Dependency of control parameters on a rate coefficient giving the potential curvature in a reaction cascade model

Many chemical reactions in vivo are self-controlled by fluxes of chemical energy and matter through biological systems, so the induction of such reactions can be governed by changes in the control parameters of the rate equation. A potential of a system is assumed to be given by Gibbs' functionG(T, P, x), which is continuously differentiable, and the rate equation can be derived from the differential (–G/x) of Taylor's expansion ofG _(T,P)(x) for the order parameterx, which corresponds to the product number, at around the critical pointC(T _C, P_C). The equation is described bydx/dt=(x)–k₁x–k₂x³, andk ₂>0. In this equation,k ₁ andk ₂ are functions of the control parameters, temperatureT and pressureP, andk ₁ is allowed to have a positive or negative values as (T, P). Thenk ₁ is an important factor that decides the induction conditions of the reactions with a phase transition in the steady statex=0. Because bothk ₁ (the transition parameter) andG are the quantity of state, they are given by the total differential, and functions that decideG andk ₁ are related to a mutual inverse function. From the above relation, the rate of change ink ₁ by G, which corresponds to the reaction energy of the system, is uniquely determined by a function ofk ₁, [f(k ₁ ^± )] andf(k ₁ ^± ) is described approximately by ±₁ k ₁ ^± in the transient process thatk ₁ approaches zero, where ₁ implies 1/RT. These results indicate that internal driving forces caused by a stimulus in a system are proportional tok ₁ ^± and that the system is regulated by competition of the forces. an approximate function fork ₁ in the transient process is described by tanh (G/RT) and Arrhenius' law is elucidated from this theory.Decreased January 19, 1992     

The bicritical phase diagram of two-dimensional antiferromagnets with and without random fields

Neutron scattering measurements have been made of the phase diagrams of the nearly two-demensional antiferromagnets Rb₂MnF₄ and Rb₂Mn_0.7Mg_0.3F₄ in a magnetic field applied along thec-axis. In Rb₂MnF₄ there is at low temperatures a spin-flop phase at fields above 5.5 T which has long range order. The observation of true long range order rather than the algebraic decay of the order characteristic of the two-dimensional XY model is presumably due to subtle anisotropy effects in the plane as well as weak three-dimensional coupling. The phase boundaries of the uniaxial and transverse phases are shown to be consistent with renormalization group predictions for two-dimensional systems. The two lines become exponentially close to each other at low temperatures. The weak three-dimensional coupling moves the bicritical point fromT=0 to a non-zero temperature. The situation is more complex in Rb₂Mn_0.7Mg_0.3F₄ because of Ising random field effects. At low fields we observe typical random field metastable behavior with a sharp metastability boundary and a gange of length scales which are time independent below that boundary. At higher fields there are substantial uniaxial fluctuations. The transverse phase boundary and the metastability line appear to intercept atT=0 showing that the random field fluctuations do have a large effect on the phase diagram. The theory of the phase diagrams has been extended to include the random field fluctuations and good agreement is obtained with the observed transverse phase boundary. Unfortunately, there is as yet no theory of the metastable uniaxial phase with which to compare our results.