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.