On 5/29/17 9:55 PM, Phillip Helbig (undress to reply) wrote:
A smooth distribution corresponds to high entropy and a lumpy one to low
entropy if gravity is not involved. For example, air in a room has high
entropy, but all the oxygen in one part and all the nitrogen in another
part would correspond to low entropy.

If gravity is involved, however, things are reversed: a lumpy
distribution (e.g. everything in black holes) has a high entropy and a
smooth distribution (e.g. the early universe) has a low entropy.

Let's imagine the early universe---a smooth, low-entropy
distribution---and imagine gravity becoming weaker and weaker (by
changing the gravitational constant). Can we make G arbitrarily small
and the smooth distribution will still have low entropy? This seems
strange: an ARBITRARILY SMALL G makes a smooth distribution have a low
entropy. On the other hand, it seems strange that the entropy should
change at some value of G.

I think the mistake here is thinking about "smooth" and "lumpy"
as a binary choice. What G affects is *how* lumpy the maximum
entropy system is.

Suppose first that there are no forces except gravity. As soon
as you turn on G, a smooth system becomes unstable -- the Jeans
length is zero. Thermodynamically, the gravitational potential
energy can become arbitrarily negative, and at fixed energy it's
entropically favorable for the system to collapse a little more,
lowering the gravitational energy, and kick out a particle with
extra kinetic energy. The classic analysis of this is Lynden-Bell
and Wood, "The Gravo-Thermal Catastrophe in Isothermal Spheres and
the Onset of Red-Giant Structure for Stellar Systems," MNRAS 138
(1968) 495.

Now suppose there are other forces that are not purely attractive.
Dynamically, the Jeans length is now finite, and this determines the
typical size of lumps. If you turn up G, the Jeans length decreases,
and you get more, smaller lumps. Thermodynamically, you can still
increase entropy by collapsing and kicking out particles with high
kinetic energy, but this process is now limited, since the collapse
will eventually be stopped by other forces. Bigger G allows more
collapse before this equilibrium is reached, and more lumpiness.

I don't know of anywhere this has been worked out, but I suspect
that if you found a measure of the amount of lumpiness in the
maximum entropy state you'd find that it varies smoothly with G.
(There's probably some nice way to use the Jeans length for this.)

Steve Carlip