[warning: great amount of entropy and little information following. premise: bad english writing. disclaimer: no scientific accuracy. synopsis: just personal considerations, which will probably change in time]

There’s some discussion over the web (here, here, here and here) about thermodynamics and gravitation, a discussion that comes out regularly. I think some takes on the argument are confused, as well pointed out by The Statistical Mechanic. Many among cosmologists and string theory phenomenologists are trying to make sense of expressions such as "growth of entropy", "low entropy states", "loss of information" and fanciful "boltzmann’s brains", "manyworlds", yet knowing little about the basic principles of SM (Statistical Mechanics, not Standard Model!). And it must be said that little is known about TD and SM in General Relativity and for far-from-equilibrium systems, both of which are relevant for cosmology (yes, they both are). I happen to know a little about non-equilibrium, having worked on fluctuation theorems for my graduate thesis, where I strongly focused on interpretational and foundational issues.

Two questions

Let me analyse these two questions:

1) if the Universe was a hot ball around the Big Bang age and now there’s plenty of structures, and having a notion of entropy as a measure of information (actually, ignorance), how can one make sense of the second law? [I would call this the growing information paradox]

2) we think of order as a complex structured state of affairs and of disorder as a simple homogeneous mess. How can it then be that gravitational systems’ equilibrium is the collapsed one, with all matter crushed toghether in a very "ordered" fashion? [I call this the tidy equilibrium paradox]

I wrote some speculative paragraphs for my thesis work about these questions, but I did not included them in the final version due to lack of space. Here an excerpt (in italian).

Simplifying a bit

First of all, for the sake of simplicity some working hypothesis, which are not at all commonly agreed:

A) I strongly believe that the thermodynamics (statistical mechanics) of the Universe and in general the large scale behavior of systems is entirely classical. I also think, but that’s irrelevant, that quantum and classical formalisms and the questions they raise (like the measure problem in QM) are totally equivalent and interchangeable for this scope*.

B) As a conseguence, I think black holes are irrelevant for the discussion, and in general for the topic of global conservation of information. I think there’s too much of an effort to understand their thermodynamics as if it were basilar for understanding the TD of the Universe. My assumption is that their existence does not affect cosmological evolution (they still might contain most of the entropy of the Universe).

C) OK, I sound like a taliban. The fact is that I refer to a universe not much like the physical Universe around us but as a global phase-space containing generic classical physical systems, because I think this makes things clearer and is sufficient to answer Atdotde’s questions. One really doesn’t need to go into the history of the Universe or into the subtleties of the privileged reference frame given by the CMB or cosmological gauge transformations to understand basic TD. I repeat: I do not care for the thermodynamical history of the Universe (with capital U), but concentrate on general TD properties of "universes". Our Universe is just a model of "universes", and I think cosmology oversimplyfies it.

Now some preliminary unnecessary considerations:

A’) In non-equilibrium systems temperature is not well-defined. One could still try to describe the system as locally at equilibrium and define a local temperature, but as subsystems interact and correlate it becomes difficult if not impossible**. In any case, it does not preserve the meaning of average kinetic energy and equipartition does not hold. Also statistical ensembles like micro or macro or grand-canonical do not exist, since we have not yet maximized entropy. But the first and the second law of thermodynamics hold (in GR the first hols locally).

B’) The second law of thermodynamics is of a statistical nature and does not at all depend on the details of the microscopic evolution, as well pointed out (in a much too arrogant way, though) by Motl. But I think Motl left some questions behind.

C’) There are at least two different nonequivalent definitions of entropy. Gibbs’s one is related to information content and stays costant for an isolated system evolving under microscopic deterministic hamiltonian dynamics, by the Liouville theorem (the same for von Neumann’s entropy in QM). Boltzmann’s entropy is a measure of the available state space for a given macro-state and therefore is measured a posteriori. The first never exceeds the second, and the two coincide at equilibrium: this already gives a proof of the second law if one consider a system starting at equilibrium and evolving. The only way to also have Gibb’s entropy grow towards equilibrium is to put the system in contact with an environment. But one can include the environment into the description… and so forth up to the whole universe.

D’) Most importantly: entropy and information are subjective quantities, they depend on what one decides to measure and with what precision. They are defined in an intrinsically coarse-grained state space. In fact Boltzmann entropy depends on the observables one uses to define which is the equivalence relation among microstates that give the same macrostate. Gibbs entropy is defined throught probability distributions, which depend on how one decides to resolve its system (constituents: molecules, atoms, strings… interactions: spins, charges…). Equilibrium distributions from which non-equilibrium ones evolve depend on macroscopic information one has about the system, that is with which Lagrange multipliers one maximizes entropy.

E’) There is no equilibrium in nature, but just local tendency to equilibrium. There is no freely expanding gas in a box, because there cannot be a box. There only is the perfect gas in free space (more about this later…)

The growing information paradox

Now my answers.

A1) It has already been made clear by Motl that the size of the Universe near the Big Bang makes entropy density huge. As the universe expands locally one sees entropy diminish and information come out in structure formation. At "equilibrium" (completely homogeneous freely expanding universe) entropy within a comoving volume stays costant. Expansion explains all. But I think there is still an opaque point, which is the one that upsets Atdotde:

1′) if information is conserved or at most diminishes, how can it be that we have a one parameter system (temperature) at the Big Bang era and marvelous gothic cathedrals now? Where does all the information come from?

A1′) The problem is that information content depends on coarse-graining. Imagine very tight large fluctuations in a small volume not resolved by your definition of entropy.

As volume increases, fluctuations smooth out, and the density distribution will start to resolve tinier fluctuations, so things will start to look much more complex.

These are two competeting mechanisms. You have an initial gain of information from "inside", and then as fluctuations spread you will start loosing it, eventually evolving towads to heat death as information leaks towards the "outside".

It is much like if the microscopic "inside" of a system were part of the environment: in stocastic modelling, noise can both come from outside or from inside, that is noise might be the effect of external perturbations or of some renormalization procedure of internal unknown interactions and degrees of freedom. That’s why one sees information grow initially, reach a maximum and then diminish.(*)

Notice that this reasoning solves a much deeper question:

1′’) is information globally conserved? If so, how do we make sense of local entropy growth?

A1′’) Information can be globally constant, and actually it has to be so if microscopic evolution is classical hamiltonian with no sources of stocastic noise, or quantum deterministic with no intrinsically classical measure apparati. A deterministic isolated system has no means to reach equilibrium***, but one can have local tendency to equilibrium if the state space available increases. That is, subsystems of the universe can each play the role of the environment for other subsystems if there is some loss of entropy at the borders… but by the Copernican Principle there are no borders: there’s expansion. Expansion of the universe explains why microscopic deterministic and reversible laws of motion with global conservation of information (I = Gibbs negetropy) are consistent with the observation of local entropy growth and tendency to equilibrium, it solves Zermelo and Lochschmidt paradoxes once and for all and explains quantum decoherence without appealing to classicla measure apparati. That’s pretty cool.

The tidy equilibrium paradox

As to the second problem

A2) It is simply fallacious to think of order-disorder as of information-entropy. Highest entropy states are not necessarily homogeneous, they simply are the most probable. Take a messy and dirty room as compared to a tidy and clean one, both containing the same amount of "stuff" (including the dirt). If I resolve single grains of dust, I will have the same amount of information in both (not considering fileds, radiation, cosmic rays etc.). If I do not resolve every single detail, for example dust, the messy room will have higher entropy, since there will be a lot more realizations of "dust all around the room" than of  "dust in the rubbish". If I do not even resolve single objects, I will have even higher entropy for the messy room, because there are many more realizations of "dresses and toys all around" than of "dresses in the closets, toys in the boy’s room". Take a gravitational system: it will collapse into I don’t know what kind of unresolved unique final state, whichever the initial conditions: you can map the initial conditions to the final state, but you cannot map back the final state into the initial condition, because of irreversibility and loss of information. The final state is not homogeneous. Is it tidy?(**)

Non-equilibrium cosmology

Now, all of this does not imply or assume that the Universe will reach heat death, with gravitationally collapsed objects totally decorrelated one from another(***), or that the expansion of the Universe is accelerated… this is independent from the above discussion.

But I strongly disagree with one point in Lubos’s post. Of course FRW cosmology models the Universe as being at equilibrium, and most of the stuff in the Universe really is at equilibrium. Yet there are many cosmologists who are starting to think that fluctuations in the matter distribution might be relevant for further comprehension; in particular they might explain accelerated expansion, be it real or only percieved (links will follow). So I think non-equilibrium thermodynamics is very relevant to the subject, and there also really is the need for a complete statistical treatment of GR.

Constructive comments that either help me polish my english or my physics understanding are welcome.

- - -

* Actually, but this would take us too far away, I believe the Cosmological Problem doesn’t need answers from QFT such as new fields, inflatons, string theory vacua… they might work for the modelling, but the underlying physics should be classical.

** In a brownian motion at the diffusive regime (= equilibrium, in a sense… I will write about this and the perfect gas in a box), average kinetic energy is given by Einstein relation

temperature = diffusion^2 / drift

where the diffusion coefficient is the sensibility of the system to the environment and drift is a damping action that allows reaching a steady state. The higher the temperature, the faster the diffusion. The higher the damping, the lower the temperature needed reach the diffusive state. That makes sense.

One can use this as a definition of temperature far from equilibrium, and generalize to the case where drift and diffusion depend on the phase-space microstate but are isotropic and diagonal, that is no correlations are estabilished between perpendicular components of the velocities. This is a kind of "local equipartition". In any case, temperature is no more an average kinetic energy, and similar mechanisms should happen in the deterministic description with interactions (as well explained by TSM). I’ll try to develop this ideas.

*** I will write about this sometime. The free-particle gas in a box cannot exist, because physically the box cannot be isolated and because mathematically the box is too singular an object. The only system that tends to some kind of equilibrium is the expanding gas in free space. Much like our Universe.

(*) Actually this reasoning is classical, not relativistic: in a curved spacetime it is not clear to me what "space" and "volume" exactly are when thinking of measurement. In GR things might be quite a bit complicated, and I think that here lies the key to understanding the Cosmological Problem (as of jan. 12th, 2009, at a too much young an age: forgive me tomate the senior!). The thing is: your measure apparatus might be expanding too… at the BB you could have been a little infinitesimal piece of energy with your tiny ruler measuring very tight fluctuations. And moreover closeby matter is virialized… so things are difficult… I’m trying to make some thinking about it… but this doesn’t change our reasoning so far, at worst it strenghtens it.

(**) Another good example is the saucepan on the stove. If there is no information poured into the system, that is no temperature gradient (heat off), than primitive chaos will reign. If a critical temperature gradient is reached ordered convection cells will form. But if too much information is poured, we will have postmodern chaotic motions, which we do not know how to resolve. That is pretty much like the (reversed) history of the universe, with the Middle Age having the more constructive role.

(***) Would it make sense a model where single collapsed objects re-BigBang independently one from another?