Exponentially stable equilibrium state

Since (8.22) has an exponentially stable equilibrium point at the origin, all solutions that start from initial conditions, satisfying r /( + tan l ku 0) + cii 0)) < 0 and m(0) > —Q and do not touch the 77 = 0 line, reach the origin (steady-sliding state) exponentially. If any of these trajectories reach the N = 0 line say at f = q, then the motion stops instantaneously and starts from the rest at u ti), —Q). This pattern continues and may result in a limit cycle at steady state. 

Now, let the equilibrium state x = 0 of the reduced system (9.1.2) be stable in the sense of Lyapunov. By definition, this means that for the system (9.1.2 and 9.1.3) in the standard form, the x-coordinate remains small in the norm for all positive times, for any trajectory which starts sufficiently close to O, provided y remains small. At the same time, the smallness of x implies the inequality (9.1.4) for the y-coordinate, i.e. y t) converges exponentially to zero. Thus, we have the following theorem. 

The problems at lower temperatures happen because reactions are slow and systems are often far out of equilibrium. (The thermodynamically stable state for all the flora and fauna in any well-aerated lake should be H2O and CO2, for example). Reaction rates increase exponentially with temperature however, and at temperatures of 350°C or higher, even reactions involving molecular oxygen can equilibrate in several days time. This is fortunate, because free energy data are sparse for aqueous species at elevated temperatures and it becomes difficult to measure or calculate Eh much above 100°C. Instead, we can use other reactions between redox-sensitive species to calculate redox-related parameters at higher temperatures. This is the subject of the following section. 

A more general treatment of detailed reaction rates is available in the activated complex theory of Eyring, which assumes that there is an intermediate state between the reactants and the products, called the activated complex or transition state which can be regarded as at least somewhat stable and which is in thermodynamic equilibrium with the reactants, thus permitting thermodynamics to be applied. Instead of an energy, we must use the free energy G (because the pressure is constant) in the exponential. This treatment yields... 

Both in the Coulomb and dipole problems analytic solutions of the equilibrium equations (2.6a) and evaluation of the associated Hessians (2.6b) becomes tedious for as few as four interacting objectsAt present, the only practical way of surveying the locally stable states of the Coulomb systems for larger values of N is to use computers to find energy minima. However, since the number of minima appears to grow exponentially with N, the energy surface (, 0, ... becomes pro-... 

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