7: Classical Statistical Mechanics

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Dynamics of particles is given by Newton’s laws or, if we include quantum effects, the quantum mechanics of point-particles. Thus, if we have a large number of particles such as molecules or atoms which constitute a macroscopic system, then, in principle, the dynamics is determined. Classically, we just have to work out solutions to Newton’s laws. But for systems with large numbers of particles, such as the Avogadro number which may be necessary in some cases, this is a wholly impractical task. We do not have general solutions for the three-body problem in mechanics, let alone for \(10^{23}\) particles. What we can attempt to do is a statistical approach, where one focuses on certain averages of interest, which can be calculated with some simplifying assumptions. This is the province of Statistical Mechanics.

If we have \(N\) particles, in principle, we can calculate the future of the system if we are given the initial data, namely, the initial positions and velocities or momenta. Thus we need \(6\;N\) input numbers. Already, as a practical matter, this is impossible, since \(N ∼ 10^{23}\) and we do not, in fact, cannot measure the initial positions and momenta for all the molecules in a gas at any time. So generally we can make the assumption that a probabilistic treatment is possible. The number of molecules is so large that we can take the initial data to be a set of random numbers, distributed according to some probability distribution. This is the basic working hypothesis of statistical mechanics. To get some feeling for how large numbers lead to simplification, we start with the binomial distribution.

7.1: The Binomial Distribution
This is exemplified by the tossing of a coin. For a fair coin, we expect that if we toss it a very large number of times, then roughly half the time we will get heads and half the time we will get tails. We can say that the probability of getting heads is 1/2 and the probability of getting tails is 1−1/2=1/2 . Thus the two possibilities have equal a priori probabilities.
7.2: Maxwell-Boltzmann Statistics
Now we can see how all this applies to the particles in a gas. The analog of heads or tails would be the momenta and other numbers which characterize the particle properties. Thus, we can consider N particles distributed into different cells, each of the cells standing for a collection of observables or quantum numbers which can characterize the particle.
7.3: The Maxwell Distribution For Velocities
The most probable distribution of velocities of particles in a gas is given by Equation 7.2.9. Thus we expect the distribution function for velocities to be as shown in Equation 7.3.1. This is known as the Maxwell distribution. Maxwell arrived at this by an ingenious argument many years before the derivation we gave in the last section was worked out.
7.4: The Gibbsian Ensembles
Interatomic or intermolecular forces are not so straightforward. In principle, if we have intermolecular forces, single particle energy values are not easily identified. Further, in some cases, one may even have new molecules formed by combinations or bound states of old ones. Should they be counted as one particle or two or more? So, one needs to understand the distribution from a more general perspective.
7.5: Equation of State
The equation of state, as given by Equation 7.4.19, requires the computation of the grand canonical partition function. This page shows explicitly the first correction to the ideal gas equation of state. The van der Waals equation is, at best, a model for the equation of state incorporating some features of the interatomic forces. Here we have a more systematic way to calculate with realistic interatomic potentials.
7.6: Fluctuations
We will now calculate the fluctuations in the values of energy and the number of particles as given by the canonical and grand canonical ensembles. In the end, the fluctuations are small compared to the average value as N becomes large.
7.7: Internal Degrees of Freedom
Many particles, such as atoms, molecules have internal degrees of freedom. The internal dynamics has its own phase-space and, by including it in the integration measure for the partition function, we can have a purely classical statistical mechanics of such systems. The question is: How do we do this? The simplest strategy, as discussed in this page, is to consider the particles in different internal states as different species of particles.
7.8: Examples
An example of the use of the idea of the partition function in a very simple way is provided by the osmotic pressure. Here one considers a vessel partitioned into two regions, say, I and II, with a solvent on one side and a solution of the solvent plus a solute on the other side. The separation is via a semipermeable membrane which allows the solvent molecules to pass through either way, but does not allow the solute molecules to pass through. Thus the solute molecules stay in region II.