State spatially homogeneous

When the steady state becomes unstable, the system moves away from it and often undergoes sustained oscillations around the unstable steady state. In the phase space defined by the system s variables, sustained oscillations generally correspond to the evolution toward a limit cycle (Fig. 1). Evolution toward a limit cycle is not the only possible behavior when a steady state becomes unstable in a spatially homogeneous system. The system may evolve toward another stable steady state— when such a state exists. The most common case of multiple steady states, referred to as bistability, is of two stable steady states separated by an unstable one. This phenomenon is thought to play a role in differentiation [30]. When spatial inhomogeneities develop, instabilities may lead to the emergence of spatial or spatiotemporal dissipative stmctures [15]. These can take the form of propagating concentration waves, which are closely related to oscillations. 

A phenomenogical expression for the hydrodynamic force F may be constructed by assuming that this force is linear in the flux velocities and in the strength of any applied flow field. We consider a system that is subjected to a macroscopic flow field v(r) characterized by a spatially homogeneous macroscopic velocity gradient Vv. We assume that Fa vanishes for all a = 1in the equilibrium state, where the flux velocities and the macroscopic... 

The simplest and often most suitable modeling tool is the one-box model. One-box models describe the system as a single spatially homogeneous entity. Homogeneous means that no further spatial variation is considered. However, one-box models can have one or several state variables, for instance, the mean concentration of one or several compounds i which are influenced both by external forces (or inputs) and by internal processes (removal or transformation). A particular example, the model of the well-mixed reactor with one state variable, has been discussed in Section 12.4 (see Fig. 12.7). The mathematical solution of the model has been given for the special case that the model equation is linear (Box 12.1). It will be the starting point for our discussion on box models. 

The production of species i (number of moles per unit volume and time) is the velocity of reaction,. In the same sense, one understands the molar flux, jh of particles / per unit cross section and unit time. In a linear theory, the rate and the deviation from equilibrium are proportional to each other. The factors of proportionality are called reaction rate constants and transport coefficients respectively. They are state properties and thus depend only on the (local) thermodynamic state variables and not on their derivatives. They can be rationalized by crystal dynamics and atomic kinetics with the help of statistical theories. Irreversible thermodynamics is the theory of the rates of chemical processes in both spatially homogeneous systems (homogeneous reactions) and inhomogeneous systems (transport processes). If transport processes occur in multiphase systems, one is dealing with heterogeneous reactions. Heterogeneous systems stop reacting once one or more of the reactants are consumed and the systems became nonvariant. 

Linearize about spatially homogeneous steady state and perform... 

In a chemical system, spatial patterns can spontaneously arise from an initial spatially homogeneous concentration profile by the selection from noise and amplification of perturbations with wavelengths in the neighborhood of some preferred wavelength. To determine the conditions under which this occurs, the behavior of the system in the vicinity of the spatially homogeneous steady state A" = Xs, Y = T, where F(XS, Ys) = G(XS, Ys) = 0, is analyzed using a standard linearization procedure. 

In the following we consider perturbations around the spatially homogeneous state given above. 

Let us start our considerations by looking at the spatially homogeneous state. In a sample with parallel alignment the apparent viscosity is V3, which can easily be seen from the force on the upper boundary ... 

The polymeric fluid that we investigate with the state variables (58) is thus static (i.e., without any macroscopic flow) and spatially homogeneous. The only time evolution that takes place in it is the evolution of the internal structure characterized by two scalars q, p. 

Besides the hypothesis of spatially homogeneous processes in this stochastic formulation, the particle model introduces a structural heterogeneity in the media through the scarcity of particles when their number is low. In fact, the number of differential equations in the stochastic formulation for the state probability keeps track of all of the particles in the system, and therefore it accounts for the particle scarcity. The presence of several differential equations in the stochastic formulation is at the origin of the uncertainty, or stochastic error, in the process. The deterministic version of the model is unable to deal with the stochastic error, but as stated in Section 9.3.4, that is reduced to zero when the number of particles is very large. Only in this last case can the set of Kolmogorov differential equations be adequately approximated by the deterministic formulation, involving a set of differential equations of fixed size for the states of the process. 

Sometimes, even spatially homogeneous chemical systems can cause bistability and show complex behavior in time. For example, autocatalysis may occur due to the particular molecular structure and reactivity of certain constituents, and reactions may evolve to new states by amplifying or repressing the effect of a slight concentration perturbation. 

A typical example of nonequilibrium spatially homogeneous systems is an isotropic system where a chemical reaction occurs. The apphcation of nonequibbrium thermodynamics for the consideration of chemically reactive systems has a few peculiarities. Indeed, heat and mass transfer pro cesses are characterized usually by continuous variations in temperature and concentration (see Section 1.5). On the other hand, the chemical transformations imply transitions between the discrete states that pertain to the individual reaction groups. 

The steady state spatial correlations in reaction-diffusion systems involving many reversible chemical reactions are examined. It has been cJready discussed that the spatial correlations are related to the breaking of detailed balance in chemical kinetics for both one species and for two species reversible reactions. Here, we focus our attention on how the spatial correlations of concentration fluctuations in a macroscopically homogeneous systems approach to the instability point. The spatial correlations depend strongly on the stability of systems for two species reactions compared to one species reactions. 

The population balance equation is employed to describe the temporal and steady-state behavior of the droplet size distribution for physically equilibrated liquid-liquid dispersions undergoing breakage and/or coalescence. These analyses also permit evaluation of the various proposed coalescence and breakage functions described in Sections III,B and C. When the dispersion is spatially homogeneous it becomes convenient to describe particle interaction on a total number basis as opposed to number concentration. To be consistent with the notation employed by previous investigators, the number concentration is replaced as n i,t)d i = NA( i t)dXi, where N is the total number of particles per unit volume of the dispersion, and A(xj t) dXi is the fraction of drops in increment X, to X( + dxi- For spatially homogeneous dispersions such as in a well-mixed vessel, continuous flow of dispersions, no density changes, and isothermal conditions Eq. (102) becomes... 

It is often stated that MC methods lack real time and results are usually reported in MC events or steps. While this is immaterial as far as equilibrium is concerned, following real dynamics is essential for comparison to solutions of partial differential equations and/or experimental data. It turns out that MC simulations follow the stochastic dynamics of a master equation, and with appropriate parameterization of the transition probabilities per unit time, they provide continuous time information as well. For example, Gillespie has laid down the time foundations of MC for chemical reactions in a spatially homogeneous system.f His approach is easily extendable to arbitrarily complex computational systems when individual events have a prescribed transition probability per unit time, and is often referred to as the kinetic Monte Carlo or dynamic Monte Carlo (DMC) method. The microscopic processes along with their corresponding transition probabilities per unit time can be obtained via either experiments such as field emission or fast scanning tunneling microscopy or shorter time scale DFT/MD simulations discussed earlier. The creation of a database/lookup table of transition... 

The following solutions to the system of equations (5.114) can be distinguished (1) states independent of r and independent of t (spatially homogeneous stationary states) (2) states independent of r, dependent on t (spatially homogeneous states) (3) states dependent on r, independent of t (stationary states) (4) states dependent on r, dependent on t. 

The first step in the examination of the system (5.114) involves finding spatially homogeneous stationary states x, = const, satisfying the equations... 

The spatially homogeneous stationary states of equations (5.116) fulfil the relations... 

---