| One of our research interests
is to understand the microscopic origin of self-organization
and macroscopic pattern formation in many-body systems out of equilibrium.
This is the case of all systems in nature including biology at any level.
The second law
of thermodynamics does not allow self-organization and formation of spatial
structures in thermal equilibrium because such processes
in isolated systems reduces the entropy of the universe.
Therefore, to form macroscopic spatial structures, the system must be open,
interacting with the external world. In this case the system under
consideration (a subsystem of the entire, isolated system) can self-organize
and remain in this condition over time, reducing its entropy but increasing
the total entropy of the universe. The requirement for the system being
open is demanding and generates serious conceptual and practical
problems: classical thermodynamics deals with equilibrium systems, and
statistical mechanics concepts were developed to understand the microscopic
origin of such thermodynamics. To deal with open systems
it is first necessary to develop
a non-equilibrium statistical mechanics theory to give microscopic
basis to a non-equilibrium thermodynamics. These general theories are not trivial and
sophisticated mathematical techniques and controversial concepts (e.g.,
how to deal with the problem of temporal irreversibility in nature) have
to be dealt with. Physicists have already addressed this topic
and some non-equilibrium approaches have been developed over the last
decades. A particular approch is based on the Maximum
Entropy formalism, which gives basis to a general non-equilibrium theory.
The theory involves a generalization of the Gibbs and Boltzmann formulations combined
with a variational principle, namely, the maximization of the information entropy.
This is the general approach that connects the microscopic processes occuring
in non-equilibrium systems with the macroscopic manifestation of such interactions.
By definition, the thermodynamics arising out of this non-equilibrium statistical
mechanics theory is the non-equilibrium thermodynamics mentioned
above. Although still mathematically complex this macroscopic formalism
leads to equations that can at least be solved numerically
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