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Kramers–Moyal Expansion
In stochastic processes, Kramers–Moyal expansion refers to a Taylor series expansion of the master equation, named after Hans Kramers and José Enrique Moyal. This expansion transforms the integro-differential master equation :\frac =\int dx' x')p(x',t)-W(x', x)p(x,t)/math> where p(x,t, x_0,t_0) (for brevity, this probability is denoted by p(x,t)) is the transition probability density, to an infinite order partial differential equation :\frac = \sum_^\infty \frac \frac alpha_n(x) p(x,t)/math> where :\alpha_n(x) = \int_^\infty (x'-x)^n W(x'\mid x) \ dx'. Here W(x'\mid x) is the transition probability rate. The Fokker–Planck equation is obtained by keeping only the first two terms of the series in which \alpha_1 is the drift Drift or Drifts may refer to: Geography * Drift or ford (crossing) of a river * Drift, Kentucky, unincorporated community in the United States * In Cornwall, England: ** Drift, Cornwall, village ** Drift Reservoir, associated with the village ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Processes
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. 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, cryptography 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 pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Master Equation
In physics, chemistry and related fields, master equations are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determined by a transition rate matrix. The equations are a set of differential equations – over time – of the probabilities that the system occupies each of the different states. Introduction A master equation is a phenomenological set of first-order differential equations describing the time evolution of (usually) the probability of a system to occupy each one of a discrete set of states with regard to a continuous time variable ''t''. The most familiar form of a master equation is a matrix form: : \frac=\mathbf\vec, where \vec is a column vector (where element ''i'' represents state ''i''), and \mathbf is the matrix of connections. The way connections among states are made determines the dimension of the problem; it is either *a d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans Kramers
Hendrik Anthony "Hans" Kramers (17 December 1894 – 24 April 1952) was a Dutch physicist who worked with Niels Bohr to understand how electromagnetic waves interact with matter and made important contributions to quantum mechanics and statistical physics. Background and education Hans Kramers was born on 17 December 1894 in Rotterdam. the son of Hendrik Kramers, a physician, and Jeanne Susanne Breukelman. In 1912 Hans finished secondary education ( HBS) in Rotterdam, and studied mathematics and physics at the University of Leiden, where he obtained a master's degree in 1916. Kramers wanted to obtain foreign experience during his doctoral research, but his first choice of supervisor, Max Born in Göttingen, was not reachable because of the First World War. Because Denmark was neutral in this war, as was the Netherlands, he travelled (by ship, overland was impossible) to Copenhagen, where he visited unannounced the then still relatively unknown Niels Bohr. Bohr took him on as a Ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
José Enrique Moyal
José Enrique Moyal ( he, יוסף הנרי מויאל; 1 October 1910 – 22 May 1998) was an Australian mathematician and mathematical physicist who contributed to aeronautical engineering, electrical engineering and statistics, among other fields.Ann Moyal, ''Maverick Mathematician: The Life and Science of J.E. Moyal'', ANU E-press, 2006; , accessible online/ref> Career Moyal helped establish the phase space formulation of quantum mechanics in 1949 by bringing together the ideas of Hermann Weyl, John von Neumann, Eugene Wigner, and Hip Groenewold. This formulation is statistical in nature and makes logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two formulations. Phase space quantization, also known as ''Moyal quantization'', largely avoids the use of operators for quantum mechanical observables prevalent in the canonical formulation. Quantum-mechanical evolution in phase space is specifie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The Royal Statistical Society
The ''Journal of the Royal Statistical Society'' is a peer-reviewed scientific journal of statistics. It comprises three series and is published by Wiley for the Royal Statistical Society. History The Statistical Society of London was founded in 1834, but would not begin producing a journal for four years. From 1834 to 1837, members of the society would read the results of their studies to the other members, and some details were recorded in the proceedings. The first study reported to the society in 1834 was a simple survey of the occupations of people in Manchester, England. Conducted by going door-to-door and inquiring, the study revealed that the most common profession was mill-hands, followed closely by weavers. When founded, the membership of the Statistical Society of London overlapped almost completely with the statistical section of the British Association for the Advancement of Science. In 1837 a volume of ''Transactions of the Statistical Society of London'' were wri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integro-differential Equation
In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. General first order linear equations The general first-order, linear (only with respect to the term involving derivative) integro-differential equation is of the form : \fracu(x) + \int_^x f(t,u(t))\,dt = g(x,u(x)), \qquad u(x_0) = u_0, \qquad x_0 \ge 0. As is typical with differential equations, obtaining a closed-form solution can often be difficult. In the relatively few cases where a solution can be found, it is often by some kind of integral transform, where the problem is first transformed into an algebraic setting. In such situations, the solution of the problem may be derived by applying the inverse transform to the solution of this algebraic equation. Example Consider the following second-order problem, : u'(x) + 2u(x) + 5\int_^u(t)\,dt = \theta(x) \qquad \text \qquad u(0)=0, where : \theta(x) = \left\{ \begin{array}{ll} 1, \qq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transition Rate Matrix
Transition or transitional may refer to: Mathematics, science, and technology Biology * Transition (genetics), a point mutation that changes a purine nucleotide to another purine (A ↔ G) or a pyrimidine nucleotide to another pyrimidine (C ↔ T) * Transitional fossil, any fossilized remains of a lifeform that exhibits the characteristics of two distinct taxonomic groups * A phase during childbirth contractions during which the cervix completes its dilation Gender and sex * Gender transitioning, the process of changing one's gender presentation to accord with one's internal sense of one's gender – the idea of what it means to be a man or woman * Sex reassignment therapy, the physical aspect of a gender transition Physics * Phase transition, a transformation of the state of matter; for example, the change between a solid and a liquid, between liquid and gas or between gas and plasma * Quantum phase transition, a phase transformation between different quantum phases * Quantum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fokker–Planck Equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck, who described it in 1914 and 1917. It is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered it in 1931. When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski), and in this context it is equivalent to the convection–diffusion equation. The case with zero diffusion is the continuity equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion. The first consistent microscopic derivation of the Fokker–Planck equation in the single schem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Drift Velocity
In physics, a drift velocity is the average velocity attained by charged particles, such as electrons, in a material due to an electric field. In general, an electron in a conductor will propagate randomly at the Fermi velocity, resulting in an average velocity of zero. Applying an electric field adds to this random motion a small net flow in one direction; this is the drift. Drift velocity is proportional to current. In a resistive material, it is also proportional to the magnitude of an external electric field. Thus Ohm's law can be explained in terms of drift velocity. The law's most elementary expression is: : u= \mu E , where is drift velocity, is the material's electron mobility, and is the electric field. In the MKS system, these quantities' units are m/s, m2/( V·s), and V/m, respectively. When a potential difference is applied across a conductor, free electrons gain velocity in the direction, opposite to the electric field between successive collisions (and lose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Open Source Software
The ''Journal of Open Source Software'' is a peer-reviewed open-access scientific journal covering open-source software from any research discipline. The journal was founded in 2016 by editors Arfon Smith, Kyle Niemeyer, Dan Katz, Kevin Moerman, and Karthik Ram. The editor-in-chief is Arfon Smith (Space Telescope Science Institute), and associate editors-in-chief are Dan Katz, Kevin Moerman, Kyle Niemeyer, and Krysten Thyng. The journal is a sponsored project of NumFOCUS and an affiliate of the Open Source Initiative. The journal uses GitHub as publishing platform. The journal was established in May 2016 and in its first year published 111 articles, with more than 40 additional articles under review. They reported approximately 1200 published articles in March 2021. The journal has been discussed in several peer-reviewed papers which describe its publishing model and its effectiveness. Abstracting and indexing The journal is abstracted and indexed in the Astrophysics Data System ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
