Surely the 2nd Law of Thermodynamics is defied by gravity?

As our sun and the other stars evolved from gravitationally led aggregations of hydrogen gas which permeated our early universe then that is an example of a high entropy system becoming a low entropy system and the so-called "arrow of time etc." was reversed?

jeffinbath said:

that is an example of a high entropy system becoming a low entropy system

No, it isn't. In the presence of gravity, a system of matter uniformly spread out with no variation in density has lower entropy than a system of matter clumped together into gravitationally bound objects. That is because the number of ways for the same amount of matter to be clumped together into gravitationally bound objects is much larger than the number of ways for that matter to be uniformly spread out with no variation in density.

The reason for the above is that number of states is counted in "phase space", not just physical space alone. In the presence of gravity, particles move to higher momentum when they fall in towards each other, so gain in momentum space the states they lose in physical space. But in actual practice, the increase in the former exceeds the decrease in the latter, which is why it happens, rather than the opposite happening.

Generally speaking, a good way to track entropy is to start with a situation where there is no change of entropy as a point of comparison. The situation where entropy does not change is any reversible change to a system that also has no net heat being exchanged with its environment. So that would be a gradual contraction in force balance with no heat being lost, which is something that basically does not happen (if it is in force balance, why would it contract unless there was heat being lost). So when you do see gravitational contraction, it is either happening out of force balance so is irreversible (and entropy increasing), or if it is in force balance (as with most astronomical objects), there is heat being lost to the environment (often in the form of thermal radiation emission). So when you see an object in force balance losing heat to its environment spontaneously, you pretty much know entropy is increasing there, by asking what it would need to look like for entropy to stay the same.

What's more, we should recognize that the increase in entropy that occurs with gravitational contraction is not some kind of coincidence. It is the reason the contraction occurs in the first place: it occurs because it is more likely that it will occur, and it is more likely it will occur because it increases entropy.

Ken G said:

Generally speaking, a good way to track entropy is to start with a situation where there is no change of entropy as a point of comparison. The situation where entropy does not change is any reversible change to a system that also has no net heat being exchanged with its environment. So that would be a gradual contraction in force balance with no heat being lost, which is something that basically does not happen (if it is in force balance, why would it contract unless there was heat being lost). So when you do see gravitational contraction, it is either happening out of force balance so is irreversible (and entropy increasing), or if it is in force balance (as with most astronomical objects), there is heat being lost to the environment (often in the form of thermal radiation emission). So when you see an object in force balance losing heat to its environment spontaneously, you pretty much know entropy is increasing there, by asking what it would need to look like for entropy to stay the same.

Am I drastically oversimplifying by interpreting this thusly:

• A system that is emitting heat is likely (must be?) increasing in entropy.
• A system that is not emitting heat is likely (must be?) not be increasing in entropy.

And the corollaries:

• A system that is increasing in entropy likely (must be?) emitting heat.
• A system that is not increasing in entropy likely (must be?) not emitting heat.

?

I must be, or you woulda said that.

DaveC426913 said:

Am I drastically oversimplifying by interpreting this thusly:

• A system that is emitting heat is likely (must be?) increasing in entropy.

The system itself that is emitting heat would lose entropy because of the lost heat, but the heat wouldn't be lost if it wasn't going somewhere with even lower temperature, so the entropy rise elsewhere more than compensates for the loss to the system.

DaveC426913 said:

• A system that is not emitting heat is likely (must be?) not be increasing in entropy.

Yes, if it's only doing reversible things, which pretty much means is in force balance. Most astronomical systems are in a very good force balance, because free fall timescales are generally much faster than the system is really changing.

DaveC426913 said:

And the corollaries:

• A system that is increasing in entropy likely (must be?) emitting heat.

Think the system and its environment. Entropy is rising if heat is being is being transferred from a higher to a lower temperature.

DaveC426913 said:

• A system that is not increasing in entropy likely (must be?) not emitting heat.

I must be, or you woulda said that.

If you just look at the system, then to not change its entropy, you need it to be in force balance, and also not exchanging heat at all with its environment. But what matters is the entropy of the system and its environment. If the environment has the same temperature as the system, then heat exchange in either direction won't change the total energy. So for example in a Carnot cycle, you maintain force balance, and you make sure everything is reversible by making sure the system always exchanges heat with an environment that is at the same temperature as the system. But that requires careful engineering not present in astronomical systems, and in the latter there is usually either a pretty significant temperature difference (like the Sun and deep space), or a pretty significant departure from force balance (like a giant molecular cloud collapsing into forming stars). Those are the (total) entropy generating mechanisms that gravity creates, even though gravity is reducing the spatial volume that the system occupies.

Ken G said:

particles move to higher momentum when they fall in towards each other, so gain in momentum space the states they lose in physical space. But in actual practice, the increase in the former exceeds the decrease in the latter

It's the other way around, isn't it? The gain from momentum uncertainty is less than the loss from position uncertainty.

Bandersnatch said:

It's the other way around, isn't it? The gain from momentum uncertainty is less than the loss from position uncertainty.

It's not so much an issue of the uncertainty principle, that's what sets the phase space volume of a single state, which is the reason the number of available states is proportional to the phase space volume. So whether the entropy rises or falls just depends on the phase space volume that each particle can access, and that depends on how the contraction occurs. If we want the system itself to not change entropy, we must make it reversible, so it has to stay in force balance, which means it will obey the virial theorem. Nonrelativistically, that means the internal kinetic energy will be half the (absolute value of) the potential energy. But if heat is not being lost (and no other changes are going on), the gained internal energy will equal all of the increase in (the absolute value of) the potential energy, not half of it. So requiring no heat exchange with the surroundings precludes the system from being in gravitational force balance.

In other words, we could start out with an undervirialized nonrelavistic system, in the sense that is has less than the necessary internal kinetic energy to be in force balance. It will rapidly contract and release potential energy, and eventually end up with the virialialized balance even if there is no heat loss. But since this would be happening out of force balance, it would be nonreversible, so entropy must increase. So we must have a situation where the volume has dropped, but the momentum of the particles has increased to conserve kinetic plus gravitational energy, and this must result in an increase in entropy because it goes one way but not the other.

The Boltzmann equation for gravitationally interacting particles doesn't work in the naive way. You have to bring a part of the collision kernel to the left-hand side of the equation by separating the long-ranged parts of the interaction by describing it by "self-consistent mean fields", i.e., the usual Vlasov equation. That explains why there's "structure formation" in the universe, i.e., if you have density fluctuations in the beginning, due to the long-ranged nature of the gravitational interaction these don't average out due to the long-ranged gravitational interaction. So if you have some larger density in some region, this will stay a larger-density region with time.

I have some trouble to relate this discussion to the microwave background.

Provided the theory of cosmic inflation is correct the CMB represents a tiny patch of space which has been in thermal equilibrium before inflation and has been in a state of high entropy then. What does this mean regarding the Entropy of the CMB where after Inflation only tiny patches of space can be in thermal equilibrium, but otherwise the whole area is within a few micro Kelvin? Did inflation change the entropy or did it remain the same?

Normally dynamical influences on the scale factor have no effect on entropy, because universal expansion acts the same as adiabatic expansion of a gas, which is entropy neutral. I don't know if inflation is different in that way, I wouldn't think so. It should also be noted that this approach treats gravity as having its own external dynamical description, but there are those who think that gravitational dynamics should have their own thermodynamics and would therefore also happen because of the way they increase entropy, but I don't think such theories have been fully developed yet.

One should note that "entropy" is more like a choice of how to treat a system, relating to choices about what you care to track and what you wish to treat as statistical, than it is like some real thing in itself. For example, classically the "true entropy" is always zero, the system is in whatever state it is in and that's it, so it is more our information about the system, and what we consider important to know about the system, that controls its entropy.

timmdeeg said:

the CMB represents a tiny patch of space which has been in thermal equilibrium before inflation and has been in a state of high entropy then

No. In the presence of gravity, "thermal equilibrium" in the sense of the matter or radiation being equally spread out everywhere at the same temperature is not high entropy. It's low entropy, because there's no gravitational clumping.

PeterDonis said:

No. In the presence of gravity, "thermal equilibrium" in the sense of the matter or radiation being equally spread out everywhere at the same temperature is not high entropy. It's low entropy, because there's no gravitational clumping.

Ah I see, thanks for clarifying.