Stability of racemic and chiral steady states in open and closed chemical systems

The stability properties of models of spontaneous mirror symmetry breaking in chemistry are characterized algebraically. The models considered here all derive either from the Frank model or from autocatalysis with limited enantioselectivity. Emphasis is given to identifying the critical parameter controlling the chiral symmetry breaking transition from racemic to chiral steady-state solutions. This parameter is identified in each case, and the constraints on the chemical rate constants determined from dynamic stability are derived.     

Entropy production in nonequilibrium systems at stationary states

We present a stochastic approach to nonequilibrium thermodynamics based on the expression of the entropy production rate advanced by Schnakenberg for systems described by a master equation. From the microscopic Schnakenberg expression we get the macroscopic bilinear form for the entropy production rate in terms of fluxes and forces. This is performed by placing the system in contact with two reservoirs with distinct sets of thermodynamic fields and by assuming an appropriate form for the transition rate. The approach is applied to an interacting lattice gas model in contact with two heat and particle reservoirs. On a square lattice, a continuous symmetry breaking phase transition takes place such that at the nonequilibrium ordered phase a heat flow sets in even when the temperatures of the reservoirs are the same. The entropy production rate is found to have a singularity at the critical point of the linear-logarithm type.     

Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems

We develop canonical perturbation theory for a physically interesting class of infinite-dimensional systems. We prove stability up to exponentially large times for dynamical situations characterized by a finite number of frequencies. An application to two model problems is also made. For an arbitrarily large FPU-like system with alternate light and heavy masses we prove that the exchange of energy between the optical and the acoustical modes is frozen up to exponentially large times, provided the total energy is small enough. For an infinite chain of weakly coupled rotators we prove exponential stability for two kinds of initial data: (a) states with a finite number of excited rotators, and (b) states with the left part of the chain uniformly excited and the right part at rest.     

Free energy and entropy production in planar Couette flow far from equilibrium

The macroscopic thermodynamic stability of a system of 108 diatomic molecules undergoing planar Couette flow far from equilibrium is reported. The system is perturbed from the steady state using a nondissipative variable colour field which induces a polarization in the system. It is found that the steady state for the system corresponds to an extremum in the generalized free energy and entropy production. However, while the free energy is always a minimum, the entropy production may be either a minimum or a maximum depending upon the direction of the colour field. These results, for a molecular system, are fundamentally different from those for an equivalent atomic system.     

The role of thermodynamic representations in the study of fluids in far from equilibrium steady states

In this paper we show that the calculation of the Rayleigh peak form for a fluid driven onto a stationary state by an external temperature gradient is independent of the thermodynamic representation used only if linear terms in the gradient are kept. However, if a result to second order is required then only one representation seems to be adequate for obtaining it. The underlying reasons for such non-equivalence of different representations are discussed.     

Control of unstable steady states in neutral time-delayed systems

We present an analysis of time-delayed feedback control used to stabilize an unstable steady state of a neutral delay differential equation. Stability of the controlled system is addressed by studying the eigenvalue spectrum of a corresponding characteristic equation with two time delays. An analytic expression for the stabilizing control strength is derived in terms of original system parameters and the time delay of the control. Theoretical and numerical results show that the interplay between thecontrol strength and two time delays provides a number of regions in the parameter space where the time-delayed feedback control can successfully stabilize an otherwise unstable steady state.     

Stability and equilibrium states of infinite classical systems

We prove that any stationary state describing an infinite classical system which is stable under local perturbations (and possesses some strong time clustering properties) must satisfy the classical KMS condition. (This in turn implies, quite generally, that it is a Gibbs state.) Similar results have been proven previously for quantum systems by Haag et al. and for finite classical systems by Lebowitz et al.Supported by N.S.F. Grant MPS 71-03375 A03. Part of this work carried out at the Courant Institute where it was supported by N.S.F. Grant GP-37069X.Supported in part by AFOSR Grant #73-2430 and N.S.F. Grant MP S75-20638.Supported by N.S.F. Grant # GP33136X-2. Part of this work was carried out at the Institute for Advanced Study.     

Invariant integral and the transition to steady states in separable dynamical systems

We show that the transition between fixed points in a separable dynamical system is fully described by an invariant integral. We discuss in detail the case of a system with two temporal variables with bilinear coupling, where the new stable state is attained asymptotically through spiraling into the fixed point. Through the invariance, it is possible to establish conditions for the control parameter that permit a (targeted) transition in finite time and without relaxation oscillations.     

Entropy,information, and complexity of the steady states of open systems not satisfying the principle of local equilibrium

The entropy of the steady state of an open electronic system is determined, and its relation to the entropy arriving at the environment is established. The notion of system ordering is defined, and quantitative parameters characterizing ordering are introduced. The information and conditional complexity of the system versus the state of thermodynamic chaos are determined. These values are numerically estimated for vacuum, magnetron, and Gunn diodes.     

Convergence to equilibrium in a system of reacting polymers

Chemical kinetics of a system of reacting polymers is modelled by an equation which shares certain properties with Boltzmann's equation. Being more tractable, however, this evolution may be of an illustrative value for the latter. The existence and uniqueness of solutions are analysed. We derive an entropy production inequality which is used to prove global exponential decay of the free energy. With its aid a uniform rate for strong convergence to equilibrium is proven. The generators of the linearlized flow at the vicinity of the equilibria are diagonalized.Work partly supported by U.S. National Science Foundation grant MCS 75-21684 A02     

Classical properties of quantum systems in minimum uncertainty states

We derive a necessary condition for the equivalence of the Schrödinger equation with the evolution equation for the marginal coordinate probability density of the Liouville equation and propose an analogon of the quantum Fourier transformation in classical statistical mechanisms. It is shown that this quantum-classical equivalence holds for simple systems in minimum uncertainty states.     

Irreversibility and entropy production in transport phenomena,III—Principle of minimum integrated entropy production including nonlinear responses

A new variational principle of steady states is found by introducing an integrated type of energy dissipation (or entropy production) instead of instantaneous energy dissipation. This new principle is valid both in linear and nonlinear transport phenomena. Prigogine’s dream has now been realized by this new general principle of minimum “integrated” entropy production (or energy dissipation). This new principle does not contradict with the Onsager–Prigogine principle of minimum instantaneous entropy production in the linear regime, but it is conceptually different from the latter which does not hold in the nonlinear regime. Applications of this theory to electric conduction, heat conduction, particle diffusion and chemical reactions are presented.     

Translation invariant equilibrium states of ferromagnetic Abelian lattice systems

The structure of the set of all translation invariant equilibrium states is determined for all temperatures, for which the free energy is differentiable. Models with several phase transitions are discussed rigorously.     

Strangeness production in proton collisions close to threshold at COSY-TOF

The K-meson production in elementary proton induced reactions is studied exclusively at the external COSY beam using the time-of-flight spectrometer TOF. The complete measurement of all primary and decay particle tracks allows the extraction of total and differential cross sections as well as Dalitz plots and invariant mass spectra of the subsystems for the channels pp → K⁺ Δp, K⁰Σ⁺p, K⁺Σ⁰p and K⁺Σ⁺n. For all channels the full phase space is covered from the reaction threshold up to the COSY-limit of about 3.5 GeV/c. Especially the analysis of the Dalitz plots of the channel pp→K⁺Δp show a strong influence of N₋^* resonances.     

Theory of metastable states close to instability

A theory of relaxation of metastable states close to the spinodal line is proposed. The statistics of the process is derived on the basis of the equation of motion of the phase transition parameter.     

Statistics of entropy production in linearized stochastic systems

We consider a wide class of linear stochastic problems driven off the equilibrium by a multiplicative asymmetric force. The force breaks detailed balance, maintained otherwise, thus producing entropy. The large deviation function of the entropy production in the system is calculated explicitly. The general result is illustrated using an example of a polymer immersed in a gradient flow and subject to thermal fluctuations.     

Deviations from the Poisson behavior of equilibrium fluctuations in a closed system

A molecular dynamics simulation of dimer formation in an adiabatic, hardcore, square-well, dilute two-dimensional gas yields a non-Poissonian dimer number distribution. Its ratio of variance to mean deviates from unity in a manner recently predicted theoretically.