Entropy of a system can be understood as "the number of states that are macroscopically indistinguishable from the state its currently in". (Sean Carroll, "The Big Picture")  Given that the early universe was such a tiny thing with all matter and energy compressed in it, why does it make sense to assume that the universe had extremely low entropy in the past?  How do we know that it was highly non-uniform at that time. I mean, intuitively speaking, how can we even imagine things in such a small space to be "in separate chunks"?
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Let me start by saying that I have no scientific background whatsoever. I am very interested in science though and I'm currently enjoying Brian Greene's The Fabric of the Cosmos. I'm at chapter 7 and so far I understand most of general ideas he has talked about. Or at least, I think I understand them :-)  There is however one part, at the end of chapter 6 that I can't grasp.  It is about entropy and the state of the universe a few minutes after the big bang.  On page 171 he says:     Our most refined theories of the origin of the universe -our most refined cosmological theories- tell us that by the time the universe was a couple of minutes old, it was filled with a nearly uniform hot gas composed of roughly 75 percent hydrogen, 23 percent helium, and small amounts of deuterium and lithium. The essential point is that this gas filling the universe had extraordinarily low entropy.   And on page 173-174:     We have now come to the place where the buck finally stops. The ultimate source of order, of low entropy, must be the big bang itself. In its earliest moments, rather than being filled with gargantuan containers of entropy such as black holes, as we would expect from probabilistic considerations, for some reason the nascent universe was filled with a hot,   uniform, gaseous mixture of hydrogen and helium. Although this configuration has high entropy when densities are so low that we can ignore gravity, the situation is otherwise when gravity can't be ignored; then, such a uniform gas has extremely low entropy. In comparison with black holes, the diffuse, nearly uniform gas was in an extraordinarily low-entropy state.   In the first part of the chapter Brian Greene explains the concept of entropy with tossing the 693 pages of War and Peace in the air.  At first, the pages are ordered. The specific order they are in make sense and are required to recognize the pages as a readable book called War and Peace. This is low entropy. It is very highly ordered and there is no chaos.  Now, when you throw the pages in the air, let them fall and then pick them up one by one and put them on top of each other, the chances you get the exact same order as the initial state are extremely small. The chance you get another order (no matter what order, just not the one from the beginning) is extremely big. When the pages are in the wrong order, there is high entropy and a high amount of chaos. The pages are not ordered and when they are not ordered you would not notice the difference between one unordered state and another one.  However, should you swap two pages in the ordered, low entropy version, you would notice the difference.  So I understand low entropy as a highly ordered state with low chaos in which a reordering of the elements would be noticeable.  I hope I'm still correct here :-)  Now, what I don't understand is how a uniform mixture of hydrogen and helium can by highly ordered? I'd say you wouldn't notice it if some particles traded places. I'd say that a uniform mixture is actually in a state of high entropy because you wouldn't notice it if you swapped some hydrogen atoms.  Brian Greene explains this would indeed be the case when gravity plays no important role, but that things change when gravity does play a role; and in the universe right after the big bang, gravity plays a big role.  Is that because a reordering of the particles would change the effects of gravity? Or is there something else that I'm missing here?
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I was recently reading some of the nice answers to the (related) question:   From many of the answers their appears to be a symmetry between:   A universe in which all matter-things (plus light) remain the same size, while the space between them expands. (The one we appear to live in) A universe where the space remains the same size and all objects in the universe shrink. Along with an appropriate change in physical length-scale constants. (One equally compatible with our observations?)   Said another way: At one point in time your ruler fit between those two galaxies $10^{50}$ times, now it fits between them $10^{51}$ times. This is usually explained as the space growing, I believe it could equally well be described by the ruler shrinking.  Is this symmetry obeyed? Are their any cosmological models presented in terms of matter shrinking?
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Starting from the assumption of a cosmology in which  dominate over evolved ones, it is not immediately obvious to me that there is a real problem, since only Boltzmann brains indistinguishable from evolved brains will be posing the question in the first place.  In other words, the vast majority of Boltzmann brains will not be troubled by the paradox because their memories will not be consistent with a stable/large physical universe, and so they would never ask such a question in the first place.  So we must only consider the subset of Boltzmann brains whose experiences would be so consistent with a universe like ours that they would pose the question "why are we not Boltzmann brains?" And in such cases, even though we may be Boltzmann brains, our experiences would be indistinguishable from evolved brains.  Is this a criticism of the Boltzmann brain paradox that has been dispatched with, or is it a legitimate cause to be suspicious of whether it is really a paradox at all?
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OK...before everyone blasts this with references to the relativistic invariance of the physical laws, time dilation, etc let me add some context. Also, I am an amateur with an interest in physics, so I don't know the details of the physics. However, I have read enough about the cosmological principle and cosmic time to develop this "confusion" .  Based on evidence gathered to-date, it appears that the universe is both , as evidenced by observations and the . This principle, when combined with the FLRW metric, allows space-time to be divided into non-intersecting slices and therefore establish a "cosmic time."   Now, the existance of such a time in no way suggests that it is the "right" time or a true "now", hence violating relativity -- although some presentist philosophers (e.g., W.L. Craig) have tried to make this argument, but it is not generally accepted in philosophical circles.  However, this leaves me confused about an apparent disconnect between relativity and the cosmological principle:  Isotropy only holds if we are at rest relative to the cosmic microwave background radiation (universal rest frame), o/w anisotropy is present - yet each reference frame is supposed to allow equally valid observations. How can these two be reconciled when most reference frames would lead us to conclude that the universe is NOT isotropic?  It seems that only by appealing to the idea of being at rest relative to "universal rest frame" can we explain away any discrepancies from isotropy as due to "peculiar motion". However, doesn't this give this "universal rest frame" and its associated time an empirically privileged status, even though physical laws work just fine in every reference frame?   For example (pardon any abuse of astronomy): if look off in some part of space and see only quasars, and then in another part of space and see only brown dwarfs, but I measure both as apparently the same distance from me, then can I conclude that we are in motion, since otherwise we would have an empirical contradiction (kind of like finding dinosaur and human remains in the same strata)?  Any help on where I am going wrong would be helpful. Intuitively, I don't think there should be a way to establish absolute ordering between space-like separated events, but the above reasoning suggests otherwise.
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I also asked in SO  a few days ago, thought it may be also interesting for physics-related answers.  I would like to model a network as a system.  A particular topology (configuration of edges between vertices) is a state-of-order of the system (a micro-state). I am trying to compute the entropy of a specific topology as a measure of the complexity of information embedded in that topological structure.   I don’t have a degree in physics, I would like to have answers that can help in creating a concept of entropy applied to networks (particularly small-world networks), as systems embedding information in their topology.  Below, I share my reasoning and doubts.   I first thought to make an analogy with Shannon entropy applied to strings: Here entropy is a measure of the randomness of a string as a sum of probability to have certain digits happenings.  Similarly, I then thought that entropy may hold for an Erdős–Rényi random network, and the measure could reflect the randomness of an edge between a pair of vertexes.   Does Shannon entropy hold for non-random types of networks?  As a second approach, I thought that according to Boltzmann’s definition, entropy is the multiplicity of equivalent states.    How could equivalent topologies be modelled (or how to can we compute similarity between two networks)?  How to measure how much a state of order of a particular topology is “uncommon”, with respect to all other possible configurations?  Should I attempt to model a topology as a probability over all possible distributions of edges (complete network)?
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On John Baez's website, , he discusses the problem of how entropy increases when a cloud of ideal gas collapses gravitationally (no black holes - keeping it simple).  But he doesn't give away the answer - instead he implies that, I presume, either gravitons or photons are released during collapse and these increase the overall system entropy.  Can someone please answer which it is, gravitons or photons?  If gravitons (or gravity waves), then does this mean that pre-20th century physicists, with no notion of gravity waves, would have concluded that gravitational attraction violates the 2nd law?  And if photons, then what about if the gas particles were totally neutral - wouldn't they not emit any EM radiation in that case?  Or am I wrong that gravitons or photons are the answer? Thanks!
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Possible Duplicate:          I love to dabble with science, I'm by no means a scolar in this field. One thing that haven't seen proven yet (maybe I missed it) is that there was only 1 singular Big Bang where it all started.   As I understand, when Hubble discovered that the universe was expanding, the conclusion was also that it was much smaller in the past. That I follow, but the theories seem to automatically agree on there being 1 Big Bang where everything started.   Couldn't there have been multiple points where matter etc 'flipped' into existence roughly around the same place and time? Is there any evidence for that? Does all the red shift for instance point to 1 spot?   Thanks for your time and knowledge in advance,  Gert-Jan
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How do you prove the  from statistical mechanics? To prove entropy will only increase with time? How to prove? Please guide.
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So i was reading about  on Wiki and I found a interesting thing written there .    It says that the S. radius of the universe is as big as the size of the universe?   How is this possible?  Since most the universe is empty space shouldn't the S. radius of our universe be significantly smaller then 13.7 light years?
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My question has a silly formulation, but I want to know if there is some sensible physical question buried in it:   Suppose an exact copy of our Universe is made, but where spatial distances and sizes are twice as large relative to ours. Would this universe evolve and function just as ours?   Since mass is proportional to volume which is distance cubed, but strength of a rope is only distance squared, I think not everything would scale proportionally.   Does this mean that a universe with our physical laws that evolves like ours can only have one particular size? Or would the physical laws scale proportionally so that it evolves in the same way? If it has a particular size, what is it relative to? Another variation: suppose another exact copy of our Universe is made, but where everything happened twice as fast relative to ours. Would it evolve in the same way as ours?
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The many-worlds interpretation of quantum mechanics has always been explained to me at a high level using examples of binary events (e.g. atom either did or did not decay at any given moment in time), which leads to a conceptually clean idea of "branching" into two distinct universes. But how does branching work if you have, say, a 70% probability of something happening (e.g. measuring the spin of an electron having gone through an SG apparatus oriented at an arbitrary angle)?  Do you say that 7 universes got to spin up and 3 that got spin down (to account for the 70% probability)? Does that mean those 7 universes are in-every-way identical copies of each other? But then what if you had something with a 71.87154819...% chance of happening? You would need an uncountably infinite number of branches to be able to represent arbitrarily precise probability ratios, and then subsets of those branches would contain uncountably infinite universes that are 100% degenerate and identical to each other. Is this what the standard many worlds interpretation assumes?  If not this, then what? You can't say that there's a 70% chance of universe A happening and a 30% chance of universe B happening if you're saying both happen. How does the many-worlds interpretation put a "weighting function" on different branches or outcomes?
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I am interested in knowing if some one here knows book/notes for statistical mechanics from the information theoretic viewpoint.  Additional Request from user83014  "Jaynes wrote a paper called Information theory and statistical mechanics (1957). I tried to read this but found it somewhat hard to follow, and I suspect it's because it's one of the earlier works, and that a modern text might be more refined since we've had time to work it out. On the other hand, what I've read from physics textbooks don't explicitly talk about information theoretic concepts like shannon entropy.  "Is there a good modern introduction to the information theoretic view of entropy?"
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There are people proposing the possibility of using entropic force to explain the gravity force between objects. The emphasis is that entropy is more fundamental than energy. It is the closest study as entropy is closely related to information. What is Mathematical formulation of ?     ... It implies that gravity is not a fundamental interaction, but an emergent phenomenon which arises from the statistical behavior of microscopic degrees of freedom encoded on a holographic screen...
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Is there a physical reason why not to think that instead of space expanding, all physical constants and parameters are shrinking (including of course the instruments we use to measure the constants) and space is static, or is it a case of Occam's razor?
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It is a simple fact the entropy in the exact meaning in dynamical system does not change over time if the system is measure-preserving and ergodic. However, it is often said that the coarse-grained entropy increases over the time. I was wondering to know is there any rigorous proof for this statement?     The coarse-grained entropy increases over the time.
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Just watched the 'How Big is the Universe?' TV program and it indicated that the universe is believed to be infinite in size. They used an approach involving measuring the internal angles of triangles. Or something like that anyway.  But hang on, the Universe has a known starting time and size followed by a finite time expanding at a finite speed. So surely, by definition, it cannot be infinite? I can accept it is honking big but by definition it cannot be infinite. Are they misleading me?
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I am having trouble in understanding the following concepts :  Pg 231 Appendix B of the link   which is of the book Chaos and the Evolving Universe by Sally J. Goerner mentions that Entropy $S$: $$ S = \ln V $$ where $V$ is the phase space volume.  According to the book, this equation is from the concept of Boltzmann's entropy.   (Q1) How is This equation coming? How can we say that entropy = log of phase space volume? References and explanation would be appreciated. According to the book, the Bolzmann's constant, $k_B$, is taken to be $1$. But actually there is a value to the constant. Can I take the Boltzmann constant to be equal to $1$?  Also, if the entropy increases, does this mean that the volume decreases?  (Q2) Secondly, can Kolmogorov entropy, from Information theory be stated as logarithm of phase space volume that is equivalent to entropy from statistical mechanics? I am unsure if I can replace Boltzman with Kolmogorov Sinai (KS) entropy.   (Q3) What is the difference between Gibb's entropy and Shannon's entropy since the formula  is the same.