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It is understood that all members of the ensemble have the same number of particles N and the same volume V. Suppose f(p, q) is a measurable property of the system, such as energy or momentum. When the system is in equilibrium, the observed value of f(p, q) must be the result obtained by averaging f(p, q) over the microcanonical ensemble in some manner. If the postulate of equal a priori probability is to be

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The refined list of candidate use cases follows:

useful, all manners of averaging must yield essentially the same answer. Two kinds of average values are commonly introduced: the most probable value and the ensemble average. The most probable value of f(p, q) is the value of f(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average of f~p, q) is defined by

U)=-------

pd 3Nqf(p, q)p(p, q)

(6.8)

Change password Browse account balances List transactions Download transactions Transfer funds Edit profile Pay bill Buy security Sell security

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Google News is a news-gathering service that identifies, assembles, and dis plays the latest news headlines from thousands of different news organizations. As you can see in Figure 6-14, Google News organizes its stories by category and lists hundreds of related stories under each lead headline. Click the link to access the originating news source and read the story, or click the Related link to view other sources take on the story. You can also click the category links (World, U.S., Business, Sci/Tech, Sports, Entertainment, and Health) to view more stories in that category. You access Google News by clicking the News tab on the Google home page, or by going directly to news.google.com.

pd 3Nqp(p, q)

The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e., if

(6.9)

In the UML, actors are represented by a stick figure, and use cases are shown as ellipses A use case diagram simply shows the structural relationships between the actors and the use cases, not the dynamic relationships The relationship between actors and use cases is shown via a directional association indicating the source of invocation Figure 7 - 1 shows the Browse account and Transfer funds use cases for the HomeDirect system Both are invoked by the customer

observed value of f may be calculated. When it is not, we should question the validity of statistical mechanics. In all physical cases we shall find that mean square fluctuations are of the order of liN. Thus in the limit as N ~ 00 the ensemble average and the most probable value became identical. Strictly speaking, systems in nature do not obey classical mechanics. They obey quantum mechanics, which contains classical mechanics as a special limiting case. Logically we should start with quantum statistical mechanics and then arrive at classical statistical mechanics as a special case. We do this later. It is only for pedagogical reasons that we begin with classical statistical mechanics. From a purely logical point of view there is no room for an independent postulate of classical statistical mechanics. It would not be logically satisfactory even if we could show that the postulate introduced here follows from the equations of motion (6.2), for, since the world is quantum mechanical, the foundation of statistical mechanics lies not in classical mechanics but in quantum mechanics. At present we take this postulate to be a working hypothesis whose justification lies in the agreement between results derived from it and experimental facts.

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