Power-law potentials

At shorter distances the repulsive forces start to dominate. The repulsive interaction between two molecules can be described by the power-law potential l/rn (n>9) caused by overlapping of electron clouds resulting in a conflict with the Pauli exclusion principle. For a completely rigid tip and sample whose atoms interact as 1/r12, the repulsion would be described by W-l/D7. In practice, both the tip and the sample are deformable (Fig. 3d). The tip-sample attraction is balanced by mechanical stress which arises in the contact area. From the Hertz theory [77,79], the relation between the deformation force Fd and the contact radius a is given by ... 

Which potential can lead to inflation within this context If one considers power law potentials like V = m2(f>2 or V = j (f>4, then the slow roll condition impose that... 

Focusing on the inverse power law potential V = MA+a/[Pg.143]

In this case, regardless of the value of a, the value of the equation of state parameter w is always around —0.8 when Qq = 0.7. Let us note that this is far from the end of the story. In order to build a realistic model, we have in particular to (i) find a framework which can naturally account for an inverse power law potential, (ii) explain why the particle associated to this field have never been observed since V" is extremely low today, one expects that the associated particles are very light, so that if they are not detected experimentally, then they must be extremely weakly coupled to ordinary matter, a situation which may necessitate some new fine tuning in the model. For a much deeper discussion about all this, see for examples Refs. Brax Martin 1999 Brax et al. 2000 Brax et al. 2001 and references therein. 

The interesting case of potentials with tails falling off like 1 /i was recently treated by Mortiz et al. [59]. They reported many aspects of the near-threshold properties for these potentials. However, they did not give values of the critical parameters Xc and a. There is no general solution of the Schrodinger equation for power-law potentials, but for tails going to zero faster than 1/r2 and E(XC) 0, the solutions are Bessel functions [60] that can be normalized for... 

The simulation of the temperature dependence was performed assuming a thermally activated tunneling process, described by a Bell type of tunneling. The high temperature rate in the tunnel model was chosen as 4 x lOi which is expected from the Eyring equation. Since the observed increase in kj2 at low temperatures is not obtainable by a simple one-dimensional Bell model an effective power law potential was employed ... 

Let us now investigate the accuracy of the asymptotic relationship shown in equation (12). For power-law potentials, a comparison of equation (12) with numerical calculations is given in ref. [18]. Even for the ground state (/ = 0), the precision of this simple expression is surprisingly good, especially at 3 = 1 and 3 = 4 (the anharmonic oscillator). Note that in this case, the coefficients change by many orders of magnitude ... 

Thus, for power-law potentials, the asymptotic expression with p = 0 heis good precision for a wide range of s values but the precision decreases as as the number of nodes, p, increases. 

If, on the other hand, p and q are both large, then ApiAq, n. For example, for ns-states in the power-law potential... 

Figure 3. The function /(/>) in equation (53) for power-law potentials. The values of the index s are shown on the curves. The dashed curve corresponds to s = oo, i.e. the square well potential.

However, microgels are able to interpenetrate and shrink at high packing density. An effective volume fraction that takes the influence of temperature on particle size into account can still be employed in order to obtain temperature-independent curves for, for example, rheological properties however, the absolute values are no longer correct. The interaction potential between microgel particles in such dense suspensions depends on the aoss-link density and was described not only by a power-law potential but also by a bmshlike interaction. ... 

Power-law potentials exhibit simple scaling properties [17,23]. If... 

The word potential comes from containing the harmonics that are present in the expansion of the potential itself. For power-law potentials, is independent of For general potentials, depends on but the method remains applicable [53]. 

The convergence of the hyperspherical expansion for baryons was studied in some detail ref. [58]. Table 5.1 shows the result of a calculation with equal masses = 1 and the simple power-law potential... 

It has been noticed [37] that A is positive for any attractive power-law potential e(a)r , where e is the sign function. In particular, if the popular Coulomb-plus-linear potential is split into its best harmonic approximation (for N = 2) and a correction to it, say... 

The translation-invariant decomposition (9.38) was first written by Post [95] and was rediscovered independently in refs. [85,86], The result (9.39) clearly constitutes an improvement with respect to the previous inequality (9.18) because the constituent mass in is decreased by a factor 3/4, and therefore the energy E. is algebraically increased. For an attractive power-law potential e(/3)r, this provides a factor (4/3). A numerical comparison is shown in table 9.1, where are listed the naive lower limit (9.18), the improved lower limit (9.39), the exact energy obtained by a hyperspherical expansion, and the variational bound derived from a Gaussian trial wave function. It is worth noticing that the new lower limit (9.39) becomes exact in the case of the harmonic oscillator. This is true for an arbitrary number A of bosons and the harmonic oscillator is the only potential for which the inequality is saturated. A beautiful proof of this property has been given by Wu and is included in ref. [ ]. 

Ground-state energy of qqQ compared with the lower limits of eqs. (9.5) and (9.6) for some power-law potenti-als e P)r and quark mass ratios M/m. The exact result corresponds to a hyperspherical expansion pushed up to a grand orbital momenrnm L = 8. 

Fig. 11.2. Binding energy of qqQ, as a function of the inverse mass Wq for the power-law potentials E rl with =0.1 and = 1.

Distant-dependent pair energies are expected to be less sensitive to variations in shapes than simple contact models, in which inter-residues interactions are assumed to be constant up to a certain cutoff distance and are set to zero at larger distances. A number of distance-dependent pairwise potentials have been proposed in the past [29,30]. We consider both simple contact models and distance-dependent power law potentials and compare their performance with that of novel profile models. 

---