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.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

Computational Analysis of Microarray Data

Most of our knowledge of biodiversity and its causes in the deep-sea benthos derives from regional-scale sampling studies of the macrofauna. Improved sampling methods and the expansion of investigations into a wide variety of habitats have revolutionized our understanding of the deep sea. Local species diversity shows clear geographic variation on spatial scales of 100-1000 km. Recent sampling programs have revealed unexpected complexity in community structure at the landscape level that is associated with large-scale oceanographic processes and their environmental consequences. We review the relationships between variation in local species diversity and the regional-scale phenomena of boundary constraints, gradients of productivity, sediment heterogeneity, oxygen availability, hydrodynamic regimes, and catastrophic physical disturbance. We present a conceptual model of how these interdependent environmental factors shape regional-scale variation in local diversity. Local communities in the deep sea may be composed of species that exist as metapopulations whose regional distribution depends on a balance among global-scale, landscape-scale, and small-scale dynamics. Environmental gradients may form geographic patterns of diversity by influencing local processes such as predation, resource partitioning, competitive exclusion, and facilitation that determine species coexistence. The measurement of deep-sea species diversity remains a vital issue in comparing geographic patterns and evaluating their potential causes. Recent assessments of diversity using species accumulation curves with randomly pooled samples confirm the often-disputed claim that the deep sea supports higher diversity than the continental shelf. However, more intensive quantitative sampling is required to fully characterize the diversity of deep-sea sediments, the most extensive habitat on Earth. Once considered to be constant, spatially uniform, and isolated, deep-sea sediments are now recognized as a dynamic, richly textured environment that is inextricably linked to the global biosphere. Regional studies of the last two decades provide the empirical background necessary to formulate and test specific hypotheses of causality by controlled sampling designs and experimental approaches.

Environmental influences on regional deep-sea species diversity

Chapter 8 - A Stochastic Computational Model for Solute Transport in Porous Media

A mathematical model of synaptotagmin 7 revealing functional importance of short-term synaptic plasticity

We discuss the quantification of molecular fluctuations in the biochemical reaction systems within the context of intracellular processes associated with gene expression. We take the molecular reactions pertaining to circadian rhythms to develop models of molecular fluctuations in this chapter. There are a significant number of studies on stochastic fluctuations in intracellular genetic regulatory networks based on single cell-level experiments. In order to understand the fluctuations associated with the gene expression in circadian rhythm networks, it is important to model the interactions of transcriptional factors with the E-boxes in the promoter regions of some of the genes. The pertinent aspects of a near-equilibrium theory that would integrate the thermodynamical and particle dynamic characteristics of intracellular molecular fluctuations would be discussed, and the theory is extended by using the theory of stochastic differential equations. We then model the fluctuations associated with the promoter regions using general mathematical settings. We implemented ubiquitous Gillespie's algorithms, which are used to simulate stochasticity in biochemical networks, for each of the motifs. Both the theory and the Gillespie's algorithms gave the same results in terms of the time evolution of means and variances of molecular numbers. As biochemical reactions occur far away from equilibrium—hence the use of the Gillespie algorithm—these results suggest that the near-equilibrium theory should be a good approximation for some of the biochemical reactions.

Computing molecular fluctuations in biochemical reaction systems based on a mechanistic, statistical theory of irreversible processes.

The effect of fluctuations in the drift velocity on dispersion by a porous medium is investigated. An analytical model is developed which represents the effect of a single discrete step in the velocity of a 1dimensional flow as a multiplicative factor that modifies the underlying linear growth in solute variance predicted by the standard advection-dispersion equation. The algebraic structure of the model identifies two variable combinations 6. and ex that characterize the step and the rate of stochastic dispersion respectively, in terms of which a simple formula for the downstream effect of the step on dispersion is obtained. This formalism is next applied to a sequence of 3 steps representing a velocity fluctuation, and it is shown that while kinetic compression effects cancel out across such a fluctuation, the stochastic dispersion increases for any plausible combination of 6. and cx. This implies that a dispersion enhancement factor f is associated with a fluctuation, and a simple formula is obtained for this in terms of variables that describe the fluctuation length and amplitUde. Moreover, the algebra leads to the definition of a natural length scale A related to the Peelet number of the flow. Repeated application of this formula is used to find the cumulative dispersion enhancement by a sequence of identical fluctuations, leading to an expression for dispersivity as a function of the distance traversed by a solute plume. Key features of the model are that the dispersivity behaves differently for traversal lengths above and below A, and that above this transition it is proportional to a fractional power of the traversal length. These features are in agreement with experimental observation of scale-dependent dispersivity, but quantitatively the observed growth in di~persivity over several orders of magnitude is not obtained for any reasonable choice of parameter values.

/pdf/solute-dispersion-by-1d-stepped-velocity-fluctuations-3lp3ge133g.pdf

Don Kulasiri

Papers

Chapter 8 - A Stochastic Computational Model for Solute Transport in Porous Media

A mathematical model of synaptotagmin 7 revealing functional importance of short-term synaptic plasticity

Computing molecular fluctuations in biochemical reaction systems based on a mechanistic, statistical theory of irreversible processes.

Solute dispersion by 1D stepped velocity fluctuations

A review of molecular mechanisms linked to potential renal injury agents in tropical rural farming communities.