Diverging length scale

G. Biroli and J. P. Bouchaud, Diverging length scale and upper critical dimension in the mode coupling theory of the glass transition. Europhys. Lett. 67, 21 (2004). 

The new phase transition (a critical point) is to be characterized by its own set of exponents. An important quantity is the length scale behaviour. The flow equation around the fixed point for d > 2 shows that one may define a diverging length-scale associated with the critical point as... 

In all these cases, if the nontrivial fixed point is stable, then it represents a critical phase with characteristic exponents, while if it is unstable it represents a critical type phase transition with its own characteristic exponents. The reunion behaviour in Sec. 4.2.3 and the reunion exponents[13] are examples of nontrivial exponents at a stable fixed point. The unstable fixed point will be associated with a diverging length scale with an exponent = l/ e as in Eq. (32). 

Fig. 2a), R oc [csp(r) c] However, there is no clear evidence for such a divergent length scale when one carries out quenches into the region of the phase diagram where the spinodal is expected to occur [3]. Rather, there is compelling evidence from both experiments and computer simulations that the transition from nucleation to spinodal decomposition is completely gradual [3] (Fig. 2a). 

Point 7 has very special properties. It occurs at random close-packing density and provides a definition of it. Point / is an isostatic point, where the number of particle contacts equals the number of force balance equations to describe them. As a result, it is a purely geometrical point the properties of the state at point / are independent of potential. Also, point / appears to be a zero-temperature mixed phase transition, with a discontinuity in the number of contacts, characteristic of first-order phase transitions, and diverging length scales, characteristic of second-order phase transitions. 

There are four parameters which must be fixed to use the MLE algorithm Embedding dimension, de, maximum scale, Sm, minimum scale, Sm and evolution time, O. Basically, de is the attractor dimension where the orbits were embedded, Sm is the estimate value of the length scale on which the local structure of the attractor is not longer being proved. Sm is the length scale in which noise is expected to appear. O is fixed for compute of divergence measurements which is the necessary time to renormalize the distances between trajectories (for more details see [50]). 

For more complex fluids, one expects < 0. Then, mean-field behavior can result from two different processes. First, the long-range nature of the intermolecular forces may cause u to be small, while A is not small. Second, may be large or even diverging. Then, A and Nqi will be small, while u is not necessarily small. This case is expected to give a sharp or even nonmonotonous crossover, because a second length scale is present. 

Vector fields whose divergence vanishes are sometimes referred to as solenoidal. A more comprehensive discussion of the conditions for approximating the velocity field as solenoidal has been given by Batchelor.8 These imply that, in cases in which the fluid is subjected to an oscillating pressure, the characteristic velocity in the Mach number condition should be interpreted as the product of the frequency times the linear dimension of the fluid domain, and that the difference in static pressures over the length scale of the domain must be small compared with the absolute pressure. Because our subject matter will frequently deal with incompressible, isothermal fluids, we shall often make use of (2-20) in lieu of the... 

This implies for example that when a(t) has the Mott minimum value (e ifafit), the dc or large length scale (L — ) conductivity is zero. Thus, the conductivity goes continuously to zero at the Anderson transition, due to interference effects not envisaged in the Mott approximation. The exponent with which a vanishes is the same as that with which the localization length diverges. 

The problem of divergence we face here is ideal for a renormalization group approach. Let us introduce an arbitrary length scale L in the transverse direction and define a dimensionless running coupling constant... 

We note from Eq. 9 that is weakly singular for 0 < 71 and diverges only in a small region with length scale of order d away from the wedge tip. Since the electric pressure scales as Pe the pressure gradient that arises as... 

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