# Meaning of thermodynamic probability

• Saptarshi Sarkar
In summary, the conversation discusses Boltzmann's entropy relation, which states that entropy is equal to the natural logarithm of the thermodynamic probability, denoted by ##Ω##. The concept of thermodynamic probability refers to the number of microstates corresponding to a macrostate, and at equilibrium, the probability of a system being in various accessible states does not change. The ##Ω## in the entropy relation signifies the number of microstates corresponding to the relevant macrostate, and entropy is an extensive property of a thermodynamic system.
Saptarshi Sarkar
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 anyone 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?

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

Saptarshi Sarkar
Orodruin said:
##\Omega## is the number of microstates corresponding to the relevant macrostate.

So, S is the entropy of that macrostate?

Saptarshi Sarkar said:
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]

Saptarshi Sarkar
Orodruin said:
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!

## 1. What is the meaning of thermodynamic probability?

Thermodynamic probability refers to the likelihood or chance of a particular state or configuration occurring in a thermodynamic system. It is a measure of the number of ways in which a system can be arranged or distributed among its possible energy states.

## 2. How is thermodynamic probability related to entropy?

Thermodynamic probability is directly related to entropy, which is a measure of the disorder or randomness in a system. As the thermodynamic probability increases, the entropy also increases, indicating a higher degree of disorder in the system.

## 3. What factors affect thermodynamic probability?

The factors that affect thermodynamic probability include the number of particles in the system, the energy levels available to the particles, and the temperature of the system. These factors determine the number of possible arrangements or configurations that the system can take on.

## 4. How is thermodynamic probability calculated?

Thermodynamic probability is calculated using the Boltzmann equation, which relates the probability of a particular state to the energy of that state and the temperature of the system. It is represented by the symbol W and is often expressed in terms of the natural logarithm.

## 5. What is the significance of thermodynamic probability in thermodynamics?

Thermodynamic probability is a fundamental concept in thermodynamics, as it helps to explain the behavior of systems at the microscopic level. It is used to calculate important thermodynamic quantities such as entropy and free energy, which are crucial for understanding the behavior of macroscopic systems.

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