# Logarithm and statistical mechanics

guma1204
Hello, I'll try to get right to the point.

Why and how does logarithmic dependence appear in statistical mechanics? I understand that somehow it is linked with probabilities, but I can not quite understand.

Homework Helper
Hello, I'll try to get right to the point.

Why and how does logarithmic dependence appear in statistical mechanics? I understand that somehow it is linked with probabilities, but I can not quite understand.
I am not sure exactly what you are referring to. But I think this is an example of what you are asking about:

The function for the probability function of a normal distribution is shown here:

It decreases exponentially as |(x-u)/sigma| increases.
It's easy to see why. Every increase in a standard deviation from mean compounds the unlikelihood - an exponential process.

Mentor
What logarithm are you talking about?

In some cases, it is simply a question of convenience. Instead of working with the multiplicity ##\Omega##, we most often use entropy instead, ##S = k \ln \Omega##.

$$p=\prod_i p_i$$
$$\ln p=\sum_i \ln p_i$$