Themes > Science > Physics > Statistical Mechanics > Statistical Mechanics Topics > Classical Statistical Mechanics

We know for a fact that all matter is composed out of small particles called atoms. Statistical mechanics is that branch of physics which explains the thermodynamic properties of nature starting from a microscopic point of view. In particular, we will attempt to apply classical mechanics to a large collection of particles, and in this way derive all the results of thermodynamics. We know that classical (Newtonian) mechanics cannot explain why atoms even exist, let alone explain its properties for which quantum mechanics is necessary. So can we, at all, classically analyze a large collection of atoms before understanding quantum mechanics? The answer, surprisingly enough, is yes. The reason is the following. In studying statistical mechanics, we will be concerned with the object's bulk (macroscopic) properties such as temperature, energy and so on. These properties result mainly from the interaction of atoms (and molecules) with each other. Recall atoms are electrically neutral, and are composed of a positively charged nucleus and negatively charged electrons which are distributed outside the nucleus. Let there be two atoms at positions ${\bf r_1}$ and ${\bf r_2}$. The distance between them is then given by $r=\vert{\bf r_1}-{\bf r_2}\vert$; the so called Lennard-Jones potential results from the quantum mechanical interaction of the charges and angular momentum that is carried by the atoms. The potential due to a typical atom or molecule is given by
\begin{displaymath} U_{LJ}(r)=U_0\{(\frac{R_0}{r})^6-(\frac{R_0}{r})^{12}\} \end{displaymath} (10.1)

where $U_0$ is a constant which depends on charge, and $R_0$ is the Lennard-Jones (LJ) radius, and is shown in Figure 10.1. Note that there is a minimum value in the inter-atomic potential at a distance of $R_0$ from the atom.
Figure 10.1: Lennard-Jones Potential
\begin{figure} \begin{center} %% \input{core/lj.eepic} \end{center} \end{figure}

As long as the atoms are moving slowly, and are farther away from each other than distance $R_0$, they can be treated as hard spheres of radius $R_0$ that behave as classical particles. For typical atoms and molecules the LJ-radius is around 3 to 5A (A= Angstrom =$10^{-10}m$). For example, for the argon atom, the LJ radius is 3.5A, and is 5A for a large molecule such as propane. However, in some cases the LJ radius is not suitable for determining the effective classical size of an atom. For example the $H_2$ molecule has an LJ radius of 0.7A, and is too small a distance to be taken as the classical radius of the $H_2$ molecule. As long as the object being analyzed is at temperatures and densities that are not very high or very low, the atoms are not squeezed together closer than the distance of the LJ radius, and we can treat the atoms as classical billiard balls. However at very low temperatures and high densities, this is not true and the classical analysis needs to replaced by quantum mechanics. At very high temperatures, the inner structure of the atoms, composed as it is out of a nucleus and electrons, needs to be taken into account, and requires an analysis which goes beyond classical mechanics.


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