Complexity Analytics and Public Policy : Cautions and Opportunities Going Forward
State space is a representation of all possible combinations of all possible values of all state variables , and the relationships among them . A point in state space is a complete description of the system at that point . Phase space is a snapshot in time of the simultaneous movement of the system through the values of all state variables . A mapping of phase space displays a manifold of phase lines which show the combined values of state variables where phase changes in the system occur . Phase space is also useful in that excluded regions of state space are displayed . Fractal geometry is used to describe the phase space of complex adaptive systems because it is the only mathematical language we know that can encompass the phase lines of such systems which do not overlap , do not repeat , and do not settle into a steady state ( Mandlebrot & Hudson , 2004 ).
A phase change is a discontinuous change in the state of the system . It indicates the crossing of a physical boundary between two different regions of phase space ( one of those manifold lines referred to earlier ). Phase changes are reversible and moving from one phase to another fundamentally alters the energy relationships among the system components or agents . What phases a system can occupy is a function of the physicality of system . We are accustomed to solid , liquid , and vapor phases of elements ; but there is also plasma , glass ( colloidal ), and crystalline phases . There seems to be a growing acceptance of phase change in social systems as associated with system collapse . How have we defined phase in these instances and what additional manifestations does that definition allow ? Is it possible to imagine a definition of phase on a social space where a discontinuous change would lead to an energetically different , yet still ordered state ? Are we looking at the right variables in our efforts to understand phase in social systems ?
Phase space refers to currently accessible regions of state space . It captures the coupling of state variables . Interaction among or coupling of system parameters could create excluded regions of state space . A phase space mapping would identify these excluded regions and their changes over time . Phase space maps the evolution of system properties with collective changes in the values of state parameters . In simple mechanical or chemical systems , points in phase space correspond to manifestations of physical characteristics like the arc of a pendulum , or the liquid , solid , or gaseous state of water . In social systems , this correspondence is not so clear both because of lack of precision in the definition of system parameters , and because of the historical practice of representing social systems as linear and static . A phase change is not just a change of position in state space . A phase change is the crossing of boundaries in phase space that mark the limits of the combination of values of state variables associated with the manifestation of specific system behavior and properties . Inside those boundaries , the system can experience a change in state without undergoing a phase change . The familiar example of phase change is the transition of water from
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