We study a simple swarming model on a two-dimensional lattice where the self-propelled particles exhibit a tendency to align ferromagnetically. Volume exclusion effects are present: particles can only hop to a neighboring node if the node is empty. Here we show that such effects lead to a surprisingly rich variety of self-organized spatial patterns. As particles exhibit an increasingly higher tendency to align to neighbors, they first self-segregate into disordered particle aggregates. Aggregates turn into traffic jams. Traffic jams evolve toward gliders, triangular high density regions that migrate in a well-defined direction. Maximum order is achieved by the formation of elongated high density regions--bands--that transverse the entire system. Numerical evidence suggests that below the percolation density the phase transition associated with orientational order is of first order, while at full occupancy it is of second order.

If a point particle moves chaotically through a periodic array of scatterers the associated transport coefficients are typically irregular functions under variation of control parameters. For a piecewise linear two-parameter map we analyze the structure of the associated irregular diffusion coefficient and current by numerically computing dimensions from box-counting and from the autocorrelation function of these graphs. We find that both dimensions are fractal for large parameter intervals and that both quantities are themselves fractal functions if computed locally on a uniform grid of small but finite subintervals. We furthermore show that there is a simple functional relationship between the structure of fractal fractal dimensions and the difference quotient defined on these subintervals.

We treat the time evolution of states on a finite directed graph, with singular diffusion on the edges of the graph and glueing conditions at the vertices. The operator driving the evolution is obtained by the method of quadratic forms on a suitable Hilbert space. Using the Beurling-Deny criteria we describe glueing conditions leading to positive and to submarkovian semigroups, respectively. IntroductionThe intentions of this paper are twofold. The first aim is to present a treatment of one-dimensional "singular" diffusion in the framework of Dirichlet forms. The second is to present suitable boundary or glueing conditions on graphs (quantum graphs) leading to positive or submarkovian C 0 -semigroups.Concerning the first topic we assume that µ is a finite Borel measure on a bounded interval [a, b]. We assume that particles move in [a, b] according to "Brownian motion" but are only allowed to be located in the support of µ, and in fact are slowed down or accelerated by the "speed measure" µ. (Incidentally, the support of µ is allowed to have gaps, what sometimes is referred to by "gap diffusion".) More generally, instead of starting with Brownian motion, one also can include a drift in the diffusion. This leads to including a scale function. The treatment of the corresponding process has a longer history (cf. [1,3-5,12-14,19,20,22]), but there appears to be no treatment of the arising evolution in the context of Dirichlet forms.Concerning the second topic, we assume that finitely many intervals, with diffusion as described above, are arranged in a graph, and we treat the question how boundary conditions (glueing conditions) at the vertices can be posed in a way to describe diffusing particles. These topics have also been treated in a recent paper by Kostrykin et al. [16]. Since we pose the glueing conditions in a different form (following [17]) it does not seem evident to us to establish the connection between the conditions given in [16] and our conditions.There are many motivations from applications for this work. Since we were not primarily motivated by specific applications we refer to [17,18,23] for some of these Mathematics Subject Classification (2000): 47D06, 60J60, 47E05, 35Q99, 05C99

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