As we've said, science is not a fixed body of 'facts', and there are disagreements. The relation between Shannon's entropy and thermodynamic entropy is one of them. Whether it is meaningful to view thermodynamic entropy as negative information has been a controversial issue for many years. The scientific disagreements rumble on, even today, and published, peer-reviewed papers by competent scientists flatly contradict each other.
What seems to have happened here is a confusion between a formal mathematical setting in which 'laws' of information and entropy can be stated, a series of physical intuitions about heuristic interpretations of those concepts, and a failure to understand the role of context. Much is made of the resemblance between the formulas for entropy in information theory and thermodynamics, but little attention is paid to the context in which those formulas apply. This habit has led to some very sloppy thinking about some important issues in physics.
One important difference is that in thermodynamics, entropy is a quantity associated with a state of the gas, whereas in information theory it is defined for an information source: a system that generates entire collections of states ('messages'). Roughly speaking, a source is a phase space for successive bits of a message, and a message is a trajectory, a path, in that phase space. In contrast, a thermodynamic configuration of molecules is a point in phase space. A specific configuration of gas molecules has a thermodynamic entropy, but a specific message does not have a Shannon entropy. This fact alone should serve as a warning. And even in information theory, the information 'in' a message is not negative information-theoretic entropy. Indeed the entropy of the source remains unchanged, no matter how many messages it generates.
There is another puzzle associated with entropy in our universe. Astronomical observations do not fit well with the Second Law. On cosmological scales, our universe seems to have become more complex with the passage of time, not less complex. The matter in the universe started out in the Big Bang with a very smooth distribution, and has become more and more clumpy -more and more complex -with the passage of time. The entropy of the universe seems to have decreased considerably, not increased. Matter is now segregated on a huge range of scales: into rocks, asteroids, planets, stars, galaxies, galactic clusters, galactic superclusters and so on. Using the same metaphor as in thermodynamics, the distribution of matter in the universe seems to be becoming increasingly ordered. This is puzzling since the Second Law tells us that a thermodynamic system should become increasingly disordered.
The cause of this clumping seems to be well established: it is gravity. A second time-reversibility paradox now rears its head. Einstein's field equations for gravitational systems are time- reversible. This means that if any solution of Einstein's field equations is time-reversed, it becomes an equally valid solution. Our own universe, run backwards in this manner, becomes a gravitational system that gets less and less clumpy as time passes -so getting less clumpy is just as valid, physically, as getting more clumpy. Our universe, though, does only one of these things: more clumpy.
Paul Davies's view here is that 'as with all arrows of time, there is a puzzle about where the asymmetry comes in ... The asymmetry must therefore be traced to initial conditions'. What he means here is that even with time-reversible laws, you can get different behaviour by starting the system in a different way. If you start with an egg and stir it with a fork, then it scrambles. If you start with the scrambled egg, and very very carefully give each tiny particle of egg exactly the right push along precisely the opposite trajectory, then it will unscramble. The difference lies entirely in the initial state, not in the laws. Notice that 'stir with a fork' is a very general kind of initial condition: lots of different ways to stir will scramble the egg. In contrast, the initial condition for unscrambling an egg is extremely delicate and special.
In a way this is an attractive option. Our clumping universe is like an unscrambling egg: its increasing complexity is a consequence of very special initial conditions. Most 'ordinary' initial conditions would lead to a universe that isn't clumped -just as any reasonable kind of stirring leads to a scrambled egg. And observations strongly suggest that the universe's initial conditions at the time of the Big Bang were extremely smooth, whereas any 'ordinary' state of a gravitational system presumably should be clumped. So, in agreement with the suggestion just outlined, it seems that the initial conditions of the universe must have been very special -an attractive proposition for those who believe that our universe is highly unusual, and ditto for our place within it.
From the Second Law to God in one easy step.
Roger Penrose has even quantified how special this initial state is, by comparing the thermodynamic entropy of the initial state with that of a hypothetical but plausible final state in which the universe has become a system of Black Holes. This final state shows an extreme degree of dumpiness - though not the ultimate degree, which would be a single giant Black Hole.
The result is that the entropy of the initial state is about l030 times that of the hypothetical final state, making it extremely special. So special, in fact, that Penrose was led to introduce a new time-asymmetric law that forces the early universe to be exceptionally smooth.
Oh, how our stories mislead us ... There is another, much more reasonable, explanation. The key point is simple: gravitation is very different from thermodynamics. In a gas of buzzing molecules, the uniform state -equal density everywhere -is stable. Confine all the gas into one small part of a room, let it go, and within a split second it's back to a uniform state. Gravity is exactly the opposite: uniform systems of gravitating bodies are unstable. Differences smaller than any specific level of coarse-graining not only can 'bubble up' into macroscopic differences as time passes, but do.
Here lies the big difference between gravity and thermodynamics. The thermodynamic model that best fits our universe is one in which differences dissipate by disappearing below the level of coarse-graining as time marches forwards. The gravitic model that best fits our universe is one in which differences amplify by bubbling up from below the level of coarse-graining as time marches forwards. The relation of these two scientific domains to coarse-graining is exactly opposite when the same arrow of time is used for both.
We can now give a completely different, and far more reasonable, explanation for the 'entropy gap' between the early and late universes, as observed by Penrose and credited by him to astonishingly unlikely initial conditions. It is actually an artefact of coarse-graining.
Gravitational clumping bubbles up from a level of coarse-graining to which thermodynamic entropy is, by definition, insensitive. Therefore virtually any initial distribution of matter in the universe would lead to clumping. There's no need for something extraordinarily special.
The physical differences between gravitating systems and thermodynamic ones are straightforward: gravity is a long-range attractive force, whereas elastic collisions are short-range and repulsive. With such different force laws, it is hardly surprising that the behaviour should be so different. As an extreme case, imagine systems where gravity' is so short range that it has no effect unless particles collide, but then they stick together forever. Increasing dumpiness is obvious for such a force law.