Spatially uniform perturbations

The general equations governing the growth or decay of different components of a disturbance are again given by (10.45) and (10.46). For simplicity we can take the case of equal diffusivities, so / = 1. Thus, for the nth mode  

Apart from the common factor y, these are the same as the equations governing growth or decay of small perturbations in the well-stirred system. The eigenvalues A1 2 are given by 

We have specified that the conditions of interest here are those lying within the Hopf bifurcation locus, so tr(U) will be greater than zero. This means that the eigenvalues appropriate to this uniform state must have positive real parts. The system is unstable to uniform perturbations. We must exclude this n = 0 mode from any perturbations in the remainder of this section. 

It is convenient to collect here the definitions of tr(U) and det(U), for later reference  

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The stability of the uniform stationary states in system (3,4) to small spatially uniform perturbations (SUPs) was studied in detail by the theory of perfectly stirred reactors (see reference 9). It will be recalled that the system... 

The linear stability analysis developed in Chapter 2 implies that the steady state will be stable to spatially uniform perturbations if, and only if. 

Thus, when the stirring stops, the uniform state remains a stationary solution of the system. Diffusion does not affect the existence of the uniform state, but it may influence its stability. In particular we are interested in determining whether this state can become unstable to spatially non-uniform perturbations. 

Fig. 10.4. A typical spatially dependent perturbation about a uniform stationary-state solution.

If det(J) is positive for all n, then the amplitudes of all the components of any perturbation will decay back to the spatially uniform stationary state. As mentioned above, det(J) is positive for n = 0, and clearly will always be positive for sufficiently large n when the last term dominates. However, eqn (10.48) is a quadratic in n2 a completely stable uniform state arises if there are no real solutions to the condition det(J) = 0- We can write the... 

An alternative way of portraying the pattern formation behaviour in systems of the sort under consideration here is to delineate the regions in chemical parameter space (the h k plane) over which the uniform state is unstable to non-uniform perturbations. We have already seen in chapter 4, and in Fig. 10.3, that we can locate the boundary of Hopf instability (where the uniform state is unstable to a uniform perturbation and at which spatially uniform time-dependent oscillations set in). We can use the equations derived in 10.3.2 to draw similar loci for instability to spatial pattern formation. For this, we can choose a value for the ratio of the diffusivities / and then find the conditions where eqn (10.48), regarded as a quadratic in either y or n, has two real positive solutions. The latter requires that... 

We can recognize three different ranges of behaviour. First, if k > e 2, the uniform system can never become unstable by Hopf bifurcation tr(U) and, hence, tr(J) are always negative and the determinants can never change sign. All spatially dependent perturbations decay back to the stable uniform state. Secondly, if k is smaller but lies in the range... 

Thus the response of a spatially uniform system in thermodynamic equilibrium is always characterized by translationally invariant and temporaly stationary after-effect functions. This article is restricted to a discussion of systems which prior to an application of an external perturbation are uniform and in equilibrium. The condition expressed by Eq. (7) must be satisfied. Caution must be exercised in applying linear response theory to problems in double resonance spectroscopy where non-equilibrium initial states are prepared. Having dispensed with this caveat, we adopt Eq. (7) in the remainder of this review article. 

In the presence of two external perturbations, L e., the spatially uniform time-independent electric and magnetic fields E and B, and of a permanent dipole moment on nucleus I, the energy of the molecule, evaluated in the singlet electronic state k /fl) is, employing Buckingham notation (1, 2) to denote molecular tensors,... 

A transition from the conditions approximating a perfectly mixed (lumped) system to the realistic conditions with both the diffusion and conductive factors being involved in the process of attaining a steady state (in a distributed system) may bring about a destabilization of the previously stable uniform state. A qualitative picture of the destruction of the spatially uniform mode of a catalytic process appears as follows. A small local perturbation resulting in a local overheating is accompanied by an increase in the reaction... 

Here we also assume that the reaction term does not depend explicitly on the spatial coordinate, therefore the dynamics of the medium is uniform in space. It is easy to see that the spatially uniform time-periodic oscillation is a trivial solution of the full reaction-diffusion-advection system, so the question is whether this uniform solution is stable to small non-uniform perturbations and more generally, if there are any persistent spatially non-uniform solutions in which the spatial structure does not decay in time. 

The analysis above refers to time-independent velocity fields and equal diffusion coefficients for all species, but Straube et al. (2004) have shown that similar behavior applies to mixing in time-dependent chaotic flows. It was shown numerically that A is a linear function of the reaction rate and the transition in the stability of the spatially uniform state to non-homogeneous perturbations takes place when the positive Lyapunov exponent of the local dynamics is equal to the exponent describing the decay rate of the dominant eigenmode. 

Traceless forms of the higher multipole operators have been given by Buckingham.12 Much of the work reviewed will refer to the case of a spatially uniform applied field, when the perturbed hamiltonian reduces to... 

We assume that the molecule is in the presence of multiple perturbations independent of time a spatially uniform magnetic field and a non-uniform electric field with uniform gradient. The electronic reference state la) = and the excited states I/) = of the molecule are eigenfunctions of the unperturbed time-independent Hamiltonian The natural... 

We now consider the effects of diffusion on an unstirred, spatially distributed system, where the steady state of the homogeneous system satisfies Eq. [60]. This corresponds to a spatial system that is stable in the presence of small, uniform perturbations. The spatially homogeneous steady state remains [X]s, [Y]s however, we now allow for the possibility of this state being destabilized by the effects of diffusion. We consider a one-dimensional system, which, after linearizing around the steady state, is described by... 

The magnetic Hamiltonians describe the interaction of electrons with the intramolecular perturbation, that is, the intrinsic magnetic dipoles m/, via the vector potential XXi A , and with an external, spatially uniform and time-independent magnetic field B = V x A ,... 

Fig.7 A. A Tp-periodic pacemaker is created near X = 0 by a localized initial perturbation when 0 0.2. This oscillation does not remain localized as a pacemaker the medium tends to the spatially uniform bulk oscillation.

In the second aspect, we again picture the catalytic material itself to have spatially invariant properties, but now we ask questions about the stability of a spatially uniform reaction state to spatial perturbations. This stability question is similar to that posed in studies of hydrodynamic stability and of the other reaction-diffusion problems considered by Turing [62], Prigogine [63,64], Nicolis [63], Othmer and Scriven [65,66] and their co-workers. Prigogine and his co-workers labeled this phenomena "symmetry-breaking" instabilities. The key idea is that since there is a finite rate of transport, the complex interactions between the rate of communication by diffusive transport and the rate of chemical change may make it dynamically impossible for a spatially uniform state to be sustained. 

In order to derive a quantum mechanical expression for the frequency-dependent polarizability we can make use of time-dependent response theory as described in Section 3.11. We need therefore to evaluate the time-dependent expectation value of the electric dipole operator (4 o(i (f)) Pa o( (t))) in the presence of a time-dependent electric field, Eq. (7.11). Employing the length gauge, Eqs. (2.122) - (2.124), which implies that the time-dependent electric field enters the Hamiltonian via the scalar potential in Eq. (2.105), the perturbation Hamilton operator for the periodic and spatially uniform electric field of the electromagnetic wave is given as... 

It is expedient to employ the Rayleigh-Schrbdinger perturbation theory, or equivalent theoretical tools (e.g., equation-of-motion and propagator methods) to evaluate explicit expressions for the properties appearing in equations (4) and (5). In the presence of time-independent, spatially uniform external perturbations (e.g static homogeneous fields) or intramolecular perturbations (nuclear magnetic dipole or electric quadrupoles), described via first- and second-order interaction Hamiltonians and the expressions for the contributions to the energy up to fourth order can be written as... 

The value o+l <0.4 found for H2 shows that even in the lowest state the molecules are rotating freely, the intermolecular forces producing only small perturbations from uniform rotation. Indeed, the estimated (3vq<135° corresponds to Fo <28 k, which is small compared with the energy difference 164 k of the rotational states j = 0 and j= 1, giving the frequency with which the molecule in either state reverses its orientation. The perturbation treatment shows that with this value of Fo the eigenfunctions and energy levels in all states closely approximate those for the free spatial rotator.9... 

As a simple example, we might impose a perturbation with a cosine distribution as illustrated in Fig. 10.4. If the uniform state is stable to such a perturbation, the amplitude will decay to zero if the uniform state is unstable, the amplitude will grow. We could ask this question of stability with respect to any specific spatial pattern, but non-uniform solutions will also have to satisfy the boundary conditions. This latter requirement means that we should concentrate on perturbations composed of cosine terms, with different numbers of half-wavelengths between x = 0 and x = 1. 

Each component of the perturbations has been separated into two terms a time-dependent amplitude An and Tm, and a time-dependent spatial term cos (nnx). If the uniform state is stable, all the time-dependent coefficients will tend in time to zero. If the uniform state is temporally unstable even in the well-stirred case, but stable to spatial patterning, then the coefficients A0 and T0 will grow but the other amplitudes Ax-Ax and 7 1-7 0O will again tend to zero. If the uniform state becomes unstable to pattern formation, at least some of the higher coefficients will grow. This may all sound rather technical but is really only a generalization of the local stability analysis of chapter 3. 

Fig. 10.5. Representation of the conditions for which the uniform stationary state is unstable to perturbations with a particular number-of half-wavelengths (a) n = 2 (b) the loci for n = 1,2,3, and 4, showing how these overlap to create regions in which more than one spatial perturbation may cause departure from the uniform state.

Fig. 10.7. Representation of conditions in the K-fi parameter plane for instability of the uniform stationary state with respect to spatial perturbations for a system with f = 10. Also shown (broken curve) is the Hopf bifurcation locus, within which the uniform state is itself unstable. The two loci cross at some point on their upper shores.

Fig. 10.9. The neutral stability curve in the n n plane below the curve parametrized by tr(J) = 0 the uniform state is unstable to perturbations of appropriate spatial form...

If, however, we start the system with a given non-uniform distribution, corresponding to n = 2 say, and a value for ji such that the initial point lies beneath the neutral stability curve, then the spatial amplitudes will not decay. Rather the positive real parts to the eigenvalues will ensure that the perturbation waveform grows. The system may move to a state which is varying both in time and position—a standing-wave solution. 

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