28th April 2006 - 01:11 AM
Hello, I'm wondering a couple things: Is there a quantum-mechanical basis to the fundamental laws of thermodynamics (i.e., the 0th-2nd laws; I know the 3rd can be derived)? Also, is there a quantum-mechanical basis to the "a priori" equal-probability postulate of statistical mechanics? Thanks.
28th April 2006 - 03:01 AM
28th April 2006 - 03:14 AM
Ohhhhhhhhhhhh. (And, of course, explanations are nice.)
13th May 2006 - 01:26 PM
Yeah, I remember reading about it somewhere. But according to me it is a yes.
19th May 2006 - 09:19 PM
The short answer is that the laws of thermodynamics are consistent with quantum physics. For instance, the Schrodinger wave equation can be used to arrive at the conclusion of the 2nd law... basically, given a large number of particles, they're going to end up arranged such that they're at the most probable energy levels. This is true in a probabilistic sense, so with a sufficiently large number of particles, the probability of it not holding goes to effectively zero. This, of course, depends on statistical mechanics... the equal probability issue is, to my knowledge, a basic assumption rather than anything that's been derived. If we were to have some more fundamental basis for deriving it, I that basis itself would be empirical or assumed. Of course, there's obviously always going to be some level at which we just have to say "we're pretty sure this is the way it works because of our observations" - not being the creator/designer of the universe ourselves, we can only learn about it by observation...
The second law of thermodynamics can actually be violated on extremely small scales (of both length and time) where a continuum approach isn't valid, since it's probabilistic in nature. The second law tells us that if we have two chambers, one at a (complete) vacuum, and the other holding a gas at some finite pressure, and break a barrier between them, we'll end up with an equal pressure of gas in both chambers, and it won't return to its original (more ordered) state where all the gas is in one chamber. But suppose there are only two atoms in the high pressure chamber... in that case, there's a significant likelihood that at some point in time, both will return to the original chamber, rather than one being in each. The second law on such scales is a measure of probability, not an absolute statement. On a macroscopic scale, it can be taken to be absolute, since the probability of it not holding is neglibly small.
Some pretty counterintuitive things can happen thermodynamically at such scales - there are some laser-material interaction situationss where adding energy as heat due to laser absorption can result in a temperature drop rather than increase... basically, the standard definitions of temperature, etc. just aren't any good at such scales.
As for the first and "zeroth" laws... the first is simply conservation of energy - it's as much a fundamental assumption in QM as anywhere else, by my understanding. The basic conservation laws (energy, mass, momentum, etc.) are about as fundamental as we can get - any change has to have come from somewhere. The "zeroth law" is as much a logical/mathematical statement as anything else - if A = B and A = C then B = C. I would think QM could be used as an example for supporting that analysis, but if you reduce it to a microscopic scale, the principle still applies. I don't think either of these is really derivative of quantum mechanics so much as consistent with it. Could be wrong here... I'm an engineer, not a quantum physicist, but that's my take on it.
30th May 2006 - 04:40 AM
Just posting to basically agree with Mithrandir and add a couple of small comments:
Yes, not only does the radomness in statistical mechanics have a quantum mechanical origin, statistical mechanics itself is simply the application of quantum mechanics to systems of very many particles.
As to the zeroth law, there is a little more to it than A=B, B=C, therefore A=C, although that is certainly the way the law sounds. What it asserts is that the state of equilibrium is transitive, so that if system A is in thermal equilibrium with B, and B with C, then A is in equilibrium with C. This is a physical idea as much as a mathematical one, since it is not completely obvious that A and C will not have a net energy exchange just because neither one exchanges energy with B. There are lots of systems where similarly formulated properties are not true: for example, hydrogen will not react with helium, so a mixture of the two is in equilibrium, and oxygen will also not react with helium, so again their mixture is in equilibrium; but oxygen and hydrogen -- boom! So the zeroth law is a genuine physical statement that thermal equilibrium really does behave like equality. It is this fact that makes it possible to define temperature; since equilibrium is behaving the way we would expect mathematical equality to behave, we can postulate that there is a quantity that is equal when two systems are in equilibrium and then name this quantity temperature.
I have never seen any justification for the equal probability of states assumption either. It seems to be empirically correct, since the theory gives the right answers, but I would also like to see some deeper justification for it. It seems so simple and clear that one feels there must be some reason why things logically HAVE to be that way, some reason more satisfying than "they just are that way."