Universidad Texas A&M
“Entropy vs. Mixing: What Determines Complexity”
Martes 27 de Mayo
Salón de actos, 12:00
William Seitz, Regents Professor
Texas A&M University at Galveston
“Entropy vs. Mixing: What Determines Complexity”
Martes 27 de Mayo
Salón de actos, 12:00
Entropy vs. Mixing: What Determines Complexity
William Seitz, Regents Professor
Texas A&M University at Galveston
Abstract to Lecture in Madrid
It is universally accepted that systems achieve maximum entropy at equilibrium,
usually characterized as a state of maximum disorder. Our concern is with systems
that evolve to maximum entropy by spontaneous transitions from lower to higher
entropy states. In general, spontaneous changes are predicted by comparing state
properties of initial and final states. Mixing character is such a state property (but
not a function) related to, but different than, entropy. We find there are many
different spontaneous paths between minimum and maximum entropy in isolated
systems. A careful examination of mixing leads to a definition of a complexity
function. Complexity arises from the fact that many states that are comparable
entropically are not comparable by mixing. This view of complexity is
complementary to the complexity in information theory related by Shannon to
entropy.
We also examine the postulate that increasing mixing character determines
spontaneity in isolated systems. Young diagrams, well known in mathematics to be
partially ordered by majorization, represent the mixing character state property. All
spontaneous processes that increase mixing character also involve an increase in
entropy so our view is fully consistent with classical thermodynamics and the
second law. Nevertheless, there are many new implications. For example, paths
that lead from order to equilibrium are significantly shorter than the path
determined by entropy alone – a fact of potential interest in studies of ecological
timescales. Also, mixing paths regularly occupy some entropy states while other
states are rarely occupied. Because the concepts here are widely employed in
diverse fields such as information theory, ecology, and economics, our approach will
likely have widespread, fundamental implications.