L.D. Edmonds All measurement "devices" collapse quantum wavefunctions by putting the quantum system that is being measured in contact with the "devices" thermal bath. The quick answer to your question is the thermal bath entropy of any measurement "device."

Thank you George Soli . Great answer. The next step though is going to the classical limit. We know that clocks that we have at home and carry around with us agree with each other (under those conditions for which relativity allows for agreements) so they all seem to have the same arrow of time. This implies some kind of consistency regarding the time direction of increasing entropy. Is there an explanation for that consistency, in view of the fact that some subsystems can have decreasing entropy compensated by increasing entropy in other interacting subsystems?

Hi L.D. Edmonds !

You are asking a deep question.

The second law of thermodynamics states that the entropy of ISOLATED SYSTEMS is non-decreasing in time. Of course, the only truly isolated system is the Universe. However, in practice, you can create "almost isolated" systems, by putting a chunk of matter inside an ideal box with perfectly reflecting walls. Actually, one can do even better. Imagine having an isolated system comprised of a hypothetical mechanical harm with no microscopic degrees of freedom, plus your material of interest. Then, the second law for the total system "harm+material" implies that the entropy of the material alone is non-decreasing (since the entropy of the harm is zero by construction). Since the evolution of the material should not depend on the detailed structure of the harm, we have discovered that the entropy of THERMALLY INSULATED systems is non-decreasing in time. Here, "thermally insulated" means that the interaction with the environment happens only through macroscopic mechanical work (no heat).

If you like having rigorous mathematical definitions of things, here there are.

ISOLATED SYSTEM: A system to which you can associate a time-independent Hamiltonian.

THERMALLY INSULATED SYSTEM: A system to which you can associate a (possibly time-dependent!) Hamiltonian.

You can find these two definitions in Landau-Lifshitz, volume 5, section 11.

The proof that a system with a time-dependent Hamiltonian obeys the second law was given by Jaynes in section IV of this beautiful paper https://bayes.wustl.edu/etj/articles/gibbs.vs.boltzmann.pdf

I remark that the second law only specifies the DIRECTION of growth of time, and not its rate. The rate at which time flows is dictated by the fundamental constants of nature (e.g. have a look at the modern definition of 1 second).

I hope this helps!

Thank you Lorenzo Gavassino . I need to study the link you sent but look forward to it because a proof that a time-dependent Hamiltonian produces the second law is very profound, I think.

Maybe this is starting to come together. One of my posts asked George how to explain consistency with different clocks that we use on a macroscopic level even if a clock is in a subsystem with decreasing entropy (not an isolated system, the environment has an increasing entropy). I think George already answered this. It is not the entropy of the subsystem that matters, it is the entropy of the clock that matters. The question of consistency now becomes a question of consistency when comparing the time-direction of increasing entropies of the different clocks. This consistency is established (I guess) by time-dependent Hamiltonians producing the second law (per a link from Lorenzo).

Well. I think one needs to be careful here.

An ideal clock must be governed by dissipationless evolution. For example, Einstein's clock is a photon bouncing between two perfect mirrors. Another possible clock is a friction-free pendulum. You need to assume that no irreversible process occurs in the clock, otherwise after a while the clock stops (no perpetual motion). This is true also for our standard mechanical or digital clocks: you don't want them to get old, because they would get worse and worse until they stop.

Therefore, a genuine idealized clock knows nothing of the second law! For this reason, a time-reversed clock works just as well as the same clock before being reversed. Their only task is to split the continuum of events into a discrete sequence of countable intervals, and both are equally good at it.

You only need the second law of thermodynamics to break the symmetry between past and future, not within the clock itself, but within the whole THERMALLY INSULATED system that contains the clock.

To be concrete. Consider a sealed (i.e. adiabatically insulated) room, and there is a clock in the room. Independently from thermodynamics, the "tic-tic-tic..." of the clock defines the length "Dt" of an interval of time. The sign of the total change in entropy DS of the WHOLE ROOM defines the orientation of this interval (i.e. whether this Dt is plus or minus Dt). If no irreversible process occurs in the room (i.e. DS=0), there is no difference between past and future, and you can pick whatever direction you want, since the system is time-symmetric. For example, a system in full thermodynamic equilibrium does not know the difference between past and future, and the orientation of time is a mere convention, with no physical consequence.

Thank you Lorenzo Gavassino for the correction and for explaining it so clearly. I am getting closer to an understanding. I think the concepts are different on a classical level (ideal instruments) than on a QM level. I like George's QM-level explanation. This might explain why a macroscopic quantity like time appears in microscopic QM equations. I'm still a little fuzzy on the classical level but I think that your explanation "You only need the second law of thermodynamics to break the symmetry between past and future, not within the clock itself, but within the whole THERMALLY INSULATED system that contains the clock." is a key point. I need to think about it some more to completely get it.

What I learned from Lorenzo on the classical level is:

"You only need the second law of thermodynamics to break the symmetry between past and future, not within the clock itself, but within the whole THERMALLY INSULATED system that contains the clock."

Let me try to figure the next one out myself. Consider a more common clock instead of an ideal clock. One kind runs on a battery. A battery is a source of irreversibility. Another kind uses a wind-up spring. Damping forces that make ratcheting work are sources of irreversibility. If one of these clocks is put in an isolation room, the prediction is that it will continue to run in the same direction even if the room contains no other sources of irreversibility. The prediction agrees with common-sense expectations. I like that. I also like George's QM version.

The thread question

“The entropy that controls the arrow of time is the entropy of what physical system?”

- when yeah, in mainstream physics indeed a tenet that “the arrow of time is controlled by increasing entropy” appeared yet in 1800s and exists now, despite that a physical thermodynamic parameter “entropy” by any means differs from other physical parameters in that by some mystic way entropy has some relation to the fundamental phenomenon/notion “Time”, while other parameter [what is true] don’t have.

This strange tenet exists in physics since in the mainstream philosophy and sciences, including physics, all really fundamental phenomena/notions, first of all in this case “Matter”, “Space”, “Time”, “Energy”, “Information”, and so everything in Matter, i.e. “particles”, “fundamental Nature forces” – and so “fields”, etc., are fundamentally completely transcendent/uncertain/irrational, and so logically inevitably if physics addresses to some correspondingly fundamental points, the result is something transcendent, in this case – to entropy some mystic impact on time is assigned.

The fundamental phenomena/notions above can be, and are really scientifically defined only in framework of the Shevchenko-Tokarevsky’s really philosophical 2007 “The Information as Absolute” conception, recent version of the basic paper see

https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed

- and more concretely in physics, how “Space”, “Time”, “Energy”, “Information” are actualized in Matter , in the SS&VT Planck scale informational physical model, 3 main papers are

https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics

https://www.researchgate.net/publication/367397025_The_Informational_Physical_Model_and_Fundamental_Problems_in_Physics

https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force

More about what is Matter see the links above, here only note that some Matter’s “arrow in time” indeed fundamentally exists, because of all/every material objects fundamentally constantly and always change their states, mostly because of the energy conservation law,

- whereas by scientific definition of “Time” this fundamental phenomenon is actualized in Matter [and in whole “Information” Set, though] first of all as one universal for everything “time dimension”, so that at every change of/in anything, including clocks, this anything moves in this dimension on some “time interval”; clocks are specific humans-made material structures that change their states regularly, and have some pointers that show for humans their stable changings.

Eventually everything in Matter exists and happens as always constantly moving disturbances in the Matter’s ultimate base – the (at least) [4+4+1]4D dense lattice of primary elementary logical structures – (at least) [4+4+1]4D binary reversible fundamental logical elements [FLE], which is placed in the Matter’s fundamentally absolute, fundamentally flat, fundamentally continuous, and fundamentally “Cartesian”, (at least) [4+4+1]4D spacetime with metrics (at least) (cτ,X,Y,Z, g,w,e,s,ct), FLE “size” and “FLE binary flip time” are Planck length, lP, and Planck time, tP.

So Matter as a whole constantly and always moves in the time dimension Planck time interval-by-Planck time interval; rather probably being always in one Planck time interval;

- and this motion fundamentally by scientific definition of “Time” is only in positive time dimension direction, any motion “back in time” are fundamentally impossible; and just that is real “arrow of time”.

Cheers

If its about "what physical system?" then it is necessary to keep "physical parameters" first instead of using abstract units(either clock or time) for analysing the results. We should use "entropy" itself instead of "time".We should use term increasing or decresing instead of "controlling".

Also,"thermodynamic arrow of time" may or may not be same to "cosmological arrow of time".

As per discussion there is no provided or said boundary conditions associated with other things given. We even do not know it is related to direct or indirect transformation.So it is necessary to consider some boundary conditions or particular example first.Any kind of undefined parameter or direct transformations both things are barrier, here.

Only as per defined boundary conditions(parameters)that any physical system carries further,we can say the entropy that controls the arrow of time belongs to which type physical system(local, non-local or extended).

Else,here we can only say "entropy is part of any undefined system" which is receiving or transfering it mystically further in any subsystem.

When we say arrow of time is controlled by increasing entropy then it simply means,flow of system energy(either molecular, atomic or beyond) is "redefined" for particular period of time and observer.While,there can be many timelines or observers as per boundary conditions itself.

If its about entropy then its about physical processes and their correlations between different spacetime as per needs.

Also, instead of considering any kind of entropy directly part of any subsystem it is necessary to define its affecting parameters first with main physical system.

I would be really happy with the idea that time has its own existence (nothing to do with entropy), but entropy increases (in spite of microscopic equations in physics being time-reversible) because initial conditions of the universe started with such a small entropy everywhere throughout the universe so that its approach to equilibrium has the entropy increasing in all isolated systems. But I don't think that is the mainstream theory, and I'm not qualified to assert that one theory is correct and not the other. I know from previous experience with RG threads that people like to challenge mainstream theories. Since I am on the fence on this, I will enjoy reading both pros and cons regarding the mainstream theory.

“…I would be really happy with the idea that time has its own existence (nothing to do with entropy), but entropy increases (in spite of microscopic equations in physics being time-reversible) ….”

Again, see SS post on page 1 – “entropy” is nothing else than some parameter of some physical systems, and – as all other parameters of the systems – exists only in humans’ theories, while everything, including humans, absolutely fundamentally exists at every time moment – i.e. in some concrete point of absolutely infinite time dimension; and at every change moves from given point to other point on some “time interval”,

- at that, because of the self-inconsistence of “Logos” set element “Change”, these intervals are always more than zero, and positive by definition of the “Logos” element “Time”.

Since in Matter everything always and constantly changes, everything, and Matter as a whole, always and constantly moves in time dimension, at that moves mostly in given state into most probable state, so, say, deterministic systems are changing with probability =1, random systems are changing in concrete cases/time moments with other probabilities – including that are described by entropy,

- but, again, “describe” and “control” are quite different notions, including in this case entropy describes, but fundamentally by no means controls, the “time arrow”.

At that practically only in mechanics processes are reversible, say, when one photon in a nucleus electric field creates e± pair, the pair annihilates creating two photons, and that is only a simple example of innumerous other ones.

Real physical task is description of what exists and happens in Matter, for what it is necessary to discover/know/use much more traits and parameters of material objects/systems, as that was discovered after middle of 1800s, when entropy was really defined only for rather simple thermodynamic systems, and so some mystic “thermal death of Universe”, “entropy control of time”, etc., appeared.

Really evaluation of entropy even in rather simple systems is rather hard task – while the answers really mostly have negligible physical sense, and so really, say, in reliable cosmology entropy really isn’t used at all.

Though till now in some physical communities entropy conserves the mystic 1800s flair, in mainstream philosophy the “terribly scientific” and “weird” word “entropy” is very popular, etc., but that isn’t a unique strange thing in mainstream science.

Cheers

Thank you Sergey Shevchenko but I don't understand what you said. I have respect for mainstream science, even though some topics can be controversial, and I can understand discussions of controversial topics better if the discussions use the vocabulary of main stream science (e.g., entropy is recognized as a definable quantity). Admittedly, that is my limitation.