Monday, May 07, 2007

Starting with the parts and ending with the whole

The focus of today's blog post is going to be statistical mechanics. In statistical mechanics, one starts with the properties of the atoms/molecules/ions in a system and try to understand how the whole system will behave. So, one starts with the microscopic properties of the components of a system and tries to understand the macroscopic properties of the whole system. The macroscopic properties here implies properties such as volume or temperature or pressure that one measures in an experiment physically. I will provide an example to make one understand how difficult this task is:

Now lets consider a box or cylinder filled with gas molecules. As each individual gas molecule is small in volume, to fill up the whole container, one would require a very large amount of gas molecules, lets just say, something in the order of a mole (A mole contains 6.023 * 10^23 molecules of the gas. This might sound enormous but is actually a small number in terms of molecules. To place things in perspective, 1 mole of water is contained in only 18 grams of water, and a liter of water typically contains 55.556 moles of water). Let us assume for simplicity that these molecules obey Newton's laws of motion and do not undergo any quantum effects.

Even under these conditions, the molecules are all moving and the total energy of the system would be the sum of each molecule's individual kinetic energy and potential energy. In addition, there will be forces acting on each molecule due to the neighboring molecules as well as the ends of the container. So, by Newton's law of motion, each particle will have a unique acceleration induced on it and the position and velocity of each particle continuously changes within the box. It becomes a hopeless situation to even try to follow an individual particle's position and velocity as the position of the other particles affect the potential energy and force of the particle we are interested in.

Hence, what one does is try to come up with a probabilistic approach as to how the system's macroscopic properties are affected by it's microscopic properties. Most of the theory that is dealt with in statistical mechanics are valid only when there is a sufficiently large number of particles (as the derivations that will come up in the coming weeks will show) and will not hold true under other conditions. Using statistical mechanics, one can go beyond the simple Newton's laws of motion and try to derive/explain the laws of thermodynamics that one can measure experimentally.

Books to understand Statistical Mechanics:
Chandler, David (1987). Introduction to Modern Statistical Mechanics.
McQuarrie, Donald (2000). Statistical Mechanics.
R.K.Pathria (1996). Statistical Mechanics.

Book to understand Thermodynamics:
Above books and
Callen, Herbert B (2001). Thermodynamics and an Introduction to Thermostatistics.

Some of the above discussion was also inspired from the Wikipedia article on statistical mechanics.


Ashutosh said...

I would also add a very simple, elegant, and lucid book: Therodynamics by Enrico Fermi

Anonymous said...

I've been thinking about this alot... It does lead to other issues...

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