# Comments on entropy and the direction of time in Landau and Lifshitz's Statistical Mechanics

In Landau and Lifshitz's Stat Mech Volume I is the comment:

However, despite this symmetry, quantum mechanics does in fact involve an important non-equivalence of the two directions of time. This appears in connection with the interaction of a quantum object with a system which with sufficient accuracy obeys the laws of classical mechanics, a process of fundamental significance in quantum mechanics. If two interactions A and B with a given quantum object occur in succession, then the statement that the probability of any particular result of process B is determined by the result of process A can be valid only if process A occurred earlier than process B?

Thus in quantum mechanics there is a physical non-equivalence of the two directions of time, and theoretically the law of increase of entropy might be its macroscopic expression. In that case, there must exist an inequality involving the quantum constant $\hbar$ which ensures the validity of this law and is satisfied in the real world. Up to the present, however, no such relation has been at all convincingly shown to exist.

Has there been such convincing work relating the direction of time and $\hbar$ since these comments were first made (somewhere between 1937 and 1975 by the dates in the prefaces)?

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+1, cheers for bringing this up. This comment struck me as profound when I first read it, and still does. Looking forward to reading some good answers to this. – Mark Mitchison Feb 15 '13 at 21:16

Landau's argument is basically a swindle for two reasons. The first reason is that he's trying to present this without talking about measurement or wavefunction collapse, which would make it sound like a fundamental statement about physical laws rather than a statement about one possible philosophical interpretation of quantum mechanics.

To see that this is misleading, note the use of the word "probability." Probability and randomness are tricky to define. One of the standard approaches is to talk about a sample space. For example, WP has:

In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment.

Note those words like "experiment," "trial," and "outcome." To give meaning to the word "probability," we would typically do something like the following: prepare an ensemble of systems in the same state, do an experiment in which we measure some observable, and count how many times we get different results. But this is clearly a measurement process. The most common definitions of the words Landau is using occur in the context of measurement, and if he had in mind some definition that didn't require a notion of measurement, the burden would be on him to spell out what those definitions were.

In fact, Landau's statement about events A and B is really a more general statement about what happens when we make measurements or observations on any random process, not just a process that's random because of quantum mechanics. Let's take a classical system like a flipping coin. The coin is described to essentially perfect precision by Newton's laws, but the outcome is random because of its extreme sensitivity to initial conditions. Let A be the event that when the coin comes to rest on the tabletop after the flip, it's heads-up. Let B be the event that, one second after that, it's heads-up. Landau's argument applies just as well to this story as to his story of a quantum-mechanical measurement; all we have to do is replace "quantum object" with "coin" and "a system which with sufficient accuracy obeys the laws of classical mechanics" with "a system whose behavior is predictable because it is not too sensitive to initial conditions."

Landau says:

In that case, there must exist an inequality involving the quantum constant $\hbar$ which ensures the validity of this law and is satisfied in the real world. Up to the present, however, no such relation has been at all convincingly shown to exist.

Here Landau is simply demonstrating that his argument isn't quantum-mechanical.

Now we come to the second element of the swindle: the words "is determined by." These have not been defined. In everyday life, a person asked to describe the coin flip would say that B "is determined by" A, since A was where the coin "decided how to lie," and B just describes the coin "staying where it was." But of course we can time-reverse Newton's laws, in which case B caused A. There's a dash of mystery here, but it's a classical mystery, not a quantum-mechanical one: it would be consistent with the laws of physics for the coin to jump off of the table, rise into the air flipping end over end, and land on your thumb -- and yet we never observe such processes in nature.

In summary, there is nothing quantum-mechanical about Landau's argument, and it really just boils down to an idea about measurement in general, which is that measurement is not time-reversible. One way of understanding why measurement is not time-reversible is that measurement involves the processing and recording of information, e.g., by a human brain. Thermodynamically, a human brain wouldn't be able to exist if the universe were in a state of maximum entropy. For reasons unknown to us, the big bang in our universe was a low-entropy big bang, and therefore the universe is not yet in a state of thermal equilibrium. This is the origin of the psychological arrow of time and the notion that measurement isn't time-reversible.

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Peeter, there is a paper on the dissipation in quantum theory by Callen and Welton from 50ies:

Irreversibility and Generalized Noise. H. B. Callen and T. A. Welton. Phys. Rev. 83 no. 1, pp 34-40. Caltech e-print.

where they derive the rate of power dissipation (irreversible) from other ideas of quantum theory. It is not rigorous though - they use the "Fermi golden rule" which itself is not time reversible, so the question then becomes how does the irreversible golden rule follow from Schrödinger's equation, which is reversible.

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The key here is that quantum mechanics is incomplete and cannot describe the measurement process still less 'the arrow of time' associated to measurements.

The Brussels School has developed an extension of quantum mechanics that addresses such questions. Next you can find an introduction

http://www.ph.utexas.edu/~gonzalo/3bgraphs.html

The 'microscopic' version of the second law is given as $\Im(\Theta)\leq 0$ where $\Theta$ is the complex extension of the unitary evolutor $U$ for dissipative systems.

$\hbar$ does not play any special role in the theory, because irreversibility is also present at the classical level, i.e. when $\hbar \to 0$. The fundamental constant here is $k_\mathrm{B}$.

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$k_B = [k_B] = 1$ in any "fundamental" system of units. Please could you explain heuristically why such an apparently mundane number should be relevant to physics? – Mark Mitchison Feb 17 '13 at 2:21
@MarkMitchison A fundamental constant of the universe such as the Boltzmann constant is not an "apparently mundane number". This constant appears in the microscopic version of the H-theorem and defines fundamental irreversibility. $h$ --> quantum; $c$ --> relativity; $k_\mathrm{B}$ --> irreversibility. The utexas link provided above already explains what is the relevancy of this recent extension of physics... – juanrga Feb 17 '13 at 16:15
Sorry, but that's completely incorrect. $k_B$ is just a conversion factor between two (arbitrary) measures of energy, e.g. Joules and Kelvin in the SI system, as explained in Landau and Lifshitz. The assignment of units energy/temperature to entropy is artificial: entropy is a measure of information so is clearly dimensionless. An example of a true fundamental constant of the universe is $c$ (the others are $\hbar$ and $G$). It is easy to see the difference with a simple thought experiment. – Mark Mitchison Feb 17 '13 at 19:16
(contd.) Imagine communicating with an alien species by (say) radio without any sort of common visual or other references. You want to describe how fast a typical human runs. You can do this because the universe has a natural velocity scale: the speed of light. So you just give them the speed as a fraction of $c$ and they can know exactly and quantitatively what speed you are talking about. Now imagine trying to do the same thing about, for example, a quantity of entropy, by expressing it in units of $k_B$. – Mark Mitchison Feb 17 '13 at 19:21
(contd.) It is easy to see that the value of $k_B$ you use doesn't just depend on the unit system, it depends on what information you naturally have access to. Aliens that could naturally perceive the microscopic motion of atoms in an ideal gas would have evolved a completely different notion of temperature to the one formulated by big-thumbed apes like us, and therefore their concept of $k_B$, if it existed at all, would also be wildly different. This is pretty well-known, I think. – Mark Mitchison Feb 17 '13 at 19:26

Each process of scattering is accompanied with infinite number of soft photons, i.e., it is generally an inelastic process. It is extremely difficult (impossible) to reverse such a process. Even in a thermal bath (i.e. with presence of soft photons in the initial state) it is impossible to reverse exactly scattering, so real processes are irreversible due to being inelastic.

Unfortunately, people reason in terms of elastic processes, even in QED where it is proven that no elastic processes are possible. This is the root of not only "time direction issue", but also of UV and IR problems in calculations.

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Surely this depends on what you know? When you say "it is extremely difficult to reverse such a process", do you actually mean "it is extremely difficult to track all the degrees of freedom corresponding to scattered photons"? After all, this scattering is surely described by a unitary (reversible) operator in a large enough Hilbert space? – Mark Mitchison Feb 15 '13 at 21:17
@MarkMitchison: Yes, that's right, Hilbert space and all that, but the problem is not in "tracking" but in preparing convergent flux of the same photons from external sources instead of divergent flux, for example. – Vladimir Kalitvianski Feb 16 '13 at 13:04
Could you explain exactly what you mean, perhaps in more general terms than a scattering experiment? I don't see why divergence of flux in itself is a problem, you just put detectors to cover every steradian of solid angle around the volume of space you are interested in, including the "external sources". The problem is in actually designing experimental apparatus that can acquire every last bit of information from the system after the scattering events; this problem is clearly rooted in measurement, hence the comments from Landau above. – Mark Mitchison Feb 16 '13 at 16:19
@MarkMitchison: No, I do not see it as a problem of measurement. One can look at it as at breaking an item into many pieces. When the number of particles is large, it is difficult to reproduce an exactly reverse process. – Vladimir Kalitvianski Feb 16 '13 at 17:01
Obviously it is difficult because you need to "track" all of the wavefunctions of the many pieces (particles). Or are you referring to something else? – Mark Mitchison Feb 17 '13 at 2:23