Logarithm and statistical mechanics

In summary, logarithmic dependence appears in statistical mechanics due to the use of logarithms to transform products into sums for easier computation. This can be seen in the function for the probability function of a normal distribution and in the calculation of entropy.
  • #1
guma1204
1
0
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
guma1204 said:
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
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
guma1204 said:
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
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)
 

1. What is a logarithm and how is it used in statistical mechanics?

A logarithm is a mathematical function that is used to measure the relative size of numbers. In statistical mechanics, it is used to calculate the probability of a system being in a certain state. This probability is then used to determine the behavior and properties of the system.

2. What is the relationship between logarithms and entropy in statistical mechanics?

In statistical mechanics, entropy is a measure of the disorder or randomness in a system. The relationship between logarithms and entropy is that the logarithm of the number of microstates in a system is directly proportional to the entropy of the system. This means that as the number of microstates increases, the entropy also increases, and vice versa.

3. How are logarithms used to calculate thermodynamic quantities in statistical mechanics?

Logarithms are used in statistical mechanics to calculate thermodynamic quantities, such as energy, temperature, and pressure. By taking the logarithm of the partition function, which describes the possible states of a system, these quantities can be determined and used to understand the behavior of the system.

4. What is the difference between natural logarithms and base 10 logarithms?

The main difference between natural logarithms and base 10 logarithms is the base number used in the calculation. Natural logarithms use the number e (approximately equal to 2.718) as the base, while base 10 logarithms use the number 10 as the base. In statistical mechanics, both types of logarithms are used, depending on the specific application.

5. Can logarithmic functions be used to describe real-world systems in statistical mechanics?

Yes, logarithmic functions are commonly used to describe real-world systems in statistical mechanics. This is because they provide a useful way to measure and analyze the behavior of complex systems, such as gases, liquids, and solids. Logarithmic functions allow scientists to understand and predict the behavior of these systems, making them a valuable tool in statistical mechanics.

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