11. Nevertheless, the two equations can be combined to produce the Poisson Boltzmann equation . 12. Villani received the Fields Medal for his work on Landau damping and the Boltzmann equation . 13. A special form of the Boltzmann equation 14. The corresponding Boltzmann equation is given by 15. If the charge density follows a Boltzmann distribution, then the Poisson-Boltzmann equation results. 16. Keizer also provided fluctuating generalizations of the Boltzmann equation and of hidrodynamics ( fluctuating hydrodynamics ). 17. In biophysics and certain surface chemistry applications, it is known simply as the Poisson Boltzmann equation . 18. However, for an ideal gas the standard Chapman Enskog solution of the Boltzmann equation is highly accurate. 19. As with any approximate model, the Poisson Boltzmann equation is an approximation rather than an exact representation. 20. The Poisson Boltzmann equation plays a role in the development of the Debye H�ckel theory of dilute electrolyte solutions.
