Maxwellian Probability Distribution Function
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The Distribution Function and its Cumulative
The Maxwellian probability distribution function (PDF) is proportional
to
where
is the molecular speed and
is a parameter that depends on the temperature and molecular mass.
The normalization factor is given by
So a straightforward definition of the Maxwellian PDF is
We have defined MaxwellPDF
so that, if it is called with just one argument, a default value
of 1 is used for .
It is sufficient to consider just
since
For molecular dynamics
has the value
where
is the mass of the molecular species,
is the Boltzmann constant, and
is the temperature (in Kelvin). This corresponds to,
To calculate random numbers with a probability distribution that
conforms to the Maxwellian PDF, we need to compute the cumulative
distribution function and invert it.
Here is the cumulative distribution function.
So we can define the cumulative Maxwellian distribution function
via
Similar to the case with MaxwellPDF,
we have
So we will just work with
to generate random numbers. The result can always be rescaled by
.
Random Numbers
To generate random numbers according to a given probability distribution
function, one can use the inverse of the cumulative distribution
function acting on random numbers generated from a uniform distribution.
In the case of MaxwellCDF
there is no closed form inverse.
However, we can build a numerical function that provides us with
an inverse. There are many ways to do this. We describe several
possibilities in the pages that follow.
Interpolation Function Method
Inverse Series Method
Inverse Series Method with Padé Approximants
[Parts of this example are based on an example
given in The Mathematica Journal.]
