Entropy unstable

The equilibrium is evidently stable when the entropy is a maximum, for then every possible change would diminish the entropy. The equilibrium will be unstable when the entropy is a minimum for a given value of the energy. This implies that if there are several conceivable neighbouring states with the same energy, that with the least entropy will correspond with a state of unstable equilibrium, whilst the others with more entropy will be essentially unstable states, except the one with the greatest amount of entropy, which will be the state of stable... 

The condition that the equilibrium shall be stable, unstable, or neutral is that the entropy shall be a maximum, a minimum or stationary respectively ... 

Steady states may also arise under conditions that are far from equilibrium. If the deviation becomes larger than a critical value, and the system is fed by a steady inflow that keeps the free energy high (and the entropy low), it may become unstable and start to oscillate, or switch chaotically and unpredictably between steady state levels. 

If the entropy production in the stationary state falls, a negative fluctuation occurs, and the system becomes unstable the stable stationary state of the system is disturbed or destroyed. The system reacts by changing its composition until a new stable state is reached. In this new state, the system is characterised by a lower entropy content than that present prior to the fluctuation, since only a negative entropy change can occur. However, the lower entropy corresponds to a higher degree of order in the system. 

Figure 5.3 Entropy of liquid and crystalline aluminium in stable, metastable and unstable temperature regions [12]. The temperatures where the entropy of liquid and crystalline aluminium are equal are denoted Tf and 7 jm crySt, respectively.

Figure 3 depicts the spectmm of Lyapunov exponents in a hard-sphere system. The area below the positive Lyapunov exponent gives the value of the Kolmogorov-Sinai entropy per unit time. The positive Lyapunov exponents show that the typical trajectories are dynamically unstable. There are as many phase-space directions in which a perturbation can amplify as there are positive Lyapunov exponents. All these unstable directions are mapped onto corresponding stable directions by the time-reversal symmetry. However, the unstable phase-space directions are physically distinct from the stable ones. Therefore, systems with positive Lyapunov exponents are especially propitious for the spontaneous breaking of the time-reversal symmetry, as shown below. 

Figure 5.21 Bifurcation far from equilibrium, (a) Primary bifurcation is the distance from equilibrium, at which the thermodynamic branching of minimal entropy production becomes unstable. The bifurcation point or critical point corresponds to the concentration (b) Complete diagram of bifurcations. As the non-linear reaction moves away from equilibrium, the number of possible states increases enormously. (Adapted, with permission, from Coveney and Highfield, 1990).

In this reaction there is a considerable increase in the entropy, three mols of a gaseous compound, 3A1C1( ), being formed from one, (A1C13). In the reactions at low temperature, when all the compounds are solid, there is no increase in the entropy if A1CI is formed, and the compound is unstable. 

Analogous methyl azidoformate forms with norbornene a thermal unstable triazoline.251 The decomposition products are 40% aziridine and 55% imide. Furthermore it has been observed that the rate of nitrogen evolution of the triazoline from methyl azidoformate increases threefold when triglyme and 20-fold when dimethyl sulfoxide are substituted for 1,1-diphenylethane as solvents. This fact supports a betaine intermediate in the thermal decomposition reaction. The triazoline from 2,4-dinitrophenyl azide and norbornene could just be isolated, but from picryl azide only the aziridine was obtained.252-254 Nevertheless, the high negative value of the activation entropy (—33.4 eu) indicates a similar cyclic transition state for both reactions. 

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