2 hours ago, MigL said:

The ultimate bound state is a Black Hole.
BHs do not lower entropy; information is 'stored' in Planck-sized domains on the surface of the Event Horizon, in accordance to the Bekenstein-Hawking relation.
It is this dynamic equilibrium ( making Studiot happy ) that accounts for BHs having a characteristic temperature, and therefore, black body ( Hawking ) radiation, as there is a flux through the surface of the EH.
The 'final' equilibrium is when the BH evaporates.

Similarly, should protons decay into fundamental particles, so will neutrons as they have nothing to keep them happy in nucleii. So no atoms.
Yeah, no transistor radios at the end of the universe.

But this is all supposition we have no way of proving.

3 hours ago, MigL said:

Are we not discussing maximal entropy, as in the 'heat death' of the universe ?
A system in a state without flux of energy through its boundary, that is in its lowest possible energy would tend to, and eventually have no choice but, to be in equilibrium.
For such a state to be dynamic it would need an influx of energy.
For such a state to be unstable it would not be in its lowest possible energy.

The OP make no mention of entropy at all.

You introduced this in post#3.

Here is an example of what I mean about 'equilibrium'

Consider an infinite ocean with infinite atmousphere above it.

I stipulate that the system is in equilibrium.

By this I mean the is a measurable property that does not change with time.

This property is the relative humidity of the atmousphere.

But this is a dynamic equilibrium since motion has not ceased since there is molecular interchange between the air and the water, which can only occur if the molecules are in motion.

Furthermore this motion introduces self balancing momentum and kinetic energy exchange between the air and the water.

I also think it is wise to distinguish between heat and general energy.

Heat is a quantity that is transferred across a system boundary.

This is inherent in the original definition of both First and Second Laws.

General energy includes energy that is (cannot be) not transferred across a system boundary.

I'm afraid you may have misinterpreted what I'm insisting.

1 hour ago, sethoflagos said:

There are near-infinitely more ways of redistributing the hadrons of the earth-moon system than their current state.

I'm insisting that these energy states that the hadrons of the Earth could be redistributed in, are simply not available states, because the hadrons would require an input of energy to move to those infinitely many states.
Consider a book with 1000 pages.
There are many more ways to arrange the pages of that book so that they are not in order, than the one way in which they are ordered.
Yet when the book is 'bound', only the ordered state is available; to access the other state you need to supply energy, and rip out all the pages.

32 minutes ago, studiot said:

The OP make no mention of entropy at all.
You introduced this in post#3.

That is correct.
I am of the opinion that 'heat death' of the universe would imply maximal entropy, as there is no more useable energy.
( also, the thread was split after the fact, due to my mention of entropy with respect to time )

39 minutes ago, studiot said:

Consider an infinite ocean with infinite atmousphere above it.

If you need to start introducing infinities ...

But if you consider an actual system, even one as improbable as the causal, observable universe, parts of it are continuously moving out of that causal frame, and the 'spacing' of the frame is getting stretched by inflation.
Energy and mass are leaving the causal universe and radiation is stretching to longer wavelengths.
Now give it a Billion, Billion, Billion, Billion years.

What would happen to your infinite sea with infinite atmosphere if it had a causal boundary like the universe does ?

Edited Wednesday at 08:27 PM1 day by MigL

28 minutes ago, MigL said:

I'm afraid you may have misinterpreted what I'm insisting.

  2 hours ago, sethoflagos said:

There are near-infinitely more ways of redistributing the hadrons of the earth-moon system than their current state.

I'm insisting that these energy states that the hadrons of the Earth could be redistributed in, are simply not available states, because the hadrons would require an input of energy to move to those infinitely many states.
Consider a book with 1000 pages.
There are many more ways to arrange the pages of that book so that they are not in order, than the one way in which they are ordered.
Yet when the book is 'bound', only the ordered state is available; to access the other state you need to supply energy, and rip out all the pages.

Firstly, good response! Game on!

You introduced the term 'Maximal Entropy' without specifying what exactly you meant by this. There are as many different interpretations as there are permutations of possible constraints on system development, and external influence.

When @swansont interprets it as the maximum possible entropy available to n_i=1,k free particles of k different species at some arbitrary V, T, you challenge my response to him on the grounds that it is inconsistent with your preferred interpretation, which seems to constrain the system to one of bound particles in an effectively closed system.

This is hardly fair argument. You are trying to have your cake and eat it.

Please explain exactly what you intended by the term 'Maximal Entropy' so we can make some positive progress.

If you like, we can digress on the apparent symmetry of the evolutionary paths of system Free Energy and system Lagrangian. I don't have a good fix on that one and would appreciate a clear explanation from someone.

2 hours ago, MigL said:

I'm afraid you may have misinterpreted what I'm insisting.

I'm insisting that these energy states that the hadrons of the Earth could be redistributed in, are simply not available states, because the hadrons would require an input of energy to move to those infinitely many states.
Consider a book with 1000 pages.
There are many more ways to arrange the pages of that book so that they are not in order, than the one way in which they are ordered.
Yet when the book is 'bound', only the ordered state is available; to access the other state you need to supply energy, and rip out all the pages.

That is correct.
I am of the opinion that 'heat death' of the universe would imply maximal entropy, as there is no more useable energy.
( also, the thread was split after the fact, due to my mention of entropy with respect to time )

If you need to start introducing infinities ...

But if you consider an actual system, even one as improbable as the causal, observable universe, parts of it are continuously moving out of that causal frame, and the 'spacing' of the frame is getting stretched by inflation.
Energy and mass are leaving the causal universe and radiation is stretching to longer wavelengths.
Now give it a Billion, Billion, Billion, Billion years.

What would happen to your infinite sea with infinite atmosphere if it had a causal boundary like the universe does ?

You are still missing the core of my argument.

The infinity was just to avoid complication, take it out if you like, the core point still remains.

The point about the 'heat death' definitely refers to Second Law entropy reaching a stationary point, which may only be a local maximum.

This has wrongly been regarded as meaning that 'all motion ceases' and I think @julius2 is asking if that also implies that time looses its meaning.

This reinterpretation has been renamed 'the big freeze' in modern times.

The core of my argument is that there is no requirement per se for motion to cease so I agree with the original question that time does not loose its meaning.

My vapour pressure equilibrium example demonstrates how such a system can maintain time and motion and not die a heat death.

1 hour ago, studiot said:

The core of my argument is that there is no requirement per se for motion to cease so I agree with the original question that time does not loose its meaning.

If you go back to my 3rd post on the 1st page of this thread, @studiot , you will read me stating the same thing.
All motion cannot cease; it is still, after all, a Quantum system, and quantum Uncertainty has to be obeyed ( even that far in the future, and maybe only fundamental particles ); energy and time in particular, meaning time is still needed.

By maximal entropy, @sethoflagos , I mean the largest entropy available to the system at that time in the future.
And I propose that there will be no degrees of freedom, or independent states, that will be available for the system to occupy, at that time in the far future, as proton decay might be the very last source of available free energy, at that time.
Can you suggest another source of free energy, in that far future, that will keep processes going ?

As to whether this is equivalent to Maxwell Entropy, I'm not sure.
( Statistical Thermodynamics was hard enough without memorizing definitions )

1 hour ago, studiot said:

My vapour pressure equilibrium example demonstrates how such a system can maintain time and motion and not die a heat death.

Okay, a word on thermodynamic equilibrium and my cryptic invitation:

2 hours ago, sethoflagos said:

If you like, we can digress on the apparent symmetry of the evolutionary paths of system Free Energy and system Lagrangian. I don't have a good fix on that one and would appreciate a clear explanation from someone.

The thermodynamic Free Energy path leads ultimately to the well known partition function of an equilibrium canonical ensemble:

Z = sum(states) exp(-E_state/(k_BT))

The Lagrangian path leads to the general quantum path integral, which after Wick rotation can be expressed as:

Z = integral(paths) exp(-S_path/h) D(path)

The similarity is striking, and suggests some inverse correspondence between temperature and imaginary time along the lines of:

1/(k_BT) ~ it/h

I suspect this may be behind the recently announced unification of quantum and classical statistical mechanics though I've yet to get around to downloading that paper to confirm or otherwise. Intriguing though. (The embedding of maximum system entropy into the realms of quantum field theory, that is).

26 minutes ago, MigL said:

By maximal entropy, @sethoflagos , I mean the largest entropy available to the system at that time in the future.

By definition, thermal equilibrium at temperatures approaching absolute zero. I'll buy it.

32 minutes ago, MigL said:

Can you suggest another source of free energy, in that far future, that will keep processes going ?

Since I'm not going to buy into proton decay, then the very occasional collision of structures may continue indefinitely.