Abstract
The violation of particle-number conservation by some spherical solutions of the Boltzmann equation is known to exist. The reason stems from the fact that the Boltzmann equation itself includes a contradiction: The velocity of a particle is independent of the position of the particle, but the acceleration, namely the time derivative of velocity, is related to the position. Considering the velocity as a function of position and time, we introduce a new equation. Under the same initial condition as for the Boltzmann equation, a strict solution of the new equation does not violate the particle-number conservation law.
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