Dynamic universality classes

The concept of universality classes, mentioned in the previous section, can be extended so as to be applicable to the characterization of the asymptotic critical behavior of dynamic properties (Hohenberg Halperin 1977). Two systems belong to the same dynamic universality class when they have the same number and types of relevant hydro-dynamic modes. Thus the asymptotical critical behavior of the mutual mass diffusivity D 2 and of the viscosity rj of liquid mixtures near a consolute point will be the same as that of the thermal diffusivity a and the viscosity j of one-component fluids near the vapor-liquid critical point (Sengers 1985). Hence, in analogy with equation (6.16) for liquid mixtures near a consolute point it can be written... 

All of these algorithms deserve systematic study, in particular of their dynamic critical behavior. One would like to know whether they all lie in the same dynamic universality class, and whether the conjecture t is exact, approximate or wrong. 

However, the inclusion of the long-range correlations places the Zimm model into a different dynamic universality class than the Rouse model. Although there exist sophisticated renormalization- oup... 

The phase separation process at late times t is usually governed by a law of the type R t) oc f, where R t) is the characteristic domain size at time t, and n an exponent which depends on the universality class of the model and on the conservation laws in the dynamics. At the presence of amphiphiles, however, the situation is somewhat complicated by the fact that the amphiphiles aggregate at the interfaces and reduce the interfacial tension during the coarsening process, i.e., the interfacial tension depends on the time. This leads to a pronounced slowing down at late times. In order to quantify this effect, Laradji et al. [217,222] have proposed the scaling ansatz... 

Thus, there is a continuous variation in the dynamical exponent for 1 < a < 3, while the attachment-detachment universality class holds for a < 1 and the step-edge universality class holds for a > 3. 

Comparing this with Eq. (21) demonstrates that the dynamics of the first step is in the step-edge universality class, and (from Eq. (19))... 

Thermal phase transitions show that dynamic properties like transport coefficients or relatmtion times may have di ient exponents for different materials and models even if the static equilibrium properties have the same exponents. Thus the static univeisality classes are split into smaUer dynamic universality groups. Conversely, certain etqxinent ratios like y/v or filv may remain constant even ifp,y, and v are a hmetion of a parameter . Nothing seems to be known yet about whether or not gelation and percolation exhibit simitar effects. 

The Universal Modeling Language is used to describe a software system [4, 5], Several kinds of diagrams exist to model the diverse properties of the system. Thus a description of the system can be developed that enables the systematic and uniform documentation of the system. The class diagram, for example, represents the classes and their relationships. But also interacting diagrams exist, to describe the dynamic behavior of the system and its objects. 

Michael Thompson is currently Editor of the Royal Society s Philosophical Transactions (Series A). He graduated from Cambridge with first class honours in Mechanical Sciences in 1958, and obtained his PhD in 1962 and his ScD in 1977. He was a Fulbright researcher in aeronautics at Stanford University, and joined University College London (UCL) in 1964. He has published four books on instabilities, bifurcations, catastrophe theory and chaos, and was appointed professor at UCL in 1977. Michael Thompson was elected FRS in 1985 and was awarded the Ewing Medal ofthe Institution of Civil Engineers. He was a senior SERC fellow and served on the IMA Council. In 1991 he was appointed director of the Centre for Nonlinear Dynamics. 

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