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Notes: We develop a second-order response theory to investigate the effects of external periodic perturbations on a chemical reaction at a stable steady state in an open reactor. We apply the theory to the quadratic Schlögl model, a single-variable nonlinear reaction. In the presence of oscillating reactant or product concentrations or oscillating rate coefficients, the average intermediate concentration, the fluxes, and the dissipation are each a Lorentzian function of frequency with midpoint at the inverse relaxation time of the system. Thus even very short relaxation times can be determined by measuring average rates as a function of frequency of the perturbation. The amplitude of the Lorentzian depends on the chemical mechanism of the reaction and is proportional to the square of the amplitude of the applied perturbation. We also show that energy from the perturbation can be used to drive the reaction in a direction opposite of that predicted by the Gibb's free energy difference of reactants and products, even under circumstances where the overall affinity is independent of the perturbation.

Notes: Periodic perturbations are applied to the input fluxes of reactants in a system which exhibits autonomous oscillations, the combustion of acetaldehyde (ACH) and oxygen, and a system which exhibits damped oscillations, the combustion of methane and oxygen. The ACH system is studied by experiments and numerical analysis and the methane system is studied by numerical analysis. The periodic perturbations are in the form of a two-term Fourier series. Such perturbations may generate multiple attractors, which are either periodic or chaotic. We discuss two types of bistable responses: a new phase bistability, in which a subharmonic frequency is added to a sinusoidal perturbation at different phases relative to the periodic response; and jump phenomena, in which the resonant frequency of a nonlinear oscillator depends on the amplitude of the periodic perturbation. Both the ACH and the methane systems confirm the phase bistability. The additional complex behavior of bistability due to jump phenomena is seen only in calculations in the methane system. In both types of bistability a hysteresis loop is formed as we vary the form of the periodic perturbation. In the methane system, we find period doubling to chaos occuring on one branch of the hysteresis loop while the other branch remains periodic. The methane system has been studied in the context of the efficiency of power production. We calculate the efficiency corresponding to each bistable attractor and find one branch of each pair to be the more efficient mode of operation. In the case of the coexisting periodic and chaotic attractors the chaotic attractor is the more efficient mode of operation.

Notes: We consider chemical reactions occurring in a compartment separated by semipermeable membranes from reservoirs of reactant and product, both held at constant pressure. In earlier work, we have developed a nonequilibrium thermodynamic theory applicable to systems with a single reactive intermediate, and we have established a link between the thermodynamic and stochastic analyses of such systems. Here we show that our results generalize directly to cases with two or more reactive intermediates, if the reaction mechanism is nonautocatalytic, or if the system is evolving toward an equilibrium steady state in the reaction compartment without first exhausting the reactant or product reservoir. Starting with nonautocatalytic mechanisms, we identify effective driving forces for each intermediate; we then obtain the driving force for an arbitrary system by mapping to an instantaneously equivalent nonautocatalytic system. The driving force can be cast thermodynamically in terms of the difference between the actual chemical potential of the intermediate and its chemical potential at a reference state (the steady state of the instantaneously equivalent nonautocatalytic system); it can also be cast kinetically in terms of reactive fluxes in the instantaneously equivalent system. Taking the product of the driving force and the net flux of each intermediate and then summing over the species gives a term in the dissipation that is specific to the intermediates. This term is minimized at nonequilibrium steady states, unlike the total dissipation (or entropy production). For the nonautocatalytic or equilibrating systems, an integral of the driving forces yields a Liapunov function for the evolution of the reaction chamber toward the steady state. The same integral also determines the stationary solution of the birth–death master equation for the species numbers of intermediates in the reaction compartment; this generalizes the Einstein relation for the probability of equilibrium fluctuations to far-from-equilibrium conditions.

Notes: We compare theory and experiments on relative stability of the two stable stationary states in optically bistable ZnSe interference filters. We examine the equistability predictions of the master equation, the Fokker–Planck equation, and a deterministic analysis. For this effectively one-variable system there exists a potential for each of these descriptions. We derive a master and Fokker–Planck equation for the fluctuations in temperature expected in our system and find that the three theoretical potentials differ significantly for homogeneous geometries of the system, but the stochastic predictions agree with the deterministic equistability point for inhomogeneous systems. We also examine the effect of external noise, which is independent of the state variable, and find that a very small amount of noise reduces the equistability prediction of the homogeneous master and Fokker–Planck equations to that of the deterministic potential. Noise in the constraints (such as light intensity) results in a new effective potential.

Notes: Coupling between continuous-flow, stirred tank reactors (CSTR's), each having multiple steady states, can produce new steady states with different concentrations of the chemical species in each of the coupled tanks. In this work, we identify a kinetic potential ψ that governs the deterministic time evolution of coupled tank reactors, when the reaction mechanism permits a single-variable description of the states of the individual tanks; examples include the iodate-arsenous acid reaction, a cubic model suggested by Noyes, and two quintic models. Stable steady states correspond to minima of ψ, and unstable steady states to maxima or saddle points; marginally stable states typically correspond to saddle-node points. We illustrate the variation in ψ due to changes in the rate constant for external material intake (k0) and for exchange between tanks (kx). For fixed k0 values, we analyze the changes in numbers and types of steady states as kx increases from zero. We show that steady states disappear by pairwise coalescence; we also show that new steady states may appear with increasing kx, when the reaction mechanism is sufficiently complex. For fixed initial conditions, the steady state ultimately reached in a mixing experiment may depend on the exchange rate constant as a function of time, kx(t) : Adiabatic mixing is obtained in the limit of slow changes in kx(t) and instantaneous mixing in the limit as kx(t)→∞ while t remains small. Analyses based on the potential ψ predict the outcome of mixing experiments for arbitrary kx(t). We show by explicit counterexamples that a prior theory developed by Noyes does not correctly predict the instability points or the transitions between steady states of coupled tanks, to be expected in mixing experiments. We further show that the outcome of such experiments is not connected to the relative stability of steady states in individual tank reactors. We find that coupling may effectively stabilize the tanks. We provide examples in which coupled CSTR's can be operated stably with one of the tanks at or beyond the single-tank marginal stability point.

Notes: Prior work has shown that an excess work is necessary to displace a chemical or physical system from a stationary state, and this excess work determines the stationary distribution of a stochastic birth–death master equation. We derive the augmentation of this master equation for a one-variable system in the presence of external noise. When this noise is much larger than internal noise, but still small compared to macroscopic averages, then the stationary distribution reduces to a form suggested by Landau and Schlögl, which is the integral of the flux of the deterministic kinetic equation. A similar result was obtained on the basis of an assumed Fokker–Planck equation. Hence, in the presence of external forces exceeding in intensity the internal fluctuations, fluxes are proportional to forces without linearization in concentrations.

Notes: In this work, we test a hypothesized form for the stationary solution Ps(X,Y) of the stochastic master equation for a reacting chemical system with two reactive intermediates X and Y, and multiple steady states. Thermodynamic analyses and the exact results for nonautocatalytic or equilibrating systems suggest an approximation of the form Pas(X,Y)=N exp(−φ/kT), where the function φ is a line integral of a differential "excess'' work Fφ, which depends on species-specific affinities. The differential Fφ is inexact. In a preceding paper, we have given an analytic argument for the use of the deterministic kinetic trajectory, connecting (X,Y) to the steady state (Xs,Ys) as the path of integration for Fφ. Here, we show that use of the deterministic trajectories leads to a potential φdet which is continuous across the separatrix between the domains of attraction of the two stable steady states in the model studied.We compare the approximate form of Ps(X,Y) thus generated with numerical solutions of the time-dependent master equation in the limit of attainment of a stationary distribution. Because the time required for convergence to the stationary distribution scales as eN with the particle number N in cases with two stable steady states, the numerical work is limited to systems with O(10–100) X and Y particles. System size affects the accuracy of the approximation. To isolate system-size effects, we compare numerical solutions and the corresponding approximations to Ps(X) for two single-intermediate master equations, since the approximation becomes exact in the limit of large particle number for such equations. Based on these comparisons, for the systems with two intermediates, the agreement between the approximation and the numerical solutions is reasonable. The agreement improves as the number of particles increases in those test cases where it has thus far been possible to vary the system size over an order of magnitude. The results obtained by integrating along deterministic trajectories are better than those from straight-line or line-segment paths. The numerical work on small, single-variable systems with two stable steady states leads to two new observations: (i) the relative heights of the steady state peaks in the exact stationary distribution may invert as the system size increases and (ii) an approximation used commonly for particle counting may give results inconsistent with the exact stationary distribution when the particle number is small, while an alternative approximation improves the agreement.