Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral.
TL;DR: The exact simulation algorithm used here to illustrate the zero-\ensuremath{\tau} limit theorem is derived for the Ornstein-Uhlenbeck process X(t) and its time integral Y(t).
Abstract: A numerical simulation algorithm that is exact for any time step \ensuremath{\Delta}tg0 is derived for the Ornstein-Uhlenbeck process X(t) and its time integral Y(t). The algorithm allows one to make efficient, unapproximated simulations of, for instance, the velocity and position components of a particle undergoing Brownian motion, and the electric current and transported charge in a simple R-L circuit, provided appropriate values are assigned to the Ornstein-Uhlenbeck relaxation time \ensuremath{\tau} and diffusion constant c. A simple Taylor expansion in \ensuremath{\Delta}t of the exact simulation formulas shows how the first-order simulation formulas, which are implicit in the Langevin equation for X(t) and the defining equation for Y(t), are modified in second order. The exact simulation algorithm is used here to illustrate the zero-\ensuremath{\tau} limit theorem. \textcopyright{} 1996 The American Physical Society.
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TL;DR: Van Kampen as mentioned in this paper provides an extensive graduate-level introduction which is clear, cautious, interesting and readable, and could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes.
Abstract: N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.
3,647 citations
TL;DR: This work investigates growing interfaces of liquid-crystal turbulence and finds not only universal scaling, but universal distributions of interface positions, which obey the largest-eigenvalue distributions of random matrices and depend on whether the interface is curved or flat, albeit universal in each case.
Abstract: Stochastic motion of a point – known as Brownian motion – has many successful applications in science, thanks to its scale invariance and consequent universal features such as Gaussian fluctuations. In contrast, the stochastic motion of a line, though it is also scale-invariant and arises in nature as various types of interface growth, is far less understood. The two major missing ingredients are: an experiment that allows a quantitative comparison with theory and an analytic solution of the Kardar-Parisi-Zhang (KPZ) equation, a prototypical equation for describing growing interfaces. Here we solve both problems, showing unprecedented universality beyond the scaling laws. We investigate growing interfaces of liquid-crystal turbulence and find not only universal scaling, but universal distributions of interface positions. They obey the largest-eigenvalue distributions of random matrices and depend on whether the interface is curved or flat, albeit universal in each case. Our exact solution of the KPZ equation provides theoretical explanations.
275 citations
Cites background or methods from "Exact numerical simulation of the O..."
...8r̄ [24, 27], where r̄ is the mean distance of the particles on the interface from the origin, whereas in our model it is determined by the purely geometrical consideration, for which there is no risk of false inactivation....
[...]
...’s model [22, 24, 27], a random number for the angular position of the new particle is generated within [π/6, φ − π/6], where φ is defined in the sketch and π/6 is due to the finite radius of the new particle....
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...(31) using the standard numerical scheme for the Ornstein-Uhlenbeck process [27], which is exact for any finite time step ∆ t....
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...[22] (also described in [24, 27]) except for the criterion of the inner particles and for the way a new particle is added to the randomly chosen ancestor....
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...A 45 638–653 [27] Alves S G, Ferreira Jr S C and Martins M L 2008 Braz....
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TL;DR: It is shown that single-cell time-lapse microscopy, by revealing time lags due to regulation, can discriminate between active regulatory connections and extrinsic noise.
Abstract: Gene regulatory interactions are context dependent, active in some cellular states but not in others. Stochastic fluctuations, or 'noise', in gene expression propagate through active, but not inactive, regulatory links. Thus, correlations in gene expression noise could provide a noninvasive means to probe the activity states of regulatory links. However, global, 'extrinsic', noise sources generate correlations even without direct regulatory links. Here we show that single-cell time-lapse microscopy, by revealing time lags due to regulation, can discriminate between active regulatory connections and extrinsic noise. We demonstrate this principle mathematically, using stochastic modeling, and experimentally, using simple synthetic gene circuits. We then use this approach to analyze dynamic noise correlations in the galactose metabolism genes of Escherichia coli. We find that the CRPGalS-GalE feed-forward loop is inactive in standard conditions but can become active in a GalR mutant. These results show how noise can help analyze the context dependence of regulatory interactions in endogenous gene circuits.
242 citations
TL;DR: In this paper, the authors proposed concatenated continuous dynamical decoupling, which can overcome not only external magnetic noise but also noise due to fluctuations in driving fields, and demonstrate experimentally that the most basic implementation of this concept yields an order of magnitude improvement to the decoherence time for the electron spin of nitrogen vacancy centers in diamond.
Abstract: The loss of coherence is one of the main obstacles for the implementation of quantum information processing. The efficiency of dynamical decoupling schemes, which have been introduced to address this problem, is limited itself by the fluctuations in the driving fields which will themselves introduce noise. We address this challenge by introducing the concept of concatenated continuous dynamical decoupling, which can overcome not only external magnetic noise but also noise due to fluctuations in driving fields. We show theoretically that this approach can achieve relaxation limited coherence times, and demonstrate experimentally that already the most basic implementation of this concept yields an order of magnitude improvement to the decoherence time for the electron spin of nitrogen vacancy centers in diamond. The proposed scheme can be applied to a wide variety of other physical systems, including trapped atoms and ions and quantum dots, and may be combined with other quantum technologies challenges such as quantum sensing and quantum information processing.
175 citations
TL;DR: Takeuchi et al. as discussed by the authors provide a comprehensive report on scale-invariant fluctuations of growing interfaces in liquid-crystal turbulence, for which they recently found evidence that they be- long to the Kardar-Parisi-Zhang (KPZ) universality class for 1 + 1 dimensions.
Abstract: We provide a comprehensive report on scale-invariant fluctuations of growing interfaces in liquid-crystal turbulence, for which we recently found evidence that they be- long to the Kardar-Parisi-Zhang (KPZ) universality class for 1 + 1 dimensions (Takeuchi and Sano in Phys. Rev. Lett. 104:230601, 2010; Takeuchi et al. in Sci. Rep. 1:34, 2011). Here we investigate both circular and flat interfaces and report their statistics in detail. First we demonstrate that their fluctuations show not only the KPZ scaling exponents but be- yond: they asymptotically share even the precise forms of the distribution function and the spatial correlation function in common with solvable models of the KPZ class, demonstrat- ing also an intimate relation to random matrix theory. We then determine other statistical properties for which no exact theoretical predictions were made, in particular the tempo- ral correlation function and the persistence probabilities. Experimental results on finite-time effects and extreme-value statistics are also presented. Throughout the paper, emphasis is put on how the universal statistical properties depend on the global geometry of the inter- faces, i.e., whether the interfaces are circular or flat. We thereby corroborate the powerful yet geometry-dependent universality of the KPZ class, which governs growing interfaces driven out of equilibrium.
171 citations
References
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TL;DR: Van Kampen as mentioned in this paper provides an extensive graduate-level introduction which is clear, cautious, interesting and readable, and could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes.
Abstract: N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.
3,647 citations
Book•
01 Jan 2010
TL;DR: In this article, a discussion of some of the basic physical concepts and methods useful in the description of situations involving systems which consist of very many particulars is presented for the junior-senior thermodynamics course given in all departments as a standard part of the curriculum.
Abstract: This book is designed for the junior-senior thermodynamics course given in all departments as a standard part of the curriculum. The book is devoted to a discussion of some of the basic physical concepts and methods useful in the description of situations involving systems which consist of very many particulars. It attempts, in particular, to introduce the reader to the disciplines of thermodynamics, statistical mechanics, and kinetic theory from a unified and modern point of view. The presentation emphasizes the essential unity of the subject matter and develops physical insight by stressing the microscopic content of the theory.
3,171 citations
1,544 citations
Book•
31 Oct 2012
TL;DR: In this paper, the authors present an approximate solution procedure for "Open" Moment Evolution Equations for continuous Markov processes with continuous states and jump Markov Processes with discrete states.
Abstract: Random Variable Theory. General Features of a Markov Process. Continuous Markov Processes. Jump Markov Processes with Continuum States. Jump Markov Processes with Discrete States. Temporally Homogeneous Birth-Death Markov Processes. Appendixes: Some Useful Integral Identities. Integral Representations of the Delta Functions. An Approximate Solution Procedure for "Open" Moment Evolution Equations. Estimating the Width and Area of a Function Peak. Can the Accuracy of the Continuous Process Simulation Formula Be Improved? Proof of the Birth-Death Stability Theorem. Solution of the Matrix Differential Equation. Bibliography. Index.
572 citations