There is something about the time reversal principle that I dont completely understand.

Consider an isolated gas of molecules that has undergone expansion in volume. Now, we know that Newtonian dynamics obeys time reversal symmetry, ie if I reverse the velocities of all particles in the system at some instant, the evolution from then on would be backward. In other words the system would retrace its path. Atleast , this definitely holds when there are no velocity dependent interactions, which we can assume here for the sake of simplicity.

However, we also know for sure (from both second law of thermodynamics and common experience) that isolated gas molecules would never contract (statistically speaking) and decrease in overall volume.

How then does one account for this apparent contradiction?

**Posted By :**Venkateshan

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It is not true that an isolated gas,undergone expansion can never contract. It do have a finite probability of contraction and going back in the smaller volume but the probability is so less that it may take even age of universe to come back in initial volume after expansion.

Second thing is that in thermodynamics the observables are average of the quantity over all the possible states. If time taken by system in passing through all the points in the phase space is very large compared to the time duration over which our measuring instrument averages the quantity then there will be error in measurement. And this error will depend on the time duration of averaging (something like energy-time uncertainty in quantum mechanics). If we try to observe any state in which let us say all the atoms are in small volume by measuring let us say pressure(that will be momentarily seems to be zero as no particle is colliding to surface of probe) then there will be following problem: the response time of probe can’t be zero and even if we keep response time very small then error in our measurement will be very large. So observation of occurrence of such state in common experiences is not easy.

Well.. i couldn’t understand your statement that second low of thermodynamics doesn’t allow contraction of gas(in the given perspective). Entropy is related to number of possible states that system can have. And second low of thermodynamics says that for an isolated system entropy(and hence number of possible states that system can have) can never decrease. One of these state can be the one in which all the particles are in small volume.

Please correct me if i am wrong somewhere.

About the point that there is a finite non-zero probability that gases can contract, yes, indeed that is correct and that is why I mentioned ‘statistically speaking’ in my post.

The fact here is, if time reversal symmetry is perfecty obeyed then the system must necessarily contact when the velocities are reversed ((in fact retract its path in phase space) and not merely have a small probability that it will come together at some point of time.

And your second point dosent apply here for similar reason. I am talking about time scales of the order in which the macroscpoic expansion takes place and what you have mentioned is the typical time over which the different microstates are averaged(during measurement) over and the latter is much smaller than the former. So even if the system does instantaneously exist in a contracted state it still wouldnt explain the expected ( if TRS holds) macroscpoic behavior which is just watching the whole process in reverse.

If this isnt clear, say so.

Venkateshan

In short I must confess that I myself do not understand these things completely, but one thing that we can say is that obviously the system

willretrace it’s path if all velocities are reversed exactly. What one has to then ask is why the ordinary distribution of velocities is so much more probable than the reverse one, and that is where the small probability of this kind of contraction comes from.And as for your initial question I think Poincare proved a theorem that a classical system will pass arbitrarily close to any point in it’s phase space given sufficient time. (Please correct me if I’m wrong here, and I would still like to get a clearer intuitive idea of this statistical breaking of Time reversal symmetry)

P.S.: Added the wikipedia link to what is known as Poincare Recurrence Theorem. -Tom.

Venky, I will return to this sometime later – when I understand it a bit better and when I think I can come up with a coherent answer- but for those who may want to explore this question, this is a classical argument by Loschmidt and is known as Loschmidt’s reversibility paradox .

What shanth alludes to in the previous comment is probably what has come to be known as Zermelo’s Recurrence Paradox. I think this teminology is due to Ehrenfest, if I remeber correctly.

Those who want to discuss about this should first understand the distinction between the two paradoxes.

The first one is the paradox created between the time-reversal invariance of classical mechanics and the Irreversibility in statistical mechanics.

Whereas, the second one alludes to the paradox that arises between the poincare recurrence property of classical mechanics with the monotonic increase of entropy demanded by the second law of thermodynamics.

Keeping this distinction in mind, I think, would greatly contribute to the clarity of discussion. đź™‚

I read the wikipedia article on Loscmidt’s paradox and also the possible resolution of it using the fluctuation theorem. However, I am not sure whether any of that pertains to my question directly.

The article on fluctuation theorem is not very clear but from whatever I understand, I think the initial state they are considering is specificed only upto macroscopic parameters. THe theorem then asserts that the ensemble average of the time average of entropy production will always be positive.

But in my problem, the state of the system is fully specified and if you assume deterministic physics there is no need for ensemble averages at all. After all the system is going to have exactly one state after time t.

Well, allright, after some thought it occured to me that the problem would reduce to something that is satisfactorily answered by the fluctuation thorem provided what Shanth had claimed can be assumed to be true. I’ll repeat it here :

If we exactly reverse the velocities the system retraces

its path in phase space.

Now is this an undisputed statement?

Well, the statement as stated is true for free particles. It is not true, if there is an external magnetic field, for example – you should reverse the magnetic field too(which you can think of as reversing the external currents which produce that field). In general, you’ve to time-reverse the whole system with its environment.

Or was your question – do all systems have time-reversal invariance at a fundamental level ? Well, there are systems which violate CP and hence T (assuming CPT symmetry ).

And I am yet to read enough to be comfortable with fluctuation theorem – in fact as my comments elsewhere make it clear, I’ve a lot to learn in non-equilibrium stat.mech.