Configurational average statistical quantities

The statistical average of any configuration-dependent tensorial quantity A(xw, t) is defined as... 

The statistical mechanical average of the quantity enclosed, taken over all configuration of the chain. 

According to statistical physics, physical quantities are represented by averages over configurations distributed according to a certain statistical ensemble. A trajectory obtained by molecular dynamics provides such a set of configurations. Therefore, a measurement of a physical quantity by simulation is simply obtained as an arithmetic average of the various instantaneous values assumed by that quantity during the MD run. 

In multiphase systems, there are many possible configurations of the interacting phases. Following the concepts of statistical mechanics, the rheological functions must be volume-averaged [Hashin, 1964]. The averaged quantities are sometimes known as bulk quantities. For example, the bulk rate of strain tensor, (y ), is expressed as ... 

Th6 Freely Jointed Chain. This type of idealized chain consisting of n links of length I is represented in Figure 7, where r is the end-to-end distance. When the chain is completely extended, R = nl. The chain may assume many configurations, each associated with an end-to-end distance Vj. In statistics, it is equivalent to consider a molecule at different times or an assembly of N molecules at the same time. An average quantity describing the assembly is the mean-square end-to-end distance defined by... 

A basically different approach is used in the MC method. The states of a system of particles are taken to be stochastic and one has to select the most probable configurations over which various properties are then averaged. Therefore, this method is suitable only for calculating equilibrium quantities since it cannot, in principle, give an answer to the question how the system achieves its equilibrium because, instead of a genuine dynamic evolution of the system, an artificial stochastic process is modelled. A possibility arises of taking into account by statistical methods the effect of the temperature on the properties of a solution. 

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