© 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 timeinvariance.
Notice, however, that here
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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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 Spring '09
 Thermodynamics, Energy, Statistical Mechanics, Entropy, Fundamental physics concepts, M. S. Shell

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