Dear Marc,

If you read carefully Karo's posts, you will see that he does not claim that the space expansion is large at microscopic scales. My understanding is that he simply says that space expansion exists at all scales. Therefore, no matter how small the effect is at microscopic scales (compared to other local interactions), it could be nonetheless the source of the arrow of time. Why ? Because, in contrast to all other local interactions, and regardless of their overwhelmingness, space expansion includes a time asymmetry.

The key point here is the time-asymmetry property of the effect and not its size.

TMO it is clear that the present expansion of our universe (which is a thermodynamically isolated system) is in accordance with the second law of thermodynamics as such an expansion allows the entropy of our universe to increase. Thus, such an expansion is totally in accordance with the second law of thermodynamics.

A common argument about the presence of an arrow of time (a problem deeply related to the second law) in the universe is the special extreme low entropy initial state of the universe at the big bang. To my knowledge this is seen as a possibility to get an arrow of time despite the fact that the fundamental laws are time reversal invariant (there have been processes identified which are not time inversion symmetric, but they could not account for the arrow or time). So in light of this, I think one would say that the special initial conditions of the universe are related to the second law of thermodynamics or more precisely that states of the universe after the big bang have higher entropy than the initial state and thus the universe is evolving towards increasing entropy.

Dear Christian,

I agree with you that "the presence of an arrow of time (a problem deeply related to the second law) in the universe is the special extreme low entropy initial state of the universe at the big bang".

Then a crucial question arises: how can such an extremely low entropy of the initial state of our universe be explained?

Sean Carroll gives the following explanation: the beginning of our universe, the so-called "Big Bang", was not the beginning of everything. There was a more primordial state: "Instead let us suppose that the universe started in a high-entropy state, which is its most natural state. A good candidate for such a state is empty space. Like any good high-entropy state, the tendency of empty space is to just sit there, unchanging. So the problem is: How do we get our current universe out of a desolate and quiescent

spacetime?

The secret might lie in the existence of dark energy. In the presence of dark energy, empty space is not completely empty. Fluctuations of quantum fields give rise to a very low temperature—enormously lower than the temperature of today’s universe but nonetheless not quite absolute zero. All quantum fields experience occasional thermal fluctuations in such a universe. That means it is not perfectly quiescent; if we wait long

enough, individual particles and even substantial collections of particles will fluctuate into existence, only to once again disperse into the vacuum (these are real particles, as opposed to the short-lived “virtual” particles that empty space contains even in the absence of dark energy).

Among the things that can fluctuate into existence are small patches of ultradense dark energy. If conditions are just right, that patch can undergo inflation and pinch off to form a separate universe all its own—a baby universe" (Carroll S. "The Cosmic Origins of Time's Arrow" Scientific American, June 2008).

Dear Christian,

If the number of microstates available to the universe increases with the expansion of the universe (like the removal of constraints on a thermodynamic system), we would not need the assumption of special initial conditions to get the second law of thermodynamics.

Dear Marc,

Carroll's arguments seem to complicate even more the situation and to only push the crucial questions back to earlier times (if the word "time" even makes sense). They don't sound very intuitive to me but I will read the article you cite before making up my mind. Thanks for the reference.

I guess the initial state would still be special if we let the "movie run backward" but it may explain the origin of the asymmetry Interesting thought.

The second law stems from straightforward large-number statistics

There are many formulations of the Second Law of Thermodynamics, and they are all, in essence, statements regarding the allowed changes for the entropy of any physical system.

The second law is a statement that all processes go only in one direction, which is the direction of greater and greater degradation of energy, in other words, to a state of higher and higher entropy. For example, when we stir a cup of tea, the smooth and swirling motion that we make with a spoon soon disappears. The swirling motion is converted - from conservation of energy - into a very tiny increase in the temperature of the tea. However, no matter how long we wait, the still tea in the cup will never suddenly start to swirl accompanied by a tiny drop in its temperature. Similarly, if a glass shatters, no amount of waiting will ever see the glass suddenly re-assemble itself, although the glass in tact and the shattered glass, up to some minute differences, have the same energy.

Since in all these examples and many others in nature, energy and momentum are conserved, clearly it is not these considerations that are responsible for events in time not being able to reverse themselves. The question is the following: why are certain physical phenomenon, allowed by conservation laws such as energy, nevertheless forbidden from occurring? The second law of thermodyamics is the underlying reason that unlikely events do not occur. Entropy is a measure of the likelihood for some event to occur, and only those events can occur for which entropy increases, since they are more likely. In other words, an isolated system always goes from a less probable to a more probable configuration. We hence have the following statement for the second law: In any physical process, the entropy S for an isolated system never decreases; that is, we have entropy change is always larger than or equal to zero.

As to the origin of the second law of thermodynamics, It is important to note that the second law of thermodynamics cannot be derived from any more basic underlying principle. Rather, it is an experimental law in that every experiment to date has confirmed the validity of this law in nature. The second law therefore is rather unique in that unlike other laws such as gravitation or quantum mechanics, it is not something that results from some deep mathematical formulation of nature's laws.

The second law is one of the few fundamental laws of physics that historically arose from very practical questions, in particular the need to understand the theory of heat engines.

http://srikant.org/core/node10.html

This is how Arthur Eddington characterized the Second Law more than eighty years ago:

"The Law that entropy increases—the Second Law of Thermodynamics—holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations—then so much for Maxwell’s equations. If it is found to be contradicted by observation—well, these experiments do bungle things sometimes. But if your theory is found to be against the Second Law of Thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation."

The expansion of the universe comply with the second law of thermodynamics otherwise be sure there will be no expansion.

I agree with both Christian and Mark.

The singularity at the big bang is the state of lowest entropy because the probability of finding any part of the universe there is by definition one. Therefore its entropy P*Ln(P) is zero.

Because it is a state of maximal compression, any change from this state must be one of expansion. However in my opinion the increase in entropy of the expanded state is due to the uncertainty principle - the wave function of the universe (nicknamed "Tigger'" in another thread), now has probabilities of location within it of less than one, therefore P*Ln(P) in aggregate must be greater than zero, and it must further increase as the universe expands.

This increase in entropy with expansion is also entirely consistent with the second law of thermodynamics applied to an isolated system.

It does not matter if you run time forwards or backwards from the singularity, the universe expands, and entropy increases - choose whatever arrow of time you like at the singularity and entropy increases. So even though the laws are symmetrical in time, this is countered by the fact that symmetry from a singularity is always an expansion, never a contraction!

The universe must expand from the singularity because the uncertainty principle requires the expansion velocity to be non zero because its position is perfectly known i.e. it must have a non zero energy - and also zero entropy.

Now the curious thing is that this answer alludes to Quantum Mechanics being important to entropy and the second law, whereas the question postulates an uncanny relationship to General Relativity!

The need for an extension of thermodynamics to relativity arises in two ways:

1-Classical thermodynamics was-perhaps unintentionally but nevertheless actually-only developed for systems which were tacitly assumed to be at rest with respect to the observer, and further investigation is necessary for the treatment of thermodynamic systems which are moving relative to the spatial coordinates in use. This further investigation must be carried out with the help of those principles-for measurements made by observers in uniform relative motion to each other-which form the subject matter of special theory of relativity.

2-Original thermodynamics tacitly assumed that the behavior of thermodynamic systems could be described with the help of ideas as to the nature of space and time which we now know to be approximately valid only for a limited range of space-time and in the absence of strong gravitational fields. The consideration of classical thermodynamics were thus actually limited to small enough systems and weak enough gravitational fields so that Newtonian theory of gravitation could be applied as a close enough approximation. This treatment is not adequate for investigating the thermodynamic behavior of large portions of the universe in connection with cosmological problems and in order to obtain more precise expressions of thermodynamical effects in regions of strong gravity, it becomes necessary to extend thermodynamics to general relativity.

Issam,

‘The second law of thermodyamics is the underlying reason that unlikely events do not occur. Entropy is a measure of the likelihood for some event to occur, and only those events can occur for which entropy increases, since they are more likely. In other words, an isolated system always goes from a less probable to a more probable configuration.’’

Unlikely events do not occur because that they are unlikely. The second law is not the cause of this but simply the expression of this basic mathematical law of big numbers.

``As to the origin of the second law of thermodynamics, It is important to note that the second law of thermodynamics cannot be derived from any more basic underlying principle. Rather, it is an experimental law in that every experiment to date has confirmed the validity of this law in nature. ``

The law has been confirmed experimentally but its certainty in the eyes of the physicist stem from its root in brute statistic which is a very strong underlying principle.

Louis

I agree, we are both saying the same thing but expressed in different ways.

The SLT is simply an elaboration of Newton's First Law of Motion. It is a law of divergence, while its complement is a law of convergence (Borchardt, 2008).

Borchardt, Glenn, 2008, Resolution of SLT-order paradox ( http://scientificphilosophy.com/Downloads/SLTOrder.pdf ).

Louis and Issam,

The second law of thermodynamics is not just a "law of big numbers". This was Boltzmann's interpretation. He assumed some kind of "chaos". We now call this "a priori equal probability of the microstates. This indeed seems to be the case for many systems, but not all. However, the important question is "given reversibility of the dynamical laws (classical or quantum), where does this a priori equal probability of the microstates come from? Prigogine claims that the dynamical laws are really not reversible due to persistent interactions and mathematical chaos (non-linearity). There is still much debate on this subject but most scientists are in agreement that it is not as simple as Boltzmann suggested. I believe (and many others also) that the expansion of the universe plays a role, but thats basically intuition with little theory. Try to convince yourselves that an isolated system with reversible microscopic dynamics will not arrive at a stable thermodynamic equilibrium.

There is also the problem that much of the discussion must be philosophical in nature if one considers the fact that for macroscopic systems recurrence in the spirit of Poincare will appear at timescales which are so vast that they are irrelevant from any practical perspective (that may include the lifetime of the universe).

The nature of the physical world, Eddington

http://henry.pha.jhu.edu/Physworld.pdf

''Now the primary laws of physics taken one by one all declare that

they are entirely indifferent as to which way you consider time to be

progressing, just as they are indifferent as to whether you view the

world from the right or the left. This is true of the classical laws, the

relativity laws, and even of the quantum laws. It is not an accidental

property; the reversibility is inherent in the whole conceptual

scheme in which these laws find a place. Thus the question

whether the world does or does not 'make sense' is outside the

range of these laws. We have to appeal to the one outstanding law

- the second law of thermodynamics - to put some sense into the

world. It opens up a new province of knowledge, namely, the study

of organization; and it is in connection with organization that a

direction of time flow and a distinction between doing and undoing

appears for the first time.``

Dear Karo,

By your last assertion "Try to convince yourselves that an isolated system with reversible microscopic dynamics will not arrive at a stable thermodynamic equilibrium" do you mean that the entropy of an isolated system (such as our universe) won't increase over time towards its thermodynamic equilibrium?

Hi Marc,

No, I'm saying that macroscopic irreversibility for an isolated system cannot be derrived form reversible laws. There is something else that is happening, eg. expansion of the universe, or persistent interaction (Prigogine), or something else.

Nature likes randomness or disorder. That is nature's order with time.

Dear Karo,

You asked for the origin of the second law. T

Dear Karo,

History is the origin of the first and second laws of thermodynamics. It is centuries of observation that first order (2 heat sources) and second order (1 heat source) perpetual movements could not be set-up that lead the scientific community to decide they are impossible. These impossiblities translate into the first and second law.

Laws or principles are decisions on a topic. They are no proofs, they can be some time falsified. They are like axioms in math, one constructs coherent theories on they ground.

sorry for the mess above.

Layzer (1975) :

Suppose that at some early moment local thermodynamic equilibrium prevailed in the universe. The entropy of any region would then be as large as possible for the prevailing values of the mean temperature and density. As the universe expanded from that hypothetical state the local values of the mean density and temperature would change, and so would the entropy of the region. For the entropy to remain at its maximum value (and thus for equilibrium to be maintained) the distribution of energies allotted to matter and to radiation must change, and so must the concentrations of the various kinds of particles. The physical processes that mediate these changes proceed at finite rates; if these "equilibrium" rates are all much greater than the rate of cosmic expansion, approximate local thermodyanamic equilibrium will be maintained; if they are not, the expansion will give rise to significant local departures from equilibrium. These departures represent macroscopic information; the quantity of macroscopic information generated by the expansion is the difference between the actual value of the entropy and the theoretical maximum entropy at the mean temperature and density.

According to Chaisson, the very expansion of the Universe, then, provides the environmental conditions needed to drive order from chaos; the process of cosmic evolution itself generates information. How that order became manifest specifically in the form of galaxies, stars, planets, and life has not yet been deciphered in detail; that is the subject of many specialized areas of current research. We can nonetheless identify the essence of the development of natural macroscopic systems — ordered physical, biological, and cultural structures able to assimilate and maintain information by means of local reductions in entropy — in a Universe that was previously completely unstructured.

(ibid., p.128-9)

''The second law say that isolated systems increase their randomness, becoming more disordered and less complex and structured as time moves forward. This is the opposite of what we see happening in the history of our universe, in which complexity increases as structures form on many scales, with the most intricate structures being the most recent.

...

So if we consider the solar system as ansolated system, the fact that parts ofit are undergoing self-organization is compatible with an overall increase of its entropy. the system as whole is trying to come to equilibrium and will increase its entropy where it can. The second law is doing its best to drive the solar system to equilibrium, but as long as there's a big star radiating hot photons into cold space, that equilibrium is postponed. While it's postponed, molecules can ride the energy flow to greater and greater states of organization and complexity. And stars burn for billions of years, so there's lots of time for complexity to proliferate. The existence of stars has much to do with why the universe is far from equilibrium almost 14 billion years after its formation.''

p. 222, Time Reborn, Lee Smolin

It is generally argued (do not know the correctness) that universe is expanding. In that sense it provides a direction of future time to expanding universe. If that always happens, I mean Universe continues to expand, the total entropy may continue to increase. But what happens when a catastrophe takes place and there is a sudden shrinkage of the universe as was when it began? Such cycles perhaps occur in nature.

Dear karo,

What do you think of this explanation (from Wikipedia "Irreversible process")?

"All complex natural processes are irreversible. The phenomenon of irreversibility results from the fact that if a thermodynamic system, which is any system of sufficient complexity, of interacting molecules is brought from one thermodynamic state to another, the configuration or arrangement of the atoms and molecules in the system will change in a way that is not easily predictable. A certain amount of "transformation energy" will be used as the molecules of the "working body" do work on each other when they change from one state to another. During this transformation, there will be a certain amount of heat energy loss or dissipation due to intermolecular friction and collisions; energy that will not be recoverable if the process is reversed".

Dear Marc,

Phenomena occurring in nature are definitely irreversible and description above from the Wikipedia is sufficiently accurate to describe the process, but the question to be addressed here is "What is at the roots of the irreversibly in nature". Newton´s laws and the Schrödinger equation are invariant under time reversal (i.e. there is no asymmetry evident at this level of description of nature). The chaos arising from non-linear dynamics is deterministic for an isolated system, so chaos cannot be argued as the source of the second law (as Boltzmann argued). Maybe, as Christian pointed out above, the Poincare recurrence time is just so great, and the Universe started in a very special low entropy state, that we will never see the macroscopic spontaneous reduction of entropy. If this were the case, then irreversibly would be just an illusion (something that Einstein believed). However, one woiuld still have to explain the extraordinary low entropy of the universe at the time of the big bang. There are many that maintain that this is not the answer, for example Prigogine who maintains that the real fundamental laws of nature must be irreversible, or others that maintain the the second law derives from the expansion of the universe. I'm inclined to believe the latter (although Penrose dismisses it) simply because of the extraordinary relation between fundamental laws of thermodynamics and general relativity, see for example, Robert Wald's book on general relativity. In a classical (non-quantum) description, the surface area of the event horizon of a black hole always increase in time (or remains constant) and associating the entropy with the surface area of the event horizon of a black hole thus gives the second law of thermodynamics (see Wald). With quantum mechanics we can have particle creation at the event horizon and an increase in the entropy of the rest of the universe at the expense of the surface area of the event horizon, but the second law still holds for the complete system of black hole plus universe. I believe that there is still a lot to be investigated concerning the connection between general relativity and thermodynamics and that it will be this that leads us to a quantum theory of gravity.

Time Reborn, Lee Smolin, p.223:

"Systems held together by gravity, stars, solar systems, galaxies, and black holes are all anti-termodynamic. They cool down when you put energy into them. This means that all these systems are unstable. The instanbilities drive them away from uniformity and stimulate the formation of patterns in space and time.

This has a lot to do with why the universe is not in equilibrium 13.7 billion years after its origin. The increasing structure and complexity that characterize the universe's history are largely explained by the fact the gravitationally-bound systems filling it, from clusters of galaxies to stas, are anti-thermodynamic.

It's easy to understand why such systems are anti-thermodynamic. Two basic features differentiate gravity from the other forces: The gravitational force is (1) long-range and (2) universally attractive. Consider a planet in orbit around a star. If you put energy in, it will move to an orbit farther from the star, where it move slower. So utting energy in decreases the speed of the planet, and this lowers the system"s temperature - because temperature is just the average speed of things in the system. Conversely, if you take energy out of the solar system, the planet must respond by falling closer to the star, where it moves faster. Hence, taking energy out heats up the system."

Dear karo, Newton´s laws and the Schrödinger equation are the conservation equations of the energy, and correspond to the first law of thermodynamics (invariant under time reversal). In fact, we don’t know if a dynamic phenomenon itself be symmetrical under time reversal only by these dynamic equations, it is the symmetry of the equations, however, we're not sure if it is also the symmetry of a dynamic phenomenon itself.

In general, these dynamic equations don’t involve the theme about dissipation, I call them “the first law physics.” We cannot find the origin of the irreversibly from “the first law physics”.

Dear Karo,

Sorry but I find your speech rather unclear.

According to Wikipedia explanation the root of the irreversibility of processes related to an increase of entropy is in the amount of heat energy loss or dissipation due to intermolecular frictions and collisions as this energy won't be recoverable if the processes are reversed.

Hi Marc,

Sorry for the long delay in responding. What you call a "Wikipedia explanation" of irreversibility is really a "Wikipedia description" of irreversibility. What the description you quote is saying is that if an amount of energy is originally contained in one, or a few, degrees of freedom, it will eventually be distributed over many degrees of freedom. But it doesn't tell us WHY that would happen given reversible dynamical laws. The "collisions" are reversible.

There is a theorem, that has been proved, due to Poincare (the Poincare Recurrence Theorem) that says that if one starts with an isolated system with reversible dynamical laws and a given microscopic initial condition, then if we wait long enough, the system will eventually return arbitrarily close to that given initial condition. For example, if I place a drop of ink in a glass of water, if I wait long enough the drop of ink should recur. Maybe "long enough" is a very long time (longer than the age of the universe) so even though in principle it should recur we don't see it? But, we never see it, for any system anywhere at any time, so either the universe was in a very special initial condition of extremely low entropy (Penrose) but this is just as difficult to explain, or the dynamical equations are really not reversible (Prigogine), or the expansion of space-time has something to do with it (I don't know who said this first, but I tend to see this as the most realistic solution).

Dear Karo,

I expect you agree that dissipative processes are irreversible. Most dynamical systems with some complexity show chaos deterministic behavior (e.g. cosmologic systems composed of at least three large bodies such as the sun, the earth and Jupiter).

As you probably know, chaos deterministic systems involve dissipitative processes (cf. Prigogine). Thus irreversibility is at least the consequence of such dissipitative processes.

Marc,

If we treat the sun, earth and Jupiter as a simple mathematical points representing the centers of their gravitational attraction then we have deterministic chaos, but absolutely no dissipation. If we treat them as finite sized bodies made up of particles then we find tidal forces in which some of their global kinetic energy gets dissipated into heat energy of the material in these bodies which is eventually radiated to space in the form of infrared photons.This is dissipation and the second law in action. I like to state it as the "dispersion of the the conserved thermodynamic quantities, i.e. energy, momentum, angular momentum, charge, etc. over ever more microscopic degrees of freedom". Given reversible dynamical laws, deterministic chaos cannot be considered as the cause of irreversibility.

Dear Karo,

I agree with your comment.

Actually, in statistical mechanics, the second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy (it was the case of our universe).

The first part of the second law, which states that the entropy of a thermally isolated system can only increase is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy (E) is:

S=Kb log[Ω (E)]

Where Kb is the Boltzmann constant and Ω (E) is the number of quantum states in a small interval between E and E + δE. δE is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of δE. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on δE.

Besides the entropy of an isolated system that is not in equilibrium (like our universe) can be defined as:

S= - Kb Σ Pi log (Pi)

Where Pi is the probabilities for the system to be found in the states labeled by the subscript i. In thermal equilibrium, the probabilities for states inside the energy interval δE are all equal to 1/Ω , and in that case the general definition coincides with the previous definition of S that applies to the case of thermal equilibrium.

From an equilibrium situation there are a number of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of 1/Ω.

In the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the entropy will increase continuously as a function of time during the intermediate out of equilibrium state as it demonstrates an irreversible increase in entropy, despite microscopically reversible dynamics.

However you may object that it should not be possible to deduce an irreversible process from time-symmetric dynamics and a time-symmetric formalism.

The explanation is that Boltzmann's equation is based on the assumption of "molecular chaos", i.e., that it is acceptable for all the particles to be considered independent and uncorrelated. This in fact breaks time reversal symmetry.

Basically correct Marc, but the "molecular chaos" of Boltzmann is an ad hoc assumption. There is no justification for its utilization. One can invoke "deterministic chaos" with external perturbation (but then the system is not isolated) or resonances which take part of the conserved quantities to non-dynamical degrees of freedom, as Prigogine suggested in his final book "The end of certainty".

Since the universe is expanding uniformly, space must also be expanding at the level of inter-atomic distances. This, coupled with deterministic chaos, I think, leads to irreversibility in isolated systems. This is my preferred view (without having studied it in detail) of irreversibility, I don't know if anyone has published anything similar?

Karo,

If space between everything including inter-atomic distance would increase then our measurement standard would increase in proportion with the universe and so we would not detect any expansion. We detect an expansion because the atomic distances do not change.

Dear Louis and Karo,

Sorry, but my last comment is wrong and thus shouldn't be taken into account except that I agree with Louis.

Actually, Karo, when you assert that space between everything including inter-atomic distance is increasing do you mean that, for instance, the sizes of bodies like planets or stars or even the distances between the bodies in our solar system, or even the distances between the stars in a given galaxy, are increasing with our universe expansion?

Marc, if you wish to edit or delete your comment, just move the mouse to the upper right corner of your post then click on the arrow that will show up.

Hope this helps -)

BTW, please do NOT upvote my present post. I have already received 3 upvotes somewhere else for providing the same information and it made me quite uncomfortable.

I am not sure how to jump into this discussion, but maybe it is of interest to point at the Loschmidt's paradox. At the end of the 19th century there was a debate about the question how time-reversible laws (for example Newton's laws) could lead to irreversible processes. Maybe I take a shortcut, but the main issue is probability. In principle on a microscopic level the equations of motion are reversible implying that it is possible that a system returns to a highly unfavorable state (f.e. all gas molecules at one spot in a big vessel). However, the probability that this will happen is extremely small. So, the 2nd law (irreversibilty and entropy increase) is telling you something about the macroworld. It is based on an observation that something is highly improbable. That does not mean a process can reverse.

Dear Louis,

Please note that I stated "Since the universe is expanding uniformly, space must also be expanding at the level of inter-atomic distances." I did not state that inter-atomic distances were changing. I don't know if they are changing. That would depend on whether or not the fundamental constants were changing. The consensus is that they are not, but it is not proved beyond reasonable doubt (as far as I understand).

However, there is no reasonable way to deny the validity of my quote above. We can't have space expanding just over very large distances. Therefore, if space is indeed expanding at the very small scale (inter-atomic distance scale) and if the inter-atomic distances are not changing, then there must be some kind of continual re-adjusting of the particles and this, I claim, coupled with non-linearity (a small change of initial conditions leads to an exponentially large divergence) may perhaps lead to the observed irreversibility in nature. Thanks for your criticism.

Hi Marc,

No, I did not "assert" that, please see my reply to Louis. The fact that we can detect a red-shift at large distances would seem to imply that inter-atomic distances are not changing, but this does not mean that space is not expanding locally at the very small scale.

karo,

If you are confine to a room and that you are given a measuring tape and that you constantly measure the dimensions of that room. If the room double in size while your measuring tape stay the same then your measurements tell you that the room double in size. If your measurement tape increase in size in the same proportion as the room then your measurements tell you that the room have a constant size.

If everything in the universe would increase in size in exactly the same proportions then we would never know about that because all we know are our measurments.

We thus kind of conclude from that that physicists do not consider that the distances at our scales and at the atomic scales are expanding. How do they assume expansion of the universe and not expansion of space at our scales iI cannot say but I am sure that any physicits can answer this basic question.

Dear H. Huinink,

The probability argument for irreversibility (Einsteins favored solution) could only be valid if the universe was created in (and we are still in) an extremely low entropy state (something, apparently without explanation). Otherwise, there should exist at least some macroscopic processes for which entropy would continually decrease in time....but we don't see this ever. We discussed some of this above, please see previous comments.

Dear Louis,

Space is expanding on all scales. Please be sure of this, I am a physicist. Please read my previous answer carefully. Thanks.

While space is theoretically expanding at all scales, the effect is observable only at extremely large scales. At smaller scales, the effect is negligible because of the existence of overwhelming local binding forces that hold together the components of a system..

Dear Karo,

The expansion of universe and the origin of the second law may be two questions. “Big Bang” only tell us that the previous entropy is small, but can not tell us why entropy increase in that the equation of the second law does not involve a factor in the expansion of universe. There is no evidence that can support that “Big Bang” is the origin of the second law, and itself may be a process of the second law.

Dear Louis,

About how to measure the distance between two galaxies what do you think of the following thoughts from Dave Rothstein in "Curious About Astronomy: What is the universe expanding into?"?:

"My favorite analogy involves imagining the universe as a gigantic blob of dough. Embedded in the dough are a bunch of raisins, spread throughout. The dough represents space, and the raisins represent the galaxies. (To the best of my knowledge, this analogy was originally proposed by Martin Gardner in his 1962 book Relativity for the Million.) We have no idea how big the dough is at this point - all we know is that it is very big, and we, sitting on some raisin somewhere inside it, are so far away from the "edge" that the edge can't possibly have any effect on us or on what we see".

Then: "Just get yourself a giant tape measure and clip it to a faraway galaxy, then come back to our galaxy and hold on tight. As the galaxy moves away, it will pull on the tape measure, and you will easily be able to read off the distance as the tape measure unwinds... one billion light-years, one and half billion light-years, two billion light-years, etc.

In our new picture of the universe, however, with the raisins and the dough, the tape measure will not unwind at all as the universe expands, because the galaxies are not actually moving with respect to each other! Instead, it will read one billion light-years the whole time. You could be perfectly justified in saying that the distance between the galaxies has not changed as time goes on. When you bring the tape measure back in, however, you will notice something unusual; due to the stretching of space, your tape measure will have stretched as well, and if you compare it to an identical tape measure which you had sitting in your pocket the entire time, you will see that all the tick marks on it are twice as far apart as they used to be. Using the tape measure from your pocket as a reference, you would now say that the galaxy is two billion light-years away, even though the first tape measure said it was one billion light-years away. As you can see, the concept of "distance" in this new picture of the universe is somewhat more complicated than in the old picture! It is unclear whether the universe as a whole is really "expanding" - all that we really measure is a stretching of the space between each pair of galaxies (Note that we might have to have an "imaginary" tape measure whose atoms aren't actually being held together by intermolecular forces in order for the scenario described above to actually take place as described).

By the way, this analogy of the tape measure is pretty similar to what actually happens to light when it travels between galaxies. When light is emitted from one galaxy and travels through space to another galaxy, during its trip through space it also will be stretched, causing it to have a longer wavelength and therefore causing its color to appear more towards the red end of the spectrum. This is what leads us to see redshifted light when we look at faraway galaxies, and it is measurements of this redshift that allow us to estimate the distances to these galaxies".

And: "Finally, I should point out that not everything in the universe is "stretching" or "expanding" in the way that the spaces between faraway galaxies stretch. For example, you and I aren't expanding, the Earth isn't expanding, the sun isn't expanding, even the entire Milky Way galaxy isn't expanding. That's because on these relatively small scales, the effect of the universe's stretching is completely overwhelmed by other forces (i.e. the galaxy's gravity, the sun's gravity, the Earth's gravity, and the atomic forces which hold people's bodies together). It is only when we look across far enough distances in the universe that the effect of the universe's stretching becomes noticeable above the effects of local gravity and other forces which tend to hold things together. (That is why, in the analogy of the tape measure I discussed above, the tape measure that you keep in your pocket does not get stretched, while the one that goes between two galaxies does get stretched)".

This last paragraph is very similar to what H.E. Lehtihet says above.

Dear Karo,

You assert to Louis: "Space is expanding on all scales. Please be sure of this, I am a physicist".

Then, what do you reply to the following assertions from both H.E. Lehtihet and Dave Rothstein: "not everything in the universe is "stretching" or "expanding" in the way that the spaces between faraway galaxies stretch. For example, you and I aren't expanding, the Earth isn't expanding, the sun isn't expanding, even the entire Milky Way galaxy isn't expanding. That's because on these relatively small scales, the effect of the universe's stretching is completely overwhelmed by other forces (i.e. the galaxy's gravity, the sun's gravity, the Earth's gravity, and the atomic forces which hold people's bodies together). It is only when we look across far enough distances in the universe that the effect of the universe's stretching becomes noticeable above the effects of local gravity and other forces which tend to hold things together"?

Dear Marc,

I already answered that question. Please see my second to last response to Louis, particularly the second paragraph. Regards.

karo,

I have read all contributions carefully and this question of the expansion of space at all scales is still puzzling to me. Since it is not important for the thread question maybe this question should deserve a thread of its own.

Regards

Dear Karo,

Of course I read your answer, i.e., "Therefore, if space is indeed expanding at the very small scale (inter-atomic distance scale) and if the inter-atomic distances are not changing, then there must be some kind of continual re-adjusting of the particles and this, I claim, coupled with non-linearity (a small change of initial conditions leads to an exponentially large divergence) may perhaps lead to the observed irreversibility in nature" but I don't understand it.

It doesn't seem necessary to look for such a complex answer to explain why "not everything in the universe is 'stretching' or 'expanding' in the way that the spaces between faraway galaxies stretch". It seems that, according to Einstein's theory of general relativity, the explanation is simply that on these relatively small scales, the effect of the universe's stretching is completely overwhelmed by other forces (i.e. the galaxy's gravity, the sun's gravity, the Earth's gravity, and the atomic forces which hold people's bodies together)".

By the way could you develop your complex theory a little bit of a continual re-adjusting of the particles (to prevent the streching on relatively small scales) that, coupled with non-linearity (a small change of initial conditions leads to an exponentially large divergence), would lead to the observed irreversibility ?

Dear Louis,

I said that space was expanding on all scales, including on the inter-atomic distance scale. I never said that the "inter-atomic distance is expanding". I believe, but I am not sure, that inter-atomic distances are not changing. Therefore, atoms must be continually moving toward each other to compensate. Knowing the Hubble expansion constant, you can determine the rate of expansion of space at the inter-atomic distance scale, and therefore the rate at which atoms are moving towards each other in order to compensate space expansion. It turns out to be extremely small, not measurable directly. But, maybe, we see an indirect affect which is manifest in the irrevesibility of nature.

Dear Marc,

Ok, let me try to put my theory in point form; (I can't really call it "my theory" because I don't know if anybody has published this previously, or even if it is worth publishing, it may be wrong. It doesn't seem, as you say "complex" to me but so simple that others must have thought about it.)

1) Space is expanding on all scales, including the inter-atomic, or even inter-quark, scales.

2) Inter-atomic distances don't seem to be changing. Therefore, there are inter-atomic forces that we all know about which keep the particles at the same distances.

3) Point 2) with 1) implies that particles at distances of the inter-atomic scale must be continually moving towards each other.

4) The rate of movement towards each other is so small that it cannot (at this time) be directly measurable. However, they must have a small but finite momentum do to the expansion of space.

5) However, maybe there is an indirect affect observable which is the irreversibility of nature. Let's see how this may come about.

6) For non-linear systems, extremely small changes in microscopic initial conditions leads to great changes on macroscopic scales (e.g. the butterfly effect).

7) The finite momentum of particles at the inter-atomic scale due to the expansion of space introduces an irreversibility into the motion which, mathematically, without the universe expansion, should be reversible (Netwon's equations, the Schrodinger equation, etc.).

8) The time asymmetric expansion of the universe therefore leads to the time asymmetry observed in all macroscopic processes in nature.

Please let me know which one of these points is not understandable or wrong, or needs more work. Regrads.

Dear Karo,

SIC = sensibility to initial conditions

AoT = Arrow of Time

SEM = Space Expansion at Microscopic scales

My understanding is that you would like to put forward the SIC to link the AoT to the SEM.

The point I have tried to make in my last post is that the effect of SEM is negligible compared to that of other local interactions that exist at these microscopic scales.

Consequently, if I should use the SIC to link AoT to something, why would I consider that this thing must be SEM (whose effect is totally negligible) and not something else (whose effect is far more important)? And your answer to this question would be that only SEM is time asymmetric. Have I understood you correctly ?

Dear H.E. Lehtihet,

It is said that “metric expansion is a key feature of Big Bang cosmology and is modeled mathematically with the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. This model is valid in the present era only on large scales (roughly the scale of galaxy clusters and above). At smaller scales matter has become bound together under the influence of gravitational attraction and such bound objects clumps do not expand at the metric expansion rate as the universe ages, though they continue to recede from one another. The expansion is a generic property of the universe we inhabit, though the reason we are expanding is explained by most cosmologists as having its origin in the end of the early universe's inflationary period which set matter and energy in the universe on an inertial trajectory consistent with the equivalence principle and Einstein's theory of general relativity (that is, the matter in the universe is separating because it was separating in the past)”.

Moreover “the expansion of space is sometimes described as a force which acts to push objects apart. Though this is an accurate description of the effect of the cosmological constant, it is not an accurate picture of the phenomenon of expansion in general. For much of the universe's history the expansion has been due mainly to inertia. The matter in the very early universe was flying apart for unknown reasons (most likely as a result of cosmic inflation) and has simply continued to do so, though at an ever-decreasing rate due to the attractive effect of gravity” (from Wikipedia “Metric expansion of space”).

So it seems clear that, as opposed to what Karo asserts, the metric expansion of our universe is only valid on large scales (i.e., on gravitationally unbound objects: roughly the scale of galaxy clusters and above).

Thus it should be said that there is not at all any stretching of the space metric at smaller scales as the expansion due to the initial inertia is totally eliminated either by gravitation or atomic interaction.

What do you think about it?

Dear Marc,

If you read carefully Karo's posts, you will see that he does not claim that the space expansion is large at microscopic scales. My understanding is that he simply says that space expansion exists at all scales. Therefore, no matter how small the effect is at microscopic scales (compared to other local interactions), it could be nonetheless the source of the arrow of time. Why ? Because, in contrast to all other local interactions, and regardless of their overwhelmingness, space expansion includes a time asymmetry.

The key point here is the time-asymmetry property of the effect and not its size.

Dear H.E.,

As regards the inertial part of universe expansion it seems to be the consequence of an initial cosmic inflation. Such an inflation clearly includes a time asymmetry. Hence, what is crucial is to explain the cause of this inflation, isn't it?

According to Sean Carroll ("The Cosmic Origins of Time's Arrow" Scientific American, 2008) "the very early universe (or at least some part of it) was filled not with particles but rather with a temporary form of dark energy, whose density was enormously higher than the dark energy we observe today. This energy caused the expansion of the universe to accelerate at a fantastic rate, after which it decayed into matter and radiation, leaving behind a tiny wisp of dark energy that is becoming relevant again today".

And "For the process to work as desired, the ultradense dark energy had to begin in a very specific configuration. In fact, its entropy had to be fantastically smaller than the entropy of the hot, dense gas into which it decayed. That implies inflation has not really solved anything: it “explains” a state of unusually low entropy (a hot, dense, uniform gas) by invoking a prior state of even lower entropy (a smooth patch of space dominated by ultradense dark energy). It simply pushes the puzzle back a step: Why did inflation ever happen?".

Then, "One of the reasons many cosmologists invoke inflation as an explanation of time asymmetry is that the initial configuration of dark energy does not seem all that unlikely. At the time of inflation, our observable universe was less than a centimeter across. Intuitively, such a tiny region does not have many microstates, so it is not so improbable for the universe to stumble by accident into the microstate corresponding to inflation. Unfortunately, this intuition is misleading. The early universe, even if it is only a centimeter across, has exactly the same number of microstates as the entire observable universe does today. According the rules of quantum mechanics,

the total number of microstates in a system never changes: entropy increases not because the number of microstates does but because the system naturally winds up in the most generic possible macrostate. In fact, the early universe is the same physical system as the late universe. One evolves into the other, after all. Among all the different ways the microstates of the universe can arrange themselves, only an incredibly tiny fraction correspond to a smooth configuration of ultradense dark energy packed into a tiny volume. The conditions necessary for inflation to begin are extremely specialized and therefore describe a very low entropy configuration.

If you were to choose configurations of the universe randomly, you would be highly unlikely to hit on the right conditions to start inflation. Inflation does not, by itself, explain why the early universe has a low entropy; it simply assumes it from the start etc.".

Of course, I recommend you very much to read the whole paper and would appreciate your comments on it.

Actually I asked Karo to read Carroll's paper. At that time he promised me to do so but, since then, he has never told me whether he did or not. Anyway I have never received his comments ... if any.

Dear Halim,

Your last post finally allowed me to see some light at the end of this tunnel and it makes sense to me that the time asymetry might be directly felt at the small scales of the universe as a space expansion effect. At the cosmic scale, the assymetry of time is asserted by the expansion of the universe directly observable. The farther we observe the cosmos and the further into the past we observe it and it is not the same as today. The furthest we can look is the CMB and we have a surprising homogenous. Surprising because according to the solution of Einstein field equations it could not have been so homegenous. It is why the theory of cosmic inflation was invented.

Marc,

You mentioned the surprising low enthropy of the early universe which was firsty point out I think by Penrose. We are here coming close to the cosmic anthropic question. The fine tuning of the basic laws of physics in the early universe. Some solve this problem by an untestable theoretical solution: the (various) multiverse scenario which transform this unlikely early state into a likely late state of the multiverse. My personal solution is to remove the theoretical thinking leading to the problem: the idea that nothing is ever created in the cosmos and that each next state is a degradation of the previous one. If the cosmos is not based on a timeless realm of physical laws but that everything, everything apparent laws, every structures, every particule has been created at some point in the cosmic evolutioh and that the phase transition of this cosmic evolution goes along the acceleration of this evolion and that there exist regularities in the emergence process then there is no surprisig low enthropy, not cosmic anthropomorphic problem, no multiverse scenario. The only problem remaining and it is not small is to elaborate this scenario,

Dear Louis,

Yes, I want to underscore the incredibly low enthropy of our early universe and, given that low-entropy states are so rare, the too low probability of such a state.

As Carroll, I consider that we must find a solution to this issue. The incredible low entropy of our universe at its beginning, which explains the dramatic time asymmetry of our observable cosmos, "seems to be offering us a clue to something deeper—a hint to the ultimate workings of space and time".

Actually, as Carroll says, "if the observable universe were all that existed, it would be nearly impossible to account for the arrow of time in a natural way. But if the universe

around us is a tiny piece of a much larger picture, new possibilities present themselves".

Then, "perhaps the distant past, like the future, is actually a high-entropy state. If so, the hot, dense state we have been calling 'the early universe' is actually not the true beginning of the universe but rather just a transitional state between stages of its history".

I understand that you cannot accept the idea of a multiverse because, presently, it is not testable. But why not searching for possible ways to test it?

You suggest "that everything, everything apparent laws, every structures, every particule has been created at some point in the cosmic evolution": I am sure you don't support a kind of creationism: could you develop your idea a little bit?

Finally, I agree with the physicist Edward Tryon who wrote (according to Sean Carroll) that "the big bang is easier to understand if it is not the beginning of everything but just one of those things that happens from time to time".

Marc,

Creationists believe that all the creative steps in the evolution of the cosmos are surnatural events perform by God. I do not read creationist non-sense so maybe I do not understand their position. My basic assumption is the universe which the name indicate is ALL THAT EXISTS is an evolution. I do not accept a theory which posits a platonic theoretical framework in a timeless platonic world pre-existing to the UNIVERSE because since the universe is ALL THAT EXISTS there is nothing outside or before it, it is all encompassing in all sense. If someone provide a theoretical framework then it has to explain how it is embodied in the universe and how in came to be in the evolution of the universe. The universe could not have been created since it is ALL THAT EXISTS. The universe changed because it is an evolution.

My second assumption is that the universe is creative . Do not ask me what is this creative part of the universe because if it could be said then it would not be creative. I say that the universe is creative because we observe the universe is an evolution and the steps of this evolution cannot be predicted in intrinsic way. There are multiple creative processes. In biological evolution, we have seen that natural selection is very effective. In Quantum mechanic there seem to be an intrinsic randomness or creativeness at the level of individual quantum processes which force quantum physics to be probabilitic. So the universe has always been in a process of being created.

My third assumption is that the universe could not have a beginning. What a beginning could look like? Whatever theory positing a beginning, I can ask where this theory come from? Any answer assuming a timeless theoretical framework is not acceptable to my previous assumtions. There can be nothing outside ALL THAT EXIST , no outside platonic framework, no outside surnatural, no outside time, nothing by definition. In this way of thinking, the laws of physics are not truly fixed and accurate and do not pre-exist to the universe but are slowly varying relations which have been stabilized at each step of the universe evolutioh. Physics has thus evolve like life has evolved. The stabilize aspect of biology did not pre-exist the evolution of life. The laws of society did not pre-exist societies. Societies have fixed some rules to harmonize social life, biology has done the same. Laws are habits of the universe

that evolved at some point.

Louis,,

Your basic assumption is that our universe is “all that exists” and thus that you do not accept that anything could exist before the emergence of our universe (or after?).

In addition you introduce the notion of “evolution of our universe”.

I would like to first address this last consideration.

I don’t know if you read the last paper I co-wrote with Guy Hoelzer “On the thermodynamics of multilevel evolution” but I think it addresses the topic.

In this paper we consider that the term “evolution” should be better defined. In Biology the term “evolution” refers to the changes in heritable information within populations over time and the dynamics of population origins and extinctions. This view of evolution naturally focuses on the evolution of living systems, which fundamentally involves reproduction of bio-molecules, cells and whole organisms. Therefore, our understanding of processes affecting biological evolution (e.g., natural selection, drift. . .) usually center on patterns of reproductive success. In physics, however, the evolution of a specific system can be more generally defined as its unfolding change in form and/or function over time. Thermodynamically speaking, any ensemble composed of a certain quantity of matter and energy (i.e.,a physical system) in an initial state at time t0 will change over time toward a new state at time t1.

Then we define different levels of “evolution”. For instance we define a level-1 evolution for a given isolated system which corresponds to the fact that its structure will progressively vanish over time and thus level-1 evolution involves monotonic and deteriorative change.

Actually the evolution of our universe involves all the levels of evolution, from level-1 to the highest possible levels (even higher levels will perhaps appear in the future on Earth or anywhere else in our universe). In particular of course it involves level-4 evolution which shows a mechanism(s) for the storage of hereditary information across many generations (i.e., the lowest level of Darwinian evolution).

However it is very likely that the ultimate fate of our universe will be to continue to expand forever and thus to go to a state of maximum entropy in which everything is evenly distributed with no gradients anymore.

Regarding the hypothesis of a multiverse most multiverse proponents are careful scientists who are quite aware of the problem (i.e., we have no information presently about regions beyond our universe) but think we can still make educated guesses about what is going on out there.

For instance (from Ellis, 2011):

1. Space has no end: “Few dispute that space extends beyond our cosmic horizon and that many other domains lie beyond what we see”. Then “It is then easy to imagine more elaborate types of variation, including alternative physics occurring out where we cannot see”;

2. Known physics predicts other domains: “Proposed unified theories predict entities such as scalar fields, a hypothesized relative of other space-filling fields such as the magnetic field. Such fields should drive cosmic inflation and create universes ad infinitum. These theories are well grounded theoretically”. However “the nature of the hypothesized fields is unknown, and experimentalists have yet to demonstrate their existence, let alone measure their supposed properties. Crucially, physicists have not substantiated that the dynamics of these fields would cause different laws of physics to operate in different bubble universes”.

3. The theory that predicts an infinity of universes passes a key observational test: “the cosmic microwave background radiation reveals what the universe looked like at the end of its hot early expansion era. Patterns in it suggest that our universe really did undergo a period of inflation” even if “not all types of inflation go on forever and create an infinite number of bubble universes”;

4. Fundamental constants are far too much finely tuned: “A remarkable fact about our universe is that physical constants have just the right values needed to allow for complex structures”;

5. Fundamental constants match multiverse predictions: e.g., observed density of dark energy.

Finally I cannot follow your assumption that from a scientific point of view our universe "is all that exists” and that “there is nothing outside or before it".

Have you any references that allow you to assert such definitive truths?

Reference:

Ellis GFR. Does The Multiverse Really Exist? Scientific American 2011.

Marc,

Your paper sound interesting. I will give it a look. Are you familiar with Smolin cosmic natural selection theory?

You may be interested in :

Is information a proper observable for biological organization?

G Longo, P-A Miquel, C Sonnenschein, A M Soto

ABSTRACT In the last century, jointly with the advent of computers, mathematical theories of information were developed. Shortly thereafter, during the ascent of molecular biology, the concept of information was rapidly transferred into biology at large. Several philosophers and biologists have argued against adopting this concept based on epistemological and ontological arguments, and also, because it encouraged genetic determinism. While the theories of elaboration and transmission of information are valid mathematical theories, their own logic and implicit causal structure make them inimical to biology, and because of it, their applications have and are hindering the development of a sound theory of organisms. Our analysis concentrates on the development of information theories in mathematics and on the differences between these theories regarding the relationship among complexity, information and entropy.

https://www.researchgate.net/publication/229151563_Is_information_a_proper_observable_for_biological_organization

Article Is Information a proper observable for biological organization?

Louis,

I am indeed familiar with Smolin cosmic natural selection theory as I have been taken by the evolutionary viewpoint of the model with possibilities of "reproduction" and "mutation" of universes. Moreover the model seems fasifiable as it leads to specific predictions such as a maximum for the neutron star mass (no more than two solar masses) or that inflation, if true, must only be in its simplest form, governed by a single field and parameter. Both predictions have held up until now.

Besides I thank you very much for the paper by Giuseppe Longo et al. ″Progress in Biophysics and Molecular Biology″.

I fully agree with G Longo et al.'s viewpoint, i.e., that ″the adoption of the information theoretical approach failed to provide biology with a pertinent observable for understanding and measuring organization″ and that ″ the concepts linked to information such as code, program, signal, etc, have hindered the comprehension both of physical dynamics and of biological organization″. It is clear to me that the dynamical angle is crucial to understand how self-organizing systems can use the free energy they obtain from accelerating the production of entropy of their environment to move still further from their thermodynamic equilibrium and thus to build more order.

Karo,

Do you remember our discussion about the reversibility of Newton laws, that govern the interactions between particles at the microscopic scale, which seem to be in conflict with the irreversibility observed at the macroscopic scale in accordance with Boltzmann’s “H-theorem” (the entropy must increase under the time evolution of the equation) explaining why the solutions should be driven towards the equilibrium of Maxwell?

I don’t know if you are aware of the recent demonstration by two French mathematicians, Stéphane Mischler and Clément Mouhot, of the deep idea of “stosszahlansatz” (molecular chaos) proposed by Boltzmann to explain how his irreversible equation can emerge from Newton’s laws of the dynamics of the particle system ?

If not, you will find attached the link to their full paper pubilshed in arXiv math (see also ″Kac’s program in kinetic theory″ in Inventiones mathematicae 2013; 93:1-147).

In short:

1) Let a microscopic system composed of a huge number of particles for which the interactions are sufficiently weak to allow the particles to continue to interact between themselves. While at the microscopic scale particles interact according to Newton laws (i.e., reversible laws with time-symmetric dynamics) at the macroscopic scale Boltzamnn's equation leads to irreversibility : how is it possible to explain this phenomenon ?

2) In 1956 Mark Kac formalised the problem mathematically in 3 questions :

a. Is it possible to prove the propagation of chaos for physically realistic models?

b. Is it possible to show that the relaxation to equilibrium, true for an infinite number of particles, also arises with a uniform relaxation rate regardless of the number of particles?

c. Is it possible to deduce Boltzmann’s “H-theorem” from the microscopic description of the system?

3) At first Stéphane Mischler and Clément Mouhot tried to answer Kac’s questions one after another ;

4) Finally they succeed by combining different mathematical tools coming from different mathematical domains : partial differential equations, probability, theoretical statistics, and geometry ;

5) Then, they found that their qualitative analysis applied to fluid dynamics as it helps to understand the behavior of many swirls interacting with each other.

Your comments?

http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CDAQFjAA&url=http%3A%2F%2Farxiv.org%2Fabs%2F1107.3251&ei=4X5GUrn_GoKLtAaJh4CQAw&usg=AFQjCNHmu-3kr1E9d8OjL4rWc2hQlF05Xg

Marc, I had a look at the paper but its a little too mathematical for my physical brain. Can you shed more light on the suggestion? For example, is Kac saying what Prigogine is saying (in "The end of certainty"), i.e. deterministic chaos with persistent interaction? Have you read Prigogines book? Regards.

Karo,

I must admit I was too optimistic when I asserted that Stéphane Mischler and Clément Mouhot have solved the problem. Their approach is based on Mark Kac’s formalization of the problem. But Kac’s formalization started from a system which is already random and irreversible.

Thus, the specific issue of starting from the microscopic dynamics, which is reversible and satisfies the Poincaré recurrence theorem to reach the irreversibility inherent to the Boltzmann dynamics is not solved by Mischler and Mouhot demonstration.

By contrast you will find attached the very recent demonstration (2013) by Isabelle Gallagher, Laure Saint-Raymond and Benjamin Texier “FROM NEWTON TO BOLTZMANN: HARD SPHERES AND SHORT-RANGE POTENTIALS” which really deals with the issue.

Of course this paper is also very much mathematical for any physical brains (and in particular for my physician brain too!).

Nevertheless you can find all what you want in this outstanding paper (which is actually a book with 15 chapters!).

If you have any specific questions about their demonstration of the term-by-term convergence to the solution to the Boltzmann equation in the case both of the hard spheres dynamics and of a smooth interaction potential, don’t hesitate to ask as I have the possibility to discuss these with my son who is a researcher in mathematics at the CNRS (actually he informed me of the existence of this very recent demonstration).

Dear Karo, your idea of the correlation between the metric expansion of the Universe and the second law of thermodynamics/arrow of time is very intriguing.  A couple of time ago, I had the same idea and I started to write a paper about that.  However, I had to stop because I found a problem.  Imagine that the cosmic expansion "dilutes" the energy, giving rise to a so called "increase of  entropy".  However, if this were true,  the atoms, that are a concentration of matter (due to nuclear forces locally stronger than the dark energy),  would not respond to the second law of thermodynamics...  Indeed, the place of the Universe where an atom lies is a sort of local contraction of the Universe... therefore, if your claims are true, in an ensemble of atoms the second law of thermodynamics cannot hold....  In other words, the strong atomic forces are locally stronger than the expanding forces of the Universe, therefore, it would mean that locally the second law of thermodynamics does not work...  If you were true, the time must be reverted, at the level of the matter, where a contraction takes place...

Therefore, the explanation of the origin of the thermodynamic arrow must be another, because the expansion of the Universe is due to a force too weak to justify the ubiquitous second law...       

Entropy: A concept that is not a physical quantity

https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity

Marc, the Gallagher et al. paper appears to be an impressive analysis of the derivation of the Boltzmann equation from Hamiltonian dynamics, but it does not seem to offer a simple explanation of the source of irreversibility.

Shufeng, how does your assertion add to the explanation of irreversibility?