## Sunday, August 22, 2010

### Time Reversal (p170)

So I’m trying to work out the concepts of time-reversibility and time-reversal invariance in a descriptive visual sense.

Working from the idea of the two plates proposed in the text book.
For a time-reversal invariant process you can use the example of rubber between the plates. If you gave the top plate a push to the right the rubber would spring back to the left. And the same motion occurs in reverse. With the rubber ‘supplying’ the initial push to the right.

However if you have a viscous fluid in between the sheets you have time-reversibility. If you push the top plate to the right, it moves to the right and stops as the movement is dissipated by friction. In reverse you can’t use the heat that is formed from the friction to propel the plate back to the left; instead you must supply the force yourself.

Is this accurate? What happens when you have a ‘turbulent flow’ liquid between the plates what happens then? Would it also be time-reversal, with a longer deceleration time due to inertia?

1. Just to clarify, I thought a viscous liquid between two sheets exhibited a irreversible process.

I agree. I think with a turbulent liquid it would act like a time irreversible process, with a longer deceleration time. Imagine pushing a board floating on water. It would float along, experiencing very little friction, but would eventually stop.

I think the main difference between time reversible and irreversible processes is that usually there is a potential field involved in the time reversible processes, like gravity (ball thrown through air), elastic potential energy (rubber between two sheets), or electromagnetic fields. The potential fields store the energy, which then release the energy later in the process. Thus the process is symmetric, the energy is stored then released, which looks the same in reverse. Processes which convert energy to other forms of energy, like heat, are not reversible.

2. Events in a low Reynolds number regime have time-reversibility when either the reversed force is constant with time or, if the force is time dependent, the applied force must be negligible compared to the viscous critical force.

The book says that to be in the turbulent regime you need a Reynolds number of ~1000, the equation for the Reynolds number is found by dividing eqn 5.8 by 5.10 this means to be in the turbulent regime the inertial force must be 1000 times larger than the viscous frictional force. So we have an extra criteria for time-reversibility. I wonder what the characteristics of the fluid in terms of the time-reversibility very close to the turbulent regime is?

The answer your question Heather turbulent flow is not reversible as the inertial effects (tendency to keep on moving) are more important (1000 times more!) so I guess on simple arguments it would take 1000 times longer to slow that particle moving under that force through friction. By the time it has slowed down it is also likely to have diffused to a different place not on the path caused by the applied force.

3. Sorry, you're right Mitch. Viscous friction is: not time-reversal invariant (I got confused).
That's a good point about the potential fields too. I was struggling to think of a process that didn't involve gravity or elasticity. Although how would you classify the car crash example? Elastic?

Interesting way of thinking Calvin. I hadn't even considered diffusion as the particle slows, but that is where the energy would dissipate to. Thanks

4. I was also considering self diffusion i.e. all the particles have random-walked off somewhere.

5. Yes, the time-reversability seems to also heavily rely on the rate of diffusion, because in the text book, the time reversible example showed that when the blob was reformed, there had been little diffusion, yet in the non-time reversible example, diffusion had readily occured. Obviously this can be related back to the viscosity of the fluid, and the sample added in, but still it was a point worth noting. Also, the reversibility would come down also to the magnitude of the force applied to the sliding plates. If you force them away quickly, you're more likely to spread the sample further causing more diffusion and reducing chances of it being time reversed, if I have thought of this correctly.