Chemistry Reference
In-Depth Information
1.1.
Complexity — Classical & quantum
The science of complexity is a relatively new field. Over decades one has
seen that systems involving several agents or components with conflicting
interests exhibit certain unique characteristics and surprises. Often a hier-
archy of new and unexpected qualitative features emerge
[
5
]
.Amongthe
many problems addressed there are many clearly defined, but
hard problems
in the sense of (i) exponentially long time needed to solve the problem and
(ii) finding hierarchical structures on the road to final solutions.
An example well known to condensed matter physicist is the problem
of spin glass, as illustrated by the Sherrington Kirkpatrick model. In-
sightful and dicult analysis by Parisi and coworkers have introduced the
concept of
replica symmetry breaking
and
unltrametricity structure
in the
solution space and several unexpected features. Physical consequences and
connection to unexpected mathematics such as p-adic number theory are
fascinating. In the spin glass problem, the underlying variable are a simple
collection of Ising variables. It is the competing interactions which frustrate
any attempts to find the ground state. In models which are constructed to
extract generic features of these type of complexity problems, randomness
is a key element. It is not necessary, of course, to have several compet-
ing agents to get complexity and surprises. In classical mechanics, even in
3 body dynamics of a generic non integrable systems, there is chaos and
complexity.
In what follows we define complexity for our purposes as follow:
that
which cannot be handled with ease, which does not yield to straight forward
analysis by known theoretical methods and where you do not know what
to expect next
. Looking back at quantum condensed matter systems that
have been studied and understood in the last century, many of them can
be called complex. Quantum mechanics underlies the complexity. The
basic laws of quantum mechanics and rules for computations, necessary
to understand condensed matter systems have been discovered. Quantum
mechanics has not failed us so for in understanding the day today material
world. Quantitative understanding and predictions we wish to make may
be di
cult, because of technical di
culties; not because of our ignorance
about the laws of nature, as it existed before the invention of quantum
mechanics.
Quantum mechanics is complex. Quantum mechanics has revealed novel
and
complex quantum worlds
in the last several decades, through simple sys-
tems. The structure of quantum mechanics with the possibility of linear
