Noise Induced Transitions

Horsthemke, Werner

doi:10.1007/978-3-642-68038-0_11

Cited by 80 publications

(143 citation statements)

References 12 publications

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“…Again, equations (31), (32) for the expected fractions n a x /N a agree with equations (18), (19) for the probabilities P a ( x, t) if making the identification (26). However, the approximate equations (31), (32) are only valid under the condition n a x n b y ≈ n a x n b y , which obviously corresponds to the factorization assumption (13). This condition is fulfilled if the absolute values of the (co)variances…”

Section: Mean Value Equations (Macroscopic Equations)mentioning

confidence: 58%

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Interrelations between stochastic equations for systems with pair interactions

Helbing¹

1992

Physica A: Statistical Mechanics and its Applications

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Several types of stochastic equations are important in thermodynamics, chemistry, evolutionary biology, population dynamics and quantitative social science. For systems with pair interactions four different types of equations are derived, starting from a master equation for the state space: First, general mean value and (co)variance equations. Second, Boltzmann-like equations. Third, a master equation for the configuration space allowing transition rates which depend on the occupation numbers of the states. Fourth, a Fokker-Planck equation and a "Boltzmann-Fokker-Planck equation". The interrelations of these equations and the conditions for their validity are worked out clearly. A procedure for a selfconsistent solution of the nonlinear equations is proposed. Generalizations to interactions between an arbitrary number of systems are discussed.

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Section: Mean Value Equations (Macroscopic Equations)mentioning

confidence: 58%

“…be found in chemical kinetics [1], laser theory [2] and biological systems [3], but they also become increasingly important in quantitative social science [4,5,6,7,8,9,10,11]. Without stochastic models certain phenomena like phase transitions could not be correctly understood [12,13,14].…”

Section: Introductionmentioning

confidence: 99%

Interrelations between stochastic equations for systems with pair interactions

Helbing¹

1992

Physica A: Statistical Mechanics and its Applications

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“…Equation (6) is the evolution equation describing the dynamics of worm propagation. It belongs to a general class of stochastic differential equations, being able to describe, with success, the evolution of dynamical systems in the mesoscale (e.g., [46][47][48]). In this form, the stochastic differential model provides a quantitative estimate for the inherent randomness.…”

Section: A Stochastic Model For Worm Propagationmentioning

confidence: 99%

Toward early warning against Internet worms based on critical‐sized networks

Magkos

Avlonitis

Kotzanikolaou

et al. 2012

Security Comm Networks

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In this paper, we build on a recent worm propagation stochastic model, in which random effects during worm spreading were modeled by means of a stochastic differential equation. On the basis of this model, we introduce the notion of the critical size of a network, which is the least size of a network that needs to be monitored, in order to correctly project the behavior of a worm in substantially larger networks. We provide a method for the theoretical estimation of the critical size of a network in respect to a worm with specific characteristics. Our motivation is the requirement in real systems to balance the needs for accuracy (i.e., monitoring a network of a sufficient size in order to reduce false alarms) and performance (i.e., monitoring a small‐scale network to reduce complexity). In addition, we run simulation experiments in order to experimentally validate our arguments. Finally, based on notion of critical‐sized networks, we propose a logical framework for a distributed early warning system against unknown and fast‐spreading worms. In the proposed framework, propagation parameters of an early detected worm are estimated in real time by studying a critical‐sized network. In this way, security is enhanced as estimations generated by a critical‐sized network may help large‐scale networks to respond faster to new worm threats. Copyright © 2012 John Wiley & Sons, Ltd.

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“…General treatments of SDEs may be found in many textbooks (e.g., Gardiner 2004;Horsthemke and Lé fè ver 1984;Paul and Baschnagel 1999;Kloeden and Platen 1992;van Kampen 2007). The starting point is the Stratonovich SDE…”

Section: Appendixmentioning

confidence: 99%

Extreme Events and the General Circulation: Observations and Stochastic Model Dynamics

Sura

Perron

2010

Journal of the Atmospheric Sciences

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This study explores the dynamical role of non-Gaussian potential vorticity variability (extreme events) in the zonally averaged circulation of the atmosphere within a stochastic framework. First the zonally averaged skewness and kurtosis patterns of relative and potential vorticity anomalies from NCEP-NCAR reanalysis data are presented. In the troposphere, midlatitude regions of near-zero skewness coincide with regions of maximum variability. Equatorward of the Northern Hemisphere storm track positive relative/potential vorticity skewness is observed. Poleward of the same storm track the vorticity skewness is negative. In the Southern Hemisphere the relation is reversed, resulting in negative relative/potential vorticity skewness equatorward, and positive skewness poleward of the storm track. The dynamical role of extreme events in the zonally averaged general circulation is then explored in terms of the potential enstrophy budget by linking eddy enstrophy fluxes to a stochastic representation of non-Gaussian potential vorticity anomalies. The stochastic model assumes that potential vorticity anomalies are advected by a random velocity field. The assumption of stochastic advection allows for a closed expression of the meridional enstrophy flux: the potential enstrophy flux is proportional to the potential vorticity skewness. There is some evidence of this relationship in the observations. That is, potential enstrophy fluxes might be linked to non-Gaussian potential vorticity variability. Thus, extreme events may presumably play an important role in the potential enstrophy budget and the related general circulation of the atmosphere.

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Noise Induced Transitions

Cited by 80 publications

References 12 publications

Interrelations between stochastic equations for systems with pair interactions

Interrelations between stochastic equations for systems with pair interactions

Toward early warning against Internet worms based on critical‐sized networks

Extreme Events and the General Circulation: Observations and Stochastic Model Dynamics

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