On 15 November 2014 11:23, meekerdb <[email protected]> wrote: > On 11/14/2014 1:29 PM, John Clark wrote: > > On 13 November 2014 18:57, LizR <[email protected]> wrote: > > >> > There appears to be a discrepancy between entropy as it is ascribed >>> to black holes and entropy in the form of configurations of mass-energy far >>> from thermodynamic equilibrium. Black hole entropy appears to be a >>> fundamental feature of physics, while the other sort only emerges due to >>> coarse graining. I'd be interested to know if anyone can shed any light on >>> this apparent discrepancy. >>> >> > I'm not sure what you mean that there are 2 types of Entropy, it always > works the same way. The Entropy of a Black Hole (and the Entropy of > anything else) is Boltzmann's constant time the logarithm of the number of > ways the Black Hole could have gotten into the state it's in now. The > reason we use a logarithm in the definition is we want to be able to say > that the total Entropy of the combined system X and Y is the Entropy of X > PLUS the Entropy of Y, if we didn't use logarithms it would be X times Y. > For example, if system X could have gotten to the way it is now in 3 > different ways and system Y could have gotten to the way it is now in 5 > different ways then the combined system could have gotten to the way it is > now in 3*5 =15 different ways, but ln 3 + ln 5 = ln 15. > > Any constant could be used but it is convenient to use Boltzmann's > constant because it's nice if Entropy is in units of energy/temperature. > > > "The numbers of ways the system could have gotten to the way it is" isn't > the usual formulation and I think it's ambiguous. In general there are > arbitrarily many possible histories and different possible starting > points. Boltzmann's formulation was the logarithm of the numbers of > possible states consistent with constraints defining the system, e.g. its > total kinetic energy or its temperature and volume. In the case of a BH > the constraints are its classical defining parameters: mass, angular > momentum, and electric charge. Classically there is no finer grained > description, so that's what seems to make BH entropy more fundamental that > the usual thermodynamic system. > > Thanks, Brent, I think you have seen the source of my puzzlement and pointed out the answer. As an exercise I will now attempt to respond to my question in a way that even a bear of little brain such as myself can understand. Please let me know if I'm anywhere in the right ballpark (or not, of course)...
On the one hand there is "classical" entropy which comes down to the arrangement and movement of some type of objects (e.g. molecules in a box). The individual molecular interactions are time-symmetric, but once one looks at the system above a certain scale (applying "coarse graining") we can see that certain arrangements - e.g. a certain distribution of velocities and positions - are more likely to be occupied than others, because there are "Vastly" more configurations that look identical above a certain scale. A black hole, however, has only one state, according to GR, described by those parameters you mentioned - according to GR there are no traces remaining of the way it got that way (unless it's imprinted in the gravity waves it radiated during its formation?) So it appears meaningless to talk about entropy for an object with only one possible state. However Hawking used the idea of BH entropy to show that BHs radiate, by adding some quantum theory to the mix, implying that the GR picture is incomplete. (The "BH information paradox") I think most theorists now suspect that the information that described the objects that originally went to create the BH is preserved somehow, and ultimately will reappear imprinted in the Hawking radiation. Modulo how the BH stores information, it seems likely that we can apply statistics to work out a similar coarse-grained entropy for it. For example if it's in the state of Planck scale units of space-time in some hypothetical TOE, we can reasonably assume those states will tend to "thermalise" (or whatever Planck cells do) towards what are (macroscopically) more likely configurations. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
