Paul M. Hunt

Large fluctuations and optimal paths in chemical kinetics

The eikonal approximation (instanton technique) is applied to the problem of large fluctuations of the number of species in spatially homogeneous chemical reactions with the probability density distribution described by a master equation. For both autocatalytic and nonautocatalytic reactions, the analysis of the distribution about a stable stationary state and of the transitions between coexisting stable states comes, to logarithmic accuracy, to the analysis of Hamiltonian dynamics of an auxiliary dynamical system. The latter can be done explicitly in a few cases, including one‐species systems, systems with detailed balance, and systems close to the bifurcation points where the number of the stable states changes. In the last case, the fluctuations display universal features, and, for saddle‐node bifurcation points, the logarithm of the probability of escape from the metastable state (per unit time) is proportional to the distance to the bifurcation point (in the parameter space) raised to the power 3/2. We compare the eikonal approximation for the stationary distribution of a master equation to Monte Carlo numerical solutions for two chemical two‐variable systems with multiple stationary states, where none of the cited restrictions exists. For one of the systems in the pattern of optimal paths we observe caustics emanating from the saddle point.

Thermodynamics far from equilibrium: Reactions with multiple stationary states

We present a thermodynamic analysis of global validity for effectively one‐variable, irreversible chemical systems with multiple steady states. A hypothetical reaction chamber is held at constant temperature and volume and is connected by selectively permeable membranes to reservoirs of reactant(s) and product(s), each at constant selected pressures. An appropriate free energy function, which yields criteria of evolution to equilibrium for the composite system of reaction chamber and reservoirs, is a hybrid of Gibbs and Helmholtz free energies. The one variable in the reaction chamber is the pressure of a chemical intermediate which varies in time according to a given reaction mechanism. With the hybrid free energy, the kinetics for a given mechanism, and a concept of instantaneous indistinguishability of systems with different mechanisms, we establish a thermodynamic driving force, or species‐specific affinity, for the intermediate. The species‐specific affinity vanishes at steady states, and upon its di...

Thermodynamic and stochastic theory for nonequilibrium systems with multiple reactive intermediates: The concept and role of excess work

We continue our development of a global thermodynamic and stochastic theory of open chemical systems far from equilibrium with an analysis of a broad class of isothermal, multicomponent reaction mechanisms with multiple steady states, studied under the assumption of local equilibrium. We generalize species‐specific affinities of reaction intermediates, obtained in prior work for nonautocatalytic reaction mechanisms, to autocatalytic kinetics and define with these affinities an excess free energy differential Fφ. The quantity Fφ is the difference between the work required to reverse a spontaneous concentration change and the work available when the same concentration change is imposed on a system in a reference steady state. The integral of Fφ is in general not a state function; in contrast, the function φdet obtained by integrating Fφ along deterministic kinetic trajectories is a state function, as well as an identifiable term in the time‐integrated dissipation. Unlike the total integrated dissipation, φd...

Interference structure in franck—condon overlap functions

Abstract A semi-classical analysis of interference structure in bound—continuum Franck—Condon overlap functions is presented. It is shown that the Condon reflection principle is applicable only in the limit of a steeply repulsive continuum potential; the transition is explored from this limit to a shallow potential limit in which the spectrum reflects the structure of the continuum wavefunction. Two new approximations are presented, and compared with exact results for a simple model.

Thermodynamic and stochastic theory for nonequilibrium systems with more than one reactive intermediate: Nonautocatalytic or equilibrating systems

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 ...

A new (Laguerre) uniform approximation for vibrational transition probabilities

A mapping between the exactly soluble forced oscillator and the general vibrationally inelastic scattering problem is shown to yield a new uniform approximation based on generalized Laguerre polynomials. Computations are reported for collinear He-H2 collisions in which H2 is represented by harmonic and Morse oscillators. The results show that the Laguerre approximation avoids the known failings of the existing Airy and Bessel uniform approximations.

Dissipation in steady states of chemical systems and deviations from minimum entropy production

According to the linear thermodynamic theory of irreversible processes, entropy production is minimized at the steady states of nonequilibrium systems. The principle of minimum entropy production is obtained if the dissipation is expanded in the deviation δ from equilibrium, truncated to lowest order (δ2), and then differentiated. At this level of apprximation, the derivative of the dissipation is linear in δ and vanishes at the steady state. For a simple chemical reaction mechanism in a well-defined model system, we have derived the corrections to this approximation, by first evaluating the dissipation and its derivative exactly, and then expanding as a series in δ. To leading order in δ, the ratio of the derivative to the dissipation at the steady state is 12 (in dimensionless units), and this ratio does not decrease as equilibrium is approached. The steady state coincides with the state of minimum entropy production only at equilibrium. Given sufficient accuracy in measurements of species concentrations, the breakdown of the principle of minimum entropy production can be detected experimentally when the relative standard deviation in measurements of the dissipation is less than δ. Our example shows that the dissipation in a reaching chemical system may increase in time, in the later stages of relaxation toward a near-equilibrium steady state.

Stationary solutions of the master equation for single and multi-intermediate autocatalytic chemical systems

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‐depe...

Flame temperature determination by dual laser ionization

Abstract A dual laser ionization (DLI) apparatus was used to determine ionic diffusion and mobility coefficients for sodium and lithium in a stoichiometric H 2 /O 2 /Ar flame. The flame temperature was calculated using the Einstein relation. The sodium and lithium DLI temperature estimates are in agreement with each other and with temperatures measured by the line-reversal technique in similar flames.

Thermodynamic and stochastic theory of nonequilibrium systems: Fluctuation probabilities and excess work

For a nonequilibrium system described at the mesoscopic level by the master equation, we prove that the probability of fluctuations about a steady state is governed by a thermodynamic function, the ‘‘excess work.’’ The theory applies to systems with one or more nonequilibrium steady states, for reactions in a compartment that contains intermediates Xj of variable concentration, along with a reactant A and product B whose concentrations are held constant by connection of the reaction chamber to external reservoirs. We use a known relation between the stationary solution Ps(X) of the master equation and an underlying stochastic Hamiltonian H: to logarithmic accuracy, the potential that gives Ps(X) is the stochastic action S evaluated along fluctuational trajectories, obtained by solving Hamilton’s equations of motion starting at a steady state. We prove that the differential action dS equals a differential excess work dφ0, and show that dφ0 can be measured experimentally in terms of total free energy change...