On Sun, Oct 18, 2020, 11:55 PM Bruce Kellett <[email protected]> wrote:
> On Mon, Oct 19, 2020 at 3:31 PM Jason Resch <[email protected]> wrote: > >> On Sun, Oct 18, 2020, 10:33 PM Bruce Kellett <[email protected]> >> wrote: >> >>> On Mon, Oct 19, 2020 at 1:09 PM Jason Resch <[email protected]> >>> wrote: >>> >>>> On Sun, Oct 18, 2020, 8:47 PM Bruce Kellett <[email protected]> >>>> wrote: >>>> >>>>> >>>>> Remember that entropy is basically related to the volume of phase >>>>> space, not of ordinary space. And phase space relates to the number of >>>>> particles (hence mass-energy). Spatial volume is essentially irrelevant >>>>> for >>>>> volumes greater than that of the corresponding black hole. >>>>> >>>> >>>> No. Consider an infinite length. With a single atom you can encode >>>> infinite information through placement of the atom along that length. This >>>> is with finite mass energy, but unrestricted spatial volume. >>>> >>> >>> That does not encode infinite information. There is, after all, only one >>> particle, and it can have only one position. If you want to encode more >>> information, you need more particles. You might need an infinite number of >>> bits to encode the position of one particle as a real number, but the >>> single particle cannot encode this. >>> >> >> This is plainly false. Every 1 mile distance that particle is placed >> along the line encodes a unique number. Travel up to 2^N miles and you can >> encode N bits. With infinite range there's no upper bound. >> > > A single particle can be in only one place, and encode on ly one bit. > If I were building a hard drive and could only write a fixed number of crosses, let's say 5 crossed, on a LxL grid, the number of possible combinations would be (L^2 choose 5). So for L = 3 that is (9 choose 5) = 126. Which means I can encode Log2(126) = 6.97 bits For L = 4 that is (16 choose 5) = 4368. Which means I can encode Log2(4368) = 12.09 bits My total number of crosses (let's say I use a single atom to represent each, is 5 in both cases). It is constant. But if I have more space, I have more ways of arranging them, and can use them to encode more bits, or conversely it takes more information to describe the system. Why does this analogy not extend to a quantum system of particles in larger or smaller regions of space? Or think of a grid of naughts and crosses, with a larger grid but fixed >> number of crosses, the number of possible combinations for drawing a fixed >> number of crosses still increases with more spaces to place them. >> > > Each combination encodes only one combination. > More unique combinations mean more states a system can possibly be in, meaning it takes more information to uniquely define the state the system can be in, or alternatively the more information the system may encode. > An arbitrary volume can only hold a limited amount of energy, or entropy, >>> as given by the Bekenstein bound. >>> >> >> Energy isn't the same thing as entropy. >> > > Bekenstein relates them. > >> But the maximum entropy for a particular mass is given when that mass >>> forms a black hole -- which saturates the Bekenstein bound. >>> >> >> The bound is always satisfied. Black holes just reach the maximum of the >> bound at a given VOLUME. >> > > I said saturated, not 'satisfied'. The bound gives the maximum possible > enclosed mass for a given volume, or the volume is that for which entropy > is maximum for a given mass which saturates the bound. > I think you're thinking of the black hole entropy equation, which is related to, but distinct from, the Bekenstein bound. >> Increasing the volume does not increase the actual entropy unless you >>> simultaneously increase the mass. >>> >> >> You keep saying this but don't provide any justification or sources. I >> implore you to read the wikipedia article and if it is wrong, please point >> me to a source with the right/corrected equation. >> > > The justification is that it is impossible to increase the mass of a black > hole without at the same time increasing its radius (volume). For a black > hole, the radius is 2M, in natural units. So the mass and radius are > directly related. Any greater volume for the same mass does not saturate > the bound. > Forget about saturating the bound, that's not the point. Saturating the bound requires maximizing entropy for a given volume. On that we agree. My point is that the bound implies that a larger amount of volume, for fixed energy, allows for higher entropy. Put your black hole in a larger volume and now the black hole has a very well defined position in that volume, which is more information than you had before. It's a generally accepted in computer science that a turing machine allowed to use infinite space could store infinite information, even with fixed total mass/energy. > > >> As explained on that page, the bound is not limited to black holes, it >> says something more general which relates entropy bounds to the product of >> spherical radius and mass. >> > > The entropy bound you are talking about is > > S <= 2pi RE. > > This is saturated when the radius and energy are related as for a black > hole: > > R = 2M, for which S = 4pi M^2. > > Nothing mysterious here. I was talking about maximum possible entropy, > which occurs when the bound is saturated, as for a black hole. > > That is really all that the Bekenstein bound says. It is a bound, after > all, and has information about the entropy only when that bound is > saturated. > If the bound strictly depends on energy, why is R included in the formulation? > So for a fixed amount of mass, the entropy is maximized when that mass is > in the form of a black hole. Increasing the volume surrounding the BH makes > no difference to the entropy maximum for that mass. > For the system as a whole it does. Now the black hole has coordinates in a larger volume which did not exist before, and must be included in any description of that system. A non-collapsed relativistic gas sits right on the edge of becoming a black hole and satisfying the bound. Consider such a relativistic gas confined to a 1 meter volume. Now considering that gas is given more space to occupy, it is placed in a sphere of 1 light-year. Are there not now many more degrees of freedom possible for that same mass energy in a 1 light-year space than when it was confined to 1 meter? Are not more bits and precision required to specify the coordinates of each particle? Jason > Bruce > > Jason >> >> In terms of the cosmological problem, the initial state has a particular >>> total mass, and that does not increase with the expansion of the universe. >>> Consequently, the maximum possible entropy does not increase either. The >>> point of the Past Hypothesis is that the initial state of this mass was of >>> low entropy since the gravitational degrees of freedom were not saturated >>> (it did not form a black hole), so there is a large amount of room >>> available for the entropy to increase. >>> >>> Bruce >>> >> -- > 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 view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/CAFxXSLTohsYxw1TObr0MurptwzKcp%3D9pG3j_2oWOQC2SsZ8eRg%40mail.gmail.com > <https://groups.google.com/d/msgid/everything-list/CAFxXSLTohsYxw1TObr0MurptwzKcp%3D9pG3j_2oWOQC2SsZ8eRg%40mail.gmail.com?utm_medium=email&utm_source=footer> > . > -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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