<i>Selected Papers on Noise and Stochastic Processes</i>

Wax, Nelson; Cohen, Marvin

doi:10.1063/1.3062012

Cited by 279 publications

(192 citation statements)

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“…The starting point was to formulate the macroscopic dynamics in terms of elementary or elementary complex (Keizer 1987) reactions where the stoichiometries of the different reactions are made explicit. Special attention has been paid to the stochastic properties of chemical networks that are in macroscopically stable stationary states, where the LNA corresponds to a linear Fokker-Planck equation for the movement of a Brownian particle in a multidimensional harmonic potential (Wax 1954). Its solution (Risken 1984) evolves to a multidimensional normal distribution for deviations in molecule numbers from their macroscopic values.…”

Section: Discussionmentioning

confidence: 99%

Fast Evaluation of Fluctuations in Biochemical Networks With the Linear Noise Approximation

Elf

Ehrenberg²

2003

Genome Res.

371

463

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Biochemical networks in single cells can display large fluctuations in molecule numbers, making mesoscopic approaches necessary for correct system descriptions. We present a general method that allows rapid characterization of the stochastic properties of intracellular networks. The starting point is a macroscopic description that identifies the system's elementary reactions in terms of rate laws and stoichiometries. From this formulation follows directly the stationary solution of the linear noise approximation (LNA) of the Master equation for all the components in the network. The method complements bifurcation studies of the system's parameter dependence by providing estimates of sizes, correlations, and time scales of stochastic fluctuations. We describe how the LNA can give precise system descriptions also near macroscopic instabilities by suitable variable changes and elimination of fast variables.

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Section: Discussionmentioning

confidence: 99%

Fast Evaluation of Fluctuations in Biochemical Networks With the Linear Noise Approximation

Elf

Ehrenberg²

2003

Genome Res.

371

463

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“…The high-frequency behavior associated with fluctuations around either of the two equilibrium states x L or x H can be approximated by an OrnsteinUhlenbeck process (Wax 1954;Bryan and Hansen 1993;Cessi 1994); the corresponding spectra S x L and S x H are…”

Section: Atmospheric Low-frequency Variability As a Random Telegraph mentioning

confidence: 99%

A mechanistic model of mid-latitude decadal climate variability

Kravtsov

Dewar

Ghil

et al. 2008

Physica D: Nonlinear Phenomena

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A simple heuristic model of coupled decadal ocean-atmosphere modes in middle latitudes is developed. Previous studies have treated atmospheric intrinsic variability as a linear stochastic process modified by a deterministic coupling to the ocean.The present paper takes an alternative view: based on observational, as well as process modeling results, it represents this variability in terms of irregular transitions between two anomalously persistent, high-latitude and low-latitude jet-stream states. Atmospheric behavior is thus governed by an equation analogous to that describing the trajectory of a particle in a double-well potential, subject to stochastic forcing. Oceanic adjustment to a positional shift in the atmospheric jet involves persistent circulation anomalies maintained by the action of baroclinic eddies; this process is parameterized in the model as a delayed oceanic response. The associated sea-surface temperature anomalies provide heat fluxes that affect atmospheric circulation by modifying the shape of the double-well potential. If the latter coupling is strong enough, the model's spectrum exhibits a peak at a periodicity related to the ocean's eddy-driven adjustment time. A nearly analytical approximation of the coupled model is used to study the sensitivity of this behavior to key model parameters.

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“…Equation (4) has been known since Ornstein and Uhlenbeck [9][10][11][12], but it is typically averaged over the thermal distribution of the velocity since, until recently, it has not been possible to measure the instantaneous initial velocity of a Brownian trajectory. The short-time behavior (t τ ) of x|v 0 is v 0 t. An average over the thermal distribution of initial velocity (or equivalently setting v 2 0 = τ ) gives Taylor's ballistic regime σ 2 = τ t 2 .…”

mentioning

confidence: 99%

Superdiffusive trajectories in Brownian motion

Duplat

Kheifets

et al. 2013

Phys. Rev. E

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The Brownian motion of a microscopic particle in a fluid is one of the cornerstones of statistical physics and the paradigm of a random process. One of the most powerful tools to quantify it was provided by Langevin, who explicitly accounted for a short-time correlated "thermal" force. The Langevin picture predicts ballistic motion, x 2 ∼ t 2 at short-time scales, and diffusive motion x 2 ∼ t at long-time scales, where x is the displacement of the particle during time t, and the average is taken over the thermal distribution of initial conditions. The Langevin equation also predicts a superdiffusive regime, where x 2 ∼ t 3 , under the condition that the initial velocity is fixed rather than distributed thermally. We analyze the motion of an optically trapped particle in air and indeed find t 3 dispersion. This observation is a direct proof of the existence of the random, rapidly varying force imagined by Langevin. Langevin [4], who introduced a stochastic differential equation for the motion of a particle in a thermal bath, with an explicit rapidly varying force f (t) constantly exerted on the particle by the fluid molecules of the bath.The Langevin equation governs the velocity v =ẋ of a free particle in a viscous fluid, in the absence of hydrodynamic memory effects:vwhere m is the particle mass, and τ = m/6πηa is the viscous relaxation time of the spherical particle of radius a in a fluid with viscosity η. The force f is "delta-correlated," meaning that its fluctuations are present down to arbitrarily short-time scales. This is equivalent to having a flat spectrum up to arbitrarily high frequencies; hence, it is referred to as white noise. The variance of the particle velocity is v 2 eq = τ at equilibrium, and thanks to the equipartition theorem is such that v 2 eq = k B T /m, thus, relating temperature T and damping rate 1/τ with the magnitude of the random force, a manifestation of the fluctuation-dissipation theorem.Langevin did not exhaust all the riches of his model but used it to compute the ensemble average squared position of the particle in the long-time limit (t τ ) and found, for particles all starting at x 0 = 0, that (and Fürth [7]) devised a similar model and reached the same conclusion regarding the existence of a diffusive régime [Eq. (2)] but also noted the existence of ballistic behavior, x 2 = τ t 2 , in the shorttime limit t τ . This result, however, still depends on an average being performed on an initial condition of particles with distributed velocities. We come back below to that crucial point, not singled-out as such by either Taylor or Langevin.Consider a particle released at t = 0 in a thermalized medium (i.e., a bath of uniform temperature T ), with constrained initial velocity v 0 (whether it is equal to zero or not) and position x 0 = 0. Solving the Langevin model [Eq. (1)] for the displacement x(t) gives a meanwhere ·|x 0 ,v 0 denotes an average over many realizations of the thermal force, with fixed initial conditions (as opposed to an average over the thermal distribution of...

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Selected Papers on Noise and Stochastic Processes

Cited by 279 publications

References 0 publications

Fast Evaluation of Fluctuations in Biochemical Networks With the Linear Noise Approximation

Fast Evaluation of Fluctuations in Biochemical Networks With the Linear Noise Approximation

A mechanistic model of mid-latitude decadal climate variability

Superdiffusive trajectories in Brownian motion

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