Problem Landau

Sunderland and Grosh (Sll) use an explicit numerical scheme for solving the Landau problem. An Eulerian coordinate system is used with the origin at the melt interface. As noted by Landau, the numerical integration is simplified by appropriate choice of the ratio of the space and time intervals. Extension to time-dependent heat flux by either a numerical or a graphical technique is indicated. 

Boundary layer approximation. The Landau problem, which was described above, is an example of an exact solution of the Navier-Stokes equations. Schlichting [427] proposed another approach to the jet-source problem, which gives an approximate solution and is based on the boundary layer theory (see Section 1.7). The main idea of this method is to neglect the gradients of normal stresses in the equations of motion. In the cylindrical coordinates (71, ip, Z), with regard to the axial symmetry (Vv = 0) and in the absence of rotational motion in the flow (d/dip = 0), the system of boundary layer equations has the form... 

The standard analytic treatment of the Ising model is due to Landau (1937). Here we follow the presentation by Landau and Lifschitz [H], which casts the problem in temis of the order-disorder solid, but this is substantially the same as the magnetic problem if the vectors are replaced by scalars (as the Ising model assumes). The themiodynamic... 

With the fomi of free energy fiinctional prescribed in equation (A3.3.52). equation (A3.3.43) and equation (A3.3.48) respectively define the problem of kinetics in models A and B. The Langevin equation for model A is also referred to as the time-dependent Ginzburg-Landau equation (if the noise temi is ignored) the model B equation is often referred to as the Calm-Flilliard-Cook equation, and as the Calm-Flilliard equation in the absence of the noise temi. 

The problem of branching of the wavepacket at crossing points is very old and has been treated separately by Landau and by Zener [H, 173. 174], The model problem they considered has the following diabatic coupling matrix ... 

In the study of (electronic) curve crossing problems, one distinguishes between a situation where two electronic curves, Ej R), j — 1,2, approach each other at a point R = Rq so that the difference AE[R = Rq) = E iR = Rq) — Fi is relatively small and a situation where the two electronic curves interact so that AE R) Const is relatively large. The first case is usually treated by the Landau-Zener fonnula [87-92] and the second is based on the Demkov approach [93]. It is well known that whereas the Landau-Zener type interactions are... 

The problem of nonadiabatic tunneling in the Landau-Zener approximation has been solved by Ovchinnikova [1965]. For further refinements of the theory beyond this approximation see Laing et al. [1977], Holstein [1978], Coveney et al. [1985], Nakamura [1987]. The nonadiabatic transition probability for a more general case of dissipative tunneling is derived in appendix B. We quote here only the result for the dissipationless case obtained in the Landau-Zener limit. When < F (Xe), the total transition probability is the product of the adiabatic tunneling rate, calculated in the previous sections, and the Landau-Zener-Stueckelberg-like factor... 

In the previous sections, we briefly introduced a number of different specific models for crystal growth. In this section we will make some further simplifications to treat some generic behavior of growth problems in the simplest possible form. This usually leads to some nonlinear partial differential equations, known under names like Burgers, Kardar-Parisi-Zhang (KPZ), Kuramoto-Sivashinsky, Edwards-Anderson, complex Ginzburg-Landau equation and others. 

Lagrange Multiplier Method for programming problems, 289 for weapon allocation, 291 Lamb and Rutherford, 641 Lamb shift, 486,641 Lanczos form, 73 Landau, L. D., 726,759, 768 Landau-Lifshitz theory applied to magnetic structure, 762 Large numbers, weak law of, 199 Law of large numbers, weak, 199 Lawson, J. L., 170,176 Le Cone, Y., 726... 

Darrieus and Landau established that a planar laminar premixed flame is intrinsically unstable, and many studies have been devoted to this phenomenon, theoretically, numerically, and experimentally. The question is then whether a turbulent flame is the final state, saturated but continuously fluctuating, of an unstable laminar flame, similar to a turbulent inert flow, which is the product of loss of stability of a laminar flow. Indeed, should it exist, this kind of flame does constitute a clearly and simply well-posed problem, eventually free from any boundary conditions when the flame has been initiated in one point far from the walls. 

The physicochemical forces between colloidal particles are described by the DLVO theory (DLVO refers to Deijaguin and Landau, and Verwey and Overbeek). This theory predicts the potential between spherical particles due to attractive London forces and repulsive forces due to electrical double layers. This potential can be attractive, or both repulsive and attractive. Two minima may be observed The primary minimum characterizes particles that are in close contact and are difficult to disperse, whereas the secondary minimum relates to looser dispersible particles. For more details, see Schowalter (1984). Undoubtedly, real cases may be far more complex Many particles may be present, particles are not always the same size, and particles are rarely spherical. However, the fundamental physics of the problem is similar. The incorporation of all these aspects into a simulation involving tens of thousands of aggregates is daunting and models have resorted to idealized descriptions. 

The modification factor plays a central role in a WL simulation and has several effects. First, its presence violates microscopic detailed balance because it continuously alters the state probabilities, and hence acceptance criterion. Only for g = 0 do we obtain a true Markov sampling of our system. Furthermore, we obviously cannot resolve entropy differences which are smaller than g, yet we need the modification factor to be large enough to build up the entropy estimate in a reasonable amount of simulation time. Wang and Landau s resolution of these problems was to impose a schedule on g, in which it starts at a modest value on the order of one and decreases in stages until a value very near to zero (typically in the range 10 5-10 8). In this manner, detailed balance is satisfied asymptotically toward the end of the simulation. 

The Ginzburg-Landau theory for the CFL phase has been derived in Refs. [10, 16-19], The authors of Ref. [19] have taken into account the rotated electromagnetism and the rotation of the star. They conclude that ordinary quantized magnetic vortices are unstable in the CFL phase, but the rotational vortices are topologically stable. They have not considered boundary problems and concluded that the CFL condensate in an external magnetic held behaves like a type I superconductor. 

K can be estimated by assuming that the system passes through the crossing region with a uniform velocity, so that the quantum one-dimensional problem can be solved to obtain the Landau-Zener form - ... 

The understanding of electronic states in atoms is to a great extent based on Schrodinger s solution of the hydrogen-atom problem. These wavefunctions have the general form (Landau and Lifshitz, 1977) ... 

The van der Waals force occurring in STM and AFM is much smaller than the van der Waals force between a neutral hydrogen atom and a proton. Actually, in STM and AFM experiments on conducting materials, the atoms near the gap are nearly neutral, which is similar to the situation of a pair of neutral hydrogen atoms rather than a proton with a neutral hydrogen atom. The van der Waals force between a pair of neutral hydrogen atoms is also a well-studied problem. The exact result at large distances is (Landau and Lifshitz, 1977) ... 

By pressing a sphere upon a planar surface, a deformation occurs near the original point of contact, see Fig. F.8. This problem was first solved by Hertz in 1881 (see Landau and Lifshitz, 1986 Timoshenko and Goodier, 1970). The derivations are complicated. We state the results without proof. 

Landau, L. D., and Lifshitz, L. M. (1977). Quantum Mechanics, third edition, Pergamon Press, Oxford. The treatment of the hydrogen molecular ion is presented as a problem on page 312. 

A good discussion of the dielectric properties of a mixture, independent of a specific model, has been given by Landau and Lifshitz (1960) we also recommend a paper by Niklasson et al. (1981) for its discussion of several aspects of the mixture problem. 

To investigate these problems, we should first devise a Ginzburg-Landau free energy and then set up dynamic equations for network and solvent taking account of both nonlinear elasticity and inhomogeneous fluctuations. Therefore, the aims of this paper are firstly to introduce such a theory [19-21], secondly to review consequences of the theory obtained so far, and thirdly to give new results. Such efforts have just begun and many problems remain unsolved. 

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