Ensemble and Dynamical Property Examples

The range of properties that can be determined from simulation is obviously limited only by the imagination of the modeler. In this section, we will briefly discuss a few typical properties in a general sense. We will focus on structural and time-correlation properties, deferring thermodynamic properties to Chapters 10 and 12. 

As a very simple example, consider the dipole moment of water. In the gas phase, this dipole moment is 1.85 D (Demaison, Hiimer, and Tiemann 1982). What about water in liquid water A zerofli order approach to answering this problem would be to create a molecular mechanics force held defining the water molecule (a sizable number exist) that gives the correct dipole moment for the isolated, gas-phase molecule at its equilibrium 

Note that, although up to this point we have described tlie expectation value of A as though it were a scalar value, it is also possible that A is a function of some experimentally (and computationally) accessible variable, in which case we may legitimately ask about its expectation value at various points along the axis of its independent variable. A good 

We may thus interpret the l.h.s. of Eq. (3.39) as a probability function. That is, we may express the probabihty of finding two atoms of A and B within some range Ar of distance r from one another as 

Note that its contribution to the probability function makes certain limiting behaviors on A Aii(r) intuitively obvious. For instance, the function should go to zero very rapidly when r becomes less than the sum of the van der Waals radii of A and B. In addition, at very large r, the function should be independent of r in homogeneous media, like fluids, 

It often happens that we consider one of our atoms A or B to be privileged, e.g., A might be a sodium ion and B the oxygen atom of a water and our interests might focus 

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The molecular dynamics and Monte Carlo simulation methods differ in a variety of ways. The most obvious difference is that molecular dynamics provides information about the time dependence of the properties of the system whereas there is no temporal relationship between successive Monte Carlo configurations. In a Monte Carlo simulation the outcome of each trial move depends only upon its immediate predecessor, whereas in molecular dynamics it is possible to predict the configuration of the system at any time in the future - or indeed at any time in the past. Molecular dynamics has a kinetic energy contribution to the total energy whereas in a Monte Carlo simulation the total energy is determined directly from the potential energy function. The two simulation methods also sample from different ensembles. Molecular dynamics is traditionally performed under conditions of constant number of particles (N), volume (V) and energy (E) (the microcanonical or constant NVE ensemble) whereas a traditional Monte Carlo simulation samples from the canonical ensemble (constant N, V and temperature, T). Both the molecular dynamics and Monte Carlo techniques can be modified to sample from other ensembles for example, molecular dynamics can be adapted to simulate from the canonical ensemble. Two other ensembles are common ... 

Graessley analyzed the equilibrium and the dynamical properties of polymers based on small, tree-like micronetworks [11,167,168]. These Gaussian micronetworks represent perfectly branched, symmetrical GGS, which grow from a central bead, see a particular example in Fig. 17 the micronetworks are finite Cayley trees (dendrimers). The peripheral beads of these micronetworks are assumed to be fixed in space. When calculating the quasiequilibrium elastic properties, the peripheral beads are taken to move affinely with the macroscopic deformations [167,168]. The evaluation of the relaxation spectrum H r) and of the relaxation modulus G(t) is done in two steps. First, the spectrum of an ensemble of isolated tree-like micronetworks with... 

This chapter introduces tools from statistical mechanics that are useful for analyzing the behavior of static and slowly driven granular media. (For fast dynamics, refer to the previous chapter on Kinetic Theory by Jenkins.) These tools encompass techniques used to predict emergent properties from microscopic laws, which are the analogs of calculations in equilibrium statistical mechanics based on the concept of statistical ensembles and stochastic dynamics. Included, for example, are the Edwards approach to static granular media, and coarse-grained models of... 

When n > 2, one can draw the reducible contributions made up of sequences of binary kernels and where states k = 0 between these kernels exist. Thus, the class associated with the skeleton of Fig. 3b contains a state k = 0 and contributes, not to Eq. (56), but to Eq. (70). In the following we shall need the relation which expresses Yg,- n) as the difference between ) and the ensemble of reducible contributions to (70) (of the type of Fig. 3b for n = 3, for example). It is necessary for us now to study systematically the points k = 0 of Eq. (70) so as to extract the reducible contributions. A study of the selection rules will permit us to solve this problem. We shall associate the appearance of the points k = 0 with the structure of the skeletons that we have introduced we shall see that the reduci-bility will be a dynamical translation of certain topological properties of the equilibrium clusters. 

In addition to the study of atomic motion during chemical reactions, the molecular dynamics technique has been widely used to study the classical statistical mechanics of well-defined systems. Within this application considerable progress has been made in introducing constraints into the equations of motion so that a variety of ensembles may be studied. For example, classical equations of motion generate constant energy trajectories. By adding additional terms to the forces which arise from properties of the system such as the pressure and temperature, other constants of motion have been introduced. 

Surface tension is one of the most basic thermodynamic properties of the system, and its calculation has been used as a standard test for the accuracy of the intermolecular potential used in the simulation. It is defined as the derivative of the system s free energy with respect to the area of the interface[30]. It can be expressed using several different statistical mechanical ensemble averages[30], and thus we can use the molecular dynamics simulations to directly compute it. An example for such an expression is ... 

Another area of supramolecular dynamics pertains to the racemization of chiral assembhes. There are numerous examples of chiral self-assembled coordination structures in which chirahty is exhibited not only on the molecular level, but also on the supramolecular level. For example, helicates are inherently chiral even when assembled from achiral components. The P and M configurations of helicates are exclusively the properties of the supramolecular ensemble. 

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