How to think about entropy microstates/macrostates for a gas in a box

In summary: ERICAN ELECTRONIC ENGINEERING CONFERENCE 2016In summary, according to Brian Greene, entropy is a measure of how many microstates there are of a system at a particular temperature and size. A low entropy system would be when all the coins are heads up, while a high entropy system would be when 50 coins are heads up. This analogy can be applied to gases in a box, where an external observer would measure a particular pressure and temperature for the gas. If the gas is spread out throughout the box, and you measure the pressure/temperature, there are a large number of other configurations of the particles that result in the same reading.
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gsingh2011
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I'm trying to relate an analogy from Brian Greene about entropy microstates/macrostates to the real world. In the analogy, you have 100 coins that you flip. The microstate is which particular coins landed heads up. The macrostate is the total number of coins that are heads up. So a low entropy configuration would be when all the coins are heads up. There is only one microstate that corresponds to that macrostate. But there is a very large number of microstates that correspond to the macrostate where 50 coins are heads up, and that is a high entropy configuration.

So relating this to a gas in a box, let me know if this understanding is correct. If all of the particles of a gas in a box are contained within a small cubic region in the corner of the box, an external observer would measure a particular pressure and temperature for the gas. And while there may be other configurations (configuration means positions/velocities of each particle) of the gas in the box that would get those same pressure/temperature readings, there aren't that many of them. However, if the gas is spread out throughout the box, and you measure the pressure/temperature, there are a large number of other configurations of the particles that result in the same reading.

Is that a correct interpretation of entropy?
 
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gsingh2011 said:
I'm trying to relate an analogy from Brian Greene about entropy microstates/macrostates to the real world. In the analogy, you have 100 coins that you flip. The microstate is which particular coins landed heads up. The macrostate is the total number of coins that are heads up. So a low entropy configuration would be when all the coins are heads up. There is only one microstate that corresponds to that macrostate. But there is a very large number of microstates that correspond to the macrostate where 50 coins are heads up, and that is a high entropy configuration.

So relating this to a gas in a box, let me know if this understanding is correct. If all of the particles of a gas in a box are contained within a small cubic region in the corner of the box, an external observer would measure a particular pressure and temperature for the gas. And while there may be other configurations (configuration means positions/velocities of each particle) of the gas in the box that would get those same pressure/temperature readings, there aren't that many of them. However, if the gas is spread out throughout the box, and you measure the pressure/temperature, there are a large number of other configurations of the particles that result in the same reading.

Is that a correct interpretation of entropy?
It can be very difficult and tricky to conceptualize entropy using the statistical mechanics micro-canonical ensemble model. If you want to use your example, compare the number of microstates that could exist for the particles of a gas in thermodynamic equilibrium at temperature T and confined to a space V to the number of microstates that could fit the same gas at temperature T and in thermodynamic equilibrium but confined to a much larger space, say 10V.

There would be many more microstates that would define the same thermodynamic equilibrium macrostate at 10V than at V because the particles would have the same speed distribution but could have many more positions available. So the 10V state has higher entropy.

But if the particles occupying the larger volume were in thermodynamic equilibrium at a lower temperature, then there might not be more microstates that would define the larger volume macrostate as the range of possible particle speeds has decreased. This illustrates the difficulty in conceptualizing entropy this way.

Note: There is the statistical mechanics/micro canonical ensemble explanation of entropy. Then there is kinetic theory which explains thermodynamic equilibrium as a statistical concept. Boltzmann figures prominently in both. But, while there are similarities, they are two distinct concepts: 1. thermodynamic equilibrium as a state in which the distribution of particle speeds/energies follows a Maxwell-Boltzmann distribution. 2. Entropy deals with systems in, and transitions between, states of thermodynamic equilibrium.

AM
 
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Related to How to think about entropy microstates/macrostates for a gas in a box

1. What is entropy and how does it relate to microstates and macrostates for a gas in a box?

Entropy is a measure of the disorder or randomness of a system. In the context of a gas in a box, it is related to the number of possible arrangements of the particles in the box. Microstates refer to the specific arrangement of particles at a given moment, while macrostates refer to the overall state of the system, which can have multiple microstates associated with it.

2. How does the number of microstates affect the entropy of a gas in a box?

The higher the number of microstates, the higher the entropy of the system. This is because a larger number of microstates means a higher degree of disorder or randomness in the system, which corresponds to a higher entropy value.

3. How does temperature affect the number of microstates and the entropy of a gas in a box?

As temperature increases, the number of microstates also increases. This is because at higher temperatures, the particles in the gas have more energy and can move around more freely, resulting in a larger number of possible arrangements. As a result, the entropy of the system also increases.

4. Can the number of microstates ever decrease for a gas in a box?

No, the number of microstates can never decrease for a gas in a box. This is due to the second law of thermodynamics, which states that the entropy of a closed system will either remain constant or increase over time. Therefore, the number of microstates can only increase or stay the same, but it can never decrease.

5. How does the concept of entropy and microstates/macrostates apply to real-world systems?

The concept of entropy and microstates/macrostates applies to all systems, including real-world systems. It is a fundamental principle of thermodynamics and is used to understand and predict the behavior of various systems, such as chemical reactions, biological processes, and even the universe as a whole.

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