# Meaning of thermodynamic probability

## Summary:

I was studying statistical mechanics when I came to know about the Boltzmann's entropy relation, $S = k_B\ln Ω$.

What does 'thermodynamic probability' actually mean and how does it give rise to the Entropy?

## Main Question or Discussion Point

I was studying statistical mechanics when I came to know about the Boltzmann's entropy relation, $S = k_B\ln Ω$.

The book mentions $Ω$ as the 'thermodynamic probability'. But, even after reading, I can't understand what it means. I know that in a set of $Ω_0$ different accessible states, an isolated system has a probability $P = \frac 1 {Ω_0}$ of being in any one of the state and that at equilibrium when entropy is maximum, the probability of the system being in various accessible states do not vary with time. Also, when two interacting systems are in equilibrium, the no of states available to the combines system is maximum.

In the Boltzmann's entropy relation, $S = k_B\ln Ω$, what does the $Ω$ signify? If it is a probability what is it the probability of and for which system are we getting the Entropy?

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Orodruin
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$\Omega$ is the number of microstates corresponding to the relevant macrostate.

$\Omega$ is the number of microstates corresponding to the relevant macrostate.
So, S is the entropy of that macrostate?

Orodruin
Staff Emeritus
Homework Helper
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So, S is the entropy of that macrostate?
Yes. This is in the first paragraph on Wikipedia’s page on entropy:
In statistical mechanics, entropy is an extensive property of a thermodynamic system. It is closely related to the number Ω of microscopic configurations (known as microstates) that are consistent with the macroscopic quantities that characterize the system (such as its volume, pressure and temperature). Entropy expresses the number Ωof different configurations that a system defined by macroscopic variables could assume.[1]

Yes. This is in the first paragraph on Wikipedia’s page on entropy:
I guess checking Wikipedia will be the first thing I do from now on. I got confused as it was called thermodynamic probability and thought that it should have the properties of probability (summation = 1).

Thanks for the help!