Stable stationary state

The demand for the presence of sharp, stable stationary states can be referred to in the language of quantum theory as a general principle of the existence and permanence of quantum numbers. 

The stable stationary states that correspond to two lower rows in Table 10.1 may be subdivided into two groups gravity and friction, depending on the dominant... 

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. 

The measurement of an enthalpy change is based either on the law of conservation of energy or on the Newton and Stefan-Boltzmann laws for the rate of heat transfer. In the latter case, the heat flow between a sample and a heat sink maintained at isothermal conditions is measured. Most of these isoperibol heat flux calorimeters are of the twin type with two sample chambers, each surrounded by a thermopile linking it to a constant temperature metal block or another type of heat reservoir. A reaction is initiated in one sample chamber after obtaining a stable stationary state defining the baseline from the thermopiles. The other sample chamber acts as a reference. As the reaction proceeds, the thermopile measures the temperature difference between the sample chamber and the reference cell. The rate of heat flow between the calorimeter and its surroundings is proportional to the temperature difference between the sample and the heat sink and the total heat effect is proportional to the integrated area under the calorimetric peak. A calibration is thus... 

Stationary-state solutions correspond to conditions for which both numerator and denominator of (3.54) vanish, giving doc/dp = 0/0, and so are singular points in the phase plane. There will be one singular point for each stationary state each of the different local stabilities and characters found in the previous section corresponds to a different type of singularity. In fact the terms node, focus, and saddle point, as well as limit cycle, come from the patterns on the phase plane made by the trajectories as they approach or diverge. Stable stationary states or limit cycles are often refered to as attractors , unstable ones as repellors or sources . The different phase plane patterns are shown in Fig. 3.4. 

If d tr(J)/d/i is positive at p, then tr(J) will be negative (corresponding to a stable stationary state) for p < p and positive (corresponding to an unstable stationary state) for p > p. In these circumstances p2 will have the opposite sign to / 2 thus if P2 is negative, p2 will be positive. The limit cycle... 

Fig. 5.3. Locus of Hopf bifurcation points in K-fi parameter plane for thermokinetic model with the full Arrhenius temperature dependence and y = 0.21. The nature of the Hopf bifurcation point and, hence, the stability of the emerging limit cycle changes along this locus at k = 2.77 x 10 3. Supercritical bifurcations are denoted by the solid curve, subcritical bifurcations occur along the broken segment, i.e. at the upper bifurcation point for the lowest k. The stationary-state solution is unstable and surrounded by a stable limit cycle for all parameter values within the enclosed region. Oscillatory behaviour also occurs in the small shaded region below the Hopf curve, where the stable stationary state is surrounded by both an unstable and...

Often the range p < p < PsU over which stable stationary state and oscillatory behaviour coexist is extremely small, but it allows the possibility of oscillations outside the region enclosed by the Hopf locus in Fig. 5.3, as... 

Fig. 5.5. A typical phase portrait for a system with ft < ft < ft, showing a stable stationary-state solution (singular point) surrounded first by an unstable limit cycle (broken curve) and then by a stable limit cycle (solid curve). The unstable limit cycle separates those initial conditions, corresponding to points in the parameter plane lying within the ulc, which are attracted to the stationary state from those outside the ulc, which are attracted on to the stable limit cycle and hence which lead to oscillations.

To all intents and purposes, the Hopf prediction is the exact result. It is not difficult to construct an intersection between the maximum and minimum which is a stable stationary state. For instance, with y = 0.02, k = 0.12, and /i = 0.2 we have 0X = 1.042, 9C = 2400, and 0SS = i/k = 5/3. The corresponding phase plane nullclines are shown in Fig. 5.10, together with a trajectory spiralling in to the stationary-state intersection. The trace of the Jacobian matrix is negative for this solution (tr(J) = —4.1 x 10 2) indicating its local stability. This is not, however, a particularly fair test of the relaxation analysis because the parameters /i and k are not especially small. In the vicinity of the origin (where is small) both approaches converge. 

Fig. 5.10. The possibility of a stable stationary-state intersection on the middle branch of the g(a, 9) = 0 nullcline the nullclines are shown as broken curves, the solid curve gives the evolution of a typical trajectory towards the stable focal state.

Fig. 5.11. Excitability in a chemical system, (a) The nullclines /(a, 0) = 0 and g(a,0) = 0 intersect just to the left of the maximum. A suitable perturbation must make a full circuit, as shown by a typical trajectory, before returning to the stable stationary state, (b), (c) The corresponding evolution of the concentration of intermediate A and the temperature excess in time showing the large-amplitude excursion.

Fig. 8.1. Indication of local stability or instability for the simple cubic autocatalytic step without decay solid curves indicate branches of stable stationary-state solutions, broken curves correspond to unstable states, (a) Stationary-state locus with no autocatalyst inflow, fl0 = 0, with one stable solution, 1 - = 0, corresponding to zero reaction (b) stationary-state locus...

Fig. 8.6. Typical arrangement of local stabilities and development of unstable limit cycle, from a subcritical Hopf bifurcation, appropriate to cubic autocatalysis with decay and no autocatalyst inflow and with 9/256 < k2 < 1/16. The unstable limit cycle grows as t, decreases below t m, and terminates by means of the formation of a homoclinic orbit at rf . Stable stationary states, including the zero conversion branch 1 — a, = 0, are indicated by solid curves, unstable states and limit cycles by broken curves.

Fig. 8.8. Phase plane representations of the birth (or death) of limit cycles through homoclinic orbit formation. In the sequence (a)-fb)-(c) the system has two stable stationary states (solid circles) and a saddle point. As some parameter is varied, the separatrices of the saddle join together to form a closed loop or homoclinic orbit (b) this loop develops as the parameter is varied further to shed an unstable limit cycle surrounding one of the stationary states. The sequence (d)-(e)-(f) shows the corresponding formation of a stable limit cycle which surrounds an unstable stationary state. (In each sequence, the limit cycle may ultimately shrink on to the stationary state it surrounds—at a Hopf bifurcation point.)...

Fig. 8.13. (a) The division of the fS0 — K1 parameter region into 11 regions by the various loci of stationary-state and Hopf bifurcation degeneracies. The qualitative forms of the bifurcation diagrams for each region are given in fi)—(xi) in (b), where solid lines represent stable stationary states or limit cycles and broken curves correspond to unstable states or limit cycles, (i) unique solution, no Hopf bifurcation (ii) unique solution, two supercritical Hopf bifurcations (iii) unique solution, one supercritical and one subcritical Hopf (iv) isola, no Hopf points (v) isola with one subcritical Hopf (vi) isola with one supercritical Hopf (vii) mushroom with no Hopf points (viii) mushroom with two supercritical Hopf points (ix) mushroom with one supercritical Hopf (x) mushroom with one subcritical Hopf (xi) mushroom with supercritical and subcritical Hopf bifurcations on separate branches. 

FlG. 8.14. The different phase plane portraits identified for cubic autocatalysis with decay (a) unique stable state (b) unique unstable stationary state with stable limit cycle (c) unique stable state with unstable and stable limit cycles (d) two stable stationary states and saddle point (e) stable and unstable states with saddle point (f) stable state, saddle point, and unstable state surrounded by stable limit cycle (g) two unstable states and a saddle point, all surrounded by stable limit cylcle (h) two stable states, one surrounded by an unstable limit cycle, and a saddle point (i) stable state surrounded by unstable limit cycle, unstable state, and saddle point, all surrounded by stable limit cycle (j) stable state, unstable state, and saddle point, all surrounded by stable limit cycle (k) stable state, saddle point, and unstable state, the latter surrounded by concentric stable and unstable limit cycles (1) two stable states, one surrounded by concentric unstable and stable limit cycles, and a saddle point. 

From the results of chapters 4 and 5, we can predict the behaviour of the system if it is well stirred. For some experimental conditions, represented by particular values for the dimensionless reactant concentration /t and the rate constant k, the system will have a uniform, stable stationary state (really only a pseudo-stationary state as /t is decreasing slowly because of the inevitable consumption of the reactant discussed previously). For other conditions, the stationary state loses its stability and stable uniform oscillations can be... 

We can recognize the first term as the trace of the matrix for the well-stirred system of chapter 4 (let us call this tr(U)) multiplied by the positive quantity y. We have specified that we are to consider here systems which have a stable stationary state when well stirred, i.e. for which tr(U) is negative. The additional term associated with diffusion in eqn (10.47) can only make tr(J) more negative, apparently enhancing the stability. There are no Hopf bifurcations (where tr(J) = 0) induced by choosing a spatial perturbation with non-zero n. 

With this identification, the stable stationary-state behaviour (found for the cubic model with 1 < A < 4) corresponds to oscillations for which each amplitude is exactly the same as the previous one, i.e. to period-1 oscillatory behaviour. The first bifurcation (A = 4 above) would then give an oscillation with one large and one smaller peak, i.e. a period-2 waveform. The period doubling then continues in the same general way as described above. The B-Z reaction (chapter 14) shows a very convincing sequence, reproducing the Feigenbaum number within experimental error. 

This is the general formula (in the linear noise approximation) for the autocorrelation function of the fluctuations in a stable stationary state. Hence it is possible to write down the fluctuation spectrum in an arbitrary system without solving any specific equations. This fact is the basis of the customary noise theory. 

Exercise. Verify that the linear noise approximation always leads to an Ornstein-Uhlenbeck process for the fluctuations in a stable stationary state. 

In particular, it is useful to define the critical point through F(nc) = 0 (the stationary state). Since multicomponent chemical systems often reveal quite complicated types of motion, we restrict ourselves in this preliminary treatment to the stable stationary states, which are approached by the system without oscillations in time. To illustrate this point, we mention the simplest reversible and irreversible bimolecular reactions like A+A —> B, A+B -y B, A + B —> C. The difference of densities rj t) = n(t) — nc can be used as the redefined order parameter 77 (Fig. 1.6). For the bimolecular processes the... 

But how does the flux evolve from its zero value at the initial moment to its quasistationary value at timescales exceeding the time tf for the formation of quasiequilibrium It is obvious that the answer may depend on initial conditions. The most natural are those corresponding to the stable stationary state of the noise-free system i.e. (q = bottom, q = 0) where bottom is the coordinate of the bottom of the potential well. We assume such an initial state here. If the... 

There is one further important practical aspect that has to be considered when taking this approach to performing an experimental bifurcation analysis impedance measurements can only be carried out with stable stationary states it is not feasible to measure unstable stationary states, or states close to a bifurcation. However, as we discussed above, N-NDR and HN-NDR systems become unstable due to ohmic losses in the circuit, whereas they are always stable for vanishing R< >. Being aware that an ohmic series resistor causes only a horizontal shift of the impedance spectrum in the complex plane, it is apparent that it is possible to infer about the existence of bifurcations from impedance measurements at sufficiently low solution resistance (or when invoking an ZR-compensation, an option many potentiostats provide). This is illustrated with the schematic impedance spectrum shown in Fig. 12, which depicts a typical impedance spectrum of an N-NDR system. The spectrum possesses two... 

The simplest situation arises for bistable systems in which the basic pattern is a front, i.e. an interfacial region that connects the two locally stable stationary states and propagates in space.10 Thereby one of the two states expands on the expense of the other one. Two experimental examples of potential fronts propagating along the... 

A typical problem in thermodynamics of systems that are far from their equilibrium is the analysis of the stability of stationary states of the system. Thermodynamic criteria of the stability of stationary states are found the same way as for systems that are far from and close to thermodynamic equilibrium (see Section 2.4) by analyzing signs of thermodynamic fluxes and forces arising upon infinitesimal deviation of the system from the inspected stationary state. If the system is in the stable stationary state, then any infinitesimal deviation from this state must induce the forces that push it to return to the initial position. 

However, equation (3.16) is not always positive. This implies a not necessarily stable stationary state of the stepwise reaction under consideration. Indeed, it was shown previously that autocatalytic reactions at certain ratios of internal parameters the reactant concentrations (thermodynamic rushes) are a spectacular example of processes with the unstable stationary state. 

We just saw that in the systems that fall into the range of linear thermo dynamics, the stable stationary state is characterized by a special point where the evolution of the system, if sHghdy deviated from this point, wiU neces sarily turn it back again to the same point. This conclusion becomes invalid as the system escapes the neighboring of the equilibrium state. 

Example 10 Multiplicity of stable stationary states at the S shaped kinetic characteristics of stepwise transformations... 

When the stationary rate of stepwise transformations is described by an S-shaped dependence, the affinity A,2 of the stepwise reaction (Figure 3.3) the properties of the reactive system can be inspected in the similar way. Like Example 9, in the system there may exist two stable stationary states at certain values of the affinity A,2 and the... 

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