Boltzmann Entropy for micro state or macro state?

In summary, the Boltzmann entropy for a given distribution of occupancy numbers in a macrostate is defined as S=k log(Ω{ni}), where omega represents the number of microstates for the given occupancy numbers. In equilibrium, omega is predominantly the number of microstates that fill up the entire 6ND gamma space. Using counting principles, omega can be calculated as the product of 1 divided by the factorial of each occupancy number. The question is whether interchanging particle labels would result in a new microstate. If yes, it means particles are no longer indistinguishable and if no, entropy becomes zero since there is only 1 microstate for the given occupancy numbers. It is important to take into account the degener
  • #1
bikashkanungo
9
0
From theory, we know that Boltzmann entropy for a given distribution, defined through a set of occupancy numbers {ni}, of the macrostate M, is given by:
S=k log(Ω{ni})
where omega is the number of microstates for the previously given set of occupancy number, {ni} . Assuming that the system is in equilibrium, we get omega to be predominantly the number of microstates which fill up the entire 6ND gamma space.

Using counting principles in the 6 dimensional mu space we get omega to be equal to product(1/[factorial(ni)]).

My question is would interchanging particle labels (such that {ni} does not change) result in a new microstate? If yes, it means that particles are no longer indistinguishable. If no, then entropy becomes zero since there is just 1 microstate for the given {ni} in gamma space.

I would appreciate if anyone clears this doubt.
 
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  • #2
I think energy does not completely define a microstate. Momentum does. So each energy state has a number of momentum states and these particles are distinguishable by their momentum (but again, not by any "identity"). You have to take this degeneracy into account.
 

1. What is Boltzmann Entropy for micro state and macro state?

Boltzmann Entropy for micro state and macro state is a statistical measure of the disorder or randomness of a system at the microscopic and macroscopic level. It is derived from the Boltzmann equation, which relates the entropy of a system to the number of possible microstates it can have.

2. How is Boltzmann Entropy related to the second law of thermodynamics?

Boltzmann Entropy is closely related to the second law of thermodynamics, which states that the total entropy of an isolated system will always tend to increase over time. This means that the disorder or randomness of a system will naturally increase, leading to a higher Boltzmann Entropy.

3. What is the difference between micro state and macro state in Boltzmann Entropy?

Micro state refers to the specific arrangement of particles in a system, while macro state refers to the overall characteristics of the system such as temperature, pressure, and volume. Boltzmann Entropy takes into account both the micro and macro states of a system to determine its overall level of disorder.

4. How is Boltzmann Entropy calculated?

Boltzmann Entropy is calculated using the formula S = k ln(W), where S is the entropy, k is the Boltzmann constant, and W is the number of possible microstates of the system. This formula can be applied to both micro and macro states to determine the overall Boltzmann Entropy of a system.

5. What is the significance of Boltzmann Entropy for understanding physical systems?

Boltzmann Entropy plays a crucial role in understanding the behavior of physical systems, particularly in the field of thermodynamics. It helps us understand the natural tendency of systems to become more disordered over time and how this relates to the laws of thermodynamics. It also has applications in fields such as statistical mechanics and information theory.

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