The total entropy of the universe is not constant and there is no heat exchange with the "outside" that we know of. The entropy of the universe in the distant past was much lower and has been increasing ever since. The second law of thermodynamics states that the entropy of an isolated system always increases. Isolated means no heat flow in or out. In fact, many physicists would say that the "arrow of time" in the universe comes from the increasing entropy.

Also, in the paper, we are clearly saying that the cosmological term (which Einstein called the cosmological "constant") is not constant, nor are Einstein's equations "accepted", they are generalized. That is the whole point of the paper.

Actually you are correct in that the holographic principle says that the information contained in a volume is encryted on the enclosing surface. However, it is not trivial. It began with black holes where it was found that the maximal entropy inside a volume scales as the radius squared, rather than the radius cubed as you would expect. Also, in the case of black holes we had the "black hole information paradox" which was essentially that since matter falling into a black hole can never emerge again and must instead, necessarily fall into the singularity, the "information" contained in that matter vanishes from the universe. The holographic principle says that the information is not lost, but resides on the event horizon. This principle, which was inspired by black hole thermodynamics, was then extended to the entire universe where it says that the maximal entropy, or information, contained in any volume of space also scales as the radius squared (rather than cubed as you would expect). By "information" we mean all matter and energy in the universe. (here is a recent paper here on ResearchGate regarding the large scale structure of the universe and an extension of holographic principle: https://www.researchgate.net/publication/46583938_Holographic_principle_and_large_scale_structure_in_the_universe)

Now, with respect to the paper you cite above, the argument is essentially that the holographic principle says that the maximal entropy contained in any volume of space is encoded in Planck-sized cells on its boundary (which is what the usual holographic principle says), however, since the universe is expanding, whereas the size of bound states, such as particles (and the size of the Planck length) is not expanding, it means that the number of Planck sized cells on the boundary of any region must be increasing! Hence the entropy of the universe must be increasing with the expansion. We called this dark entropy.

We then showed that this entropy acts to drive the expansion, whereas the energy content inside the universe acts gravitationally to try and slow the expansion. Normally, the matter content inside a volume of space tends to contract -- in some cases all the way to a black hole, where an event horizon forms at exactly the place where the entropy content of the surface equals the matter/energy content inside.

In the case of the universe as a whole, it began with a big bang, and hence a changing entropy content and an expansion. We showed that at a certain point in the expansion, the entropy begins to dominate the energy content and causes the expansion to accelerate rather than decelerate as it has done up to that point. We compared these results with the recent experimental results showing an accelerated expansion and showed that they were consistent with each other. In other words, we showed that the universe has just tipped over from an energy dominated universe to an entropy dominated one and a corresponding accelerated expansion.

Article Holographic Principle and Large Scale Structure in the Universe

Look at arXiv:1212.1711 [physics.gen-ph].

Once you cross the black hole horizon, time becomes space and space becomes time, so your argument is not really applicable.

The holographic principle together with the putative expression of a black hole are wrong (arXiv:1110.5322). The paper referred to has technical problems: their expression for the first law is a combination of the first and second laws commonly referred to as the Gibbs equation. According to the Bekenstein expression for a black hole, the energy varies inversely with the temperature, and the entropy varies as the square of the energy. Hence, there is a clear violation of the third law. The authors of the paper claim that as K (proportional to the absolute temperature) tends to zero so does the entropy. This is what one would expect from Planck's formulation of the third law, but not from Bekenstein's expression. It is also absurd to associate an entropy density with a (cosmological) constant. The entropy density of the universe varies but the entropy is constant (at least if Einstein's equations are accepted) since there is no exchange of heat with the "outside". Moreover, there is no bound on the entropy set by the black hole expression because the latter is a convex function of the energy while all known entropies are concave functions of the energy. The error in the demonstration of the Bekenstein limit is assumption that the logarithm of the partition function vanishes for a certain value of the inverse temperature. An entropy bound is also in conflict with Boltzmann's "heat death", and Clausius' principle that the entropy tends to a maximum with respect to the constraints imposed on the thermodynamic system. I explain all this in http://www.youtube.com/watch?v=qoK1H1QhJVM

The total entropy of the universe is not constant and there is no heat exchange with the "outside" that we know of. The entropy of the universe in the distant past was much lower and has been increasing ever since. The second law of thermodynamics states that the entropy of an isolated system always increases. Isolated means no heat flow in or out. In fact, many physicists would say that the "arrow of time" in the universe comes from the increasing entropy.

Also, in the paper, we are clearly saying that the cosmological term (which Einstein called the cosmological "constant") is not constant, nor are Einstein's equations "accepted", they are generalized. That is the whole point of the paper.

The entropy of an isolated system cannot increase. Planck's interpretation was to allow for fluctuations, but that is not macroscopic thermodynamics. A temperature cannot be defined for isolated systems. There is no second law for such systems: no heat exchange therefore no entropy. Clausius' inequality applies to the case where the heat is rejected at a higher temperature than that given by the sum dQ/T=0. There is no "arrow of time" for an isolated system.

Einstein's equations demand adiabaticity. Where is the adiabatic assumption removed? What looks like Gibbs equation does not mean that it is. It could be a work term and not TdS. There is no compelling reason for the association, and setting K=0 implies the entropy term vanishes is in contradiction with the Bekenstein-Hawking expression for a black hole which, in turn, is in contradiction with the third law (Planck's formulation).

Sanjay, just look at eqn (17-6) in H. P. Robertson and T. W. Noonan "Relativity and Cosmology" to see that it is Einstein's equations that tell you the universe is adiabatic. There is no generalization of Einstein's equations that would allow you to remove the adiabatic condition--it's built in. I'm not saying that Einstein's equations have anything to do with cosmology, but there is no getting around the adiabatic condition if you adopt the Einstein equations.

Bernard, you write`` The entropy of an isolated system cannot increase''

Consider the counter example of a freely expanding gas which was originally confined to a small section of a rigid container (see, for example, page 192 in H.B. Callen ``Thermodynamics ..."). One assumes that no particle can penetrate the walls and no heat or work can be exchanged with the outside, but yet, one can still calculate the entropy increase in this process, in agreement with the relation dS>=dQ/T where dQ=0 here. Can you explain to me why this reasoning can't be applied to the entire universe? Are you defining entropy differently?

Yes the Universe is (as far as we know) adiabatic, since there is no "outside" that we can exchange heat with. But that does not mean that the entropy of the universe is constant. In fact, the bulk of the increasing entropy going on around us is the increase due to irreversible processes. I can exemplify all of this using an isolated box as an analogy. Take an isolated system consisting of a rigid box (no transfer of mass or heat in or out) and begin with a bunch of gas particles confined in one corner. Then let them go. We can easily calculate the increase in entropy and no heat has flowed into or out of the box. Now think of the universe as a box, and us as the little bit of gas in the corner.

Thomas, what you are doing is comparing two states of equilibrium: a more and a less constrained states of equilibrium, and you notice that the entropy has increased in the less constrained state of equilibrium. This has nothing to do with the putative expansion of the big bang. There are no two states of equilibrium with the same volume and number of particles, etc., which can be compared.

Not worrying about how such a large energy source could be concentrated at a single point, you apply Einstein's equations to the expansion. Substitution of the stress-energy tensor into the Bianchi identity gives you dE+pdV=0, i.e. Q=0 for no heat can enter or leave the universe. Then all you get are the adiabatic relations between volume and temperature depending upon whether you are treating radiation (ultrarelativistic) or matter (nonrelativistic) dominated universes.

It is for this reason that the inflationary scenario is conceptually wrong: you can't get an increase in entropy using Einstein's equations by considering a negative pressure. In fact, negative pressures cause implosions not explosions!

Thomas, there is another point worth considering. You claim that dS>=dQ/T and that dQ can be zero and you still get irreversibility in an isolated system. However, from the first law dE=dQ+dW, and if dQ=0 you would conclude that the process is adiabatic so you would never get the Gibbs relation, dE=TdS+dW. In fact, as I show in my book, "A New Perspective on Thermodynamics" (Springer, 2009), it is a matter of choice whether you include irreversibility into the second law or into the first law. There is no clear cut distinction between the two laws as Clausius would have us believe.

Bernard, let me first reply to your earlier post: I agree that in thermodynamics textbooks free expansion is treated by comparing the entropy of two equilibrium states. However, one has to be careful with the different definitions of entropy.

The entropy we usually calculate in thermodynamics is the Clausius entropy S_C, which is only defined in equilibrium.

Unfortunately, while the concepts of irreversibility and thermodynamic entropy

were already decently well understood at the end of the 19th century by people like Boltzmann, Maxwell, Kelvin etc., confusion came up in the following 100 years. For example, you're probably referring to the Gibb entropy (which is defined

for statistical ensembles and not for individual microstates) and you're correct that

S_G stays constant for an isolated system following Hamiltonian dynamics.

However, S_G is not the correct entropy for understanding the time evolution of macroscopic systems, i.e. it's not the one the second law is referring to.

One has to use Boltzmann's entropy S_B=k*ln(Omega) which also works out of equilibrium and can be time-dependent.

Only in equilibrium all definitions agree, S_C=S_G=S_B.

Boltzmann found that S_B can be expressed in a simple form using the one-particle density f for dilute gases where pre-collisional correlations can be ignored and only binary collisions are relevant.

This formula looks almost identical to S_G which might have started confusions.

He showed that S_B is increasing, if started with ``typical'' initial conditions (Boltzmann's H-theorem, it will not always increase, but the likelihood for anti-thermodynamic behavior goes exponentially to zero with particle number)

To modify our little thought experiment: Assume, a low density gas is confined to the corner of a container and assume that the gas is not initially in a pathological microstate.

Now, remove both the constraint and the container to start completely free expansion. The expansion can be calculated by solving the Boltzmann equation (for example by using Bird's method - Direct Simulation Monte Carlo) and you can monitor the time-dependance of Boltzmann's nonequilibrium entropy. It will increase.

There is still weak points in this argument, for example the assumption of Molecular Chaos in the derivation of the Boltzmann equation is not exactly true, there is correlations which build up to the so-called long-time tails and to have this argument work for the entire universe one has to assume that the universe started at a state of very low entropy.This leads then to the hard question of why the universe had such an initial state and there is arguments by Penrose involving

gravity about a low entropy but ``natural'' initial state.

There is nice articles by JL. Lebowitz and E. Goldstein explaining these issues in detail, see Physica A 263 (1999) 516-527, cond-mat/0105242v1 and cond-mat/0304251v2.

Bernard, to your second post: In free expansion, we have dQ=dW=dE=0 thus

its adiabatic and the first law is trivially true 0=0+0. However, its an irreversible adiabatic process, that is dS>dQ/T, and hence it is not isentropic.

Since its a nonequilibrium process we just cannot use the relation dQ=TdS thus

we must apply the first law in its general form dE=dQ+dW, since neither dQ=TdS nor

dW=-pdV are valid under irreversible changes, that is under nonequilibrium conditions.

Thomas, I agree there is a great deal of confusion about entropy in the literature. The Boltzmann entropy you are referring to is what Khinchin denotes as the logarithm of the structure function [Khinchin, Mathematical Foundations of Statistical Mechanics, Dover, p. 37]. It defines the entropy in terms of the mechanical variables of E,V,N and corresponds to the microcanonical ensemble. [Khinchin makes the claim that the volume of phase space cannot increase faster than a finite power of the energy. This can be relaxed in the case of black body radiation where it increases exponentially to a power of the energy less than one. The Bekenstein-Hawking expression for a black hole takes the power to be 2, and this violates the concavity property of the entropy. This is the seat of the negative heat capacity and violates the second law.]

However, Boltzmann's principles does not correspond to any probability distribution; it is incomplete because it employs the assumption of a priori probabilities. [This assumption is contradicted by the end result that the probability distribution is an exponential decreasing function of the energy, which is derived by adding the superfluous energy constraint, in addition to the number constraint, in maximizing the multinomial coefficient or the number of ways particles can be distributed among urns. The urns do not have energy labels. ] Boltzmann's principle can be "completed" and transformed in a bona fide probability distribution by using Gauss' error law [B. H. Lavenda, Statistical Physics: A Probabilistic Approach, Wiley-Interscience, Ch. 1,4. It won't work for black body radiation because the equation of state is not logarithmic.]

As regards your second post, I don't see what's wrong with dW=-pdV for irreversible changes [B. H. Lavenda, A New Perspective on Thermodynamics, Springer, Ch. 5]

Unfortunately, there is much to criticize about how irreversibility is handled in nonequilibrium thermodynamic formulations [B. H. Lavenda, Thermodynamics of Irreversible Processes, Dover].

Sanjay, the heat content, or enthaply, is E+pV. Q is the heat absorbed or rejected by the system to an appropriate heat reservoir. The phase space volume W is the product of the configuration space volume V and the momentum phase space, proportional to the cube of the momentum. In the case of black body radiation V is fixed (the cavity) and it is the momentum space volume that is related to the electromagnetic waves in the cavity that increases with frequency. Heat is required to to keep the walls of the cavity at a constant temperature. [By the way, black body radiation cannot be used for black hole evaporation. Black body radiation is a reversible process, black hole evaporation is not.]

What is it for the big bang? The configuration volume is ever increasing, and it is for this reason that there is no definition of a thermodynamic system. All we have is an adiabatic condition from the first law, with Q=0, relating temperature to inverse to the radius of the volume (radiation) or to the square of the radius (matter). To say that the universe started in a low entropy state would necessary imply that the entropy density is for ever decreasing if the universe is expanding. This is the reason why inflation is conceptually flawed. There is no source of heat that would lead to an exponential growth in time in an adiabatic expansion. When inflation invokes Einstein's equations with a negative pressure, it doesn't realize that Einstein's equations require Q=0. This results in a physical absurdity: negative pressure causes implosion---not explosion.

Well, Sanjay and Bernard, I should get back to my own research since time is running and the entropy of the universe is increasing the minute I write on here :-) but I can't resist to point out some misconceptions. @Sanjay: I agree for the most part but adiabatic does not always mean isentropic, adiabatic just means no heat exchange, dQ=0, but S can still change as in free expansion, dS>0. @Bernard, let's not talk about black holes and the universe before much simpler examples like the ones I mentioned are agreed upon. The interpretation of Boltzmann's formula by Lebowitz is not restricted to specific ensembles or to equilibrium, it just says: tell me the macroscopic variables you think are important in a system and divide phase space accordingly into corresponding subspaces, then we find out how many microscopic states are able to reproduce each of these states and the entropy of a given macroscopic state is then more or less the logarithm of the number of microstates consistent with it.

Here is an interesting review article that came out today regarding gravity and its relation to thermodynamics. http://arxiv.org/abs/1303.1535

Let's face it, most of these arguments are quotes from old "classic" textbooks. Yes, statstical mechanics does provide a view which may be interpreted as a relation between maybe: "entropy and gravity".

Been there, done that, it's a waste of time.

The real meaning of the second law is that the total entropy production diS always increases, i.e., diS>=0, and It is not essential to consider the adiabatic condition δQ=0, or isolated system for diS>=0.

Universe, black hole and… are the generalized thermodynamic system, for these system, the equations of thermodynamics must be written with a complete form, and the equation of the second law does not involve a factor in the expansion of universe, unless one redefine the entropy.

The concept of entropy arose historically phenomenological way in thermodynamics. Pattern was seen that if the work is performed on the body, a part of it in any case, from the standpoint of useful work, is lost. This part is spent on heating. In line with this Clausius entropy is defined as the ratio of the increment of thermal energy to temperature. The fact that entropy can only increase, called the second law of thermodynamics.

In consequence of the Boltzmann in the framework of the molecular-kinetic theory was given to the statistical definition of the concept of entropy by linking it with the number of states of the system. There are other definitions of entropy, for example, KS-entropy.

Although these definitions anyway linked, each of them has its own area of application. All it means is that while the concept of entropy is in the development stage. Difficulties in the development of this concept, most likely related to the fact that in physics, there is still no clarity about the nature of the symmetry breaking time in nonequilibrium dynamical systems. Indeed, Newton's equations of motion for material points are reversible, but in reality, the dynamics of system of material points is irreversible.

In our opinion, the solution to this problem is to use instead of the equation motion of a material point, the equation of motion of the equilibrium system of the material points (structured particles). But the latter is necessary to obtained from the principle of the dualism of energy, considering it as the sum of the energy of motion and internal energy of the system. This will avoid the use of the hypothesis monogeneity forces. Then the entropy can be associated with the transformation of the energy of motion of the equilibrium system in its internal energy of the motion in a non-uniform force field [Somsikov V. M. Nonequilibrium systems and mechanics of the structured particles. Elsever. Chaos and Complex system. 2013, XV, 581 p. 31-40].

Entropy: A concept that is not a physical quantity

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