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

[gsingh2011]

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?

 

[Andrew Mason]

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