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By Eric Bertin
This course-tested primer offers graduate scholars and non-specialists with a uncomplicated knowing of the strategies and techniques of statistical physics and demonstrates their wide selection of purposes to interdisciplinary subject matters within the box of advanced procedure sciences, together with chosen points of theoretical modeling in biology and the social sciences.
Generally talking, the pursuits of statistical physics could be summarized as follows: at the one hand to check structures composed of a giant variety of interacting devices, and at the different to foretell the macroscopic, collective habit of the process thought of from the viewpoint of the microscopic legislation governing the dynamics of the person entities. those targets are primarily additionally shared through what's now known as 'complex platforms science,' and as such, structures studied within the framework of statistical physics should be thought of to be one of the easiest examples of complicated structures – whereas additionally supplying a slightly good constructed mathematical treatment.
The moment variation has been considerably revised and elevated, that includes particularly 3 new chapters addressing non-conserved debris, evolutionary inhabitants dynamics, networks, homes of either person and matched uncomplicated dynamical platforms, and convergence theorems, in addition to brief appendices that supply invaluable tricks on how one can practice uncomplicated stochastic simulations in practice.
Yet, the unique spirit of the e-book – to stay obtainable to a wide, non-specialized readership – has been stored all through: the layout is a collection of concise, modular and self-contained topical chapters, fending off technicalities and jargon up to attainable, and complemented by way of a wealth of worked-out examples, so one can make this paintings invaluable as a self-study textual content or as textbook for brief courses.
From the studies of the 1st edition:
“… an exceptional creation to easy suggestions of statistical physics and complicated structures for college kids and researchers with an curiosity in advanced platforms in different fields … .” Georg Hebermehl, Zentralblatt MATH, Vol. 1237, 2012
“… this brief textual content continues to be very clean for the mathematician.” Dimitri Petritis, Mathematical stories, factor 2012k
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Extra info for Statistical Physics of Complex Systems: A Concise Introduction
Example text
The white noise ξ(t) still satisfies ξ(t) = 0 and ξ(t)ξ(t ) = δ(t−t ). In the following subsections, we will generically consider this case when dealing with the Langevin equation. More generally, one may consider Langevin equations coupling an arbitrary number N of variables xi , dxi = Qi (x1 , . . , xN ) + ξi (t), dt i = 1, . . 50) where the N stochastic variables ξi (t) are white noises with correlations ξi (t)ξj (t ) = ij δ(t − t ) , i, j = 1, . . , N. 51) Another important generalization of the Langevin equation is the one with multiplicative noise, in contrast to additive noise.
Still, one can resort to the generic mathematical formalism of stochastic processes in order to describe statistically the time evolution of some specific variables of interest, like the position or velocity of a probe particle immersed in a fluid. This formalism is presented in Sect. 1, in the simplest case of Markov processes. The example of the random evolution of a single degree of freedom in a noisy environment (diffusive motion), leading to the Langevin and Fokker-Planck equations, is discussed respectively in Sects.
In principle, the statistics of the noise could also depend on the velocity v. We assume here that it is independent of v, but we will come back to this point below. To proceed further, it is necessary to choose a specific model for both Fcoll and ξ(t). The average force Fcoll can be interpreted as an effective friction force, which slows down the probe particle; it is thus natural to choose, as a first approximation, a linear friction force Fcoll = −γv, with γ > 0 a friction coefficient. Then, a model of the noise should be given.