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. It is the driving process of Schramm–Loewner evolution. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is a key process in terms of which more complicated stochastic processes can be described. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. The Wiener process plays an important role in both pure and applied mathematics. It is one of the best known Lévy processes ( càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.
It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown. In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. ( February 2010) ( Learn how and when to remove this template message)Ī single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process Please help to improve this article by introducing more precise citations. We also analyze the positivity of the reduced density matrix and derive the corresponding Fokker-Planck equation in the classical limit.This article includes a list of general references, but it lacks sufficient corresponding inline citations. We identify this with a corresponding transient negative entropy production rate, which is understood to be characteristic of non-Markovian dynamics. While the short-time behavior of this solution does not diverge substantially from the Markovian behavior, at intermediate times we find a resurgence of coherence, which we name lateral coherence.
QUANTUM BROWNIAN MOTION FREE
We also obtain a fully analytical solution in the free particle case. In this paper we develop a non-Markovian extension of the standard Caldeira-Leggett model, based on expanding the dynamics of the reduced system at high temperature in inverse powers of the high-frequency cutoff of the Ohmic spectral density of the environment and derive a non-Markovian master equation for the reduced density matrix for the case of a general potential. There have been many approaches to explore Markovian and non-Markovian dynamics within the framework of open quantum systems, but the richness of the ensuing dynamics is still not fully understood.
It also has important repercussions on foundational problems in physics, such as the process of decoherence and the so-called quantum measurement problem. Understanding the behavior of a quantum system coupled to its environment is of fundamental interest in the general field of quantum technologies.
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