Equilibrium and dissipative structures

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Cosma Shalizi was asked to comment on these concepts

[edit] Equilibrium

Yes, this is one of those words which gets used in ways which are just similar enough to cause epic confusion.

1. In mechanics, a system is in equilibrium when there is no net force acting at any point in space. Since F=ma, this implies that there is no acceleration, and consequently at most uniform rectilinear motion.

2. In dynamical systems theory, an equilibrium state is just a fixed point. (In fact my impression, not backed by a thorough survey, is that the term "equilibrium" is fading out of use in dynamics, in favor of "fixed point".)

• Along any given direction in the state space, a fixed point can

be stable, unstable or neutral. Stable means (roughly) that sufficiently-small displacements away from the fixed point along that direction are reduced over time; unstable that they are amplified; neutral that they don't, on average, either grow or shrink.

• A ball balanced on the rim of a bowl is neutral w.r.t.

perturbations that keep it along the rim, but unstable to others; the same ball at the bottom of the bowl is in a stable fixed point. (In both cases it is also in mechanical equilibrium.)

• The above is a rough explanation of stability because there are

lots of ways of filling in the mathematical details which lead to similar but not quite co-extensive concepts.

3. In thermodynamics, a system is in equilibrium when the forward and reverse rates for _each_ individual process altering the extensive state variables are equal (the "detailed balance" condition). There are also non-equilibrium steady states (NESS), where the _net_ rate for each state variable is zero, though the individual processes are not in balance.

• This bit of terminology is the same in physics and chemistry.
• Both thermodynamic equilibrium and NESS are fixed points, so

"equilibria" in the dynamical systems sense.

• Phase transitions are (pace your correspondent) not part of the

B-Z reaction.

• Maximum entropy _characterizes_ equilibrium (in a closed

system), but doesn't define it --- it's a result, not a definition.

• To give an example, if we have the reactions

               [H2O liquid] <=> [H2O gas] (evaporation/condensation)
               [CH4] + [O2] <=> [CO2] + [H2O gas] (combustion/synthesis)

then to have the system be in equilibrium we'd need the rate of

evaporation to match the rate of condensation, and the rate of methane

combustion to match the rate of methane synthesis. On the

other hand we could have a non-equilibrium steady state in which there

is net combustion and net condensation if the inputs and

outputs get removed at the right rates...

• If a system's state variables break up into two (or more) sets

which are nearly or fully decoupled, it can happen that one set is in equilibrium while the other isn't, or they are both in equilibrium but not in equilibrium with each other. There is some evidence that this happens in fluid turbulence, where one set of variables characterizes the ordinary thermal motion of the fluid molecules, with one temperature, and another characterizes the motion of the vortices (with a different and _negative_ temperature).

4. A Markov chain is in equilibrium when its distribution is invariant under the action of the chain.

• That is, the distribution is a fixed point. This can lead to

some confusion when Markov chains are used to model thermodynamic processes.

5. A game is in equilibrium if each agent's strategy is optimal, given the other agents' strategies.

• Thus there is no "force" pushing the agent to change.
• Because strategies are rules for decision-making over time,

contingent on what's already happened, a game in equilibrium need not be one where nothing changes.

Of all of these, (5) seems like the most nearly appropriate one for interesting social-scientific applications, though it suffers from the usual game-theory problem of demanding too much from the agents. Of course even if you had a social group which was in game-theoretic equilibrium, it would be far from thermodynamic equilibrium, because all living things are.

This isn't an exhaustive list, and if I had more time I'd say more about economics and biology. -- Cosma Shalizi

[edit] Dissipative structures

As to Prigogine claims of self-organization far-from-equilibrium through dissipative structures? That was the source of the discussion, I said my piece about this some time ago:

       http://bactra.org/notebooks/dissipative-structures.html

See "What Prigogine claims to have done..." to the end of the discussion

       http://bactra.org/notebooks/prigogine.html