Maxwell-Boltzmann equilibrium solution

The Boltzmann equation is a nonlinear, integrodifferential equation. As such it is extremely difficult to solve and, in fact, almost no exact solutions are known, apart from the Maxwell-Boltzmann equilibrium solution. Furthermore, only a few existence theorems are known notable are the theorems of Carleman, later extended by Wild and by Morgenstern, proving the existence of a solution of the nonlinear Boltzmann equation for special intermolecular potentials in the case that the system is spatially uniform, i.e., that the distribution function does not depend on r. However, there are a number of circumstances where the system is close enough to equilibrium that the distribution function may be written... 

The Debye-Hiickel theory is a study of the equilibrium properties of electrolyte solutions, where departures from ideal behaviour are considered to be a result of coulombic interactions between ions in an equilibrium situation. It is for this reason that equilibrium statistical mechanics can be used to calculate an equilibrium Maxwell-Boltzmann distribution of ions. 

When H has reached its minimum value this is the well known Maxwell-Boltzmann distribution for a gas in thermal equilibrium with a uniform motion u. So, argues Boltzmann, solutions of his equation for an isolated system approach an equilibrium state, just as real gases seem to do. Up to a negative factor -k, in fact), differences in H are the same as differences in the thermodynamic entropy between initial and final equilibrium states. Boltzmann thought that his //-theorem gave a foundation of the increase in entropy as a result of the collision integral, whose derivation was based on the Stosszahlansatz. 

Boltzmann s form of the //-theorem, dH/dt 0, is recovered whenever the boundary conditions are such that dOfdt = 0. If the molecules are specularly reflected at the walls, then this condition is certainly satisfied, as it is if the distribution function at the walls is a Maxwell-Boltzmann distribution with the same temperature for both incident and reflected particles. In addition, we can also verify that if the walls are at a uniform temperature Ty, and if the thermostat condition is satisfied, then the equilibrium distribution function, Eq. (38), is a solution of the Boltzmann equation, Eq. (36), and for this distribution df/dt = 0 and dH/dt = 0. Therefore, according to the Boltzmann equation, if in the course of time the system reaches an equilibrium state with/ given by Eq. (38), it will remain in equilibrium for all later times. 

The solution (54) represents the local equilibrium Maxwell-Boltzmann distribution over the velocity and rotational energy levels with the temperature T and strongly non-equilibrium distribution over chemical species and vibrational energy levels. The distribution functions... 

Let us suppose, that the solution to (8.2.8), (8.2.9) is a local Maxwell-Boltzmann distribution desciibiiig locally equilibrium state of the gas phase. 

The Boltzmann equation is considered valid as long as the density of the gas is sufficiently low and the gas properties are sufficiently uniform in space. Although an exact solution is only achieved for a gas at equilibrium for which the Maxwell velocity distribution is supposed to be valid, one can still obtain approximate solutions for gases near equilibrium states. However, it is evident that the range of densities for which a formal mathematical theory of transport processes can be deduced from Boltzmann s equation is limited to dilute gases, since this relation is reflecting an asymptotic formulation valid in the limit of no coUisional transfer fluxes and restricted to binary collisions only. Hence, this theory cannot without ad hoc modifications be applied to dense gases and liquids. 

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