I was browsing through a collection of Landau’s papers and came across one of the more bizarre explanations of the second law of thermodynamics that I’ve seen. The few people who frequent this blog probably don’t need this explained to them, but the reason why many philosophers think that the second law needs ‘explaining’ is that the fundamental laws of classical and quantum mechanics are time symmetric, but the second law is not. If we think that all physical phenomena are completely described by the fundamental laws, then it seems puzzling that one can observe large scale time asymmetry when said laws are time symmetric.
Another way of stating that puzzle is that if one is given an isolated system in a state of less-than-maximal entropy, then standard applications of statistical mechanics lead to the conclusion that the system is likely to be found in a state of higher entropy in both directions in time — the past and the future. Landau accepts this as well. He then takes the problem of explaining observed monotonic change in time (a popular interpretation of the second law) to be one of explaining why all isolated subsystems seem to undergo monotonic change in the same direction. He offers the following argument:
If a statistical body is isolated from its surroundings for a period which is not too large compared with the relaxation time,* its entropy changes monotonically. If two such bodies are isolated, then the monotonic changes cannot occur in mutually opposed directions. This is immediately clear, since these two systems can be connected by an arbitrarily weak coupling into one single system, at any rate when their periods of isolation are simultaneous or partly so. If we apply the proof introduced above of the monotonic entropy change in this case of two weakly linked parts of one system, then we can see immediately that the directions of the entropy changes of the parts cannot be opposed to each other. If different systems are not simultaneously isolated from their surroundings, then we can introduce to the argument any necessary intermediary steps. We can now see that either the entropy of every statistical body which is not isolated from the universe for too long is constant, or there is a particular universal time direction for entropy increase.
This is weird since the clause about ‘relaxation time’ (see footnote) is a provision for the [supposed] fact that on time scales larger than the relaxation time, there isn’t a monotonic increase in entropy. So the ‘universal time direction’ would in this argument hold only for the system’s relaxation time. Yet, it’s also supposed to hold for all isolated systems that could be potentially coupled to the system in consideration! So, all such potentially coupled systems will undergo monotonic increases in entropy during the same time scale as our model system’s relaxation time!
It gets more bizarre. Recall that in the passage above, Landau concludes that either
- ‘the entropy of every statistical body which is not isolated from the universe for too long is constant’, or
- ‘there is a particular universal time direction for entropy increase.’
Landau opts for 2. Here’s how:
Everyday experience shows, however, that [1.] is not the case. This alone is quite sufficient (even without Bohr’s hypothesis that the inner layers of the stars may only be described in terms of relativistic quantum theory) to conclude that there are at least regions in the world which do not obey the statistics.** This makes the two directions of the time axis non-equivalent; the direction which is associated with the increase of the entropy of a system isolated over not too long a period form its surroundings can be defined as the future. For classical mechanics the conceptions of past and future are entirely without meaning. Only then do we obtain the second law of thermodynamics as the law of increase of entropy. This laws, as well as the existence of past and future observed in everyday experience, is only possible because the world as a whole does not obey the laws of thermodynamics.
This is the first time I’ve seen a claim like this: that entropy increases monotonically with time only because the laws of thermodynamics don’t apply to the entire universe! I must admit that because of the problems mentioned above, I can’t take any of Landau’s suggestions seriously, but the paper was a highly amusing read, and very reflective of how physicists tend to be much more imaginative than philosophers of physics.
*This clause about ‘relaxation time’ is supposed to get around the notion that the evolution of a system’s entropy with time should appear time-symmetric on the large scale, as a series of mostly small fluctuations away from equilibrium and back again. If we look at the system on a time scale close to the relaxation time, then we should be ‘within’ that portion of a fluctuation that involves monotonic increase or decrease of entropy with time.
**Landau states earlier in the paper that ‘in relativistic conditions… thermodynamic considerations [are] no longer valid’, and that he does not ‘expect thermodynamics to be upheld in the case of relativistic quantum theory’.
M. Bronstein, & L. Landau (1933). Über den zweiten Wärmesatz und die Zusammenhangsverhältnisse der Welt im Großen Physikalische zeitschrift der Sowjetunion, 4
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