Synopses & Reviews
Synopsis
Chapter 1. Probability elements
1.1 Introduction to random variables
1.2 Axiomatic scheme
1.2.1 Conditional probability
1.2.2 Bayes' theorem
1.2.3 Statistical Independence
1.2.4 Random Variable
1.3 Frequency scheme
1.3.1 Probability density
1.3.2 Properties of the probability density
1.4 Characteristic function G(k)
1.4.1 The simplest of random walks
1.4.2 Examples of G(k) is not developable in a Taylor series
1.4.3 Characteristic function in a toroidal network
1.4.4 Function of characteristic function
1.5 Cumulants development
1.6 Central limit theorem
1.7 Random variable transformation
1.8 Correlations between random variables
1.8.1 Statistical independence
1.9 Fluctuations development
1.10 Multidimensional characteristic function
1.10.1 Diagrams development (many variables)
1.11 Terwiel cumulants
1.12 Gaussian distribution (many variables)
1.12.1 Gaussian with odd null moments
1.12.2 Novikov's theorem
1.13 Transformation for n dimensional probability densities
1.13.1 Marginal probability density
1.14 Conditional probability density
1.15 Problems and solutions
Chapter 2. Fluctuations around thermal equilibrium
2.1 Spatial correlations (Einstein's distribution)
2.1.1 The Gaussian approximation
2.2 Minimal work
2.2.1 Fluctuations in terms of P, V, T variables
2.3 Fluctuations of mechanical character
2.3.1 Fluctuations of a tight rope
2.4 Temporal correlations
2.5 Problems and solutions
Chapter 3. Elements of stochastic processes
3.1 Introduction
3.1.1 Time dependent random variable
3.1.2 Characteristic fun
Synopsis
This textbook is the result of the enhancement of several courses on non-equilibrium statistics, stochastic processes, stochastic differential equations, anomalous diffusion and disorder. The target audience includes students of physics, mathematics, biology, chemistry, and engineering at undergraduate and graduate level with a grasp of the basic elements of mathematics and physics of the fourth year of a typical undergraduate course. The little-known physical and mathematical concepts are described in sections and specific exercises throughout the text, as well as in appendices. Physical-mathematical motivation is the main driving force for the development of this text.
It presents the academic topics of probability theory and stochastic processes as well as new educational aspects in the presentation of non-equilibrium statistical theory and stochastic differential equations.. In particular it discusses the problem of irreversibility in that context and the dynamics of Fokker-Planck. An introduction on fluctuations around metastable and unstable points are given. It also describes relaxation theory of non-stationary Markov periodic in time systems. The theory of finite and infinite transport in disordered networks, with a discussion of the issue of anomalous diffusion is introduced. Further, it provides the basis for establishing the relationship between quantum aspects of the theory of linear response and the calculation of diffusion coefficients in amorphous systems.