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We present a general methodology for constructing lattice Boltzmann models of hydrodynamics with certain desired features of statistical physics and kinetic theory. We show how a methodology of linear programming theory, known as Fourier‐Motzkin elimination, provides an important tool for visualizing the state‐space of lattice Boltzmann algorithms that conserve a given set of moments of the distribution function. We show how such models can be endowed with a Lyapunov functional, analogous to Boltzmann's H, resulting in unconditional numerical stability. Using the Chapman‐Enskog analysis and numerical simulation, we demonstrate that such entropically stabilized lattice Boltzmann algorithms, while fully explicit and perfectly conservative, may achieve remarkably low values for transport coefficients, such as viscosity. Indeed, the lowest such attainable values are limited only by considerations of accuracy …