Entropy of the Universe

Chestermiller

Studiot, I think you also meant to say that an isolated system is closed, so that no mass enters or leaves the system (dm = 0).

Studiot

Perfectly true but I'm not sure I needed to say that.

An isolated system is one in which nothing, neither energy (whether in the form of heat or work or whatever) nor matter (mass) crosses the boundary.

A closed system is one in which energy (work ,heat etc) may be transferred across the boundary but matter (mass) may not.

So, of necessity an isolated system is also a closed one.

Chestermiller

I'm not sure you needed to say that either. Still, when you were talking specifically about the definition of an isolated system, I though it might be worth mentioning. No big deal.

Jano L.

I understand your point; this is how the inequality is usually derived, for irreversible process. However, I think the process still has to be quasi-static as well. This is because we refer to entropy. Entropy is a function of macroscopic state (defined by U, V...). To a state which is not described by such set of values (non-equilibrium state), there is no way to ascribe entropy. Or is there?

This seems to be the case with the Universe; what macroscopic variables would you choose to describe its state?

Chestermiller

For entropy, the only two states that matter are the initial and final equilibrium states. These two states determine the change of entropy. There are an infinite number of paths that the system take to get from the initial to the final equilibrium state. These paths do not need to be quasi-static. If the path between the initial and the final state is reversible, then the heat flow entering through the boundary divided by the temperature at the boundary (integrated over time) will equal the change in entropy between the initial and final equilibrium states. If the path between the initial and the final state is irreversible, then the heat flow entering through the boundary divided by the temperature at the boundary (integrated over time) will be less than the change in entropy. The change in entropy provides an upper bound to the integral of the heat flow entering through the boundary divided by the temperature at the boundary, over all possible paths.

Jano L.

Of course. However, is Universe in equilibrium state?

Darwin123

According to most scientists, the observable universe is not in an equilibrium state. The observable universe can't be in an equilibrium state because it is expanding.

The universe started in a high density, high temperature and low entropy state. The universe in this state is commonly referred to as the Big Bang. As time goes on, it is asymptotically approaching a zero density, zero temperature and high entropy state. This asymptotic limit is often referred to as the Heat Death.

Entropy is a well defined quantity even in a non equilibrium state. However, it is defined in terms of a hypothetical series of reversible transitions.

The Zeno "paradox" seems to have resurfaced yet again. The OP has given an argument that entropy can't be increasing because any irreversible process can be broken up into a series of reversible processes. The argument is formally similar to the argument that nothing can move because each motion can be broken up into a series of not moving steps. This argument against motion was used by the classical Greek philosopher, Zeno. He was trying to prove that logic isn't sufficient in analyzing the real world.

The thermodynamics argument formally maps onto Zeno's paradox because on a subatomic level, all irreversible processes occur due to the motion of particles.

Lots of work has been done on resolving "Zeno's" paradox. Whether you except these "resolution arguments" is up to you. However, I will rephrase Zeno's paradox in terms of thermodynamics.

Any irreversible process lasting a finite amount of time can be broken down into an infinite series of reversible processes each lasting an infinitesimal amount of time. "Infinitesmal time" means "in the limit of zero time". Since each step is reversible, there should be no such thing as an irreversible process.

"Irreversible processes occur" despite this argument. Therefore, one can claim that logic doesn't work in thermodynamics. However, I present two argument against this conclusion.

"Infinitesimal time" is not the same as "zero time". An "infinitesmal time" is a hypothetically "the limit" as the time approaches zero. The mathematical meaning of "limit" has been analyzed extensively since the time of Zeno. "Limits" are mathematical and logically defined and analyzed. "Being "the limit" is not mathematically the same as "being equal to". So an infinite series of "infinitesimal processes" can add up to a "finite process" using strict mathematical formalism.

Another argument for logic is that energy and momentum is quantized according to quantum mechanics. Therefore, there really is no such thing as an "infinitesimal reversible process".

Both arguments are anti-intuitive to many people. The last is really a cheat since no self consistent quantum mechanical theory has been found consistent with astronomical observations. However, I am done if you agree that the problem is a modified version of Zeno's paradox.

I think an in depth discussion of Zeno's paradox would within forum guidelines as long as it took place in the mathematics part of the forum. There are a whole bunch of scientific problems that can be mapped onto Zeno's paradox. Some people reject biological evolution on the basis of Zeno's paradox. We see here that someone has a problem with thermodynamics due to Zeno's paradox. Since practically every branch of science uses the mathematical concept of "limit", Zeno's paradox can turn up anywhere.

So the problem is the concept of limit, not with thermodynamics.

Studiot

There are several Zeno paradoxes.

The resolution of the one about the semidistance relies on the limit

[tex]\mathop {\lim }\limits_{n \to \infty } \frac{1}{{2n}} = 0[/tex]

Which leads to the sum

[tex]\sum\limits_1^\infty {\frac{1}{{2n}}} = 1[/tex]

What particular limit are you referring to?

Darwin123

I am talking about the limit corresponding to the distance an arrow travels, or the one about Achilles and the tortoise.

I am not sure what the corresponding mathematical expressions are.

In any case, obviously entropy is increasing in the universe. The universe is "quasistatic" only over very small time intervals. It is not static.

The sum of a large number of short interval irreversible processes is an irreversible process. A short

Jano L.

This is not true, if we want to use the words reversible and irreversible in their standard meaning in thermodynamics. Thermodynamically irreversible process cannot consist of thermodynamically reversible processes. It can only consist of microscopically reversible processes, but this is irrelevant, since there is no entropy on the microscopic level of description. I do not see any connection to Zeno paradox here.

The term "quasistatic process" does not only mean that the process is slow. It requires that the process be a sequence of states very close to thermodynamic equilibrium. Now observed part of the Universe is not in such equilibrium as a whole; its state is not described by few thermodynamic parameters.

If we want to talk about entropy of Universe, we have to define it first, and the standard sdefinition [itex]\Delta S(1~to~2) = \int_1^2 \frac{dQ}{T}[/itex] does not apply for the Universe; there is no 1 or 2 and there is no [itex]T[/itex].

Studiot

Hello Jano, why can Darwin not use the other standard definition of entropy?

S(E,V,N,α) = k lnΩ(E,V,N,α)