Stochastic Processes and Statistical Physics

Moyal, J. E.

doi:10.1111/j.2517-6161.1949.tb00030.x

Cited by 186 publications

(105 citation statements)

References 30 publications

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“…This can be achieved by assuming that the mean field r evolves much more slowly than the individual x i , so that the adiabatic approximation is applicable. This assumption implies that the random variable r obeys a Markov process, so that the equation for P (r, t) can be written in the form of a Kramers-Moyal expansion, 19), 20) as…”

Section: Adiabatic Approximationmentioning

confidence: 99%

System-Size Effects on the Collective Dynamics of Cell Populations with Global Coupling

Teramae

2003

Progress of Theoretical Physics

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Phase-transition-like behavior is found to occur in globally coupled systems consisting of finite numbers of elements, and a theoretical explanation of this behavior is given. The system studied is a population of globally pulse-coupled integrate-and-fire cells subject to a small additive noise. As the population size is changed, the system exhibits phase-transitionlike behavior, that is, there exists a well-defined critical system size above which the system remains in a monostable state with high-frequency activity and below which a new phase characterized by the alternation of high-and low-frequency types of activity appears. The mean field motion obeys a stochastic process with a state-dependent noise, and the above described phenomenon can be interpreted as a noise-induced transition characteristic of such processes. The coexistence of high-and low-frequency activity observed in finite size systems was reported by Cohen, Soen and Braun [Physica A 249 (1998), 600] in experiments on cultivated heart cells. The present report gives the first qualitative interpretation of their experimental results. §1. IntroductionCollective dynamics of coupled dynamical elements represents a central issue of nonlinear dynamics, and has served as a subject of extensive study over the last few decades. The relevant areas to which their study has been applied include chemical reactions, 2) societies of living organisms, 3)-5) lasers, 6), 7) semiconductors, 8), 9) neural networks 10)-12) and cardiac systems. 3), 13), 14) In most theoretical studies, however, the system size is assumed infinite. While this idealization could be valid for such systems like spatially extended chemical reactions, there seem to be important practical cases in which the finiteness of the system size should be taken into account explicitly. Here we cite two existing theoretical studies on the collective dynamics of populations in which the finiteness of the system size plays a crucial role. Firstly, Daido 15) investigated the collective behavior of an inhomogeneous system of oscillators, focusing on the statistics of fluctuations close to the onset of collective motion. Secondly, Pikovsky et al. 16) studied coupled bistable elements and demonstrated numerically the existence of phase-transition-like behavior existing only for finite systems. Apart from these theoretical studies, an interesting experiment was reported recently by Cohen et al. 1) They cultivated heart cells in various population sizes and found that these cells exhibit system-size-dependent behavior. Some more details of their reports are the following. Heart cells extracted from the ventricles of neonatal rats were cultivated, and time series of their spontaneous spiking activity were recorded. Initially, these heart cells were prevented from interacting by a chemical treatment, but after some time, they spontaneously assembled to form sub-

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Section: Adiabatic Approximationmentioning

confidence: 99%

System-Size Effects on the Collective Dynamics of Cell Populations with Global Coupling

Teramae

2003

Progress of Theoretical Physics

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“…Substituting this into (13), dividing by a, then taking the limit as a approaches 0, gives the Kramers-Moyal expansion (Kramers, 1940;Moyal, 1949),…”

Section: Derivation Of the Age-oriented Fokker-planck Development Equmentioning

confidence: 99%

Modeling the Effects of Developmental Variation on Insect Phenology

Yurk

Powell

2010

Bull. Math. Biol.

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Phenology, the timing of developmental events such as oviposition or pupation, is highly dependent on temperature; since insects are ectotherms, the time it takes them to complete a life stage (development time) depends on the temperatures they experience. This dependence varies within and between populations due to variation among individuals that is fixed within a life stage (giving rise to what we call persistent variation) and variation from random effects within a life stage (giving rise to what we call random variation). It is important to understand how both types of variation affect phenology if we are to predict the effects of climate change on insect populations.We present three nested phenology models incorporating increasing levels of variation. First, we derive an advection equation to describe the temperature-dependent development of a population with no variation in development time. This model is extended to incorporate persistent variation by introducing a developmental phenotype that varies within a population, yielding a phenotype-dependent advection equation. This is further extended by including a diffusion term describing random variation in a phenotype-dependent Fokker-Planck development equation. These models are also novel because they are formulated in terms of development time rather than developmental rate; development time can be measured directly in the laboratory, whereas developmental rate is calculated by transforming laboratory data. We fit the phenology models to development time data for mountain pine beetles (MPB) (Dendroctonus ponderosae Hopkins [Coleoptera: Scolytidae]) held at constant temperatures in laboratory experiments. The nested models are parameterized using a maximum likelihood approach. The results of the parameterization show that the phenotype-dependent advection model provides the best fit to laboratory data, suggesting that MPB phenology may be adequately described in terms of persistent variation alone. MPB phenology is simulated using phloem temperatures and attack time distributions measured in central Idaho. The resulting emergence time distributions compare favorably to field observations.

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“…The a j and b jk are known as the first and the second derivative moments (Moyal 1949) of X(t), respectively; they can be determined from the governing equations of the dynamical system. In Eq.…”

Section: Diffusive Markov Processes -A Primermentioning

confidence: 99%

Some recent advances in the theory of random vibration

2002

Naturwissenschaften

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A brief introduction is first given of the history of the technical field of random vibration, tracing back to the works of physicists since the beginning of the twentieth century. This is then followed by an account of more recent developments, with emphasis on nonlinear and quasi-nonlinear systems, and on analytical solutions for the associated Fokker-Planck equation and the generalized Pontryagin equation. The governing equation of a quasi-nonlinear system is linear, but with one or more randomly varying coefficients. The techniques for finding exact probability solutions, approximate probability solutions, conditions for motion stability, and the failure probability and statistics of a system are discussed.

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Stochastic Processes and Statistical Physics

Cited by 186 publications

References 30 publications

System-Size Effects on the Collective Dynamics of Cell Populations with Global Coupling

System-Size Effects on the Collective Dynamics of Cell Populations with Global Coupling

Modeling the Effects of Developmental Variation on Insect Phenology

Some recent advances in the theory of random vibration

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