved the reality of these molecules and their motions by producing the first statistical physics analysis of Brownian motion. French physicist Jean Perrin used Einstein's results to experimentally determine the mass, and the dimensions, of atoms, thereby conclusively verifying Dalton's atomic theory.

In probability theory and related fields, a stochastic (/stəˈkæstɪk/) or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.

Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were invented repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.

At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables.

A dispersion is a system in which distributed particles of one material are dispersed in a continuous phase of another material. The two phases may be in the same or different states of matter.

Dispersions are classified in a number of different ways, including how large the particles are in relation to the particles of the continuous phase, whether or not precipitation occurs, and the presence of Brownian motion. In general, dispersions of particles sufficiently large for sedimentation are called suspensions, while those of smaller particles are called colloids and solutions.

In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaussian process ${\textstyle B_{H}(t)}$ on ${\textstyle [0,T]}$, that starts at zero, has expectation zero for all ${\displaystyle t}$ in ${\textstyle [0,T]}$, and has the following covariance function:

where H is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value leading to a smoother motion. It was introduced by Mandelbrot & van Ness (1968).

The annus mirabilis papers (from Latin: annus mirabilis, lit. 'miraculous year') are four papers that Albert Einstein published in the scientific journal Annalen der Physik (Annals of Physics) in 1905. As major contributions to the foundation of modern physics, these scientific publications were the ones for which he gained fame among physicists. They revolutionized science's understanding of the fundamental concepts of space, time, mass, and energy.

1. The first paper explained the photoelectric effect, which established the energy of the light quanta ${\displaystyle E=hf}$, and was the only specific discovery mentioned in the citation awarding Einstein the 1921 Nobel Prize in Physics.
2. The second paper explained Brownian motion, which established the Einstein relation ${\displaystyle D=\mu \,k_{\text{B}}T}$ and compelled physicists to accept the existence of atoms.
3. The third paper introduced Einstein's special theory of relativity, which proclaims the constancy of the speed of light ${\displaystyle c}$ and derives the Lorentz transformations. Einstein also examined relativistic aberration and the transverse Doppler effect.
4. The fourth, a consequence of special relativity, developed the principle of mass–energy equivalence, expressed in the equation ${\displaystyle E=mc^{2}}$ and which led to the discovery and use of nuclear power decades later.

These four papers, together with quantum mechanics and Einstein's later general theory of relativity, are the foundation of modern physics.

Robert Brown FRSE FRS FLS MWS (21 December 1773 – 10 June 1858) was a Scottish botanist and paleobotanist who made important contributions to botany largely through his pioneering use of the microscope. His contributions include one of the earliest detailed descriptions of the cell nucleus and cytoplasmic streaming; the observation of Brownian motion; early work on plant pollination and fertilisation, including being the first to recognise the fundamental difference between gymnosperms and angiosperms; and some of the earliest studies in palynology. He also made numerous contributions to plant taxonomy, notably erecting a number of plant families that are still accepted today; and numerous Australian plant genera and species, the fruit of his exploration of that continent with Matthew Flinders.

The standard author abbreviation R.Br. is used to indicate this person as the author when citing a botanical name.

Jean Baptiste Perrin (French: [ʒɑ̃ batist pɛʁɛ̃]; 30 September 1870 – 17 April 1942) was a French atomic physicist who, in his studies of the Brownian motion of minute particles suspended in liquids (sedimentation equilibrium), verified Albert Einstein's explanation of this phenomenon and thereby confirmed the atomic nature of matter. For this work, he was awarded the Nobel Prize in Physics in 1926.

In mathematics, the Wiener process (or Brownian motion, due to its historical connection with the physical process of the same name) is a real-valued continuous-time stochastic process named after Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments). It occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.

The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a white noise Gaussian process, and so is useful as a model of noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and disturbances in control theory.

Louis Jean-Baptiste Alphonse Bachelier (French: [baʃəlje]; 11 March 1870 – 28 April 1946) was a French mathematician at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, as part of his doctoral thesis The Theory of Speculation (Théorie de la spéculation, defended in 1900).

Bachelier's doctoral thesis, which introduced the first mathematical model of Brownian motion and its use for valuing stock options, was the first paper to use advanced mathematics in the study of finance. His Bachelier model has been influential in the development of other widely used models, including the Black-Scholes model.