Oscillation phase transitions

T. Hill and Y. Chen, Cooperative Effects in Models of Steady-State Transport Across Membranes Oscillating Phase Transition, Proc. Natl Acad. Scl USA 66(1), 189-196 (1970). 

Because the degrees of freedom decouple in the linear approximation, it is easy to describe the dynamics in detail. There is the motion across a harmonic barrier in one degree of freedom and N — 1 harmonic oscillators. Phase-space plots of the dynamics are shown in Fig. 1. The transition from the reactant region at q <0 to the product region at q >0 is determined solely by the dynamics in (pi,qi), which in the traditional language of reaction dynamics is called the reactive mode. 

The book thus embraces an extended study on a variety of issues within the theory of orientational ordering and phase transitions in two-dimensional systems as well as the theory of anharmonic vibrations in low-dimensional crystals and dynamic subsystems interacting with a phonon thermostat. For the sake of readability, the main theoretical approaches involved are either presented in separate sections of the corresponding chapters or thoroughly scrutinized in appendices. The latter contain the basic formulae of the theory of local and resonance states for a system of bound harmonic oscillators (Appendix 1), the theory of thermally activated reorientations and tunnel relaxation of orientational... 

The free theory for the quench models is provided by the potential (4), where A = 0 and m2(t) changes signs either instantaneously or for a finite period. In the Minkowski spacetime, we can apply the LvN method simply by letting R = 1. Before the phase transition (rrii = (mg + m2)1/2), all the modes are stable and oscillate around the true vacuum ... 

Either in the instantaneous quench or in the finite quench, short wavelength modes with k > rrif = (mg — m )1/2 are still stable even after the phase transition and oscillate around the false vacuum as... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

The electrical potential across a LB film of dioleoyl-lecithin deposited onto a fine-pore membrane, imposed between equimolar aqueous solutions of NaCl and KC1, was reported to exhibit rhythmic and sustained pulsing or oscillations of electrical potential between the two solutions. These oscillations were attributed to the change of permeability of Na+ and K+ ions across the membrane, which originated from the phase transition of lecithin. 

What is described above is an idea of the so-called chemical clock, that is a reaction with periodic (oscillating) change of reactant concentrations its period could be estimated as 5t > nc/p. In the condensed matter theory a leap in densities is interpreted as phase transitions of the first order. From this point of view, the oscillations correspond to a sequence in time of phase transitions where the two phases (i.e., big clusters of A s containing inside rare and small clusters of B s and vice versa) differ greatly in their structures. 

What was said above is illustrated by Fig. 1.29 and Fig. 1.30 corresponding to the cases p > pc and p < pc respectively. To make the presented kinetic curves smooth, in these calculations the transformation rate A — B was taken to be finite. To make results physically more transparent, the effective reaction rate K (t) of the A —> B transformation is also drawn. The standard chemical kinetics would be valid, if the value of K (t) tends to some constant. However, as it is shown in Fig. 1.30, K(t) reveals its own and quite complicated time development namely its oscillations cause the fluctuations in particle densities. The problems of kinetic phase transitions are discussed in detail in the last Chapter of the book. 

Lastly, non-elementary several-stage reactions are considered in Chapters 8 and 9. We start with the Lotka and Lotka-Volterra reactions as simple model systems. An existence of the undamped density oscillations is established here. The complementary reactions treated in Chapter 9 are catalytic surface oxidation of CO and NH3 formation. These reactions also reveal undamped concentration oscillations and kinetic phase transitions. Their adequate treatment need a generalization of the fluctuation-controlled theory for the discrete (lattice) systems in order to take correctly into account the geometry of both lattice and absorbed molecules. As another illustration of the formalism developed by the authors, the kinetics of reactions upon disorded surfaces is considered. 

The effects of phase transitions, surface diffusions, and defects on surface catalyzed reactions Fluctuations and oscillations (with D.G. Vlachos and L.D. Schmidt). J. Chem. Phys. 93, 8306-8313 (1990). 

The process of catalyst oxidation and reduction can be treated as a reversible phase transition [136]. It is to this process that the authors of recent investigations [37, 47-49, 85] ascribe critical effects. When studying kinetic self-oscillations in the oxidation of hydrogen over nickel [37] and measuring CPD, the authors established that the reaction performance oscillates between the states in which oxygen is adsorbed either on the reduced or on the oxidized nickel surface. Vayenas et al. [47-49], by using direct measurements of the electrochemical activity of 02 adsorbed on Pt, showed that the isothermal self-oscillations of the ethylene oxidation rate over Pt are due to the periodic formation and decomposition of subsurface Pt oxides. A mathemati-... 

The studies of Ertl and co-workers showed that the reason for self-oscillations [142, 145, 185-187] and hysteresis effects [143] in CO oxidation over Pt(100) in high vacuum ( 10 4 Torr) is the existence of spatio-temporal waves of the reversible surface phase transition hex - (1 x 1). The mathematical model [188] suggests that in each of the phases an adsorption mechanism with various parameters of CO and 02 adsorption/desorption and their interaction is realized, and the phase transition is modelled by a semi-empirical method via the introduction of discontinuous non-linearity. Later, an imitation model based on the stochastic automat was used [189] to study the qualitative characteristics for the dynamic behaviour of the surface. 

These models consider the mechanisms of formation of oscillations a mechanism involving the phase transition of planes Pt(100) (hex) (lxl) and a mechanism with the formation of surface oxides Pd(l 10). The models demonstrate the oscillations of the rate of C02 formation and the concentrations of adsorbed reactants. These oscillations are accompanied by various wave processes on the lattice that models single crystalline surfaces. The effects of the size of the model lattice and the intensity of COads diffusion on the synchronization and the form of oscillations and surface waves are studied. It was shown that it is possible to obtain a wide spectrum of chemical waves (cellular and turbulent structures and spiral and ellipsoid waves) using the lattice models developed [283], Also, the influence of the internal parameters on the shapes of surface concentration waves obtained in simulations under the limited surface diffusion intensity conditions has been studied [284], The hysteresis in oscillatory behavior has been found under step-by-step variation of oxygen partial pressure. Two different oscillatory regimes could exist at one and the same parameters of the reaction. The parameters of oscillations (amplitude, period, and the... 

Alphen (dHvA) effect every cycle of the magnetic oscillations is accompanied by a first-order phase transition, and by the appearance of a domain structure. Figure 1, taken from [29], illustrates the splitting of the NMR frequency of Ag109 in metallic silver under the condition of the dHvA effect, which unambiguously testifies to the occurrence of diamagnetic domains. 

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