The present work explores minimal kinetic theories based on the unconventional entropy function, H _ lnf (Burg entropy). This entropy function was originally derived by Boghosian et al as a basis for discrete-velocity and lattice Boltzmann models for incompressible fluid dynamics. In the model of Boghosian et al, the velocity space is discretized using a finite set of fixed speed vectors which are symmetrically distributed over the velocity space. Using the continuous form of Burg entropy, we extend the work of Boghosian along the following lines: (1) It is shown that, in the continuous-velocity limit of the models of Boghosian, i.e. when the speed is fixed while the particles are free to move in all continuous directions, the equilibrium distribution function can be identified with Poisson kernel of the Poisson integral formula, as an explicit function of the macroscopic properties of the fluid. (2) The expansion coefficients in the Fourier series expansion of the equilibrium distribution function are the velocity moments of the equilibrium distribution function. (3) Employing the theory of functions of a complex variable, the real velocity space is mapped into complex spaces, and it is shown that the velocity moments of the equilibrium distribution function can easily be evaluated using the Poisson integral formula, in the complex plane. (4) The errors due to discretization of the velocity space, when using a hexagonal lattice, is analysed, and it is shown that these errors can be neglected at low Mach numbers. (5) In, the new model, a symmetric discretization of the velocity spaces is not necessary. (6) Rest particles have been included to rectify the speed of sound of the model. (7) A new moment system has been introduced which can be used to replace the hexagonal lattice with a more convenient orthogonal lattice. Using the moment system, one can employ multiple relaxation times, to further increase the stability of the model. To implement the streaming stage, in the present work, different numerical methods have been discussed. We have shown that both ”lattice based” and ”discontinuous Galerkin based” models can be used to solve the advection equation and stream the particles. Based on the above methods and streaming strategies, we have introduced the following new models: d 2 z 6, d 2 z 7 (RP), and d 2 z 9 (MB), which have been used to solve some benchmark problems and the results are compared with the works of others. The convergence rate of the new model is second order in space, which is similar to the convergence rate of the conventional lattice Boltzmann models. Based the simulation results, the lattice based d 2 z 6 is both more cost effective and more stable than the conventional d 2 q 9. The ”discontinuous Galerkin” based streaming schemes are less cost effective than the lattice based models, while they offer better flexibility in handling very complex geometries. Key words: Entropic kinetic models, lattice Boltzmann