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| | Can the Second Law of Thermodynamics
Be Deduced from Quantum Theory?
by David R. Schneider
Original Draft: 01/07/87
Abstract: The second law of thermodynamics has never been
known to have been violated. It retains the status
of a law because it cannot be fully deduced from
the statistical mechanics, which is itself derived
from quantum theory. It is shown that the second
law is unnecessary if any of several common
formulations are considered, and quantum theory is assumed.
The asymmetry of entropy increase is also
called into question.
1. Introduction
1.1 The second law of thermodynamics (SL) is one of the most interesting of known physical laws, for it defines a direction in time.
All other physical laws are, in principle, completely symmetric with respect to the time dimension.
The existence of an asymmetric law within a set of theories obeying symmetry principles has led to numerous attempts at reconciliation.
Recently, Prigogine [1] has advanced the idea that SL is more fundamental than quantum theory (QT).
This idea is very interesting, and has many merits. However, it seems as if physical theory would be better served if SL could be deduced from QT rather than vice versa; SL does not seem to be "rich enough" to give us a strong foundation for QT.
However, this perspective does not rule out the idea that assuming one always leads to the other.
1.2 There are several formulations of SL [2]. Several common ones are given below:
a. A transformation whose only final result is to transform into work heat extracted from a source which is at the same temperature throughout is impossible. (Lord Kelvin)
b. A transformation whose only final result is to transfer heat from a body at a given temperature to a body at a higher temperature is impossible. (Clausius)
c. For any transformation occurring in an isolated system, the entropy of the final state can never be less than that of the initial state. (Fermi)
d. It is impossible to construct a perpetuum mobile of the second kind. (General)
It is no problem to derive SL from QT for systems in which the number of particles is large.
This is done in statistical mechanics, for example. The problem is that QT specifically allows violation of SL in individual processes.
As Wheeler [3] said, "Quantum theory laughs at the second law of thermodynamics."
Even with a large group, violation is possible. Apparently, conflicts exist between SL and QT.
1.3 It will be shown here that the paradox is one of semantics. Given the usual interpretation of science, SL is redundant if QT is assumed.
The following sections will show how formulations 2, 3 and 4 above are derived from QT.
(Formulation 1 is ignored because it has been shown by Fermi and others to be equivalent to 2.)
There is still the problem of the requirement of QT that entropy increases to the future, which leads to an "arrow" in time.
This paradox can be resolved by requiring that entropy increase to the past as well.
Closer scrutiny reveals that by normal interpretations of QT, entropy does increase to the past as well as the future.
2. Observation of Violation of SL is Impossible
2.1 SL says that heat cannot be transferred from one body to a warmer body. QT predicts that such an instance is extremely unlikely, but theoretically possible.
Or does it? A better formulation, in keeping with Mach's principle (all we know is the results of experiments), is that we cannot observe a transfer of heat from a cold body to a warm body.
Assume we are measuring the temperature of a body A with N atoms (or particles, etc.).
We bring it in contact with a colder body B. At a later time, we separate the two and again measure the temperature.
(This example will be used throughout this paper. Assume an ideal apparatus.)
How many times R must we do this to have a reasonable chance of seeing an increase in the temperature in A, assuming QT only?
Let's say that we limit N so that R takes one year (assume a rate of once per minute).
This would lead us to a small system of atoms, but one which is greater than 3.
Now, reduce N towards 1. What happens? Will the likelihood of observing a violation of SL increase?
2.2 It may increase some, but at some point a new factor comes into play: the Heisenberg uncertainty relations.
As N approaches 1, our uncertainty increases. Suddenly, we start seeing that SL appears to be violated more often.
However, there is so much ambiguity in our experimental set-up (since both position and momentum are required to determine temperature when N is small) that we can no longer be sure we have actually witnessed such a violation.
Therefore, we conclude that QT predicts that unambiguous violation of SL in an actual experiment is virtually impossible.
Failure to ever observe such a violation would be consistent with QT as well, as long as the statistical predictions were within appropriate confidence levels.
On the other hand, observation of a violation would clearly show that SL is unnecessary.
Conceptually, an experiment could be devised to observe a violation of SL over R repetitions of the above test, where QT would yield a large likelihood of such violation in a relatively unambiguous way.
R might be so large as to be practically unachievable in any single person's lifetime; but suppose it was in fact carried out. If the violation occurred as predicted by QT, then SL is now proven false. If not, then either QT is wrong and/or SL is necessary.
The only problem with the above experiment is that it would appear to serve no clear purpose.
Any number of similar experiments could be imagined, in which predictions of any theory known to man are subjected to long periods of test for the purpose of observing a hypothetical exception.
Should we repeat tests of the speed of light on trillions of photons to see if any move at a velocity other than c?
It is certainly
possible than some photons would, if relativity is wrong. But what reason would be given to actually perform such a test?
3. Entropy Increase Over Time is Definitional
3.1 Entropy increases with time. This is well known from statistical mechanics [4].
Assume we are considering the "number of permutations" formulation of entropy.
By definition (and assuming QT, of course), the entropy of A after being brought into contact with B will always increase (or be equal).
One could never observe even a single case where such a situation were observed.
This is derived by Fermi in his book.
3.2 Thus, entropy always increases because QT dictates the following: if you assume the system was limited to a specific number of configurations to start with, then the number of possible outcomes will always be at least as large.
Further, the number of input configurations can really only be limited by assumption anyway.
Consequently, it must be admitted that QT predicts entropy increase. Further, such increase is really a definition: entropy is the measure which corresponds to a quantum observable.
In QT, the spread of indeterministic probability waves leads to an increase in the number of permutations of the system in the future, apparently without limit.
4. Construction of a Perpetuum Mobile is Impossible
4.1 Does QT predict that a perpetual motion machine (PMM) is impossible, in principle?
Yes, this can be directly derived from QT alone (without SL). Suppose someone were to present you with an alleged
PMM. You will allow the presenter three opportunities to operate it in such a way as to show heat extracted from an object colder than its surrounding
environment. The machine is a macroscopic device (i.e. larger than a penny, for example). Would you bet against it?
4.2 If you believe in QT, you would. It is obvious that QT would prohibit such a machine from having anything close to a 50% chance of operating successfully.
If it is large at all, it could not operate successfully in a large number of tries if QT is assumed.
If it has a very low likelihood of converting heat into work (approaching zero as the desired amount of work increases), how can it be called a PMM?
How could its perpetual nature be demonstrated anyway?
You might as well invent a law to state that no person could toss 100 coins into the air at the same time, and observe all heads upon landing. If QT does a good job of explaining observed facts, why is SL needed?
5. The Symmetry of Entropy Increase
5.1 SL is completely superfluous in modern physics. We don't need it to explain why the formulations of section 1.2 are obeyed; QT does this for us. But one paradox remains: why does entropy increase define an arrow of time?
It seems that the second law of thermodynamics is an ad hoc law, with no particular significance other than as a general tendency which results from QT.
Therefore, the "thermodynamic arrow of time" must be similarly artificial.
If entropy increases with time, or if the temperature of a set of bodies tends to approach some equilibrium point - this is only a consequence of normal quantum mechanics.
If you say that thermodynamics is asymmetric with respect to time you are really saying that QT is asymmetric with respect to time.
If QT consists of the propagation of indeterministic probability waves which expand until an observation is made, is QT really symmetric with respect to time?
Maybe it is not the second law which needs review; perhaps QT does.
5.2 According to Prigogine, there are irreversible processes in nature. Does QT contain any irreversible processes? Yes: the act of observation. In the example of Schroedinger's cat (SC), we irreversibly gain knowledge when we look inside the box. But is this knowledge asymmetric with respect to time? After all, we know it for the future, but didn't know it in the past.
No; even here there is no fundamental asymmetry. For it can be seen that for any particular initial conditions, in any state of order whatsoever, entropy increases to the past as well as to the future.
With SC, the number of possible states prior in time (which lead to the currently observed state) increases without limit as we go farther and farther back in time.
We don't know which one(s) led to the observed state.
Thus stated, observation is the act of quantizing the state of the system via one or more assumptions.
This is consistent with QT. Now SL can be seen as symmetric with respect to time, if the trivial and artificial "heat-oriented" examples (HOE) are ignored.
(Note: Such examples are clearly artificial if an analogy with a ordered deck of playing cards is considered - the order was consciously arranged, not randomly arrived at.
Further, the effect of gravity may be significant in restoring symmetry to such examples.)
The act of observation is in effect the establishment of a quantum minima of some kind: the number of theoretically possible input configurations at any prior point in time is now reduced to a minimum value.
Our uncertainty is reduced (although the Heisenberg uncertainty relations prevent us from reducing uncertainty to zero) temporarily, and then increases to the future.
So we can now see that the alleged thermodynamic arrow of time results from neglecting the tacit assumption that the initial conditions consisted of a single state at some point in time, or was derived from some single, specific prior state.
This could never be proven to be true, and it completely defies the spirit of QT. QT is as symmetric in the sense of entropy as it is in all other areas.
5.3 In conclusion, we can now see that SL serves no purpose other than as a classical approximation for certain macroscopic phenomena.
Hopefully, the paradox of time asymmetry with respect to thermodynamics is resolved.
References
1. I. Prigogine, From Being to Becoming (1980).
2. E. Fermi, Thermodynamics (1936).
3. P. Davies, The Arrow of Time, (19xx).
4. x. Tolman, Statistical Mechanics (1938).
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