Tuesday, September 25, 2007

Thermodynamic confusion: A critique

An important point concerning the nature of thermodynamics - how it developed historically and how it's taught - was mentioned earlier. I want to expand on it, because it turns out to be the key to much confusion around the subject.

Equilibrium. At the center of the confusion is the concept of thermodynamic equilibrium. It's the keystone of how thermodynamics is taught, even though in real life, equilibria are limited in extent and duration. The natural state of things is much more often disequilibrium, with "stuff" - mass, heat - flowing from one place to another. Real equilibrium is only achieved with serious control in laboratory conditions. Conceptually, it's a powerfully simplifying concept that makes it a good place to start teaching students about thermodynamics. Everyone knows what temperature and pressure are - or they think they do.

For often-forgotten historical reasons, thermodynamic equilibrium is often defined for a system "in isolation," even though "equilibrium in isolation" is an oxymoron. When you're in equilibrium, you're in equilibrium with something else. And that is true in thermodynamics. More perceptive and modern treatments start by dividing the world up into "system" and "environment" and stipulate that the system can exchange something (mass, energy, volume) with the environment.* While the totals of each of these extensive variables is exactly conserved, the value retained by the system alone fluctuates according to some statistical law. In fact, a thermodynamic system is really an ensemble of many possible systems, each with a slightly different value of volume, energy, and mass. In a thermodynamic description, one or more of these system extensive variables is replaced by an conjugate or corresponding intensive variable:
  • Volume -> pressure
  • Energy -> temperature
  • Numbers of various chemical species -> chemical potential
To be an equilibrium with its environment means that a system shares the same intensive variable with its surroundings:
  • Mechanical equilibrium: pressure
  • Thermal equilibrium: temperature
  • Chemical equilibrium: chemical potential
The statistical definition of equilibrium defines it has the "most likely state"; this requirement connects statistical mechanics with thermodynamics. For large systems (largeness is not a requirement for thermodynamics, BTW, a point often botched in textbooks), the system's extensive variables (energy, mass/number, volume), while not precisely fixed, deviate only by small fluctuations from well-defined mean values.

Nonequilibrium. If the system fails to satisfy one or more of these criteria, it is out of equilibrium with its surroundings in one or more respects. If the pressures differ, system volume moves in the direction of lower pressure. If the temperatures differ, heat flows toward the lower temperature, moving energy. If the chemical potentials differ, molecules flow toward lower chemical potential, or chemicals react so as to equalize them, changing the chemical distribution. In each case, a difference in intensive variable is associated with a flow of the corresponding extensive variable.

Historical reasons for the confusion. The invocation of "isolated" systems (which actually are incapable of having any thermodynamics) was rooted historically in the development of equilibrium thermodynamics by Gibbs, Einstein, and others. Their point was to free thermodynamics from concern with microscopic details and statistical mechanics. The real requirement is that, while the system is in exchange-contact with its environment, the details of that exchange-contact are irrelevant to defining and using the system's equilibrium state. In fact, you couldn't have a system state (pressure, temperature, chemical potential) at all if the system cannot be sharply distinguished from its surroundings.

While it was necessary to make thermodynamics its own branch of physics, chemistry, and engineering, this requirement has been a source of serious confusion to students and even experts. Along with it goes another confusion, the false belief that thermodynamic systems require a large number of degrees of freedom (like a large number of moving particles or a large volume). In fact, thermodynamics can be defined for the simplest non-trivial case, a single degree of freedom with two possible states; for example, a quantum magnetized spin with "up" and "down" states. The real requirement is that the system evolve for a "long" time in contact with an environment that exchanges energy, etc., in so many different ways that it's only practical to use a statistical and not a deterministic description. (This "long time" requirement is called ergodicity, and checking its validity in real systems is far from obvious.) A large number of possible system states is also not necessary. All that is needed is a large number of transitions among states, transitions happening so often and in so many ways that, again, a statistical treatment is all that's possible. In (ergodic) time, all system transitions - stimulated by contact with the environment - become more or less equally probable, the condition of detailed balance and a big step towards thermodynamic equilibrium.

Exact definition of equilibrium. In a modern treatment, the total entropy (related to the probability of a state of the whole system+environment) is most simply assumed to be the sum of the entropies for system and environment separately; and ditto for the energy, volume, etc. For any conserved extensive variable X, the total entropy S is

S = Ssys(Xsys) + Senv(Xenv) , X = Xsys + Xenv = fixed

The most likely state (maximum of S) is found by requiring that

dS = (δSsys/δXsys)·dXsys + (δSenv/δSenv)·dXenv = 0 , dX = dXsys+ dXenv = 0

leading to the equality of two partial derivatives for the system and environment, respectively:

δSsys/δXsys = δSenv/δXenv

The intensive variable conjugate to X is then defined in terms of the partial derivatives of S with respect to X.

Thermodynamic equilibrium globally and locally. The need for the system to be coupled to its environment becomes clearer in the nonequilibrium case. Here the system doesn't have a "long time" to wait. The details of its boundaries matter, because they control how exactly the system and environment are in exchange-contact. But as long as the changes over space and time of the intensive variables (pressure, etc.) of the system are "smooth enough," it's still possible to redefine those intensive variables as fields that vary in space and time. (In equilibrium, they would all be the constant over the entire extent of the system). This forms the basis for the local thermodynamic equilibrium (LTE) approximation used everywhere in fluid dynamics (thermohydrodynamics), including weather and climate. It allows us to talk about, say, temperature here and temperature there - they don't have to be the same, but they are defined.

The difference between equilibrium and nonequilibrium becomes clear in a different way when viewed through the prism of extensive variable fluctuations. In equilibrium, fluctuations happen all the time, but they never fail to dissipate. (The Fluctuation-Dissipation Theorem connects the two.) In nonequilibrium, some of the fluctuations do not dissipate. Flows in and out from the system boundaries keep "dissipation from dissipating," so to speak, and, precisely by that means, prevent the system from reaching equilibrium.

If both equilibrium and nonequilibrium require that a system be coupled to an environment and thus not isolated, what's the difference in the two cases? The essential difference is this: In equilibrium, the exchange of energy, etc., between system and environment is unconstrained. Nonequilibrium means the exchange-contact is constrained. In fact, different types of nonequilibria can be defined by the way in which this exchange-contact is constrained.

Real life is always nonequilibrium. Equilibrium is one of those all-important but misleading idealizations that applies only in certain laboratory situations and cannot be applied in real life without these caveats.

What's the point of all this for climate? Let's break thermodynamic equilibrium down into the three criteria and examine them one by one from the atmospheric point of view.

Mechanical equilibrium is the only one of the three that is nonproblematic. Even in a tornado, where the variation of pressure is violent and the pressure much lower than normal, the air is nowhere close to a vacuum. A tornado might carry off everything else, but it will leave local pressure equilibrium intact as a reliable assumption, and every other weather phenomenon is less violent.

Thermal equilibrium looks okay on the surface, but it really isn't. "Heat" means just that part of the atmosphere's internal energy subdivided into randomized bits at the molecular level, with temperature indicating the average subdivided bit. But the atmosphere also has kinetic energy in the form of wind and turbulence, spatially "large" modes of motion not in thermal equilibrium. Somehow the two scales (molecular and "large") are separated, although there's a continuum of turbulence and wind scales. Energy can be moved between "large" and "small" scales, but only the latter has a temperature.

In the atmosphere, the place of chemical equilibrium is taken by phase equilibrium, which is supposed to apply to the various forms of water. In fact, it applies only in a crude sense. As normally applied, phase equilibrium requires contrasting phases to be neatly separated in space into "bulk" (macroscopic) regions. Water phases are sometimes neatly separated in space (like ocean-air or ice-air), and for these, local phase equilibrium is a good approximation. But the perpetual evaporation/condensation/precipitation cycle requires the contrasting water phases to spatially interpenetrate. Phase equilibrium is generally not a good approximation here.
  • Evaporation proceeds by pulling and mixing up filaments of water vapor the way you see toffee stretched and folded on the boardwalk in summertime. It's a not a "bulk" process representable by large neighboring spatial blocks.
  • Condensation of clouds requires catalysts (dust and aerosols) and forms "blobs" of varying sizes suspended in air (nucleation). Only when the "blobs" are large enough does phase equilibrium apply to good approximation.
  • Precipitation means liquid or solid water "blobs" heavy enough to fall against upward air currents.
Each transformation of water phase also triggers a discontinuity in density (molar volume) and energy (heat of melting/freezing or of condensation/evaporation). These discontinuities are easy to approximate if the phases are neatly separated in "bulk," but not if they interpenetrate (filamentation or nucleation).

Various and uncontrolled approximations are used to "patch up" the modeling of such situations. These two failures of modeling (one connected with large-scale motion of air, the other with water phase transitions) are in turn reflected in precisely the main weaknesses of current climate modeling.
* The same system-environment distinction is needed for quantum measurement theory. A lot of mystery surrounding quantum mechanics can be dispelled if a measurement is viewed as just another interaction of environment and system. It obviates the need for mystifications like the "Copenhagen interpretation." And no mythical beasts - like hidden variables or many histories - need be invoked. The "hidden" variable is you: the observer is part of the environment.

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