# Entropy of a mass arrangement around the earth

An mind experiment, taking the entire Earth as an isolated system

Then this is the initial state:

N masses are distributed around the earth, at different height. (for example we can use a single grain of sand from one building of every city of the world)

Heights are not specified, can be choosen randomly, and those different initial dispositions microstates $\Omega1$ are equivalent to the macrostate (initial state).

the next step

The N masses are released and fall because of gravity

Finally all the N masses lie in its floor, they have become all closer each other, and the height now, relative to it's reference floor, are all equals zero. (If we take a sphere model then it will be zero height, same radial position for all)

So, at final state there are fewer possible microstates $\Omega2<\Omega1$, for the same macrostate, then if process was isolated. Why did entropy went down?

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Some comments: first of all, a microstates can only correspond to the same macrostate if they have equal macroscale properties (i.e. total energy). This puts some constraints on your initial states, not all random disributions of positions will corresponds to the same macrostate. More importantly, when the masses fall, they convert their potential energy to kinetic. In your toy system, there is nowhere for this energy to go, so they will never reach a state of rest at the floor. If you allow energy to be transferred away by friction etc then the entropy of those microscope DOF will increase. –  user2963 Jan 13 '12 at 16:02
@zephyr yes, the state of rest at the floor needs disipation, and there is an implicit air friction assumption, but about the constraints, second macrostate still have larger constraints, even for equal energy of the ensemble, many height combination gives same energy but far fewer has all energy converted to heat and same height. –  HDE Jan 13 '12 at 19:53
this is not true, you have to take into account the incredible multiplicity of microstates which correspond to the excitations created when the kinetic energy is converted to heat. –  user2963 Jan 13 '12 at 20:14
Agreed with zephyr, the entropy of the system increases. The height of each grain of sand is only a single degree of freedom. That tiny number of degrees of freedom does indeed become more ordered, but the zillions of atomic degrees of freedom become more disordered. –  Greg P Jan 13 '12 at 20:44
comments correct. somebody should make an answer so the question can be completed. My advice to @HDE: when thinking about entropy problems think about biological systems which work by diminishing entropy in the body. The law holds for closed systems and the system as you describe it is not closed. –  anna v Jan 14 '12 at 5:45
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## 1 Answer

An always attractive force, Gravity, seems to be antagonistic with an always disipating "force", Entropy, I mean the mere existence of clumps of matter (planets, stars..) again seems to propitiate the concentration of energy (the opposite of Entropy)..

Where is the trick? (Based on Zephyr/UnbanRonMaimon and Greg P comments)

When the masses fall (or become a clump), they convert their potential energy to kinetic. In the toy system of particles, there is nowhere for this energy to go, so they will never reach a state of rest at the floor. If it's allowed for energy to be transferred away by friction then entropy will increase.

So, taking into account the incredible multiplicity of microstates which correspond to the excitations created when the kinetic energy is converted to heat, this overcome the relative reduction in the height possibilities. The height of each grain of sand is only a single degree of freedom. That tiny number of degrees of freedom does indeed become more ordered, but the zillions of atomic degrees of freedom become more disordered.

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