From: kkumer@phy.hr (Kresimir Kumericki) Subject: Re: Entropy Date: 30 Aug 2000 23:08:47 GMT Newsgroups: sci.physics.research Summary: [missing] John Baez wrote: > Oh... duh! The assumptions are different: you were requiring that > the kinetic plus potential energy stay constant, whereas I was letting > energy leak off in the form of radiation or "boiling off" or something. > The calculations really apply to different regimes: yours applies to > the "fast regime" when the cloud is getting virialized but (you assume) > acting like a closed system, whereas mine applies to the "slow regime" > when it's already virialized and is slowly losing energy by radiation or > boiling off. > It all makes sense now. Great! Trying to catch up with the news after summer holidays I got interested in this thread and I tried to do a back-of-the-envelope calculation to see whether in the "slow regime", where cloud is virialized all the time, there is enough entropy produced by the radiation/evaporation to make up for the reduced entropy of the cloud as it shrinks. I know it should come out right (in fact, I think the calculation below is in effect tautologic) but this gravity-entropy business can be tricky so I wanted to make an explicit check. So, if anyone's still interested, here it is: We already know that the entropy of the cloud is S_cloud = kN [(1/2) ln N + (1/2) ln V + C] , and this means that for a small change dR of cloud radius we have entropy change dS_cloud = (3/2) kN dR/R (*) . On the other hand, the assumption that cloud is virialized means that dP = -2 dK = -2 dU , where U is the energy of radiation/vapour. If we take that energy U was radiated in a reversible process we will get the lower bound on the produced entropy: dS_rad >= dU/T = dK/T . But we know that potential energy P is proportional to -1/R and this means, again by virial theorem, that K ~ 1/2R, or that for ideal monoatomic cloud dK = -K dR/R = -(3/2) kNT dR/R . This finally means that radiated entropy is dS_rad >= -(3/2) kN dR/R , which exactly cancels (*): dS_total = dS_cloud + dS_rad >= 0 , and saves the Second Law. As an aside, note that for short-range forces V ~ -1/r^n, n>2, it is impossible to virialize the cloud and still have it shrinking because this would mean |dK|>|dV| which would violate energy conservation. -- ------------------------------------------------------------- Kresimir Kumericki kkumer@phy.hr http://www.phy.hr/~kkumer/ Theoretical Physics Department, University of Zagreb, Croatia -------------------------------------------------------------