XPost: sci.astro.research   
   From: carlip@physics.ucdavis.edu   
      
   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   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)