- #1

guma1204

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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.

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- Thread starter guma1204
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- #1

guma1204

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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.

- #2

.Scott

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I am not sure exactly what you are referring to. But I think this is an example of what you are asking about:

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.

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.

- #3

DrClaude

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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##.

- #4

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If ##p_i## are probabilities of independent events, then the total probability isWhy and how does logarithmic dependence appear in statistical mechanics? I understand that somehow it is linked with probabilities, but I can not quite understand.

$$p=\prod_i p_i$$

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.

Last edited:

- #5

morrobay

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The probability of finding N particles in volume V is w = (cV)

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