The first
conclusion is that the macroscopic and the microscopic quantities can be
connected when we look at the non-local quantities. The question that follows
is then whether we can make a similar connection regarding the local quantites,
i.e., the information density k(x) in the macro formalism and the local entropy s(i) in the micro formalism. Our
investigation of this has resulted in a negative conclusion regarding the
possibility to establish such a connection. The main reason is the following.
How information
(or entropy) is distributed over a system will depend on how one looks for
information, i.e., how one searches the system. The macroscopic formalism is
based on a procedure that involves not only space but also resolution (length
scale). here information is localised at positions where new spatial structure
appears as resolution is improved. In our microscopic approach, one can view
the search for information as a procedure where we sweep the system and
quantify information locally based on how unexpected the local configuration
is, as discussed above. These two approaches leads to different ways in
which information is allocated in the system. This is a general observation,
that holds also in other situations when one wants to charaterise structure and
complexity: how ordered or complex a specific system is depends on what one is
looking for and how.