- More Really is Different
By Mile Gu, Christian Weedbrook, Alvaro Perales, Michael A. Nielsen
Abstract: In 1972, P.W.Anderson suggested that `More is Different', meaning that complex physical systems may exhibit behavior that cannot be understood only in terms of the laws governing their microscopic constituents. We strengthen this claim by proving that many macroscopic observable properties of a simple class of physical systems (the infinite periodic Ising lattice) cannot in general be derived from a microscopic description. This provides evidence that emergent behavior occurs in such systems, and indicates that even if a `theory of everything' governing all microscopic interactions were discovered, the understanding of macroscopic order is likely to require additional insights.
Since Stephen Luttrell asked last week what I think about the paper, I thought I should come back to it.
The paper is really neat. Here is a quick summary. The authors consider a 2-dimensional spin lattice with nearest neighbor interaction. We thus have a lattice of ups and downs that can influence the other ups and downs around them according to some rule given by the Hamiltonian of the system. On this lattice, the authors put certain “designer Ising blocks” with an associated Hamiltonian, composed of several spin states. These blocks have the property that if they are in the ground state (the state of lowest energy), and one forces the spins on one side to be in a certain pattern of ups and downs by applying a field, then the other side will produce a certain output. They give specific examples in the appendix.
If one covers a 2-dimensional semi-infinite plane with these blocks and has an input on one side of the first row, the spin-spin interactions give a rule determining what is on the other side of that row, which is the input for the second row. And so on. Thus, the ground state of the system is fully determined. But what can we say about this ground state?
Now here is the clue of the paper. The authors show that with the appropriate initialization and designer blocks, one can map a cellular automaton to this spin-system. A cellular automaton operates on a one dimensional input line according to a specific rule. This rule crates a new state, on which the rule is applied again etc. This is commonly shown in a diagram with all the so created states below each other, each corresponding to a certain time step. In the spin-lattice, there are no time steps, but the ground state would be a picture of these states of the cellular automaton. Note however that for the spin-lattice this is not a time-dependent realization of this state. The ground-state just is specified according to some rules.
However, there exists cellular automata for which it can be proven that no non-trivial questions can be answered about their evolution, without actually running them and looking, thus one can never say anything about the total evolution. Because of the correspondence to the spin-lattice this then means there are questions about its ground state that can't be answered either. An example for such a question would be what the overall magnetization is of the system. There is thus no way to derive this quantity from the Hamiltonian, the question is undecidable.
It should be noted that it is important for this conclusion the spin-lattice is indeed infinite. The approximation that a system is infinite is very common in physics and often used to simplify computations. What this argument thus shows is that in this limit, there can remain questions open that fundamentally can not be answered about the whole system.
The title of the paper is a reference to Anderson's paper “More is Different,” an argument against reductionism, claiming that not every system can be understood by merely analysing its parts. Gu et al's paper provides an explicit example for which it can be proved that it indeed is not possible to understand the whole system from the behavior of its constituents. For this argument to hold however “more” isn't enough, it has to be infinitely more.
On the risk of merely expressing my utter ignorance, this doesn't surprise me much. What the authors have shown in the paper is a map from a cellular automata, commonly run on a computer, to a 2-d spin lattice. This lattice is a physical realization of a computer code and thus similar conclusions hold for both. As far as I am concerned, if I run the code and visualize it on my screen (for an infinitely long time of course) this is also a physical realization of the code, it is a state my computer's hardware is in. However, the map to the spin system is without doubt much cleaner and better to analyze.
It would be interesting to see whether one could find a possibly weaker statement for large, but finite systems.
For more on the topic, see also my post Emergence and Reductionism, NewScientist's article Why nature can't be reduced to mathematical laws, or check out Stuart Kauffman's last week talk The Open Universe: Toward a Post-Reductionist Science, PIRSA:08100058.
Mile Gu, Christian Weedbrook, Alvaro Perales, Michael A. Nielsen (2008). More Really is Different arXiv