Exploring the Link Between Entropy and the Arrow of Time in the Early Universe

[MatthewKM]

TL;DR Summary

Arrow of time and Entropy

My understanding is that a sub atomic particle has no arrow of time. Clearly the Arrow of time as understood in our macroscopic world has one direction. Entropy being an arrow in the direction toward uniform distribution of the individual and or sums of all the forms of energy in a system and , I suppose ultimately the uniform distribution of a final form of energy, begs a number of questions.

Starting from Hot Dense and Smooth at the beginning of the universe: where in physics is the convergence of the notion of Entropy and the Arrow of time?

Or put another way: assuming uniformity, where in the subatomic/nuclear/atomic scale is the “line in the sand” where Entropy has no arrow of time? ( Quarks and Gluons, Nucleons, Plasma, Atoms, Molecules?)
If that could be answered, then was there a point in the early universe before which (and the use of the word “before” is sysyphean so no need to comment) the notion of entropy is an incorrect or confusing way of referencing the early universe?
Is this “Sub Atomic Frame” fixed regardless of what model of the universe one holds with? (Inflation, Multi universe, eternal inflation, etc) Or does the Subatomic Line in the sand “smear or blur” depending.

Matt

 

[PeroK]

I'm not sure whether it answers your question, but consider mixing hot and cold water. The processes at the molecular level are reversible; whereas, the macroscopic process of ending up with uniformly warm water is not reversible.

The irreversiblity in this case is a consquence of the statistical laws of a large number of particles.

You can see similar phenomena in computer simulations where the irreversiblity is fundamentally statistical.

 

[anorlunda]

Large numbers of particles have properties that emerge only with large collections.

Following @PeroK 's hot/cold example, it would not violate the laws of physics for the water to unmix itself, or for your coffee and cream to unmix themselves into black and white. But it is probable that they won't do that. Entropy and the second law are based on statistics. Statistical mechanics is the name of the branch of physics that deals with such questions.

By the way, temperature is another of those emergent properties. It is meaningless to try to use the concept of temperature for a single particle. But for large collections of particles, temperature is proportional to the average kinetic energy.

 

[MatthewKM]

Well yes and no. Your response in a way is a restatement of my question without answering it but it is asking a question analogous to a later lumpier time in the universe. Re framing it Let me mix two cups of entropy of different amounts of time from a final state and what happens They intermix to homogeneity then resume their collective march toward maximal entropy. I'm asking a question about the irrelevance of the arrow of time inherent in sub atomic particle given that ( in an unchanging heat state) sub atomic particles look the same and behave the same regardless of where or when they are and therefore a mass of identical sub atomic particles in a uniform heat state has no arrow of time as it has no entropy to move toward. That said, where is the sub atomic particle size where this applies? The quark and gluon, the nucleon, the atom, molecule etc?

 

[anorlunda]

I'm asking a question about the irrelevance of the arrow of time inherent in sub atomic particle

It's not that straight forward. Time reversibility at the quantum level is closely associated with unitarity. Yet, there are claims that quantum mechanics also includes entropy and the 2nd law.

https://en.wikipedia.org/wiki/Quantum_statistical_mechanics
https://en.wikipedia.org/wiki/Quantum_thermodynamics#Emergence_of_the_second_law

Beware, the math is not easy to follow. They are too difficult for me.

 

[MatthewKM]

Surely yes. But is not a unitarity a "time independent" Hamiltonian? I can't follow the math well but I'm getting better by the week. Never straight forward for certain.

Entropy being the inexorable change of a system "over time" toward the most uniform possible distribution of energy, the physics (quantum or otherwise ) of a system of identical sub atomic particles becomes indistinguishable running forward or backward through time and thereby invalidates or stops time. So does the physics of entropy apply during the ambiguity of this state? I imagine perturbations as the beginning of the universe (and time).

 

[PeterDonis]

is not a unitarity a "time independent" Hamiltonian?

No. Any Hamiltonian is Hermitian, meaning that the time evolution operator $\exp(i H t)$ is always unitary. This is true whether $H$ has an explicit time dependence or not.

 

[PeterDonis]

Entropy being the inexorable change of a system "over time" toward the most uniform possible distribution of energy

No, that's not what entropy is. Entropy is the logarithm of the number of microstates of the system that are compatible with its observed macroscopic properties. Whether or not a "uniform" system has higher entropy depends on the system and its dynamics. For example, in the presence of gravity, a "uniform" system, with matter spread out equally everywhere, has lower entropy than a system with matter clumped into gravitating bodies.

 

[PeterDonis]

the physics (quantum or otherwise ) of a system of identical sub atomic particles becomes indistinguishable running forward or backward through time

More precisely, both the "forward" and the "backward" processes are valid solutions of the equations.

and thereby invalidates or stops time.

It does no such thing. Both the "forward" and the "backward" processes occur in time. They are just two processes that are each the time reverse of the other.

So does the physics of entropy apply during the ambiguity of this state?

There is no "ambiguity" involved.

 

[PeterDonis]

My understanding is that a sub atomic particle has no arrow of time.

Your understanding is incorrect.

The correct statement is that the laws of physics (with a few minor exceptions that don't matter for this discussion) are time symmetric. That means that, if a given process is a solution of the equations, its time reverse is also a solution. So solutions generally occur in pairs, each of which is the time reverse of the other. There is also the edge case of solutions which are themselves time symmetric, so they are their own time reverses. However, most solutions are not time symmetric in themselves; for example, the solution that describes our expanding universe is not. It has a time reverse which describes a contracting universe.

The term "arrow of time" has at least three meanings in the literature:

(1) The thermodynamic arrow: the direction in which entropy increases.

(2) The cosmological arrow: the direction in which the universe expands.

(3) The consciousness arrow: the direction we consciously experience as the "future" (i.e., the direction in which we anticipate, rather than remember, events).

There are also multiple questions posed in the literature regarding arrows of time:

(Q1) How can there be a thermodynamic arrow of time when the laws of physics are time symmetric?

The answer to this one is simple once we realize, as above, that time symmetry of the laws does not require time symmetry of solutions. Any individual solution which is not time symmetric can have a thermodynamic arrow of time: its initial conditions are conditions of low entropy, and as it evolves according to the underlying dynamics, entropy increases. In the time reverse of such a solution, entropy would decrease. But we can only live in one solution, and by observation we can see that it is the one in which entropy increases.

(Q2) Why does the cosmological arrow point in the same direction as the thermodynamic arrow?

The answer to this one is that the initial condition of our universe was highly uniform, and in the presence of gravity, a highly uniform distribution of matter has very low entropy. Gravitational clumping increases entropy. In our universe, the highly uniform initial condition is associated with rapid expansion; in its time reverse, it would be a highly uniform final condition, i.e., the universe would have had to start out contracting from an enormous size, with lots of gravitationally clumped matter, and all of those clumps would have to un-clump themselves in just the right way to end up in a highly uniform, rapidly contracting end point. We don't know of any way such an initial condition could come about, whereas we do have theoretical proposals for how the highly uniform, rapidly expanding state of our early universe could come about. This illustrates that, while two solutions that are time reverses of each other can both be valid solutions of the equations, that does not mean they are equally reasonable or likely physically.

(Q3) Why does the consciousness arrow point in the same direction as the thermodynamic arrow?

This is more a question of neuroscience than physics, but as I understand it, the usual answer is that forming memories requires an increase of entropy (more precisely, storing memories, which means erasing whatever information previously existed in the storage location, requires an increase of entropy), so the direction of time in which we can remember things must be the "past" direction thermodynamically, i.e., the direction in which entropy was lower than it is now.

where in the subatomic/nuclear/atomic scale is the “line in the sand”

As the above shows, there is no such "line in the sand". The arrows of time described above apply at all scales.

 

[anorlunda]

The correct statement is

Excellent post Peter. Very thorough. Thanks for educating us.

 

[MatthewKM]

Your understanding is incorrect.

The correct statement is that the laws of physics (with a few minor exceptions that don't matter for this discussion) are time symmetric. That means that, if a given process is a solution of the equations, its time reverse is also a solution. So solutions generally occur in pairs, each of which is the time reverse of the other. There is also the edge case of solutions which are themselves time symmetric, so they are their own time reverses. However, most solutions are not time symmetric in themselves; for example, the solution that describes our expanding universe is not. It has a time reverse which describes a contracting universe.

The term "arrow of time" has at least three meanings in the literature:

(1) The thermodynamic arrow: the direction in which entropy increases.

(2) The cosmological arrow: the direction in which the universe expands.

(3) The consciousness arrow: the direction we consciously experience as the "future" (i.e., the direction in which we anticipate, rather than remember, events).

There are also multiple questions posed in the literature regarding arrows of time:

(Q1) How can there be a thermodynamic arrow of time when the laws of physics are time symmetric?

The answer to this one is simple once we realize, as above, that time symmetry of the laws does not require time symmetry of solutions. Any individual solution which is not time symmetric can have a thermodynamic arrow of time: its initial conditions are conditions of low entropy, and as it evolves according to the underlying dynamics, entropy increases. In the time reverse of such a solution, entropy would decrease. But we can only live in one solution, and by observation we can see that it is the one in which entropy increases.

(Q2) Why does the cosmological arrow point in the same direction as the thermodynamic arrow?

The answer to this one is that the initial condition of our universe was highly uniform, and in the presence of gravity, a highly uniform distribution of matter has very low entropy. Gravitational clumping increases entropy. In our universe, the highly uniform initial condition is associated with rapid expansion; in its time reverse, it would be a highly uniform final condition, i.e., the universe would have had to start out contracting from an enormous size, with lots of gravitationally clumped matter, and all of those clumps would have to un-clump themselves in just the right way to end up in a highly uniform, rapidly contracting end point. We don't know of any way such an initial condition could come about, whereas we do have theoretical proposals for how the highly uniform, rapidly expanding state of our early universe could come about. This illustrates that, while two solutions that are time reverses of each other can both be valid solutions of the equations, that does not mean they are equally reasonable or likely physically.

(Q3) Why does the consciousness arrow point in the same direction as the thermodynamic arrow?

This is more a question of neuroscience than physics, but as I understand it, the usual answer is that forming memories requires an increase of entropy (more precisely, storing memories, which means erasing whatever information previously existed in the storage location, requires an increase of entropy), so the direction of time in which we can remember things must be the "past" direction thermodynamically, i.e., the direction in which entropy was lower than it is now.As the above shows, there is no such "line in the sand". The arrows of time described above apply at all scales.

Excellent Thank you Thought process now nicely shifted in the right direction. (or is it left) Thank you

 

[vanhees71]

No. Any Hamiltonian is Hermitian, meaning that the time evolution operator $\exp(i H t)$ is always unitary. This is true whether $H$ has an explicit time dependence or not.

The Hamiltonian should be self-adjoint not only Hermitian. Then if $\hat{H}$ is explicitly time-dependent the time-evolution operator in the Schrödinger picture reads
$$\hat{C}(t,t_0)=\mathcal{T}_c \exp(-\mathrm{i} \int_{t_0}^t \mathrm{d} t' \hat{H}(t').$$

 

[vanhees71]

Your understanding is incorrect.

The correct statement is that the laws of physics (with a few minor exceptions that don't matter for this discussion) are time symmetric. That means that, if a given process is a solution of the equations, its time reverse is also a solution. So solutions generally occur in pairs, each of which is the time reverse of the other. There is also the edge case of solutions which are themselves time symmetric, so they are their own time reverses. However, most solutions are not time symmetric in themselves; for example, the solution that describes our expanding universe is not. It has a time reverse which describes a contracting universe.

The term "arrow of time" has at least three meanings in the literature:

(1) The thermodynamic arrow: the direction in which entropy increases.

(2) The cosmological arrow: the direction in which the universe expands.

(3) The consciousness arrow: the direction we consciously experience as the "future" (i.e., the direction in which we anticipate, rather than remember, events).

There are also multiple questions posed in the literature regarding arrows of time:

(Q1) How can there be a thermodynamic arrow of time when the laws of physics are time symmetric?

The answer to this one is simple once we realize, as above, that time symmetry of the laws does not require time symmetry of solutions. Any individual solution which is not time symmetric can have a thermodynamic arrow of time: its initial conditions are conditions of low entropy, and as it evolves according to the underlying dynamics, entropy increases. In the time reverse of such a solution, entropy would decrease. But we can only live in one solution, and by observation we can see that it is the one in which entropy increases.

(Q2) Why does the cosmological arrow point in the same direction as the thermodynamic arrow?

The answer to this one is that the initial condition of our universe was highly uniform, and in the presence of gravity, a highly uniform distribution of matter has very low entropy. Gravitational clumping increases entropy. In our universe, the highly uniform initial condition is associated with rapid expansion; in its time reverse, it would be a highly uniform final condition, i.e., the universe would have had to start out contracting from an enormous size, with lots of gravitationally clumped matter, and all of those clumps would have to un-clump themselves in just the right way to end up in a highly uniform, rapidly contracting end point. We don't know of any way such an initial condition could come about, whereas we do have theoretical proposals for how the highly uniform, rapidly expanding state of our early universe could come about. This illustrates that, while two solutions that are time reverses of each other can both be valid solutions of the equations, that does not mean they are equally reasonable or likely physically.

(Q3) Why does the consciousness arrow point in the same direction as the thermodynamic arrow?

This is more a question of neuroscience than physics, but as I understand it, the usual answer is that forming memories requires an increase of entropy (more precisely, storing memories, which means erasing whatever information previously existed in the storage location, requires an increase of entropy), so the direction of time in which we can remember things must be the "past" direction thermodynamically, i.e., the direction in which entropy was lower than it is now.As the above shows, there is no such "line in the sand". The arrows of time described above apply at all scales.

I think (3) has nothing to do with physics, but that's an age-old debate (making the Nobel committee to explicitly give Einstein his prize NOT for relativity ;-)). It's at best a physiological arrow of time and at worst subjective.

You forgot the most fundamental arrow of time:

(0) The "causality arrow of time". That's an assumption underlying the very foundations of physics, i.e., that time is an oriented parameter defining the "causal order of events", i.e., that there are natural causal laws that describe Nature.

(1) follows from (0) via coarse graining, i.e., when you go from fundamental microscopic descriptions (QT) to effective macroscopic descriptions (quantum statistical physics), which is necessary (at least FAPP) because of the complexity of many-body systems. It's just impossible to describe $10^{24}$ particles in all microscopic details, and it's also not necessary to describe what we can really observe on macroscopic systems surrounding us.

Despite other claims in the (textbook) literature, it doesn't matter whether the microscopic laws are time-reversal symmetric or not, and indeed in Nature they are not, because the weak interaction breaks all the "discrete symmetries" of the full Poincare group except CPT, which must be a symmetry for any local (microcausal) relativistic QFT. The H-theorem only relies on unitarity, not on T and/or P symmetry!

(2) you have already explained to be identical with the thermodynamical arrow of time.

Another "arrow of time" is also the "electrodynamic arrow of time", which comes into the game when solving Maxwell's equations and we choose the retarded propagator of the d'Alembert operator, and this is well justified, because it solves the usual radiation problem, where electromagnetic waves are radiating out from a "compact source" (currents and charges). Of course also the corresponding time-reversed solution is a valid solution of Maxwell's equations, but it's very hard to realize in experiment: You'd have to create an incoming wave with some sources far away from the "compact source" (which now becomes a "compact sink") precisely such as to realize the time-reversed state of the retarded solution such that the incoming wave is completely absorbed by the "compact sink". This is FAPP impossible, i.e., again it's the "complexity" of the time-reversed situation that makes it very unlikely to ever occurring in Nature. In this sense this arrow of time is similar to the thermodynamic arrow of time (1).

 

[tade]

Summary: Arrow of time and Entropy

My understanding is that a sub atomic particle has no arrow of time. Clearly the Arrow of time as understood in our macroscopic world has one direction. Entropy being an arrow in the direction toward uniform distribution of the individual and or sums of all the forms of energy in a system and , I suppose ultimately the uniform distribution of a final form of energy, begs a number of questions.

Starting from Hot Dense and Smooth at the beginning of the universe: where in physics is the convergence of the notion of Entropy and the Arrow of time?

Or put another way: assuming uniformity, where in the subatomic/nuclear/atomic scale is the “line in the sand” where Entropy has no arrow of time? ( Quarks and Gluons, Nucleons, Plasma, Atoms, Molecules?)
If that could be answered, then was there a point in the early universe before which (and the use of the word “before” is sysyphean so no need to comment) the notion of entropy is an incorrect or confusing way of referencing the early universe?
Is this “Sub Atomic Frame” fixed regardless of what model of the universe one holds with? (Inflation, Multi universe, eternal inflation, etc) Or does the Subatomic Line in the sand “smear or blur” depending.

Matt

apparently B–Bbar oscillations, even at that subatomic level, exhibit time-asymmetry