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.

 

[.Scott]

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.

 

[DrClaude]

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, $S = k \ln \Omega$.

 

[Demystifier]

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.

If $p_i$ are probabilities of independent events, then the total probability is
$$p=\prod_i p_i$$
However, products are not easy to compute, especially if $i$ is a continuous label. Therefore we transform the product into a sum via
$$\ln p=\sum_i \ln p_i$$
That's the origin of most logarithms in statistical physics.

 

[Point Conception]

Take statistical mechanics entropy : S = k ln w. Where w is probability that system is in present state relative to all other possible states
The probability of finding N particles in volume V is w = (cV)^N So S = kN(ln c + ln V)