Abstract
In the preceding chapters we have focused attention on the statistical thermodynamics of systems which are near equilibrium or near stable, nonequilibrium steady states. There are, however, many important examples of nonequilibrium systems that are nonstationary and exhibit nonlinear transients, periodic orbits, and bounded, aperiodic motion. The time course of variables in nonstationary systems is usually studied with the aid of differential equations. For macroscopic systems these equations correspond to the conditional average, and the three types of nonstationary behavior just mentioned represent the nonstationary average trajectories in physical ensembles. In Section 5.4 we have already studied an example of this sort, namely, the nonlinear isomerization reaction with the mechanism A+B⇆2B.
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References
Nonlinear Transients
J. Kramer and J. Ross, Stabilization of unstable states, relaxation, and critical slowing down in a bistable system, J. Chem. Phys. 83, 6234–6241 (1985).
J. Keizer, A theory of spontaneous fluctuations in macroscopic systems, J. Chem. Phys. 63, 398–403 (1975).
A.C. Hindmarsh, Ordinary differential equation solver, Lawrence Livermore Laboratory, Report UCID-30001 (1974).
Limit Cycles: Oscillations and Fluctuations
W. Hahn, The Stability of Motion (Springer-Verlag, Berlin, 1967).
L.S. Pontryagin, Ordinary Differential Equations (Addison-Wesley, Reading, MA, 1962), Chapter 5.
R.H. Abraham and C.D. Shaw, Dynamics—The Geometry of Behavior. Part 1: Periodic Behavior (Aerial Press, Santa Cruz, CA, 1981).
Chemical and Electrochemical Oscillations
R.J. Field, E. Körös, and R.M. Noyes, Oscillations in chemical systems. II. Thorough analysis of temporal oscillations in the bromate-cerium-malonic acid system, J. Am. Chem. Soc. 94, 8649–8664 (1972).
I.R. Epstein, Oscillations and chaos in chemical systems, Physica 7D, 47–58 (1983).
R.J. Field and M. Burger, Oscillations and Travelling Waves in Chemical Systems (Wiley, New York, 1985).
J. Wajtowicz, Oscillatory behavior in electrochemical systems, in Modern Aspects of Electrochemistry, J. Bockris and B. Conway, eds. (Plenum Press, New York, 1973), pp. 47–120.
S.-W. Lin, J. Keizer, P.A. Rock, and H. Stenschke, On the mechanism of oscillations in the “beating mercury heart,” Proc. Nat. Acad. Sci. U.S.A. 71, 4477–4481 (1974).
J. Keizer, P.A. Rock, and S.W. Lin, Analysis of the oscillations in “beating mercury heart” systems, J. Am. Chem. Soc. 101, 5637–5649 (1979).
Chaotic Attractors
R.H. Abraham and CD. Shaw, Dynamics—The Geometry of Behavior. Part II. Chaotic Behavior (Aerial Press, Santa Cruz, CA, 1983).
J.D. Farmer, E. Ott, and J. Yorke, The dimension of chaotic attractors, Physica 7D, 153–180 (1983).
O.E. Rössler, An equation for continuous chaos, Physics Letters 57A, 397–398 (1976).
J.L. Hudson and J.C. Mankin, Chaos in the Belousov-Zhabotinski reaction, J. Chem. Phys. 74, 6171–6177 (1981).
H.L. Swinney, Observations of order and chaos in nonlinear systems, Physica 7D, 3–15 (1983).
J.-C. Roux, J.S. Turner, W.D. McCormick, and H.L. Swinney, Experimental observations of complex dynamics in chemical reaction, in Nonlinear Problems: Present and Future, A. Bishop, D. Campbell, and B. Nicolaenko, eds. (North-Holland, Amsterdam, 1982), pp. 409–422.
A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1982).
B. Mandelbrot, Fractals (W.H. Freeman, San Franciso, 1982).
H.L. Swinney and J. P. Gollub, Hydrodynamic Instabilities and the Transition to Turbulence, 2nd ed. (Springer-Verlag, Berlin, 1985).
Pancreatic β-cell
I. Atwater, CM. Dawson, A. Scott, G. Eddlestone, and E. Rojas, The nature of the oscillatory behavior in electrical activity for the pancreatic β-cell, in Biochemistry and Biophysics of the Pancreatic β-Cell (Georg Thieme Verlag, New York, 1980), pp. 100–107.
T. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic β-cell, Biophys. J. 42, 181–190 (1983).
T. Chay and J. Rinzel, Bursting, beating, and chaos in an excitable membrane model, Biophys. J. 47, 357–366 (1985).
I. Atwater and J. Rinzel, The β-cell bursting pattern and intracellular calcium, in Ionic Channels in Cells and Model Systems, R. Latorre, ed. (Plenum Publishing, New York, 1986), pp. 353–362.
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Keizer, J. (1987). Nonstationary Processes: Transients, Limit Cycles, and Chaotic Trajectories. In: Statistical Thermodynamics of Nonequilibrium Processes. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1054-2_10
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DOI: https://doi.org/10.1007/978-1-4612-1054-2_10
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