Friday, June 06, 2008

Entropy, information, and the ice cube

Looked at the right way, a humble ice cube can teach you a lot about thermodynamics. As Nicholas Taleb points out in The Black Swan, the simplest facts about how it melts contain the kernel of the Second Law of thermodynamics by implication. Elaborating upon the simplest case illuminates many other situations far more complex.

The Second Law is not deterministic, but probabilistic. It doesn't say, the system has to evolve in such and such a way. It just indicates, given where a system is now, what the most probable direction it will evolve in. Thermodynamics comes into play when we can't know the exact state of every water molecule. Instead, all we know are certain fixed totals about the system: the total energy, the total volume, the total number of molecules. In the very simplest case, total isolation, the ice cube does nothing.

In the next simplest case, the cube is isolated except for contact with a heat bath, which is defined by a temperature, which we'll assume is at or above the ice melting temperature. The Second Law tells us the overwhelmingly probable evolution of the cube: it will change in the direction that increases its entropy or disorganization. That includes heating up (acquiring heat from the heat bath). In this case, that includes melting.

When the entropy increases, information is lost. Imagine that the ice cube has some features on its surface, or that it was carved into some shape. After it's fully melted, all that's left is puddle of water. The puddle of water is the final result, regardless of whatever funky features the solid ice had in its shape.

Since its discovery in the 19th century, the Second Law has had a sad countenance, apparently nothing but a tale of decay and decline. Certainly the thought that an elaborately carved ice sculpture and a plain ice cube of identical mass might end the same way - as a large, featureless puddle of water - made thermodynamics seem like the truly dismal science.

But these are only the two simplest possibilities for the ice cube. It could be in contact with a chemical bath of some substance that reacts and binds with water (hydrates). In that case, two processes, melting and hydration, proceed simultaneously. The larger and/or faster will predominate, since it increases the entropy faster. Melting might not happen at all, because the Second Law would then have another and better avenue to satisfy itself. The Second Law doesn't tell you that melting has to happen; just that, whatever avenues of change are available, the one that gets you to higher entropy faster wins. And it only specifies a probabilistic tendency, specifying nothing in general about rate, except that it's positive.

By historical convention, systems in passive contact with external "baths" are not considered "open." They are in thermodynamic equilibrium with themselves and any "baths" in contact. Truly "open" systems are ones with "flow-through," where matter, radiation, and/or heat flow in and flow out. "Open" systems are not in equilibrium, in general, either with themselves or with the outside. In that case, the Second Law still applies, but it only applies to the whole system and its environment. Any part of the whole can see its entropy decline, so long as the entropy of the whole rises. If an open system exhibits a strong spontaneous tendency under certain conditions to lower its entropy and acquire structure, it has to expel the excess entropy outside of itself. The system is said to be self-organizing.

Hence, biology, evolution, and weather.

The competing rates of different processes become a more complex but even more critical tangle in the self-organizing case. For self-organization to succeed, the spontaneous structure has to form faster than any competing process (dissipation) importing entropy back into the system from the outside. Thermodynamic equilibrium is not valid for "open" systems as a whole, but might be valid for parts of the system. Typically, equilibrium is an excellent approximation for suitably "small" part of the system, where local temperature and pressure can be defined and local thermodynamic equilibrium (LTE) holds. But it remains true that on intermediate to the largest scales of the system, LTE is badly violated. These are exactly the scales over which flows of matter, radiation, and heat are most obvious.

One of the things that makes weather and climate prediction so hard is that on intermediate to large scales, the evolution of the atmosphere is not, on the one hand, a simple application of determinism: the system is chaotic; but on the other, not a simple application of thermodynamic arguments either. On short scales (a few to a few tens of meters), the atmosphere respects LTE pretty well, and thermodynamics can be used to predict its evolution (if boundary conditions are known). But on the scales of storms, cyclones, and fronts, the atmosphere, while "thermodynamic" in some sense, is nowhere close to thermodynamic equilibrium, and the simple probabilistic arguments of thermodynamic equilibrium don't apply. It's chaotic enough to make long-term, detailed prediction impossible; but not so chaotic that simple statistical arguments can be used instead. It's somewhere in between: highly sensitive to poorly known initial conditions and past history.

This realm - in between simple linear predictability and simple statistical equilibrium - is not only a result of chaos, but constitutes a distinct area of dynamics and physics, usually given the name complexity. It's a dynamical regime rich with unpredictable structures that repeatedly form and dissipate - like weather, or living things. (June 2)

POSTSCRIPT: The conventional global temperature index continues its recent precipitous drop. In case you've been hiding in a basement the last few months, this was the coldest spring, and the coldest May, in many years.

It should be stressed again that this conventional global temperature index (one of a handful of composite statistical indexes used by the IPCC and others) is not the temperature of anything. The Earth has no single temperature: it's not in thermal equilibrium, with either itself or a "heat bath." It's a complex weighted statistical composite of many individual temperature measurements of the air, ocean surface, and radiation. (The first two are local; the third is nonlocal, by its nature.) The attempt to pass this composite off as the unique "temperature of the Earth" is one of the many major fallacies of the climate change hysteria.

The direction and timing of the trend are not in doubt. Something has been happening in the last decade and has accelerated. That something is cooling. But the exact nature and magnitude of the trend can only be understood by disaggregating the composite index back into its originating individual temperature measurements and looking at their trends in time and space.

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