Environmental Engineering Reference
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Fig. 10
Conformation of PAA ISA 23 as a function of
ΚΈ
5 Scaling
Scaling and universality are two amazing properties that collectively have gener-
ated the modern theory of critical behaviour appearing in different areas of modern
physics, such as condensed matter, field theory, plasma physics, complex systems,
dynamical systems, and hydrodynamics (Kadanoff
2000
; Cardy
1997
). The
univer-
sality
quality means that many different systems present the same critical behaviour,
while
scaling
is concerned with the fact that in a neighbourhood of a critical point the
system is scale invariant. Preserving this symmetry in the system makes it possible to
relate physical phenomena which take place at very different length scales. As a con-
sequence, the correct description of systems near their critical points can be described
by
power laws
and this kind of behaviour might be analysed by dimensional con-
siderations known as
scaling laws
. Even though the Renormalization Group (RG)
approach is a good alternative to obtain in an accurate way the critical exponents
(Albert
1982
; Le Guillou and Zinn-Justin
1980
), in many instances the use of this
approach in complex systems is quite difficult. On the other hand, numerical simu-
lations allow one to describe in a simpler and more attractive way different complex
systems, but the possibility to use numerical simulations near the critical points of a
system is still a topic under discussion.
Coarse graining
is another common concept
when we study systems which present scale invariance and, when different scales
are involved. The coarse grained simulations have shown to be a very good alterna-
tive. DPD (Hoogerbrugge and Koelman
1992
) is one such a coarse graining method
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