By D. R. Axelrad

ISBN-10: 0080252346

ISBN-13: 9780080252346

This most up-to-date quantity within the Foundations & Philosophy of technological know-how & expertise sequence offers an account of probabilistic useful research and indicates its applicability within the formula of the behaviour of discrete media with the inclusion of microstructural results. even supposing quantum mechanics have lengthy been well-known as a stochastic thought, the creation of probabilistic techniques and ideas to classical mechanics has normally now not been tried. during this research the writer takes the view that the numerous box amounts of a discrete medium are random variables or capabilities of such variables. as a result the probabilistic mechanics of discrete media are in accordance with the mathematical idea of chance and the axiomatics of degree idea.

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**Additional resources for Foundations of the Probabilistic Mechanics of Discrete Media**

**Example text**

TisT. : mj 2,. (^u^2, · . , O = F { x i , X 2 , . . 117) can be general ized to: £ WMx^.. . x(U} = where xf.. = ^ . (A)^{x„ X, . . 119) 38 Mathematical (ii) Second-order Preliminaries characteristics of stochastic functions In the analysis o f probabihstic mechanics and particularly in the modelling of the response behaviour of discrete materials in which the field quantities are either random variables or random processes, the so-called second-order properties of random functions are of great significance.

X ( t j t + i ) - x ( í f c ) a r e mutually independent ( D o o b [ 3 3 ] ) , Feller [ 2 8 ] ) . Since there exists a simple relationship between x ( i i ) , . . , x(ifc) and their increments, the process will be specified by the density o f an increment x ( i ) - x ( i ' ) for all f < t and the first-order density Po (x) at some ίο ^ Γ . A n important representative o f this group is the Poisson process. Thus a random process with independent increments satisfying: P{x(t)-x{t') = k} = ^^^^-P^\-Mt-n (1138) for any i, t'eT, t' ^ t is called a Poisson process.

X(r, ω ο ) represents a sample of values o f the measurements o f χ at each point in this space. The random variable ω χ ( γ ο , ω ) represents the statistical outcome o f the values o f χ at points Γο in R " . Examples of physical spaces of the R " type will be considered in deahng with the quasi-statics and dynamics o f structured solids for instance or with the molecular dynamics o f discrete fluids. I f the physical field in the Euclidean space R " is characterized in terms o f a stochastic process x^: R " χ ^ C, it corresponds strictly to precise measurements at a point TQ Ε R".

### Foundations of the Probabilistic Mechanics of Discrete Media by D. R. Axelrad

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