Field Theoretic Methods

Täuber, Uwe C.

doi:10.1007/978-0-387-30440-3_200

Cited by 13 publications

(8 citation statements)

References 46 publications

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“…This section will first provide a brief overview how a coherent-state path integral representation can be constructed directly from the fundamental master equation that defines a stochastic interacting particle system [29][30][31][32][33][34][35], see also [49,50]; reference [33] provides an illustrative alternative derivation. This field-theoretic representation faithfully encodes statistical fluctuations, including those caused by discreteness and the internal reaction noise, as well as emerging correlations in spatial reaction-diffusion systems, and allows for systematic approximative analysis, as will be detailed below for two-species predator-prey models.…”

Section: Field-theoretic Analysismentioning

confidence: 99%

“…Expanding e −a d 0 b b ≈ 1 − a d 0 b b in the limit a 0 → 0 effectively replaces the 'hard' exponential constraint with a 'softened' particle number restriction, which will henceforth be used. Action (15) may now serve as a basis for further systematic coarse graining, constructing a perturbation expansion as described below, or, if required, a subsequent renormalization group analysis [35,49,50].…”

Section: Field Theory Representationmentioning

confidence: 99%

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Population oscillations in spatial stochastic Lotka–Volterra models: a field-theoretic perturbational analysis

Täuber

2012

J. Phys. A: Math. Theor.

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Field theory tools are applied to analytically study fluctuation and correlation effects in spatially extended stochastic predator-prey systems. In the meanfield rate equation approximation, the classic Lotka-Volterra model is characterized by neutral cycles in phase space, describing undamped oscillations for both predator and prey populations. In contrast, Monte Carlo simulations for stochastic two-species predator-prey reaction systems on regular lattices display complex spatio-temporal structures associated with persistent erratic population oscillations. The Doi-Peliti path integral representation of the master equation for stochastic particle interaction models is utilized to arrive at a field theory action for spatial Lotka-Volterra models in the continuum limit. In the species coexistence phase, a perturbation expansion with respect to the nonlinear predation rate is employed to demonstrate that spatial degrees of freedom and stochastic noise induce instabilities toward structure formation, and to compute the fluctuation corrections for the oscillation frequency and diffusion coefficient. The drastic downward renormalization of the frequency and the enhanced diffusivity are in excellent qualitative agreement with Monte Carlo simulation data.Population oscillations in Lotka-Volterra models 2 the biological complexity provides the opportunity to consistently incorporate stochastic fluctuations and spatio-temporal correlations, whose crucial importance has long been recognized in the field [5].This paper addresses predator-prey competition models that are defined via reaction-diffusion systems on a regular d-dimensional lattice, and whose rate equations in the well-mixed mean-field limit reduce to the two coupled ordinary differential equations originally introduced independently by Lotka [6] and Volterra [7] nearly a century ago. These stochastic spatial predator-prey models have served as paradigmatic examples for the emergence of cooperative steady states in the dynamics of two competing populations [8]-[10] (see also Ref. [11] for a fairly recent overview). The deterministic Lotka-Volterra rate equation model is characterized by a neutral cycle in phase space, describing regular undamped nonlinear population oscillations with the unrealistic feature that both predator and prey population densities invariably return to their initial values. In contrast, computer simulations of sufficiently large stochastic Lotka-Volterra systems yield long-lived erratic population oscillations [12]- [19], whose persistence can be understood through a resonant stochastic amplification mechanism [20] that drastically extends the transient time interval before any finite system ultimately reaches its absorbing stationary state, where the predator population becomes extinct [21,22]. In spatially extended systems, the mean-field Lotka-Volterra reaction-diffusion equations allow for traveling wave solutions [23]- [25]. In the corresponding stochastic lattice realizations, these regular wave crests become spreading activity fro...

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Section: Field-theoretic Analysismentioning

confidence: 99%

Section: Field Theory Representationmentioning

confidence: 99%

Population oscillations in spatial stochastic Lotka–Volterra models: a field-theoretic perturbational analysis

Täuber

2012

J. Phys. A: Math. Theor.

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“…10.2.3). Another idea tries to understand emergent complexity as arising from fundamental quantum field theories (Täuber 2008).…”

Section: )mentioning

confidence: 99%

The Two Volumes of the Book of Nature

Glattfelder¹

2019

Information—Consciousness—Reality

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The Book of Nature has been found. The mathematical description of the universe gives the human mind the power to manipulate reality and technology becomes possible. However, this miraculous knowledge generation has been very specific: fundamental aspects of reality (from the quantum realm to cosmic scales) are encoded analytically, i.e., as equations. This appears to exclude realworld complexity, for instance, the emergent property of consciousness appearing in a self-organizing biological neural network. A fluke of reality allows the human mind to also conquer this domain. What appears as complexity, turns out to be the result of simple rules. Only very recently have the fruits of technology given humans a new level of abstraction: the magic of computation. Now, complex systems can be encoded algorithmically, i.e., by utilizing algorithms and simulations running in computers. As a result, complexity can be tamed and comprehended. This new knowledge generation is understood as Volume II of the Book of Nature, whereas physical science represents Volume I. Underlying the analytical and algorithmic formal representations are two fundamental structures of mathematics: the continuous and the discrete. In this sense, all human knowledge generation is unified mathematically. Level of mathematical formality: medium to low.

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“…Таким образом, данный класс универсальности требует аккуратной физической интерпретации. В этой связи можно вспомнить, что в формализме Дои-Пелити [43], [44], где исходная микроскопическая задача формулируется в терминах операторов рождения и уничтожения, члены, квадратичные по полям отклика, могут появляться в функционалах действия с отрицательными знаками [44].…”

Section: обсуждение и заключениеunclassified

Случайный Рост Границы Раздела Фаз В Случайной Среде: Ренормгрупповой Анализ Простой Модели

Antonov¹,

Kakin²

2015

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Field Theoretic Methods

Cited by 13 publications

References 46 publications

Population oscillations in spatial stochastic Lotka–Volterra models: a field-theoretic perturbational analysis

Population oscillations in spatial stochastic Lotka–Volterra models: a field-theoretic perturbational analysis

The Two Volumes of the Book of Nature

Случайный Рост Границы Раздела Фаз В Случайной Среде: Ренормгрупповой Анализ Простой Модели

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