We propose an entropic geometrical model of crowd behavior dynamics (with dissipative crowd kinematics), using Feynman action-amplitude formalism that operates on three synergetic levels: macro, meso and micro. The intent is to explain the dynamics of crowds simultaneously and consistently across these three levels, in order to characterize their geometrical properties particularly with respect to behavior regimes and the state changes between them. Its most natural statistical descriptor (order parameter) is crowd entropy S that satisfies the Prigogine's extended second law of thermodynamics, ∂tS ≥ 0 (for any nonisolated multi-component system). Qualitative similarities and superpositions between individual and crowd configuration manifolds motivate our claim that goal-directed crowd movement operates under entropy conservation, ∂tS = 0, while naturally chaotic crowd dynamics operates under (monotonically) increasing entropy function, ∂tS > 0. Between these two distinct topological phases lies a phase transition with a chaotic inter-phase. Both inertial crowd dynamics and its dissipative kinematics represent diffusion processes on the crowd manifold governed by the Ricci flow.

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We propose the basis for a rigorous approach to modeling combat, specifically under conditions of complexity and uncertainty. The proposed basis is a tensorial generalization of earlier Lanchester-type equations, inspired by the contemporary debate in defence and military circles around how to best utilize information and communications systems in military operations, including the distributed C4ISR system (Command, Control, Communications, Computing, Intelligence, Surveillance and Reconnaissance). Despite attracting considerable interest and spawning several efforts to develop sound theoretical frameworks for informing force design decision-making, the development of good frameworks for analytically modeling combat remains anything but decided. Using a simple combat scenario, we first develop a tensor generalization of the Lanchester square law, and then extend it to also include the Lanchester linear law, which represents the effect of suppressive fire. We also add on-off control inputs, and discuss the results of a simple simulation of the final model using our small scenario.

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We present a new approach to modelling dynamics of confined crowds driven by Entropic Stochastic Resonance (ESR). The standard approach is to model confined Brownian particles using overdamped Langevin equations and corresponding linear, real-time, Fokker-Planck equations for Probability Density Functions (PDFs). Instead, we propose a new approach based on a set of (weakly or strongly) coupled Quantum Neural Networks (QNNs), which are self-organised, complex-valued nonlinear Schrödinger equations with unsupervised Hebbian-type learning. Utilising the full power of nonlinear analysis in the complex-plane, the new approach promises to be ideal for any kind of two-dimensional terrains. Besides, instead of over-simplistic Brownian particles, the new approach allows us to model crowds consisting of rigid-body-type agents.