Noninteracting/noninteraction

The free energy of a monolayer domain in the coexistence region of a phase transition can be described as a balance between the dipolar electrostatic energy and the line tension between the two phases. Following the development of McConnell [168], a monolayer having n circular noninteracting domains of radius R has a free energy... 

The fact that allowed fennion states have to be antisymmetric, i.e., changed m sign by any odd pemuitation of the fennions, leads to an interesting result concerning the allowed states. Let us write a state wavefiinction for a system of n noninteracting fennions as... 

For non-mteracting particles in a box, the result depends on the particle statistics Fenni, Bose of Boltzmamr. The state of a quanPim system can be specified by the wavefrmction for that state, Tv(Qi> Q2 . qyy). is the vth eigensolution to the Scln-ddinger equation for an A -particle system. If the particles are noninteracting, then the wavefrmction can be expressed in temis of the single-particle wavefrinctions given... 

In the discussion above, scattering from molecules is treated as a superposition of noninteracting electron... 

In the classical picture developed above, the wavepacket is modeled by pseudo-particles moving along uncorrelated Newtonian trajectories, taking the electrons with them in the form of the potential along the Uajectory. In this spirit, a classical wavepacket can be defined as an incoherent (i.e., noninteracting) superposition of confignrations, X/(, t)tlt,(r, t)... 

The results of the derivation (which is reproduced in Appendix A) are summarized in Figure 7. This figure applies to both reactive and resonance stabilized (such as benzene) systems. The compounds A and B are the reactant and product in a pericyclic reaction, or the two equivalent Kekule structures in an aromatic system. The parameter t, is the reaction coordinate in a pericyclic reaction or the coordinate interchanging two Kekule structures in aromatic (and antiaromatic) systems. The avoided crossing model [26-28] predicts that the two eigenfunctions of the two-state system may be fomred by in-phase and out-of-phase combinations of the noninteracting basic states A) and B). State A) differs from B) by the spin-pairing scheme. 

The problem of the theoretical description of biopolymer water adsorption isotherms has drawn the attention of researchers for a long time. In the works [19], [20] a rigorous statistical basis for equations describing the isotherms for the case of homogeneous adsorption surfaces and noninteracting adsorption sites of N different types has been suggested. The general equation is ... 

The pattern of orbital energies is different for benzene than it would be if the six tt electrons were confined to three noninteracting double bonds The delocalization provided by cyclic conjugation in benzene causes its tt electrons to be held more strongly than they would be in the absence of cyclic conjugation Stronger binding of its tt electrons is the factor most responsible for the special stability—the aromaticity—of benzene... 

Equation (3.16) shows that the force required to stretch a sample can be broken into two contributions one that measures how the enthalpy of the sample changes with elongation and one which measures the same effect on entropy. The pressure of a system also reflects two parallel contributions, except that the coefficients are associated with volume changes. It will help to pursue the analogy with a gas a bit further. The internal energy of an ideal gas is independent of volume The molecules are noninteracting so it makes no difference how far apart they are. Therefore, for an ideal gas (3U/3V)j = 0 and the thermodynamic equation of state becomes... 

The McCabe-Thiele approach has been developed to describe the Sorbex process (76). Two feed components, A and B, with a suitable adsorbent and a desorbent, C, are separated ia an isothermal continuous countercurrent operation. If A is the more strongly adsorbed component and the system is linear and noninteracting, the flows ia each section of the process must satisfy the foUowiag constraints for complete separation of A from B ... 

Doppler broadening arises from the random thermal agitation of the active systems, each of which, in its own test frame, sees the appHed light field at a different frequency. When averaged over a Maxwellian velocity distribution, ie, assuming noninteracting species in thermal equilibrium, this yields a line width (fwhm) in cm C... 

The strength of laminates is usually predicted from a combination of laminated plate theory and a failure criterion for the individual larnina. A general treatment of composite failure criteria is beyond the scope of the present discussion. Broadly, however, composite failure criteria are of two types noninteractive, such as maximum stress or maximum strain, in which the lamina is taken to fail when a critical value of stress or strain is reached parallel or transverse to the fibers in tension, compression, or shear or interactive, such as the Tsai-Hill or Tsai-Wu (1,7) type, in which failure is taken to be when some combination of stresses occurs. Generally, the ply materials do not have the same strengths in tension and compression, so that five-ply strengths must be deterrnined ... 

When there are two or mote variables, they might interact with one another, ie, the effect of one variable upon the response depends on the value of the other variable. Figure 1 shows a situation where two noninteracting variables, preparation type and temperature, independently affect time to mpture, ie, the effect of temperature on time to mpture is the same for both preparation types. In contrast. Figure 2 shows two examples of interactions between preparation and temperature. 

The performance of the interacting controller is almost as good as the noninterac ting controller on most processes, but the tuning rules differ because of the above relationships. With digital PID controllers, the noninteracting version is commonly used. 

A cornerstone of the analysis of vaporization processes in a vacuum is the classical theory of gases. In this theory, a gas is assumed to consist of noninteracting molecules which undergo elastic collisions with one another and... 

It is to be emphasized that, despite the formal similarity, the physical problems are different. Moreover, in general, diabatic coupling is not small, unlike the tunneling matrix element, and this circumstance does not allow one to apply the noninteracting blip approximation. So even having been formulated in the standard spin-boson form, the problem still remains rather sophisticated. In particular, it is difficult to explore the intermediate region between nonadiabatic and adiabatic transition. 

Figure 15.6 indicates that this is a relatively noninteractive column. PMMA is eluted in toluene, yet the calibration dependences surprisingly diverge in the area of high molar masses. It has the opposite tendency to that observed for various polymers with PS/DVB columns by Dawkins (5,20) and also contradicts the finding for columns No. 5 and 6. 

This is a rather noninteractive column as well (Fig. 15.7). The calibration curves converge slightly in the area of high molar masses. 

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