When modeling of tumor-driven angiogenesis, a major source of analytical and computational complexity is the strong coupling between the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network, and the family of interacting underlying fields. To reduce this complexity, we take advantage of the system intrinsic multiscale structure: we describe the stochastic dynamics of the cells at the vessel tip at their natural mesoscale, whereas we describe the deterministic dynamics of the underlying fields at a larger macroscale. Here, we set up a conceptual stochastic model including branching, elongation, and anastomosis of vessels and derive a mean field approximation for their densities. This leads to a deterministic integro-partial differential system that describes the formation of the stochastic vessel network. We discuss the proper capillary injecting boundary conditions and include the results of relevant numerical simulations.
Random number (bit) generators are crucial to secure communications, data transfer and storage, and electronic transactions, to carry out stochastic simulations and to many other applications. As software generated random sequences are not truly random, fast entropy sources such as quantum systems or classically chaotic systems can be viable alternatives provided they generate high-quality random sequences sufficiently fast. The discovery of spontaneous chaos in semiconductor superlattices at room temperature has produced a valuable nanotechnology option. Here we explain a mathematical model to describe spontaneous chaos in semiconductor superlattices at room temperature, solve it numerically to reveal the origin and characteristics of chaotic oscillations, and discuss the limitations of the model in view of known experiments. We also explain how to extract verified random bits from the analog chaotic signal produced by the superlattice.
An angiogenic system is taken as an example of extremely complex ones in the field of Life Sciences, from both analytical and computational points of view, due to the strong coupling between the kinetic parameters of the relevant branching - growth - anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. To reduce this complexity, for a conceptual stochastic model we have explored how to take advantage of the system intrinsic multiscale structure: one might describe the stochastic dynamics of the cells at the vessel tip at their natural microscale, whereas the dynamics of the underlying fields is given by a deterministic mean field approximation obtained by an averaging at a suitable mesoscale. But the outcomes of relevant numerical simulations show that the proposed model, in presence of anastomosis, is not self-averaging, so that the ``propagation of chaos" assumption cannot be applied to obtain a deterministic mean field approximation. On the other hand we have shown that ensemble averages over many realizations of the stochastic system may better correspond to a deterministic reaction-diffusion system.
In a semiconductor superlattice with long scattering times, damping of Bloch oscillations due to scattering is so small that nonlinearities may compensate it and Bloch oscillations persist even in the hydrodynamic regime. To demonstrate this, a Boltzmann-Poisson transport model of miniband superlattices with inelastic collisions is proposed and hydrodynamic equations for electron density, electric field, and the complex amplitude of the Bloch oscillations are derived by singular perturbation methods. For appropriate parameter ranges, numerical solutions of these equations show stable Bloch oscillations with spatially inhomogeneous field, charge, current density, and energy density profiles. These Bloch oscillations disappear as scattering times become sufficiently short. For sufficiently low lattice temperatures, Bloch and Gunn type oscillations mediated by electric field, current, and energy domains coexist for a range of voltages. For larger lattice temperatures (300 K), there are only Bloch oscillations with stationary amplitude and electric field profiles.
Physical systems exhibiting fast spontaneous chaotic oscillations are used to generate high-quality true random sequences in random number generators. The concept of using fast practical entropy sources to produce true random sequences is crucial to make storage and transfer of data more secure at very high speeds. While the first high-speed devices were chaotic semiconductor lasers, the discovery of spontaneous chaos in semiconductor superlattices at room temperature provides a valuable nanotechnology alternative. Spontaneous chaos was observed in 1996 experiments at temperatures below liquid nitrogen. Here we show spontaneous chaos at room temperature appears in idealized superlattices for voltage ranges where sharp transitions between different oscillation modes occur. Internal and external noises broaden these voltage ranges and enhance the sensitivity to initial conditions in the superlattice snail-shaped chaotic attractor thereby rendering spontaneous chaos more robust.
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