One-dimensional physics

The harmonic oscillator is an idealized one-dimensional physical system in which a single particle of mass m is attracted to the origin by a force F proportional to the displacement of the particle Ifom the origin... 

A crystal, or even a thin film, is a three-dimensional object, but some of their properties can be "quasi-one-dimensional"—that is, resemble, but do not coincide with, one-dimensional physics, whose characteristics are simple but often peculiar [39]. 

It follows that the evaluation of the extent to which one-dimensional physics is relevant has always played an important part in the debate surrounding the theoretical description of the normal state of these materials. One point of view expressed is that the amplitude of in the b direction is large enough for a FL component to develop in the ab plane, thereby governing most properties of the normal phase attainable below say room temperature. In this scenario, the anisotropic Fermi liquid then constitutes the basic electronic state from which various instabilities of the metallic state, like spin-density-wave, superconductivity, etc., arise [29]. Following the example of the BCS theory of superconductivity in conventional superconductors, it is the critical domain of the transition that ultimately limits the validity of the Fermi liquid picture in the low temperature domain. 

Another view is to consider the Bechgaard salts as more correlated systems as a result of one-dimensional physics which maintains relevance within a large extent of the normal phase and even affects the mechanisms by which long-range order is stabilized in these materials. 

One-Dimensional Physics Workshop, Boseman, Mont., July 1973. 

This chapter will focus on one-dimensional physical models, and on the techniques known as waveguide filters for constracting very simple models that yield expressive soimd synthesis. First weTl look at the simple ideal string, and then refine the model to make it more reahstic and flexible. At the end we will have developed models that are capable of simnlating an interesting variety of one-dimensional vibrating objects, inclnding stiff stmctures such as bars and beams. 

Remoissenet, M., Low-amplitude breather and envelope sohtons in quasi-one-dimensional physical models, Phys. Rev. B, 33 (4), 2386-2392, 1986. 

IR and microwave radiation. The one-dimensional physical model describing heat and mass transfer allows access to the temperature, moisture content and pressure fields, while energy inputs induced by electromagnetic radiation were determined by the Lambert-Beer law. A comparison of experimental and simulated results on cellular concrete demonstrated the relevance of this model to studies on combined drying. 

To begin a more general approach to molecular orbital theory, we shall describe a variational solution of the prototypical problem found in most elementary physical chemistry textbooks the ground-state energy of a particle in a box (McQuanie, 1983) The particle in a one-dimensional box has an exact solution... 

One-Dimensional Conduction The one-dimensional transient conduction equations are (for constant physical properties)... 

In (2.19), F has been replaced by P because force and pressure are identical for a one-dimensional system. In (2.20), S/m has been replaced by E, the specific internal energy (energy per unit mass). Note that all of these relations are independent of the physical nature of the system of beads and depend only on mechanical properties of the system. These equations are equivalent to (2.1)-(2.3) for the case where Pg = 0. As we saw in the previous section, they are quite general and play a fundamental role in shock-compression studies. 

A one-dimensional mesh through time (temporal mesh) is constructed as the calculation proceeds. The new time step is calculated from the solution at the end of the old time step. The size of the time step is governed by both accuracy and stability. Imprecisely speaking, the time step in an explicit code must be smaller than the minimum time it takes for a disturbance to travel across any element in the calculation by physical processes, such as shock propagation, material motion, or radiation transport [18], [19]. Additional limits based on accuracy may be added. For example, many codes limit the volume change of an element to prevent over-compressions or over-expansions. 

If all the PES coordinates are split off in this way, the original multidimensional problem reduces to that of one-dimensional tunneling in the effective barrier (1.10) of a particle which is coupled to the heat bath. This problem is known as the dissipative tunneling problem, which has been intensively studied for the past 15 years, primarily in connection with tunneling phenomena in solid state physics [Caldeira and Leggett 1983]. Interaction with the heat bath leads to the friction force that acts on the particle moving in the one-dimensional potential (1.10), and, as a consequence, a> is replaced by the Kramers frequency [Kramers 1940] defined by... 

While the goal of the previous models is to carry out analytical calculations and gain insight into the physical picture, the multidimensional calculations are expected to give a quantitative description of concrete chemical systems. However at present we are just at the beginning of this process, and only a few examples of numerical multidimensional computations, mostly on rather idealized PES, have been performed so far. Nonetheless these pioneering studies have established a number of novel features of tunneling reactions, which do not show up in the effectively one-dimensional models. 

The discussion so far has dealt with one-dimensional models which as a rule do not directly apply to real chemical systems for the reasons discussed in the introduction. In this section we discuss how the above methods can be extended to many dimensions. In order not to encumber the text and in order to make physics more transparent, we conflne ourselves to two dimensions, although the generalization to more dimensions is straightforward. 

Preparation research of SWCNT was also put forth by lijima and his co-worker [3]. The structure of SWCNT consists of an enrolled graphene to form a tube without seam. The length and diameter depend on the kinds of the metal catalyst used in the synthesis. The maximum length is several jim and the diameter varies from 1 to 3 nm. The thinnest diameter is about the same as that of Cgo (i.e., ca. 0.7 nm). The structure and characteristics of SWCNT are apparently different from those of MWCNT and rather near to fullerenes. Hence novel physical properties of SWCNT as the one-dimensional material between molecule and bulk are expected. On the other hand, the physical property of MWCNT is almost similar to that of graphite used as bulk [6c]. 

The extension of generic CA systems to two dimensions is significant for two reasons first, the extension brings with it the appearance of many new phenomena involving behaviors of the boundaries of, and interfaces between, two-dimensional patterns that have no simple analogs in one-dimensional systems. Secondly, two-dimensional dynamics make it an easy (sometimes trivial) task to compare the time behavior of such CA systems to that of real physical systems. Indeed, as we shall see in later sections, models for dendritic crystal growth, chemical reaction-diffusion systems and a direct simulation of turbulent fluid flow patterns are in fact specific instances of 2D CA rules and lattices. 

Two approaches to the attainment of the oriented states of polymer solutions and melts can be distinguished. The first one consists in the orientational crystallization of flexible-chain polymers based on the fixation by subsequent crystallization of the chains obtained as a result of melt extension. This procedure ensures the formation of a highly oriented supramolecular structure in the crystallized material. The second approach is based on the use of solutions of rigid-chain polymers in which the transition to the liquid crystalline state occurs, due to a high anisometry of the macromolecules. This state is characterized by high one-dimensional chain orientation and, as a result, by the anisotropy of the main physical properties of the material. Only slight extensions are required to obtain highly oriented films and fibers from such solutions. 

The physical properties of relaxation set by Eq. (1.37) are easy to discuss for its one-dimensional projection... 

The quasi-one-dimensional model of laminar flow in a heated capillary is presented. In the frame of this model the effect of channel size, initial temperature of the working fluid, wall heat flux and gravity on two-phase capillary flow is studied. It is shown that hydrodynamical and thermal characteristics of laminar flow in a heated capillary are determined by the physical properties of the liquid and its vapor, as well as the heat flux on the wall. 

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