Far from Equilibrium: The Gas Pendulum Leonard J. Soltzberg Simmons College, Boston MA 02115 The study of processes occurring far from equilibrium is perhaps the Ereat scientific "sleer)er"of the latter half of the 20th century. Like quantum meihanics during the first decades of this century, far-from-equilibrium dynamics and thermodynamics has the potential for explaining a great wealth of phenomena but until recently has been the domain of a small number of specialists ( I ) . T h e concepts which define this rapidly evolving body of theory have yet to find their wav into undereraduate chemistrv. let alone into the public lexicon. The oistacles that slowedthe general appreciation of wave mechanics in the chemical communitv also inhibit a wider appreciation of far-from-equilibrium processes: various parts of the theorv are surorisine and counterintuitive, certain implications are ph~losopkcallyand theologically disturbing, and the mathematics is unfamiliar to many of the current generation of chemists. In the dynamics and thermodynamics of isolated systems, processes are governed by the universal principle that the entropy of the system moves to a maximum as the system spontaneously approaches equilibrium. In a closed system in thermal contact with its surroundings, the free energy of the svstem moves to a minimum: if this anoroach to eauilihrium .. o&rs irreversibly, there wili he a net increase in t i e entropy of the universe. In contrast, the study of nonequilibrium processes focuses on systems which are neither isolated nor closed; there is exchange of both energy and matter with the "outside world" across finite potential gradients. Most of the processes that we actually encounter in evervdav . . life are of this type. For nonequilibrium systems that are near equilibrium, the behavior is not very striking. For example, a system may be subject to asmall continuing input of heat that prevents it from reaching thermal equilibrium, or there may be a small continuous input of a chemical reactant that sustains a chemical noneauilibrium. If the ~ e r t u r b a t i o nis small. such a system approaches-instead of kquilibrium-a steady state in which the flows of energy and matter conform to a minimum "entropy production", where entropy production is the ongoing increase in entropy of the universe due to the flow of energy andlor matter across a finite gradient (2). Except for the replacement of a static macroscopic equilibrium state with a dynamic but relatively quiescknt steady state, the macroscopic appearance of a near-equilibrium system is not dramatically different from that of a system actually a t equilibrium. In particular, the entropy production is a t a minimum (it would be zero a t equilibrium); and if the system is disturbed by, say, a temporary temperature or pressure fluctuation, its response to the perturhation is proportional t o the size of the perturbation and the system returns to its original state after the perturbation is over (this is true of both equililAun and near-equilil)rium states). The situation is qualitati\~elsdifferent for far-from-eauilihrium systems:' 1) As a system is driven further and further frum equilibrium by,
for example, the imposition of an increasingly large temperature -eradient. there can be a sham transition from the nearequilihrium regime to a new, far-from-equilibrium behavior. This transition is called a "bifurcation".
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This behavior also requires that the differential equations governing the system contain at least one nonlinear term.
This sharp transition is due to the fact that the near-equilibrium state has become unstable. The actual transformation is triggered by a random fluctuation or disturbance. At or near equilibrium, such a disturbance would produce only a small effect, and the system would return to its original state. Far from equilibrium, the disturbance is magnified rather than damped, and it produces a permanent transformation to the newly stable far-from-equilibrium regime. 3) If there are two stable far-from-equilibriumstates at the bifurcation paint, it is impossible to predict which will actually appear in any particular ease. 4) Among the surprising characteristicsof far-from-equilibrium states are coherent motions of large numbers of molecules, producing ordered structures in space andlor time. This spontaneous ordering implies a decrease in the entropy of the system itself, but this is compensatedby a larger overall entropy production an the system attempts to dissipate the large free energy excess imposed by the driving gradients (3). Thus, these coherent structures are called "dissipative structures". Examples of dissipative structures are convection currents (4), various nonlaminar flow oatterns in liauids (.4.) . chemical wavrs (31, periodic precipitation such as 1.ieaegang rings ( 6 ) . prriodiccondmwion r71, prriodir pa3 evolution @ I , pruhahly hurricnnesnnd rurnadurs,nnd porsiblyall livin~orgnnismsD2)
11).
5) Ifthesystem isdriven yet further from equilibrium, there may
be further bifurcationsleading tonew, more complicated dissipative structures. Eventually, a very large disequilibrium can lead to what appears to be disorderly behavior, although it is hard to tell in specific cases whether this is just avery complex periodic behavior or an aperiodic, unpredictable behavior ( 4 ) . Such a regime is called "deterministic chaos" (12). Model Systems Recognition of the pervasiveness and importance of farfrom-eauilibrium ~ h e n o m e n is a motivating intensive analvsis of the conditions under which such behavior can be expected. Many of the real systems of interest are quite complicated, so i t is useful to find simpler model systems that display bifurcation, spontaneous ordering, and deterministic chaos. A simple example that illustrates the differences among equilibrium, near-equilibrium, and far-from-equilihrium is a rigid pendulum. Hanging motionless, the pendulum is a t equilihrium (Fig. la); it is a t the minimum of both ootential and kinetic enerev. A small oush will oroduce a small oscillation; a somewhat harder push will produce a somewhat larger oscillation (Fig. lb). I n either case, removal of the excess energy by friction and air resistance will restore the pendulum to its orieinal eauilibrium Dosition (Fie. lc). T o model a far-from-e
