at which this occurs is proportional to the molecular collision frequency. The Boltzmann equation is therefore modified to the BGK form: where is the molecular collision frequency, and is the local Maxwellian distribution function given the gas temperature at this point in space. General equation (for a mixture) For a mixture of chemical species labelled by indices the equation for species is where , and the collision term is where , the magnitude of the relative momenta is and is the differential cross-section, as before, between particles i and j. The integration is over the momentum components in the integrand (which are labelled i and j). The sum of integrals describes the entry and exit of particles of species i in or out of the phase-space element. Applications and extensions Conservation equations The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy. For a fluid consisting of only one kind of particle, the number density is given by The average value of any function is Since the conservation equations involve tensors, the Einstein summation convention will be used where repeated indices in a product indicate summation over those indices. Thus and , where is the particle velocity vector. Define as some function of momentum only, which is conserved in a collision. Assume also that the force is a function of position only, and that f is zero for . Multiplying the Boltzmann equation by A and integrating over momentum yields four terms, which, using integration by parts, can be expressed as where the last term is zero, since is conserved in a collision. The values of correspond to moments of velocity (and momentum , as they are linearly dependent). Zeroth moment Letting , the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation: where is the mass density, and is the average fluid velocity. First moment Letting , the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation: where is the pressure tensor (the viscous stress tensor plus the hydrostatic pressure). Second moment Letting , the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation: where is the kinetic thermal energy density, and is the heat flux vector. Hamiltonian mechanics In Hamiltonian mechanics, the Boltzmann equation is often written more generally as where is the