There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

抗原变异可使得多种致病微生物易于逃避宿主免疫应答。表达在感染红细胞表面的恶性疟原虫红细胞表面蛋白1（PfPMP1）与感染红细胞、内皮细胞、树突状细胞以及胎盘的单个或多个受体作用，在黏附及免疫逃避中起关键的作用。每个单倍体基因组var基因家族编码约60种成员，通过启动转录不同的var基因变异体为抗原变异提供了分子基础。

疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A

https://www.tau.ac.il/%7Eklafter1/258.pdf

The random walk's guide to anomalous diffusion: a fractional dynamics approach

The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

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

The Langevin Equation

Diffusion and Reactions in Fractals and Disordered Systems

Historical Background and Introductory Concepts Methods for Solving Langevin and Fokker-Planck Equations Matrix Continued Fractions Escape Rate Theory Linear and Nonlinear Response Theory Brownian Motion of a Free Particle and a Harmonic Oscillator Rotational Brownian Motion about a Fixed Axis in a Periodic Potential Brownian Motion in a Tilted Periodic Potential: Application to the Josephson Tunnelling Junction and Ring Lasers Brownian Motion in a Double-Well Potential Isotropic and Anisotropic Rotational Brownian Motion in Space in the Presence of an External Potential with Applications to Dielectric and Kerr Effect Relaxation in Fluids and Liquid Crystals Brownian Motion of Classical Spins with Applications to Superparamagnetism Magnetic Stochastic Resonance Dynamic Hysteresis Switching Field Surfaces Inertial Langevin Equations with Applications to Orientational Relaxation in Liquids Itinerant Oscillator Model Anomalous Diffusion Continuous Time Random Walks Methods for the Solution of Fractional Fokker-Planck Equations.

The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering

The concept of a random process the Brownian motion of a free particle - the velocity distribution the Brownian motion of a free particle - the distribution of the displacements one-dimensional Brownian motion in a potential excluding inertia effects the multi-dimensional Langevin equation rotational Brownian motion application to relaxation and loss processes in electric and magnetic fields.

The Langevin equation : with applications in physics, chemistry, and electrical engineering

The reversal time, superparamagnetic relaxation time, of the magnetization of fine single domain ferromagnetic nanoparticles owing to thermal fluctuations plays a fundamental role in information storage, paleomagnetism, biotechnology, etc. Here a comprehensive tutorial-style review of the achievements of fifty years of development and generalizations of the seminal work of Brown [Phys. Rev. 130, 1677 (1963)] on thermal fluctuations of magnetic nanoparticles is presented. Analytical as well as numerical approaches to the estimation of the damping and temperature dependence of the reversal time based on Brown's Fokker-Planck equation for the evolution of the magnetic moment orientations on the surface of the unit sphere are critically discussed while the most promising directions for future research are emphasized.

/pdf/thermal-fluctuations-of-magnetic-nanoparticles-fifty-years-4i2ozzz33w.pdf

William T. Coffey

Papers

The Langevin Equation

Diffusion and Reactions in Fractals and Disordered Systems

The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering

The Langevin equation : with applications in physics, chemistry, and electrical engineering

Thermal fluctuations of magnetic nanoparticles: Fifty years after Brown