Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problems 2.12-2.14.
To treat the statistical physics of macroscopic objects, we need to deal with very large numbers of particles, typically on the order of or more. The methods we’ve used for analyzing interacting Einstein solids by calculating the number of microstates for each macrostate breaks down for such numbers, as computers aren’t able to calculate the required binomial coefficients exactly.
One way of reducing very large numbers to smaller numbers is to take the logarithm, so we’ll review a few properties of the logarithm here.
The natural logarithm, or logarithm to base is defined so that
The logarithm tends to as and to as , although the latter limit is reached much more slowly than pretty well every other elementary function. The (real) logarithm is defined only for .
A plot looks like this:
A few identities involving the logarithm can be derived from its definition. First, the logarithm of a product:
For the derivative, we can use implicit differentiation:
Since we can use a Taylor expansion to get an approximation for for small :
For , which is fairly close to 0.1. For , so the approximation is quite good here.
A more general form of this approximation can be derived for where . We get
The natural logarithm uses as the base and is the most common logarithm in physics and mathematics because its properties are especially simple. Logarithms can be defined relative to any other real number, however, and the definition is similar to 1. Base 10 logs are defined so that
To convert to natural logs, take the natural log of both sides:
A similar technique can convert bases of exponentiation as in