Infinite discontinuity

Point discontinuities, infinite discontinuities and asymptotic behaviour. 

For both first-order and continuous phase transitions, finite size shifts the transition and rounds it in some way. The shift for first-order transitions arises, crudely, because the chemical potential, like most other properties, has a finite-size correction p(A)-p(oo) C (l/A). An approximate expression for this was derived by Siepmann et al [134]. Therefore, the line of intersection of two chemical potential surfaces Pj(T,P) and pjj T,P) will shift, in general, by an amount 0 IN). The rounding is expected because the partition fiinction only has singularities (and hence produces discontinuous or divergent properties) in tlie limit i—>oo otherwise, it is analytic, so for finite Vthe discontinuities must be smoothed out in some way. The shift for continuous transitions arises because the transition happens when L for the finite system, but when i oo m the infinite system. The rounding happens for the same reason as it does for first-order phase transitions whatever the nature of the divergence in thennodynamic properties (described, typically, by critical exponents) it will be limited by the finite size of the system. 

This physical postulate has an obvious meaning if energy could vary discontinuously (by a finite amount during an infinitely small time) this would mean that the power would be infinite for an infinitely short time, which is ruled out by the postulate. 

The essential feature of the discontinuous theory appears when a trajectory reaches a point for which T xc,ye) => 0 we call this point (xe>Ve) a critical point In such a case the differential equations (6-194) lose their meaning as x and y become infinite, exhibiting thus a discontinuous jump. 

The delta function is everywhere zero except at the origin, where it has an infinite discontinuity, a discontinuity so large that the integral under it is unity. The limits of integration need only include the origin itself Equation (15.9) can equally well be written as... 

The behavior of the function U z) is shown in Fig. 1.15a. In the limiting case of surface masses the potential remains a continuous function, but its derivative, dUjdz, has a discontinuity at the plane z = 0, Fig. 1.15b. In deriving Equations (1.152 and 1.153) we used the fact that the potential is a continuous function at the layer boundaries otherwise the field would be infinitely large. 

In the examples described above the resulting probability distributions were discontinuous functions. However, it is not difficult to imagine cases in which the distributions become continuous in the limit of an infinite - or at least a very large - number of trials. Sucb is the case in the application of statistical arguments to problems in thermodynamics, as outlined in Section 10.5. 

Although -0 is continuous at the origin its first derivative is discontinuous and dip/dx 0. This peculiar behaviour is a result of the infinite potential energy at the origin. 

Continuous variables are those that can at least theoretically assume any of an infinite number of values between any two fixed points (such as measurements of body weight between 2.0 and 3.0 kilograms). Discontinuous variables, meanwhile, are those which can have only certain fixed values, with no possible intermediate values (such as counts of five and six dead animals, respectively). 

For infinitely fast kinetics, then, the temperature profiles form a discontinuity at the infinitely thin reaction zone (see Fig. 6.11). Realizing that the mass burning rate must remain the same for either infinite or finite reaction rates, one must consider three aspects dictated by physical insight when the kinetics are finite first, the temperature gradient at r = rs must be the same in both cases second, the maximum temperature reached when the kinetics are finite must be less than that for the infinite kinetics case third, if the temperature is lower in the finite case, the maximum must be closer to the droplet in order to satisfy the first aspect. Lorell et al. [22] have shown analytically that these physical insights as depicted in Fig. 6.15 are correct. 

It should be noted that the theory described above is strictly vahd only close to Tc for an ideal crystal of infinite size, with translational invariance over the whole volume. Real crystals can only approach this behaviour to a certain extent. Flere the crystal quality plays an essential role. Furthermore, the coupling of the order parameter to the macroscopic strain often leads to a positive feedback, which makes the transition discontinuous. In fact, from NMR investigations there is not a single example of a second order phase transition known where the soft mode really has reached zero frequency at Tc. The reason for this might also be technical It is extremely difficult to achieve a zero temperature gradient throughout the sample, especially close to a phase transition where the transition enthalpy requires a heat flow that can only occur when the temperature gradient is different from zero. 

Figure 1.33. Electronic densities for = 0 (continuous straight line) and (p 0 (discontinuous lines) for an infinite linear chain of period a. For simplicity it has been assumed that 5 = 0 and = 1.

In addition to initial conditions, solutions to the Schrodinger equation must obey certain other constraints in form. They must be continuous functions of all of their spatial coordinates and must be single valued these properties allow VP P to be interpreted as a probability density (i.e., the probability of finding a particle at some position can not be multivalued nor can it be jerky or discontinuous). The derivative of the wavefunction must also be continuous except at points where the potential function undergoes an infinite jump (e.g., at the wall of an infinitely high and steep potential barrier). This condition relates to the fact that the momentum must be continuous except at infinitely steep potential barriers where the momentum undergoes a sudden reversal. 

Accdg to Cook, Langweiler assumed for the plain-wave deton behind the wave front a simplified constant p(x) (density-distance) and W(x) (particle velocity-distance) contour followed by a sharp (presumably discontinuous) rarefaction. He gave as the velocity of the rarefaction front the value (D+W)/2, where (D) is detonation velocity. Then he deduced that in an explosive of infinite lateral extent, the compres-sional region or detonation head of the wave should grow in thickness in accordance with the equation ... 

The electron is not allowed outside the box and to ensure this we put the potential to infinity outside the box. Since the electron cannot have infinite energy, the wave function must be zero outside the box and since it cannot be discontinuous, it must be zero at the boundaries of the box. If we take the sine wave solution, then this is zero at =0. To be zero at x=a as well, there must be a whole number of half waves in the box. Sine functions have a value of zero at angles of nn radians where n is an integer and so... 

It was not only chemistry in the eighteenth century that illustrates the problem of discontinuity and distinct identity. Denis Diderot in the Prospectus for the great EncyclopMe, 1751, described the recalcitrant tendency of nature to resist categorization Nature presents us only with particular things, infinite in number and without firmly established divisions. Everything shades off into everything else by imperceptible nuances. ... 

In practice, this large difference is reduced by about one-quarter because the boundary condition is not infinitely hard as implied by eqn (5.1) but much softer. Within the jellium model the positive ions are smeared out uniformly within a sphere of radius, R, so that there is indeed an abrupt discontinuity in background charge density at r = R. However, the free-... 

The presence of a crack or other discontinuity presents a serious difficulty for the standard V z) theory, because the reflectance function is defined for infinite plane waves that are reflected into infinite plane waves, and this requires a reflecting surface that is uniform. If a surface contains a crack then this requirement is violated, and an incident plane wave may be scattered into a whole family of waves (Tew et al. 1988). This scattering can be described in k-space by a scattering function S(kx, k x), where the prime refers to incident waves and the unprime to scattered waves. The x-direction is taken as tangential to the surface, and at this stage the theory is confined to two dimensions in the plane normal to both the surface and the crack. The response of the microscope can then be written in terms of the scattering function by integrating over the incident and reflected waves separately... 

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