A moving boundary model motivated by electric breakdown: II. Initial value problem

An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be formulated as a Laplacian growth model regularized by a 'kinetic undercooling' boundary condition. Using this model we study both the linearized and the full nonlinear evolution of small perturbations of a uniformly translating circle. Within the linear approximation analytical and numerical results show that perturbations are advected to the back of the circle, where they decay. An initially analytic interface stays analytic for all finite times, but singularities from outside the physical region approach the interface for $t\to\infty$, which results in some anomalous relaxation at the back of the circle. For the nonlinear evolution numerical results indicate that the circle is the asymptotic attractor for small perturbations, but larger perturbations may lead to branching. We also present results for more general initial shapes, which demonstrate that regularization by kinetic undercooling cannot guarantee smooth interfaces globally in time.Comment: 44 pages, 18 figures, paper submitted to Physica

Complexity Thinking as a Tool to Understand the Didactics of Psychology

The need to establish a research field within psychology didactics at secondary level has recently been voiced by several researchers internationally. An analysis of a Swedish case coming out of secondary level education in psychology presented here provides an illustration that complexity thinking-derived from complexity theory-is uniquely placed to consider and indicate possible solutions to challenges, described by researchers as central to the foundation of a new field. Subject matter didactics is defined for the purpose of this paper as a combination of general didactics and subject matter content, and considering the international nature of research traditions coming out of psychology, the implications of the results presented here cannot be regarded as limited solely to national concerns. An online survey was sent to secondary schools in Sweden. Discussions and lectures along with teaching to the book-alternatively used as inspiration-emerged as central from the thematic analysis of the results, providing the first mapping of teaching practices secondary level psychology in Sweden. An analysis, founded on complexity thinking-combined with a model enabling a delimitation of the scope of study-focused on time use and the importance placed on self-knowledge, along with the transformation of theory into practice. The former pointed to a teacher-centered nested subsystem (e.g., asymmetric relations between teachers and students), whereas the latter pointed to student-centered nested subsystems coming out of embodied knowledge (e.g., students as node) where psychological perspectives are learnt through self-reflection, case studies, and everyday life experiences (turning theory to practice), implying a holistic approach. The analysis applied to the Swedish case provides an illustration of how complexity theory has the potential to address challenges at the micro and the macro levels to the establishment of a new research field in psychology didactics and to indicate possible solutions (drawing among other things upon teaching experiences coming out of the Swedish case study). Psychology's high relevance to everyday life, multi-causality, perspective pluralism, dynamic systems character, and scientific character make complexity thinking a relevant approach in the consideration of challenges to the establishment of a research field in didactics of psychology