Three Toed Sloth (who has been attending the complex systems summer school in China which I was supposed to attend before my life turned upside down and I ran off to Seattle) has an interesting post on Landauer’s principle. Landauer’s principle is roughly the principle that erasing information in thermodynamics disipates an amount of entropy equal to Bolztman’s constant times the number of bits erased. Cosma points to two papers, Orly Shenker’s “Logic and Entropy”, and John Norton’s “Eaters of the Lotus”, which both claim problems with Landaur’s principle. On the bus home I had a chance to read both of these papers, and at least get an idea of what the arguments are. Actually both articles point towards the same problem.
Here is a simplistic approach to Landaur’s principles. Suppose you have a bit which has two values of a macroscopic property which we call 0 and 1. Also suppose that there are other degrees of freedom for this bit (like, say, the pressure of whatever is physically representing the bit). Now make up a phase space with one axis representing the 0 and 1 variables and another axis representing these degrees of freedom. Actually lets fix this extenral degree of freedom to be the pressure, just to make notation easier. Imagine now the process which causes erasure. Such a process will take 0 to 0, say, and 1 to 0. Now look at this processs in phase space. Remember that phase space volumes must be conserved. Examine now two phase space volumes. One corresponds to the bit being 0 and some range of the pressure. The other corresponds to the bit being 1 and this same range of pressures. In the erasure procedure, we take 1 to 0, but now, because phase space volume must be preserved, we necesarily must change the values of the extra degree of freedom (the pressure), because we can’t map the 1 plus range of pressures region to the 0 plus the same range of pressures because this latter bit of phase space is already used up. What this necesitates is an increase of entropy, which at its smallest can be k ln 2.
From my quick reading of these articles, their issue is not so much with this argument, per se, but with the interpretation of this argument (by which I mean they do not challenge the logical consistency of Laundauer and other’s formulations of the principle, but challenge instead the interpretation of the problem these authors claim to be solving.) In both articles we find the authors particularly concerned with how to treat the macroscopic variables corresponding to the bits 0 and 1. In particular they argue that implicit in the above type argument is that we should not treat these macroscopic variables as thermodynamic-physical magnitudes. The author of the first paper makes this explicilty clear by replacing the phase space picture I’ve presented above by two pictures, one in which the bit of information is 0 and one in which the bit of information is 1 and stating things like “A memory cell that – possibly unknown to us – started out in macrostate 0 will never be in macrostate 1″ (emphasis the authors.) The authors of the second article make a similar point, in particular pointing out that “the collection of cells carrying random data is being treated illicitly as a canonical ensemble.”
What do I make of all this? Well I’m certainly no expert. But it seems to me that these arguments center upon some very deep and interesting problems in the interpretation of thermodynamics, and also, I would like to argue, upon the fact that thermodynamics is not complete (this may even be as heretical as my statement that thermodynamics is not universal, perhaps it is even more heretical!) What do I mean by this? Consider, for example, one of our classical examples of memory, the two or greater dimensional ferromagnetic Ising model. In such a model we have a bunch of spins on a lattice with interactions between nearest neighbors which have lower energy when the spins are aligned. In the classical thermodynamics of this system, above a certain critical temperature, in thermodynamic equibrium, the total magnetization of this system is zero. Below this temperature, however, something funny happens. Two thermodyanmic equilibrium states appear, one with the magnetization pointing mostly in one direction and one with the magnetization point mostly in another direction. These are the two states into which we “store” information. But, when you think about what is going on here, this bifurcation into two equibria, you might wonder about the “completeness” of thermodynamics. Thermodynamics does not tell us which of these states is occupied, nor even that, say each are occupied with equal probability. Thermodynamics does not give us the answer to a very interesting question, what probability distribution for the bit of stored information!
And it’s exactly this question to which the argument about Landauer’s principle resolves. Suppose you decide that for the quantities, such as the total magnetic field, you treat these as two totally separate settings with totally different phase spaces which cannot be accessed at the same time. Then you are lead to the objections to Landauer’s principle sketched in the two papers. But now suppose that you take the point of view that thermodynamics should be completed in some way such that it takes into account these two macroscopic variables as real thermodynamic physical variables. How to do this? The point, I think many physicist would make, it seems, is that no matter how you do this, once you’ve got them into the phase space, the argument presented above will procedure a Landauer’s principle type argument. Certainly one way to do this is to assert that we don’t know which of the states the system is in (0 or 1), so we should assign these each equal probability, but the point is that whatever probability assumption you make, you end up with a similar argument. in terms of phase space volume. Notice also that really to make these volumes, the macroscopic variables should have some “spread”: i.e. what we call 0 and 1 are never precisely 0 and 1, but instead are some region around magnetization all pointing in one direction and some region around magnetization pointing in another direction.
I really like the objections raised in these articles. But I’m not convinced that either side has won this battle. One interesting thing which I note is that the argument against Laundauer’s principle treats the two macrostates 0 and 1 in a very “error-free” manner. That is to say they treat these variables are really digital values. But (one last heresy!) I’m inclided to believe that nothing is perfectly digital. The digital nature of information in the physical world is an amazingly good approximation for computers….but it does fail. If you were able to precisely measure the information stored on your hard drive, you would not find zeros and ones, but instead zeros plus some small fluctuation and ones plus some small fluctuations. Plus, if there is ever an outside environment which is influencing the variable you are measuring, then it is certainly true that eventually your information, in thermodynamics, will disappear (see my previous article on Toom’s rule for hints as to why this should be so.) So in that case, the claim that these two bit states should never be accessible to each other, clearly breaks down. So I’m a bit worried (1) about the arguments against Laundauer’s principle from the point of view that digital information is only an approximation, but also (2) about arguements for Laundauer’s principle and the fact that they might somehow depend on how one completes thermodynamics to talk about multiple eqiulibria.
Of course, there is also the question of how all this works for quantum systems. But then we’d have to get into what quantum thermodynamics means, and well, that’s a battle for another day!
Update: be sure to read Cris Moore’s take on these two papers in the comment section. One thing I didn’t talk about was the example Shenker used against Laundauer’s principle. This was mostly because I didn’t understand it well enough and reading Cris’s comments, I agree with him that this counterexample seems to have problems.

My recent post on thermodynamics and computation reminded me of a very nice article by Geoffrey Grinstein that I read a while back.
Suppose we are trying to store digital information into some macroscopic degree of freedom of some large system. Because we desire to store digital information, our system should have differing phases corresponding to the differing values of the information. For example, consider the Ising model in two or greater dimensions. In this case the macroscopic degree of freedom over which we wish to store our information is the total magentization. In order to store information, we desire that the magnetization come in, say, two phases, one corresponding to the system with positive total magnetization and the other corresponding to negative total magnetization.
Assume, now that the system is in thermal equilbrium. Suppose further that there are some other external variables for the system which you can adjust. For the example of the Ising model, one of these variables could be the applied external magnetic field. Since the system is in thermal equilbrium, each of the phases will have a free energy. Now, since we want our information to be stored in some sort of robust manner, we don’t want either of the phases to have a lower free energy, since if it did, the system would always revert to the phase with the lowest free energy and this would destroy our stored information. Since we require the free energy of all information storing phases to be equal, this means that we can always solve these equality equations for some of the external variables. This means that if we plot out the phases as a function of the external variables, we will always end up with coexisting phases along surfaces of dimension less than the number of external variables. For our example of the Ising model in an external magnetic field, what happens is that the two phases (positive and negative total magnetization) only coexist where the magnetic field equals zero. If you have any magnetic field in the positive magnetic direction, then the thermodynamical phase which exists in equibrium is the phase with the postivie total magnetization. So coexistence of phases, and in particular of information storing phases, in the external variable space, is always given by a surface of dimension less than the number of external variables
What is interesting, and why this gets connected with my previous post, is Toom’s rule. Toom’s rule is two dimensional cellular automata rule which exhibits some very interesting properties. Imgaine that you have a two dimensional square lattice of sites with classical spins (i.e. +1 and -1) on each of the lattice sites. Toom’s rule says that the next state of one of these spins is specified by the state of the spin, its neighbor to the north, and its neighbor to the east. The rule is that the new state is the majority vote of these three spins (i.e. if the site has spin +1, north has spin -1, and east has spin -1, the new state will be spin -1.)
Toom’s rule is interesting because it exhibits robustness to noise. Suppose that at each time step, the cellular automata instead of performing the correct update, with some probability the site gets randomized. What Toom was able to show was that for the Toom update rule, if this probability of noise is small enough, then if we start our system with a positive magnetization (just like the Ising model, we define this as the sum of all the spin values) then our system will remain with a postive magnetization and if we start our system with a negative magnetization it will similarly retain its magnetization. Thus Toom showed that the cellular automata can serve, like the two dimensional Ising model at zero applied field, as a robust memory.
But what is nice about Toom’s rule is that it gives an even stronger form of robustness. Remember I said that the noise model was to randomize a single site. Here I meant that the site is restored to the +1 state with 50% probability and the -1 state with 50% probability. But what if there is a bias in this restoration. From the Ising model point of view, this actually corresponds to applying an external magnetic field. And here is what is interesting: for Toom’s rule the region where the two phases which store information can coexist is not just at the (effectively) external magnetic field equal zero point, but instead is a region of external magnetic field between some positive and negative value (set by the probability of noise.) So it seems that Toom’s rule violates the laws of thermodynamics!
The solution to this problem is to realize that the probability distribution produced by Toom’s rule is not given by a thermodynamic Boltzman distribution! Toom’s rule is an example of a probabilistic cellular automata whose steady state is not described by classical thermodynamics. This is exactly one of the models I have in mind when arguing that I do not know whether the eventual state of the universe is going to be in Gibbs-Boltzman thermodynamic equibrium.
Along these lines, Charlie Bennett and Geoffrey Grinstein, have, however, shown that while the steady state of Toom’s rule is not given by a Gibbs-Boltzman thermodyanmic distribution, if one considers the histories of the state of the cellular automata, instead of the state itself, then Toom’s rule is given by a Boltzman distribution over the histories of the cellular automata. It’s at this point that my brain just sort of explodes. That a system’s histories are in equibrium is very strange: normally we think about equibria being generated in time, but here we’ve already used up our time variable! I suspect that the answer to this puzzle can be achieved by refering to the Jaynes approach to entropy, but I’ve never seen this done.