ALBERT

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 prove that Liapunov functions for a single reactive intermediate evolving toward a nonequilibrium steady state can be obtained from a global potential φ. We consider reactions occurring in a chamber containing a reactant, the intermediate, and a product. Reservoirs connected to the chamber serve to hold the reactant and product concentrations constant, in nonequilibrium proportions. The Liapunov property of φ is significant because of the role it plays in the thermodynamic and stochastic analysis of nonequilibrium systems: φ is defined in terms of the reactive flux to produce the intermediate and the flux to remove the intermediate. The derivative of φ with respect to the concentration of the intermediate yields an effective chemical driving force that is specific to the intermediate, and its time derivative yields a species-specific component of the dissipation that is minimized at steady states. These results hold both near to equilibrium and far from equilibrium for systems with one intermediate, independent of the number of steady states. Local Liapunov functions are also provided by the "excess dissipation,'' the second variation in the entropy or in the Helmholtz free energy for the reaction chamber, and quadratic functions introduced in Keizer's fluctuation–dissipation theory. Linearization of the force and flux expansions for nonequilibrium systems yields an idealized model in which the dissipation decreases monotonically in time and thus provides a Liapunov function for evolution to steady states. This result does not hold for a chemical system approaching a steady state with an arbitrarily small, but macroscopic displacement from equilibrium, even though the series expansions of the reactive fluxes and conjugate thermodynamic forces are closely approximated by truncation at the linear terms. There are always small regions in the immediate vicinity of nonequilibrium steady states where the dissipation increases in time while the system relaxes.

Notes: 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 differentiation we obtain necessary and sufficient conditions for the stability of steady states and for critical points. The integral of the species-specific affinity globally provides valid Liapunov functions for the evolution of the intermediate. These results are independent of the number of steady states of the system, and they hold both near to and far from equilibrium. For a large class of mechanisms with a single intermediate, the integral of the species-specific affinity appears in the irreversible partof the time-dependent transition probability of the single-variable Master equation and in its stationary solution. Hence for these mechanisms we obtain a direct interpretation of the stochastic results in terms of thermodynamic quantities. The time rate of change in the pressure of the intermediate multiplied by its species-specific affinity yields a species-specific term in the dissipation. The total system dissipation (or entropy production) is not in general a minimum at a nonequilibrium steady state, but the species-specific term is minimized at every such state. The expression of the stationary solution of the master equation in terms of the species-specific affinity provides a generalization of the Einstein relation for the probability of equilibrium fluctuations to far-from-equilibrium conditions. The functional form of the species-specific term in the dissipation parallels a form that appears in Boltzmann's H theorem for the momentum relaxation of a dilute gas.

Notes: The dynamics of fluctuations in systems approaching a nonequilibrium steady state, with or without detailed balance, are investigated by means of a Lagrangian function, which is derived from the generator of time displacement (Hamiltonian) of the mesoscopic evolution equation. In the thermodynamic limit, the stationary probability distribution for the fluctuating variables is expressed in terms of the action of this stochastic Lagrangian along the fluctuational trajectory, the most probable path of infinite duration for the generation of a particular fluctuation away from the steady state. The fluctuational trajectory is related by a gaugelike transformation to the deterministic trajectory, which is the most probable path for the relaxation of the macroscopic system to the steady state. This framework is applied to the analysis of one-variable chemical reactions modeled by a constant step master equation, and to two-variable systems in the linearized region around the steady state, where the fluctuations are described by a linear Fokker–Planck equation. In these examples, the thermodynamic significance of the action along the fluctuational trajectory is established by relating the irreversible (odd under time inversion) part of the Lagrangian and the time derivative of a deterministic excess work. © 1995 American Institute of Physics.

Notes: 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 changes for the reaction compartment and the reservoirs. Thus we connect the probability of concentration fluctuations in an open reaction compartment to thermodynamic functions for the entire closed system containing the compartment. The excess work dφ0 is the difference between the total free energy change for a specified change in the quantities of A, X, Y, and B in the state of interest, and the free energy change for the same changes in species numbers, imposed on the same system in a reference state (A,X0,Y0,B).The reference-state concentration for species Xj is derived from the momentum pj canonically conjugate to Xj along the fluctuational trajectory. For systems with linear rate laws, the reference state (A,X0,Y0,B) is the steady state, and φ0 is equivalent to the deterministic excess work φdet* introduced in our previous work. For nonlinear systems, (A,X0,Y0,B) differs from the deterministic reference state (A,X*,Y*,B) in general, and φ0≠φdet*. If the species numbers change by ±1 or 0 in each elementary step and if the overall reaction is a conversion A→X→Y→B, the reference state (A,X0,Y0,B) is the steady state of a corresponding linear system, identified in this work. In each case, dφ0 is an exact differential. Along the fluctuational trajectory away from the steady state, dφ0(approximately-greater-than)0. Along the deterministic kinetic trajectory, dφ0≤0, and φ0 is a Liapunov function. For two-variable systems linearized about a steady state, we establish a separate analytic relation between Ps(X), φdet*, and a scaled temperature T*. © 1995 American Institute of Physics.

Notes: Consideration is given to some special features of normalized apparent-resistivity (NAR) curves, resulting from the deployment of Wenner electrode arrays on the surface above dipping earth structures. Limiting values of the potential are derived when a direct-current source is located at points on the surface above a simple two-region dipping-bed earth model and their influence on corresponding NAR curve characteristics is investigated. Particular attention is given to the exploitation of such features to provide a new and direct approach to model parameter estimation, either as an alternative to traditional curve-matching techniques or as a source of supporting information when other earth model characterization methods have been employed. Throughout, the emphasis is on the single-dipping-bed model, but application to more complex structures is discussed, including examples of two dipping beds, dipping dikes and more general tilted unconformities.