Stochastic Structure-property correlations

An outstanding problem concerns itself with the structure of a hard sphere phase. This is a special instance of the more. general difficulty of the specification of the structure of infinitely extended random media. These questions will perhaps be the subject of a future mathematical discipline-stochastic geometry. The pair correlation function g(r), even if it is known, hardly suffices to specify uniqudy the stochastic metric properties of a random structure. For a finite N and V finite) system in equilibrium in thermal contact with a heat reservoir at temperature T, the density in the configuration space of the N particles [Eq. (2)]... 

By far the most common methods of studying aqueous interfaces by simulations are the Metropolis Monte Carlo (MC) technique and the classical molecular dynamics (MD) techniques. They will not be described here in detail, because several excellent textbooks and proceedings volumes (e.g., [2-8]) on the subject are available. In brief, the stochastic MC technique generates microscopic configurations of the system in the canonical (NYT) ensemble the deterministic MD method solves Newton s equations of motion and generates a time-correlated sequence of configurations in the microcanonical (NVE) ensemble. Structural and thermodynamic properties are accessible by both methods the MD method provides additional information about the microscopic dynamics of the system. 

It can be shown that ARCH and GARCH models are able to approximate stochastic differential processes if the latter fulfil certain properties. Albeit the goodness of fit is limited, both types of methods are related and can be converted into each other. Moreover, simple stochastic processes show quite simple auto-correlation structures similar to basic ARMA models. For instance, the Ornstein-Uhlenbeck process can be seen as the continuous equivalent of the AR(1) process. In other words, an Ornstein-Uhlenbeck process measured in discrete intervals can be interpreted/modeUed as an AR(1) process (see also (2.23), (2.60), and (2.61)). ... 

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