And some of the children even know that the freezing air has something to do with a slower chaotic thermal motion of the atoms.

However, the combination of thermodynamics and gravity seems to be a mysterious lake filled with dragons where most laymen - and those physicists who have never really mastered these two disciplines of physics - drown almost instantly. They think that the combination of the two frameworks - gravity and thermodynamics - is completely confusing, paradoxical, or even inconsistent. And they tend to answer 90+ percent of the questions incorrectly - even though 2% would already be pretty bad for a real scientist.

Robert Helling was kind enough to summarize almost all of this confusion. He was even modest enough to present the confusion as his own which is pretty embarrassing but it is not too bad if you realize that the same confusions are shared by pretty much all physics fans in the blogosphere.

Clifford Johnson replied that thermodynamics and gravity fit together because we have many working examples in string theory. Well, I obviously agree with Clifford.

However, string theory is only good as a very accurate microscopic description of these physical systems: it is good as a full formulation of the "statistical physics" describing systems that gravitate (besides doing many other things). In this sense, string theory is a set of detailed laws to be inserted into detailed, accurate formulae of statistical physics.

On the other hand, thermodynamics is defined as a more rough, approximate, macroscopic description of such systems with many degrees of freedom. That's why Robert's confusion is clearly much more elementary and comments about string theory seem like a thermonuclear bomb prepared to kill a mosquito.

So instead of going into details of the string theoretical description, I will confine my attention to the thermodynamic comments about the situation. Let us follow Robert's text and react to various remarks that Robert makes:

The main puzzle I would like to understand is the question regarding the entropy balance of the universe: According to the second law of thermodynamics, entropy is never decreasing. I hope this is the correct sign, I can never remember it.Well, we had at least a two-semester course - hundreds of hours - dedicated to statistical physics and thermodynamics in the college. I believe that these are important disciplines for any physicist (or another natural scientist!) who actually wants to understand the real world. Not knowing that the entropy increases rather than decreases is a pretty bad result.

In the blogosphere that often tries to attract uneducated, ignorant, but very self-confident personalities (if I have to avoid the word "idiots"), it may look hot and sexy not to know in which way the entropy goes but as a college student in a serious college, one should have been dismissed years ago.

So if it is increasing, it should have been minimal at the big bang which seems to be at conflict with the universe being a hot soup of all kinds of fluctuations right after the big bang.The entropy is always increasing - or, more precisely, "almost" never decreasing by "macroscopic" amounts. This is an absolutely general principle of thermodynamics that can be equally generally derived from statistical reasoning, regardless of any details of the dynamics.

By the very definition of the past, the initial state(s) serve(s) as the initial condition for a subsequent evolution, and the "envelope" (including the macroscopically indistinguishable states) of the possible state(s) that evolve from the initial state(s) inevitably has a greater volume in the phase space and/or a higher number of microstates (in the quantum mechanics) than the initial state(s). We will discuss some special features of gravity later but the second law of thermodynamics is surely not open to a debate and it is not supposed to be attacked or questioned whenever we add a new term to an equation. It is a principle that holds for any system with many degrees of freedom that does or could exist in Nature.

Concerning the last blue sentence above, in a sharp contrast with Robert's proclamations, there exists no conflict between the high temperature of the early Universe and its low entropy. The temperature and the entropy are two different quantities even though the laymen tend to think that a discipline as unimportant as thermodynamics should include at most one physical quantity. ;-) While in many ordinary systems they tend to increase or decrease together, there is no universal proportionality or other law linking these two different quantities.

Robert's comment is equivalent to the proposition that every function of the form "g(x)/x" must diverge near "x=0". Well, it doesn't. Try e.g. "g(x)=1-cos(x)" to see that such a ratio can even go to zero at "x=0". We can be very quantitative about the behavior of the entropy and temperature for any system we like. And of course, we can always see details why Robert's guess is wrong. But he apparently doesn't want to see it.

For example, the majority of the entropy of the Universe today (over a googol) is stored in large black holes that are mostly located at the galactic centers (like our own). Because these black holes still grow much faster than they evaporate (and sometimes they merge), their combined entropy keeps on increasing.

Before people knew about these large black holes, they thought that most of the entropy was carried by the microwave background (10^{90} or so), the CMB. Well, by dimensional analysis, the entropy density goes like "T^3" (thanks, McGuigan, for erasing the wrong inverse!) where "T" is the temperature. A fixed spatial region in the FRW coordinates therefore carries a CMB entropy independent of the time - because the temperature goes like "1/a" where "a" is the conformal factor that stretches the spatial distances. This constancy is characteristic for states near the equilibrium.

However, there were other eras before the radiation-dominated era. In most of them, the entropy was strictly increasing because "things were happening". Whenever "things are happening", rather than being stuck in some equilibrium, the entropy is strictly increasing. I don't want to be describing dozens of dynamical systems and their entropy growth rates but be sure that in every process in Nature, the entropy is macroscopically non-decreasing, as guaranteed by the completely universal statistical argument.

Robert's comment about a "conflict" is a stinky bullshit, to put it very mildly.

Concerning the last blue sentence above, in a sharp contrast with Robert's proclamations, there exists no conflict between the high temperature of the early Universe and its low entropy. The temperature and the entropy are two different quantities even though the laymen tend to think that a discipline as unimportant as thermodynamics should include at most one physical quantity. ;-) While in many ordinary systems they tend to increase or decrease together, there is no universal proportionality or other law linking these two different quantities.

Robert's comment is equivalent to the proposition that every function of the form "g(x)/x" must diverge near "x=0". Well, it doesn't. Try e.g. "g(x)=1-cos(x)" to see that such a ratio can even go to zero at "x=0". We can be very quantitative about the behavior of the entropy and temperature for any system we like. And of course, we can always see details why Robert's guess is wrong. But he apparently doesn't want to see it.

For example, the majority of the entropy of the Universe today (over a googol) is stored in large black holes that are mostly located at the galactic centers (like our own). Because these black holes still grow much faster than they evaporate (and sometimes they merge), their combined entropy keeps on increasing.

Before people knew about these large black holes, they thought that most of the entropy was carried by the microwave background (10^{90} or so), the CMB. Well, by dimensional analysis, the entropy density goes like "T^3" (thanks, McGuigan, for erasing the wrong inverse!) where "T" is the temperature. A fixed spatial region in the FRW coordinates therefore carries a CMB entropy independent of the time - because the temperature goes like "1/a" where "a" is the conformal factor that stretches the spatial distances. This constancy is characteristic for states near the equilibrium.

However, there were other eras before the radiation-dominated era. In most of them, the entropy was strictly increasing because "things were happening". Whenever "things are happening", rather than being stuck in some equilibrium, the entropy is strictly increasing. I don't want to be describing dozens of dynamical systems and their entropy growth rates but be sure that in every process in Nature, the entropy is macroscopically non-decreasing, as guaranteed by the completely universal statistical argument.

Robert's comment about a "conflict" is a stinky bullshit, to put it very mildly.

With the popular science interpretation of entropy as a measure of disorder or negative information the early universe must have been highly ordered and should have contained maximal information, a notion which is highly counter intuitive. So this needs some clearing up.The total entropy was lower in the past - a completely general fact of thermodynamics - and there are good reasons to assume that the entropy was strictly zero at the very beginning. And the concept of the entropy as a degree of disorder is not just a popular science interpretation: it is much more accurate and correct science than anything that Robert has written in his article.

A low entropy in the past doesn't imply that the entropy density - the density of disorder per cubed meter - was small. Because the region corresponding to the currently visible Universe was very tiny, the entropy density was actually higher than today. But the entropy density is allowed to increase or decrease, depending on the context. Only the total entropy is obliged to increase at all times.

In a sharp contradiction with kilotons of bullshit printed on the blogosphere, there is absolutely nothing counter-intuitive about the entropy being lower in the past (especially not during the Big Bang which took place a long time ago) and about the past configurations' being more organized as measured by their total entropy. On the contrary, it is completely intuitive because every closed system in the world had a lower entropy in the past: this feature is essentially a defining property of the entropy.

In a sharp contradiction with kilotons of bullshit printed on the blogosphere, there is absolutely nothing counter-intuitive about the entropy being lower in the past (especially not during the Big Bang which took place a long time ago) and about the past configurations' being more organized as measured by their total entropy. On the contrary, it is completely intuitive because every closed system in the world had a lower entropy in the past: this feature is essentially a defining property of the entropy.

You shouldn't be using the word "entropy" if you don't understand this feature. Every sensible text about the entropy gets to its increasing character in the first five sentences and whoever doesn't know basic things from the first 5 sentences about a concept doesn't know the concept and should avoid the word unless he is a pseudointellectual trying to look smarter than he is.

If your intuition has a problem with this basic general fact, you should realize that your personal intuition is a worthless and useless piece of junk. Much like a burned, environmentally friendly microprocessor, you should splash it into the toilet, learn proper high school physics and college physics instead, and thank me for this extremely kind, wise, and important advice.

The simplest resolution would be that it is compatible with observation to assume that the universe has infinite volume and if it has a finite entropy density the entropy is infinite and any discussion of increasing or decreasing entropy is meaningless as it will be infinite at any time and it does not make sense to talk about more or less infinite entropy.That's nonsense, too. The second law of thermodynamics is a non-vacuous, important principle controlling the real world and it never degenerates into a meaningless "infinity minus infinity" indefinite form or a tautology. In fact, the total entropy is not infinite. The total entropy of the visible Universe is close to one googol. The second law of thermodynamics strictly applies to closed systems only but huge regions of space - with diameters counted in billions of light years - are almost closed and the second law applies to them with a huge accuracy.

Of course, one must evolve a "region of space" according to a natural "cosmic reference frame" (like the frame linked to the microwave background) so that we are comparing the corresponding regions at two different moments. This choice of the "reference frame" can be justified by the attempt to make the regions as similar to closed systems as possible. We don't want too much entropy crossing the boundaries of the region in either direction which is why we want the boundaries of the region to be aligned with the world lines of the typical entropy carriers (the microwave background photons).

However, cosmology is the very worst example of an out-of-equilibrium system that Robert could have chosen. More than 99.9999% of the Universe is filled with stuff that is so close to thermal equilibrium that the radiation that these regions send to our telescopes - the microwave background - is the most accurate black body thermal radiation that we have ever observed in Nature.

Indeed, stars and mammals have temperatures that differ from those 2.725 Kelvin degrees of the cosmic microwave background. But stars and mammals are not studied by cosmology. They are studied by astrophysics and zoology, respectively. Cosmology is the idealized description where these special local phenomena - life and burning stars - are ignored. That's why Robert's claim is the very opposite of the truth. Instead of dissing thermodynamics or cosmology, he should diss his idiosyncratic opinions.

More generally, every collection of degrees of freedom that interact with others in the same set inevitably undergoes thermalization. It means that after a finite time, the thermalization time, the degrees of freedom reach a thermal equilibrium and the entropy reaches the maximum (given by the assumption that additional, hidden degrees of freedom are not excited). Whenever things are out of equilibrium, the total entropy has to be strictly increasing. Robert has psychological problems with both concepts - the strict universal entropy increase and thermal equilibrium. However, one of them always takes place in any physical system, by arguments that don't depend on any dynamical details like having or not having gravity.

If you want to study different "aspects" of cosmology or quantum gravity than the microwave background and its temperature, the last two sentences of the previous paragraph will continue to hold. For example, the interactions of baryonic matter with the thermal de Sitter radiation is very slow and allows them to have different temperatures for long time intervals. The entropy is inevitably increasing by occasional interactions of the de Sitter photons and the matter. However, in a very far future, a new thermal equilibrium will be reached. The de Sitter space will be essentially empty and it will be filled with the extremely cold radiation emanating from the cosmic horizon only.

As I have mentioned, most of the entropy of the Universe today is carried by the large black holes at galactic centers. That term is followed by the CMB entropy. However, we can also talk about the much higher entropy of the cosmic horizon, 10^{120} or so. It's not a coincidence that the figure is close to the inverse cosmological constant in the Planck units: in these units, the entropy is the area horizon and the cosmic horizon has area that is the inverse cosmological constant (radius^2=1/Lambda).

But I don't think this [discussion of densities and volumes] is the real problem. I am much more worried about another point: I am not convinced it makes sense to apply thermodynamic reasoning to situations that involve gravity!By looking at your question mark, you seem to be pretty proud about your rudimentary ignorance, Robert. General arguments of thermodynamics and statistical physics are absolutely general. The "equations of state" and relationships between the energy, temperature, and entropy depend on the system. But certain general principles such as the second law don't. There is nothing divine about gravity that could invalidate some basic principles of thermodynamics. Gravity is just another force, just another term in some equations. Thermodynamics was designed with all these terms in mind.

Obviously, the universe as we see it is not in thermal equilibrium, all the interesting stuff we see are local fluctuations. So standard textbook equilibrium thermodynamics does not apply.These sentences represent such a complete collapse of a physics judgment that I can't believe that Robert wrote them seriously. Academically speaking, every physical system in reality is out of the thermal equilibrium because objects never quite match their idealized descriptions blah blah blah. (This ideological, unscientific bullshitting has literally become politically correct. Tens of millions of dopes want to hear that science can never calculate anything so that they don't have to learn it. Tons of scum like Woit and Smolin have literally made careers out of repeating this populist, moral lie. While the constantly repeated statements about the imperfection of science may be true if interpreted literally, they are often morally false, and very importantly untrue.)

However, cosmology is the very worst example of an out-of-equilibrium system that Robert could have chosen. More than 99.9999% of the Universe is filled with stuff that is so close to thermal equilibrium that the radiation that these regions send to our telescopes - the microwave background - is the most accurate black body thermal radiation that we have ever observed in Nature.

Indeed, stars and mammals have temperatures that differ from those 2.725 Kelvin degrees of the cosmic microwave background. But stars and mammals are not studied by cosmology. They are studied by astrophysics and zoology, respectively. Cosmology is the idealized description where these special local phenomena - life and burning stars - are ignored. That's why Robert's claim is the very opposite of the truth. Instead of dissing thermodynamics or cosmology, he should diss his idiosyncratic opinions.

Remember for example, temperature is a property of an equilibrium, the fact it is well defined is sometimes called the zeroth law and out of equilibrium situations do not have a temperature!Well, Robert could have figured out that there had to be something badly wrong with his opinions because the temperature 2.725 K seems to exist and is pretty accurate.

More generally, every collection of degrees of freedom that interact with others in the same set inevitably undergoes thermalization. It means that after a finite time, the thermalization time, the degrees of freedom reach a thermal equilibrium and the entropy reaches the maximum (given by the assumption that additional, hidden degrees of freedom are not excited). Whenever things are out of equilibrium, the total entropy has to be strictly increasing. Robert has psychological problems with both concepts - the strict universal entropy increase and thermal equilibrium. However, one of them always takes place in any physical system, by arguments that don't depend on any dynamical details like having or not having gravity.

If you want to study different "aspects" of cosmology or quantum gravity than the microwave background and its temperature, the last two sentences of the previous paragraph will continue to hold. For example, the interactions of baryonic matter with the thermal de Sitter radiation is very slow and allows them to have different temperatures for long time intervals. The entropy is inevitably increasing by occasional interactions of the de Sitter photons and the matter. However, in a very far future, a new thermal equilibrium will be reached. The de Sitter space will be essentially empty and it will be filled with the extremely cold radiation emanating from the cosmic horizon only.

As I have mentioned, most of the entropy of the Universe today is carried by the large black holes at galactic centers. That term is followed by the CMB entropy. However, we can also talk about the much higher entropy of the cosmic horizon, 10^{120} or so. It's not a coincidence that the figure is close to the inverse cosmological constant in the Planck units: in these units, the entropy is the area horizon and the cosmic horizon has area that is the inverse cosmological constant (radius^2=1/Lambda).

Again, we will encounter no paradoxes, conflicts, or violations of the second law. The entropy of the horizon may be thought of as the maximum entropy of everything that can exist behind the cosmic horizon. The situation is equivalent to the case of a black hole - however, the "interior of a black hole" is replaced by the "exterior of our visible Universe". Write down the two geometries in static coordinates and you should see the analogy very clearly. The qualitatively new thing is that the entropy of the "outer space" behind the cosmic horizon has to be bounded, much like the entropy inside a black hole horizon. But it is still a term in the entropy: it never decreases and it counts the logarithm of the number of some microstates.

Robert continues:

The question is thus When, not If. However, depending on the dynamical system, it can take very different times - and different events may occur - before the equilibrium is reached. For example, a Universe with no cosmological constant would be filled with thermal microwave radiation whose temperature would be approaching "T=0". Because we know that the cosmological constant is positive, the temperature will actually be dropping to the Hubble temperature only which is pretty low, anyway. ;-)

So these long-term predictions about equilibrium are completely universal. What's wrong are Robert's assumptions about the linearization. Deviations from an attractor decay exponentially only if you're very close to the attractor so that a linearized description of its vicinity, as a tangent space, is a good approximation. Generally, the Universe (with all its key effects) clearly doesn't satisfy this assumption. There are many "phases" and "types of objects" that co-exist and one can jump from one to the other by tunneling, phase transitions, and so on.

The crossover from the matter-dominated era to the radiation-dominated era or the later cosmological constant-dominated era are just two examples of such "qualitative" changes, and with all these changes, the naive linearized description with a universal exponential decay, with a rate derived from a particular "phase", breaks down.

Gases in empty space tend to evolve into states of uniform density and these states happen to have a higher entropy. But all sensible people, including children, know that even mundane systems such as gases in external fields already behave differently. For example, the Earth's atmosphere has a different density (and pressure) at different altitudes. It's completely natural and the air surely doesn't want to become equally dense at different altitudes. The only universal rule is that the entropy is increasing. But how such an evolution looks like certainly depends on the details of dynamics.

Robert continues:

But things are even worse: The usual systems that we are used to describe thermodynamically (steam engines, containers of gas etc) have the property that the equilibrium is an attractor of the dynamics: All kinds of small, local perturbations diffuse away exponentially fast.The equilibrium is always the ultimate attractor of any dynamics expressed by an approximate, compact, classical configuration space. Why? The entropy has to increase and (assuming no fine-tuning) there is only one point in the configuration space where the entropy is maximized.

The question is thus When, not If. However, depending on the dynamical system, it can take very different times - and different events may occur - before the equilibrium is reached. For example, a Universe with no cosmological constant would be filled with thermal microwave radiation whose temperature would be approaching "T=0". Because we know that the cosmological constant is positive, the temperature will actually be dropping to the Hubble temperature only which is pretty low, anyway. ;-)

So these long-term predictions about equilibrium are completely universal. What's wrong are Robert's assumptions about the linearization. Deviations from an attractor decay exponentially only if you're very close to the attractor so that a linearized description of its vicinity, as a tangent space, is a good approximation. Generally, the Universe (with all its key effects) clearly doesn't satisfy this assumption. There are many "phases" and "types of objects" that co-exist and one can jump from one to the other by tunneling, phase transitions, and so on.

The crossover from the matter-dominated era to the radiation-dominated era or the later cosmological constant-dominated era are just two examples of such "qualitative" changes, and with all these changes, the naive linearized description with a universal exponential decay, with a rate derived from a particular "phase", breaks down.

This is in line with our intuitive understanding of the second law: The homogeneous state is the one with the highest entropy and thus the diffusion is governed by the second law.The second law doesn't say that things converge towards a spatially uniform configuration. The second law says that the physical systems evolve towards states of a greater entropy (a measure of disorder). It's a completely different thing than uniformity. What Robert presents as "our intuitive understanding of the second law" is actually completely orthogonal to the second law. It has nothing to do with it in general.

Gases in empty space tend to evolve into states of uniform density and these states happen to have a higher entropy. But all sensible people, including children, know that even mundane systems such as gases in external fields already behave differently. For example, the Earth's atmosphere has a different density (and pressure) at different altitudes. It's completely natural and the air surely doesn't want to become equally dense at different altitudes. The only universal rule is that the entropy is increasing. But how such an evolution looks like certainly depends on the details of dynamics.

For example, in the case of the atmosphere, molecules want to be closer to the surface because there are many more velocity states (or a larger volume in the momentum space) near the surface (in the position space) that the particles can occupy without (hugely) exceeding the allowed (or typical or average) total (kinetic plus potential) energy per particle (dictated by the overall energy conservation). Thermodynamics hasn't changed or collapsed in this exercise. The only thing that has changed was the formula for the energy i.e. the Hamiltonian: and the formula for the Hamiltonian is what defines the "dynamics" (detailed time evolution). But the general principles and methods of thermodynamics are more universal than a particular Hamiltonian.

Yes, gravity tends to clump things. They evolve into "compact" configurations. But this fact is in no tension with the second law of thermodynamics whatsoever. The only reason why Robert thinks that there is a tension is that his thinking is sloppy - perhaps deliberately sloppy. The existence of the attractive gravitational force is not only consistent with the increasing entropy: one makes another more natural in this context. In fact, the maximally clumped configurations of matter tend to maximize the entropy - they're the black holes that carry the maximum entropy that can be squeezed into the same volume.

Whenever attractive gravity dominates, higher entropy becomes associated with non-uniform matter distributions, and these two descriptions therefore apply to a typical state of matter in the future. The previous sentence may sound counter-intuitive to someone but if it is so, it proves that his or her intuition is failing completely (because every sensible person knows that gravity naturally clumps things and future configurations naturally have a higher entropy so clumped objects that will actually be created or evolved in the future must have a higher entropy). The emotions show nothing wrong about cosmology, thermodynamics, or their union. Science is not about the emotions or intuition or uneducated, slow people with bad scientific intuition.

This is not the case anymore as soon as gravity is the dominating force: What is different here is that gravity is always attractive.While gravity is enough to destroy Robert's oversimplified description, as he correctly writes, it is not really necessary. His description of the future as a uniform gas is only correct if the external fields always get eliminated. For gases in any external fields, the naive picture already breaks down. But we didn't need the force to be "universally attractive": an attractive force mediated by a scalar field combined with some repulsive forces could be enough, too. And we didn't need the force to obey the equivalence principle.

Yes, gravity tends to clump things. They evolve into "compact" configurations. But this fact is in no tension with the second law of thermodynamics whatsoever. The only reason why Robert thinks that there is a tension is that his thinking is sloppy - perhaps deliberately sloppy. The existence of the attractive gravitational force is not only consistent with the increasing entropy: one makes another more natural in this context. In fact, the maximally clumped configurations of matter tend to maximize the entropy - they're the black holes that carry the maximum entropy that can be squeezed into the same volume.

Whenever attractive gravity dominates, higher entropy becomes associated with non-uniform matter distributions, and these two descriptions therefore apply to a typical state of matter in the future. The previous sentence may sound counter-intuitive to someone but if it is so, it proves that his or her intuition is failing completely (because every sensible person knows that gravity naturally clumps things and future configurations naturally have a higher entropy so clumped objects that will actually be created or evolved in the future must have a higher entropy). The emotions show nothing wrong about cosmology, thermodynamics, or their union. Science is not about the emotions or intuition or uneducated, slow people with bad scientific intuition.

Although it may sound unpopular, there are many fewer people than 6 billion in the world who can actually use their intuition to answer both deep and elementary physics questions, and Robert is unfortunately not in the lucky group. Well, I know that people prefer to hear that everyone should build on their intuition because everyone, including mediocre crackpots of Garrett Lisi's caliber, are new Einsteins. Well, people like to hear it but it is a lie. There are only 100-200 intuitive Einstein equivalents in the world and none of them thinks that Lee Smolin or Garrett Lisi is a good physicist. Everyone else - about 6.5 billion people - should mechanically learn the important insights found by the smarter people.

The ultimate cosmic equilibrium is actually also stable - and as uniform as in the non-gravitational systems: things may collapse into black holes but the black holes ultimately evaporate, leaving a uniform "microwave" behind background (whose temperature is going to drop to the de Sitter temperature). It is actually no coincidence that black holes must evaporate, because of thermodynamical reasons.

At any rate, the processes that are important in cosmology are stable and the only "unstable" processes are the short-lived periods that interpolate between them (like a collapse of a star). These periods see a fast, strict increase of the entropy.

What is true is that there are many kinds of events in the Universe and they cannot be described by a single ordinary linearized "stable" differential equation. But cosmology is surely not the first context where this observation can be made. There is absolutely no conflict between the existence of many phases of the Universe, multiple cosmological periods, and transitions in between them on one side, and the second law of thermodynamics on the other side.

The second law of thermodynamics is in conflict with naive ideas of people who don't understand science well. But that's not a specific feature of the second law of thermodynamics. Every insight in science is in conflict with he naive intuition of the ignorant people. I don't think that the importance of Robert's observation goes beyond this trivial remark.

But this means the equilibrium is unstable.Not really. Instead, it shows that the whole linearized picture of Robert is not enough to cover all the relevant phenomena. Robert uses the word "unstable" in a very ill-defined way because he doesn't say what quantities are allowed to be varied and which phenomena are included. In fact, all the important phenomena reveal the system as stable. For example, black holes are stable in the sense that they have no negative modes (like those of the long cosmic strings). The damped, quasinormal, ringing modes (of positive imaginary part of the frequency) guarantee that any deviation from the perfect black hole shape is decreasing exponentially: the half-time is usually comparable to the black hole radius (over the speed of light).

The ultimate cosmic equilibrium is actually also stable - and as uniform as in the non-gravitational systems: things may collapse into black holes but the black holes ultimately evaporate, leaving a uniform "microwave" behind background (whose temperature is going to drop to the de Sitter temperature). It is actually no coincidence that black holes must evaporate, because of thermodynamical reasons.

At any rate, the processes that are important in cosmology are stable and the only "unstable" processes are the short-lived periods that interpolate between them (like a collapse of a star). These periods see a fast, strict increase of the entropy.

But this means the equilibrium is unstable. This is at least in conflict with the naive understanding of the second law above.

The second law surely doesn't require all equilibria to be stable, so unstable equilibria have nothing to do with the second law. They have nothing to do with the previous discussion either.

The second law of thermodynamics is in conflict with naive ideas of people who don't understand science well. But that's not a specific feature of the second law of thermodynamics. Every insight in science is in conflict with he naive intuition of the ignorant people. I don't think that the importance of Robert's observation goes beyond this trivial remark.

Thermodynamics holds and is important for all these macroscopic processes, whether or not they occur in equilibrium and whether or not they damp certain perturbations exponentially. Thermodynamics actually likes and studies these things and especially the subtleties caused by the unusual signs: it surely doesn't prohibit them.

Some particular conclusions about physical systems - like the existence of long-lived, stable phases - may only hold for systems that satisfy various ("convex") inequalities but thermodynamics as a method to study and describe large physical systems is always valid and some of its principles, such as the second law, are completely universal. Again, thermodynamics is exactly the discipline that likes to ask what happens if some thermodynamical potentials fail to be convex and what other qualitative effects follow from this assumption. These are no taboos (or violations of "axioms"): these are the very features and the

Thermodynamics as a methodology and as an effective description is beautifully valid in the case of black holes in quantum gravity, too. The only thing that is invalid are wrong propositions such as those written by Robert. The idea that there is anything wrong with thermodynamics whenever gravity is nonzero is a breathtaking stupidity and Robert should return his college degree if he means is seriously because the comment is equivalent to the belief that science cannot be used for the macroscopic systems in the real world.

Some deeper inspection reveals that when you axiomatise thermodynamics you usually make some assumption on convexity (or concavity, depending on whether you use intensive or extensive variables of state) of your favorite thermodynamic potential (free energy etc). IIRC this is something you impose. Your system has to fulfill this property in order to be described by thermodynamics. And it seems that gravity does not have this property (the stability) and it quite possible (if I am not mistaken, sitting here in a train without any books or internet access) thermodynamic arguments do not apply to gravity.Again, complete rubbish. Thermodynamics is an effective description that exists for an arbitrary excited system with many degrees of freedom.

Some particular conclusions about physical systems - like the existence of long-lived, stable phases - may only hold for systems that satisfy various ("convex") inequalities but thermodynamics as a method to study and describe large physical systems is always valid and some of its principles, such as the second law, are completely universal. Again, thermodynamics is exactly the discipline that likes to ask what happens if some thermodynamical potentials fail to be convex and what other qualitative effects follow from this assumption. These are no taboos (or violations of "axioms"): these are the very features and the

*raison d'être*of thermodynamics.Thermodynamics as a methodology and as an effective description is beautifully valid in the case of black holes in quantum gravity, too. The only thing that is invalid are wrong propositions such as those written by Robert. The idea that there is anything wrong with thermodynamics whenever gravity is nonzero is a breathtaking stupidity and Robert should return his college degree if he means is seriously because the comment is equivalent to the belief that science cannot be used for the macroscopic systems in the real world.

Note well that I am talking classically (actually even only about the weak field situation in which the fluctuations are well described by Newtonian gravity), I have not even mentioned black holes and their negative heat capacity due to Hawking radiation which should make you even more uneasy about thermodynamic stability.Larger black holes are heavier and colder, so it is not shocking that the heat capacity (energy/mass per unit temperature) is negative. But that's not a "problem" of thermodynamics. This sign is one of the things whose consequences is studied by thermodynamics: again, it is one of the "features". Whether some ignorant people feel uneasy is completely inconsequential. Robert's problem is that he only knows - vaguely, with uncertain signs - the thermodynamics of linearized stable systems that are equivalent to ideal gases and he sells his ignorance about the remaining 99% of thermodynamics as a virtue and as a criticism against thermodynamics.

That's just too bad - it's the arrogance of stupidity. Has he ever heard e.g. of the Landau theory of second-order phase transitions? Does he realize that the theory belongs to thermodynamics and many other, analogous insights are included in the discipline? Why does he think that his superficial knowledge of some signs taken from the ideal gases - with a 50% chance of a mistake - are all of thermodynamics?

So the two operations never commute but the whole success of the effective theories (for the long-distance phenomena) is that they approximately commute. When we talk about unknown microscopic phenomena, we may be uncertain when an effective theory exactly breaks down. But in QFT, we know the rules. We know the rules in cosmology, too. They're calculable (and have been calculated). Gore's sick neural cell was an example of the fact that the "commutator" - the failure of the two operations to commute and the failure of effective theories to accurately describe reality - is actually linked with the "intelligence of a physical system or a civilization", if you wish. An advanced extraterrestrial civilization could do things that would not be included in our simple astrophysical or cosmological models of their galaxies. Feel free to take it as a measure how advanced a civilization is.

However, when you assume these "cultural" things to be absent, the procedure works pretty well and the effective - and even FRW - cosmology gives qualitatively correct predictions - and usually also pretty accurate quantitative ones. The Universe is nearly uniform when averaged over L = 300-MPc boxes which is why an effective theory for longer distances simply has to exist and has to be accurate, up to (relative) corrections of order "(L_star/L)^n" where "L_star" is the length scale of the new local non-uniform phenomena and "n" is a positive exponent.

It's inevitable for the Universe to be uniform at long enough scales because it was "created" in a state of a low entropy, i.e. a pretty uniform one (assuming that gravitational degrees of freedom dominated), and because of general causal arguments, it didn't have enough time for the gravitational clumping to change this structure qualitatively.

What could have been interesting would be to sketch the principles of a new, non-local theory of quantum gravity (or a description in terms of some non-local degrees of freedom) that naturally allows such an influence and that would agree with some additional things - not only with the highly speculative explanation of the cosmological constant. No such a complete picture has emerged so far - even though it could be motivated by the complementarity (of the cosmic horizon). That's why the research direction is dead.

You know, when I was a freshman, I also had a tendency to dismiss thermodynamics. It is not a fundamental science, I thought. The people who are doing it are not fundamental physicists. They're emergent and they're not as good, anyway. Blah blah blah.

Well, the sociological portion of these thoughts was pretty accurate but fortunately my thermodynamics teachers were charming and wise people who taught me a lot of stuff about all kinds of stable and unstable systems, their behavior, and the microscopic origin of the assumptions that lead to a macroscopic behavior. I have learned to think in their way.

Partly because of the knowledge that I was taught - even though I wasn't really "dreaming" about this kind of knowledge at that time - I instantly began to think that everyone who misunderstands thermodynamics - its very basic principles, its scope, and its relationships to other portions of science - as profoundly as Robert does has no right to think that he understands how the world around works at the scientific level because pretty much everything we observe are systems with very many degrees of freedom where thermodynamics is always responsible for a major part of the macroscopic behavior.

And that's the memo.

There is however a related problem my classically relativistic friends told me about: When discussing cosmology, it is usually a good first approximation that the universe is homogeneous which supposedly it is at large scales. At small scales however, this is obviously not the case with voids, galaxies, stars, stones etc. But for the evolution at large scales you average all those local fluctuations and replace everything by the cosmological fluid.That's why this "averaged" cosmology ends up being such a well-behaving, simple system where the laws of classical physics and thermodynamics can be directly applied.

The problem with the non-linear theory of gravity is however that it is by far not obvious that this averaging commutes with time evolution: That, starting from good initial conditions it does not matter if you first average and then compute the time evolution of the averaged matter density or if you first compute the time evolution and then to the spatial averaging.The averaging never quite commutes with the evolution, not even for other systems in science. For example, in meteorology, a butterfly wing can create a hurricane. If you don't believe this proposition, there's a more obviously correct one. One additional tiny dysfunctional neural cell in Al Gore's brain is enough to lower the global mean temperature in 2100 by 0.07 °C (and also destroy the world's economy, if you care). This cell obviously creates a bigger impact in the future than what you would expect from the averaging procedure.

So the two operations never commute but the whole success of the effective theories (for the long-distance phenomena) is that they approximately commute. When we talk about unknown microscopic phenomena, we may be uncertain when an effective theory exactly breaks down. But in QFT, we know the rules. We know the rules in cosmology, too. They're calculable (and have been calculated). Gore's sick neural cell was an example of the fact that the "commutator" - the failure of the two operations to commute and the failure of effective theories to accurately describe reality - is actually linked with the "intelligence of a physical system or a civilization", if you wish. An advanced extraterrestrial civilization could do things that would not be included in our simple astrophysical or cosmological models of their galaxies. Feel free to take it as a measure how advanced a civilization is.

However, when you assume these "cultural" things to be absent, the procedure works pretty well and the effective - and even FRW - cosmology gives qualitatively correct predictions - and usually also pretty accurate quantitative ones. The Universe is nearly uniform when averaged over L = 300-MPc boxes which is why an effective theory for longer distances simply has to exist and has to be accurate, up to (relative) corrections of order "(L_star/L)^n" where "L_star" is the length scale of the new local non-uniform phenomena and "n" is a positive exponent.

It's inevitable for the Universe to be uniform at long enough scales because it was "created" in a state of a low entropy, i.e. a pretty uniform one (assuming that gravitational degrees of freedom dominated), and because of general causal arguments, it didn't have enough time for the gravitational clumping to change this structure qualitatively.

The first thing is of course what we always compute while the second thing is what really happens. An incarnation of this problem was an argument that was discussed a few years ago that what looks like the cosmological constant in our local patch of the universe is just a density fluctuation with a super-horizon wave length. At first you would reject such a suggestion since something that happens over regions that are causally disconnected from us should not influence our local observations. However, due to the non-linear nature of gravity this argument is too fast and needed a more thorough inspection.The rejection "at first" is of course correct. If one assumes the laws of physics to respect locality, at least for quantities that are averaged over vast volumes of space and for non-local effects that are assumed to propagate through astronomical distances, it logically and rigorously follows that local observations are not affected by properties of trans-horizon modes. The previous equivalence is pretty much a tautology. I have no idea what it exactly means to deny it.

What could have been interesting would be to sketch the principles of a new, non-local theory of quantum gravity (or a description in terms of some non-local degrees of freedom) that naturally allows such an influence and that would agree with some additional things - not only with the highly speculative explanation of the cosmological constant. No such a complete picture has emerged so far - even though it could be motivated by the complementarity (of the cosmic horizon). That's why the research direction is dead.

My impression is that eventually it was decided that this idea does not work. I would be happy to be informed by somebody follows these things more closely.There are probably new wrong papers about the idea but Robert's attempt to convince himself that the idea is dead because of different reasons is a misguided wishful thinking. The idea is wrong for exactly the "at first" reasons that he doesn't like. These "at first" reasons are the serious, proper physics, while his alternative comments are the rubbish. He paints the whole story upside down.

To wrap up, I feel that I would need to have to understand much more basic things about thermodynamics applied to gravity before I could make sensible statements about the entropy of the universe or Boltzmann brains and the similar.Well, that's surely the case and the word "I" in the paragraph above, meaning Robert Helling, is paramount. You could also add other names next to "I". For example, Robert's reader Bruce Rout has not only abandoned thermodynamics but he has also concluded - using sloppy verbal exercises similar to Robert's own maneuvers - that the Universe couldn't possibly expand. ;-)

You know, when I was a freshman, I also had a tendency to dismiss thermodynamics. It is not a fundamental science, I thought. The people who are doing it are not fundamental physicists. They're emergent and they're not as good, anyway. Blah blah blah.

Well, the sociological portion of these thoughts was pretty accurate but fortunately my thermodynamics teachers were charming and wise people who taught me a lot of stuff about all kinds of stable and unstable systems, their behavior, and the microscopic origin of the assumptions that lead to a macroscopic behavior. I have learned to think in their way.

Partly because of the knowledge that I was taught - even though I wasn't really "dreaming" about this kind of knowledge at that time - I instantly began to think that everyone who misunderstands thermodynamics - its very basic principles, its scope, and its relationships to other portions of science - as profoundly as Robert does has no right to think that he understands how the world around works at the scientific level because pretty much everything we observe are systems with very many degrees of freedom where thermodynamics is always responsible for a major part of the macroscopic behavior.

And that's the memo.

Hey guy, though you are a little too aggressive from my point of view, I think is good to do this kind of "blog-peer-reviewed" activity.

ReplyDeleteI am not working in string theory or thing of the kind (which I will hardly call "real world" ;D ) but in robotics and AI. I found myself reading about entropy in systems with long range interaction due to some models I am doing for self-assembly systems...apparently the things aren't as settled as you may like. Here are some links I find pretty interesting.

http://www.cscs.umich.edu/~crshalizi/notebooks/tsallis.html (A critical notebook about Tsallis entropy)

http://www.ncbi.nlm.nih.gov/pubmed/16383968?dopt=Abstract (A paper I almost understood)

http://www.mdpi.com/1099-4300/10/3/380 (a new concept for out-of-equilibrium systems...though I do not know for sure if you can put the Universe here :? )

http://www.mdpi.com/1099-4300/10/3/160 (Equations for the evolution of entropy...apparently a full-open question!)

Thanks for your long memo!

Hello Lubos,

ReplyDeleteI thoroughly enjoyed your very well thought out and articulate post. You have a wonderful sense of humour. Robert appears a little confused about the thermodynamics of gravitational systems. Let's just say it's a fuzzy area with no hair. I appreciate the wink. It is very difficult to describe entropy in layman's terms. It's even difficult in anybody's terms. Nevertheless, Planck used to teach his students that entropy is the log of the temperature, so that we can't have a zero temperature, because we can't have the log of zero. However, the great scientist knew full well that entropy was the log of omega, number of degrees of freedom. He was just making a point to his students. According to a student of his, who I was very privileged to talk to, Planck was one of the best teachers of physics who ever lived. If Robert is confused, which he openly admits, don't chastise him; teach him. Lots of carrots, very small stick.

Entropy is not equated to temperature, however they are realted. Israel questions whether we can even discuss things like the temperature of Black Holes. I believe he would say we can discuss such things, but we should be careful with the math.

The interior of the event horizon is forbidden territory. We can conjecture, but that's about it. It would be very nice if information was conserved across an event horizon for the sake of completeness of quantum theory. Apparently small black holes are hot and large ones are not. Perhaps, and it is only conjecture, confusion decreases from intense gravitational forces inside an event horizon. I wonder if there is an upper limit to the size of a black hole as a result? Just pondering here.

An expanding universe is a very popular theory. I am familiar with some of its tenets. If you think Robert's statements on thermodynamics are somewhat naive, you gotta take a closer look at the Big Bang Theory. Guy, they got some whoppers in there.