Time reversal symmetry in physics

[som]

It is said that Newton's laws of motion or laws of Quantum Mechanics posses time reversal symmetry but the second law of Thermodynamics does not. What I understand by the first part of the sentence is the following.

The dynamical state of a system changes with the increase of time. The state at time 't+dt' is different than that at time 't'. If somehow time goes in the reverse direction from 't+dt' to 't' we will get back the state what we originally got. Please help me know whether this is a correct thinking.

 

[Cutter Ketch]

Yes, I think you understand what they mean by time reversal. The question is: does it look like physics when you run the movie backward?

In a simple conservative system the reversed behavior looks like normal motion and doesn’t violate any laws of physics. However in a complex ensemble system the reversed motion does violate the laws of thermodynamics. In other words running the movie backwards is distinguishable from things that happen in the real world.

For example picture a planet orbiting the sun. Now reverse time. The planets orbits in the other direction, but that orbit doesn’t break any laws of physics. Now picture a box divided in two. In one half there is a gas of molecules. In the other half there is a vacuum. At time zero the wall is removed and the gas molecules diffuse into the other half colliding and spreading. If you reverse all of the motions of all of the molecules they will exactly retrace their paths and go back into one half of the box. Even inelastic collisions will “undo” as the resulting vibrations come back together just exactly right to put energy back into the colliding particles. Obviously this spontaneously collapsing gas will not happen in the real world.

What is happening here? Thermodynamics deals with the ensemble behavior of a large number of particles, vibrational modes, etc. The condition that results in all of the particles going back into one half of the box requires all of the particles to be in one precise combination of motions. Each particle has a range of possible motions. Together the possible motions for all the particles is a gigantic set. The probability of this one particular state is infinitesimal. There may be a neighborhood of states that results in all the particles going into one half of the box, but it is still a tiny subset of all the possibilities. In fact the vast majority of possibilities are found within a small neighborhood of the gas being uniformly distributed in the box.

 

[anorlunda]

If somehow time goes in the reverse direction from 't+dt' to 't' we will get back the state what we originally got.

It is not t returning to the original t, it is the state of the physical system that returns to the original state. You must be very careful with the words you use.

But you must be very careful where to apply time reversal symmetry. In macro thermodynamic systems, it does not always apply. At the quantum level it becomes something called unitarity, which is probably too advanced for you to understand. But in classical mechanics, meaning motions of bodies and such, it does apply.

Physics
In physics, the laws of motion of classical mechanics exhibit time reversibility, as long as the operator π reverses the conjugate momenta of all the particles of the system, i.e. {\displaystyle \mathbf {p} \rightarrow \mathbf {-p} }

(T-symmetry).

In quantum mechanical systems, however, the weak nuclear force is not invariant under T-symmetry alone; if weak interactions are present reversible dynamics are still possible, but only if the operator π also reverses the signs of all the charges and the parity of the spatial co-ordinates (C-symmetry and P-symmetry). This reversibility of several linked properties is known as CPT symmetry.

Thermodynamic processes can be reversible or irreversible, depending on the change in entropy during the process.

 

[lightarrow]

..
Now picture a box divided in two. In one half there is a gas of molecules. In the other half there is a vacuum. At time zero the wall is removed and the gas molecules diffuse into the other half colliding and spreading. If you reverse all of the motions of all of the molecules they will exactly retrace their paths and go back into one half of the box. Even inelastic collisions will “undo” as the resulting vibrations come back together just exactly right to put energy back into the colliding particles. Obviously this spontaneously collapsing gas will not happen in the real world.

Ok, very right. But after the molecules have completely diffused one into another, wait enough (a lot!) time: one day (if the number of molecules is not too big) they will separate again in the two boxes. While this happens, in which direction time is flowing?

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[anorlunda]

I think you are asking about the arrow of time. https://en.wikipedia.org/wiki/Arrow_of_time

It is a tricky subject. It sounds paradoxical that a few particles have time reversibility but a lot of particles don't (micro versus macro). To truly understand the origins of the second law, and the arrow of time, takes a lot of study. Much more than can be answered in a couple of paragraphs in an Internet forum.

My advice is to go study and learn for a while instead of asking questions here. The reason I say that is asking questions prior to learning the lessons is a very poor strategy for learning. In school, you ask questions at the end of the lecture, or the course, and after reading the book, not before.

 

[lightarrow]

I think you are asking about the arrow of time. https://en.wikipedia.org/wiki/Arrow_of_time

But did you answer to me or to the OP?

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lightarrow

 

[som]

It is not t returning to the original t, it is the state of the physical system that returns to the original state. You must be very careful with the words you use.

The state returns to the original only in case of periodic motion, I guess. A particle acted upon by a constant force will never come to its 'original'. Does it mean there is no time-reversal symmetry for such case?

 

[som]

What is happening here? Thermodynamics deals with the ensemble behavior of a large number of particles, vibrational modes, etc. The condition that results in all of the particles going back into one half of the box requires all of the particles to be in one precise combination of motions. Each particle has a range of possible motions. Together the possible motions for all the particles is a gigantic set. The probability of this one particular state is infinitesimal. There may be a neighborhood of states that results in all the particles going into one half of the box, but it is still a tiny subset of all the possibilities. In fact the vast majority of possibilities are found within a small neighborhood of the gas being uniformly distributed in the box.

It sounds that a probabilistic attitude of the system, whether it is one-particle or many, breaks the time reversal symmetry. Does it?

 

[anorlunda]

The state returns to the original only in case of periodic motion, I guess. A particle acted upon by a constant force will never come to its 'original'. Does it mean there is no time-reversal symmetry for such case?

No, time reversal symmetry does not mean it will return by itself. It says that the law of physics (think of Newtons Laws) apply forwards and backwards.

Watch a film of two billiard balls coming together, colliding, then rolling away in different directions. You can check that their motion obeys Newton's laws. Now play the film backward. It looks perfectly natural. If you check, Newtons Laws are still obeyed. In fact, by looking at film, you can't tell if it is running forward or backward.

 

[som]

No, time reversal symmetry does not mean it will return by itself. It says that the law of physics (think of Newtons Laws) apply forwards and backwards.

Watch a film of two billiard balls coming together, colliding, then rolling away in different directions. You can check that their motion obeys Newton's laws. Now play the film backward. It looks perfectly natural. If you check, Newtons Laws are still obeyed. In fact, by looking at film, you can't tell if it is running forward or backward.

That means to check the validity of time reversal symmetry, the prescription is to allow the system to evolve for a certain time (t, say), then flip the momentum of all particles and check that after the same time has been elapsed we get back the original state what we started with. Is it correct?

 

[anorlunda]

That means to check the validity of time reversal symmetry, the prescription is to allow the system to evolve for a certain time (t, say), then flip the momentum of all particles and check that after the same time has been elapsed we get back the original state what we started with. Is it correct?

That is a way, but more restrictive than necessary. I prefer my words, that the classical laws of physics are valid both forwards and backwards. That is more general because I don't have to start and end at the same state.

 

[som]

That is a way, but more restrictive than necessary. I prefer my words, that the classical laws of physics are valid both forwards and backwards. That is more general because I don't have to start and end at the same state.

OK, I understand the point. But at the same time I become a bit curious that you mentioned 'classical laws of physics are valid both forwards and backwards'. Does it imply quantum laws or more specifically Quantum mechanics lacks this symmetry?