Asymmetric boundary conditions

In the case of a symmetric (or Just slightly asymmetric) potential the instanton trajectory consists of kink and antikink, which are separated by infinite time and do not interact with each other. In other words, we may change the boundary conditions, namely, suppose that the time spans from — 00 to -t- 00 for a single kink, and then multiply the action in (5.72) by factor 2. 

The basic equation and boundary conditions for the symmetrical fluctuations are the same as those for the asymmetrical fluctuations except for the superscript s. The diffusion equation is written in the form... 

Asymmetric boundary conditions [41,109,110,113] in which the two surface/interfaces of the thin films have different energies, such as substrate-supported films [66,68,114]. 

Unlike the bulk morphology, block copolymer thin films are often characterized by thickness-dependent highly oriented domains, as a result of surface and interfacial energy minimization [115,116]. For example, in the simplest composition-symmetric (ID lamellae) coil-coil thin films, the overall trend when t>Lo is for the lamellae to be oriented parallel to the plane of the film [115]. Under symmetric boundary conditions, frustration cannot be avoided if t is not commensurate with L0 in a confined film and the lamellar period deviates from the bulk value by compressing the chain conformation [117]. Under asymmetric boundary conditions, an incomplete top layer composed of islands and holes of height Lo forms as in the incommensurate case [118]. However, it has also been observed that microdomains can reorient such that they are perpendicular to the surface [ 119], or they can take mixed orientations to relieve the constraint [66]. 

There are two kinds of effects of the membrane on the enzyme behavior a specific interaction between the enzyme and the lipid membrane and a nonspecific interaction of the membrane structure by itself on the enzyme kinetics. In the case of ATPase, the enzyme in solution is working in homogeneous and isotropical conditions. At the opposite extreme, in the membrane the enzyme is working under asymmetrical boundary conditions. In the last case there is a coupling between a scalar process and the vectorial transport effect. In conclusion, the effect of the membrane on the enzyme behavior is not only a chemical effect, but also a geometrical one. 

So far almost all aspects of the stationary-state and even the time-dependent behaviour of this reaction-diffusion system differ only qualitatively from that found in the corresponding CSTR. In this section, however, we can consider a variation for which there can be no parallel in the well-stirred system—that of a reaction-diffusion cell set up with asymmetric boundary conditions. Thus we might consider our infinite slab with separate reservoirs on each side, with different concentrations of the autocatalyst in each reservoir. (For simplicity we will take the reactant concentration to be equal on each side.) Thus if we identify the reservoir concentration for p < — 1 as / L and on the other side (p + 1) as / R, the simple boundary conditions in eqn (9.11) are replaced by... 

R.J. Kee and A.A. McKillop. A Numerical Method for Predicting Natural Convection in Horizontal Cylinders with Asymmetric Boundary Conditions. Comput. Fluids, 5 1-14,1977. 

Let i//, be an asymmetric-top wave function. A convenient complete orthonormal set to use here is the symmetric-top wave functions, which are functions of the same coordinates (the Eulerian angles) and satisfy the same boundary conditions as the asymmetric-top functions ... 

Both plates are held at specified, variable temperatures. For asymmetric wall temperature boundary conditions, the lower plate may be held at the input temperature, while the top wall temperature would be at a specified variable value. 

The third electro-optical effect using calamitic nematic liquid crystals makes use of a flexoelectric effect manifested by a curved asymmetrical nematic medium. This corresponds to piezoelectricity in crystals. The existence of flexoelectricity in a nematic phase under certain boundary conditions was predicted in the late 1960s and then confirmed experimentally several years later. However, LCDs using this effect, such as bistable nematic displays are only in the development stage and as such they will not be discussed in this monograph. 

Equations like these are typically solved by Laplace transforms, of which the principles have been laid down in Lapp. 10. Elaborations for a number of diffusion problems can be found in sec. 1.6.5. In fact. ]4.6.34b) Is nothing else than Pick s second law. The final result depends on the boundary conditions which are different for the different electrokinetlc phenomena, see [4.6.19-23 and 31a-c[. In more complicated cases not considered here (asymmetrical electroljdes. electrol)rtes with different D s. xa not large, interacting particles, etc.) often only numerical solutions are available. 

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