Logarithm and statistical mechanics

  • #1
guma1204
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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.
 

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  • #2
.Scott
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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:

ffe7c5cbdecda556bf2170e31f1f9a127b74e239


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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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##.
 
  • #4
Demystifier
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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.
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.
 
Last edited:
  • #5
morrobay
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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)
 

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