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where the sum extends over all ways of distributing N distinguishable atoms over N lattice sites. If the volume of a unit cell of the lattice is chosen to be unity, then N is the volume of the system. The grand partition function is !2dz, N, T) =

FIGURE 6.23 If you have selected Show Last Value from the technical studies dialog box, then the most recent value for each study you are using will be shown in the legend.

zNaQG(Na , T)

(14.16)

f3 PG = {

~ log !2dz, N, T)

1 1 B - = - z - 10g!2 (z NT) u N Bz G"

(14.17)

To establish a correspondence between the lattice gas and the Ising model, let occupied sites correspond to spin up and empty sites to spin down. Then Na - N +. In the Ising model a set of N numbers {Sl"'" SN} uniquely defines a configuration. In the lattice gas an enumeration of the occupied sites determines not one, but N a ! configurations. The difference arises from the fact that the atoms are supposed to be able to move from site to site. This difference, however, is obliterated by the adoption of "correct Boltzmann counting." Hence (14.18) where the function g(N +, N ++) and the sum L' are identical with those appearing in (14.11). The grand partition function is e{JNPc = !2 G(z, N, T) =

(14.19)

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Comparing one chart to another is one of the most valuable ways to compare performance across markets. You might want to compare a stock to the market in general, or one commodity to another, or a mutual fund to a broad stock market index, or even multiple securities together (such as ve different stocks all at once). Being able to compare gives you a way to view relative performance between different nancial instruments. There are two basic kinds of comparison: price and percentage. A price comparison plots two or more securities into a price pane, showing their actual prices over the requested period of time. Because the prices might be very different (for example, comparing a stock of about $3 to another of about $80), this can often yield a graph that is not very useful. The other kind of comparison is a percentage comparison, which is much more common, since it illustrates the percentage change in price over a certain period of time, irrespective of the price of the securities. Once you have a single security chart, you can compare this security to one or more others by choosing Comparison Chart from the Chart Settings menu (Figure 6.24). When you do this, the Comparison Chart dialog box appears (Figure 6.25). In it are these controls: Index Selection. A list of major U.S. indexes is presented. You can double-click any one of these to add it to the list of items that should be compared. Index symbols are often strange and hard to remember, so this list makes it convenient to choose an index.

You can set a priority level for the task using the Priority . To set a completion amount, you can click the % Complete .

A comparison between (14.19) and (14.11) yields the accompanying table of correspondence. Hence a solution of the Ising model can be immediately tran-

e2{J(q- H)

1 (M[

The lattice gas does not directly correspond to any real system in nature. Ii we allow the lattice constant to approach zero, however, and then add to the resulting equation of state the pressure of an ideal gas, the model corresponds k a real gas of atoms interacting with one another through a zero-range potential Thus it may be interesting to study the phase transition of a lattice gas. The lattice gas has also been used as a model for the melting of a crystal lattice. When it is so used, however, the lattice constant must be kept finite. The kinetic energy of the atoms in the crystal lattice is appended in some ad hex: fashion. Such a model would only have a mathematical interest, because it is not clear that it describes melting.

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