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The_macroscopic_world - How the macroscopic world works...

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© M. S. Shell 2008 1/23 last modified 9/21/2009 How the macroscopic world works ChE210A Previously, we covered how the world works at a microscopic level—how atoms and molecules interact with each other and what laws govern their time evolution. Now, we take a macros- copic point of view. What happens when many, many ( ~10 g2870g2871 ) molecules come together? Microstate probabilities Remember that, from a macroscopic point of view, we care about macrostates, that is, states of the system characterized by a few macroscopic variables, like g1831,g1848, and g1840 . For any one macros- tate, there may be many microscopic configurations through which the system evolves. In other words, there may be many microstates compatible with a given macrostate. A macros- tate therefore relates to an ensemble of microstates. We will focus here on classical isolated systems, as these offer the simplest fundamental perspective. Imagine that a closed, insulated container of molecules evolves in time. The microstate of this system constantly changes, as the positions and velocities of each molecule varies under the influence of interactions with the surrounding molecules and the container walls. However, the macrostate of this system—the values g1831,g1848,g1840 —stays constant, since each of these parameters is rigorously constant per the microscopic Newtonian evolution. Imagine performing a simple test of this system: freezing time and interrogating the instanta- neous microstate. We could perform this operation a very large number of times, each time notating which microstate was viewed. If we had a list of all the possible microstates of the system, we would put a tick mark next to the current one each time we repeated the proce- dure. Eventually, we would be able to assign a probability to each of the total possible micro- states, by counting the tick marks for each and dividing by the total number of marks. We will call this probability the microstate probability. It simply gives the relative likelihood that we would see different microstates of the system, for this particular macrostate. Our microstate probabilities would change if we changed the macrostate—altered the total energy or volume, for example. At equilibrium, the microstate probabilities do not change with time. That is, it does not matter when we start making tick marks, so long as we are able to continue doing so long enough to gather accurate statistics. But regardless of the extent of our own tallying, the underlying probabilities inherent to the system remain the same. These statements may seem redundant, as the mere statement of equilibrium suggests a time-invariance. Notice, however, that here
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© M. S. Shell 2008 2/23 last modified 9/21/2009 we are talking about time invariance of a microscopic property, not a macroscopic one. Thus while the microscopic degrees of freedom continually vary—the positions and velocities change continuously—the microstate probabilities do not, at equilibrium.
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