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:<math>n(x,y,z,t) = \int f \,dv_x \,dv_y \,dv_z</math>
:<math>N(t) = \int n \,dx \,dy \,dz. </math>
Here, N is the total number of particles and ''n'' is the number density of particles - the number of particles per unit volume, or the [[density]] divided by the mass of individual particles.
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The [[Maxwell-Boltzmann distribution|basic distribution function]] uses the [[Boltzmann constant]] <math>k</math> and temperature <math>T</math> with the number density to modify the [[normal distribution]]:
:<math> f = \frac{n}{\sqrt{(2 \pi kT)^3}} \exp\left({-\frac{m(v_x^2 + v_y^2 + v_z^2)}{2kT}}\right). </math>
Related distribution functions may allow bulk fluid flow, in which case the velocity origin is shifted, so that the [[exponent]]'s [[numerator]] is <math>m((v_x - u_x)^2 + (v_y - u_y)^2 + (v_z - u_z)^2)</math>; <math>(u_x, u_y, u_z)</math> is the bulk velocity of the fluid. Distribution functions may also feature non-isotropic temperatures, in which each term in the exponent is divided by a different temperature.
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