2.2 The Maxwell-Boltzmann distribution

Now we distribute the particles over the cells, such that particles are allotted to cell no. . In a closed system with total energy the population numbers must fulfill the condition . The other condition is, of course, the conservation of the number of particles, . Apart from these two requirements the allottment of particles to cells is completely random.

We may understand this prescription as the rule of a game of fortune,
and with the aid of a computer we may actually play that game!

Applet LBRoulette: Start |

Instead of playing the game we may calculate its outcome by probability
theory.
For good statistics we require that .
A specific -tuple of population numbers
will here be called a
**partitioning**. (If you prefer to follow the literature, you
may refer to it as a *distribution*.) Each partitioning
may be performed in many different ways, since the labels of the
particles may be permutated without changing the population numbers
in the cells. This means that many specific **allottments**
pertain to a single **partitioning**. Assuming that the allotments
are elementary events of equal probability, we simply count the number
of possible allottments to calculate the probability of the
respective partitioning.

The number of possible permutations of particle labels for a given
partition
is

Since each allottment is equally probable, the most probable
**partitioning** is the one allowing for the largest number of
**allottments**. In many physically relevant cases the probability of
that optimal partitioning very much larger than that of any other,
meaning that we can restrict the further discussion to this one
partitioning. (See the previous discussion of the multinomial
distribution.)

Thus we want to determine that specific partitioning
which renders the expression 2.20 a maximum, given the
additional constraints

(2.21) |

Since the logarithm is a monotonically increasing function we may scout for
the maximum of instead of - this is mathematically much easier.
The proven method for maximizing a function of many variables, allowing for
additional constraints, is the **variational method with Lagrange
multipliers** (see Problem 2.1).
The variational equation

(2.22) |

(2.23) |

(2.24) |

(2.25) |

(2.26) |

(2.27) |

(2.28) |

(2.29) |

This density in velocity space is commonly called

So we have determined the population numbers of the cells
in space by maximizing the number of possible allottments. It is
possible to demonstrate that the partitioning we have found is not just
the most probable but **by far** the most probable one. In other words,
any noticeable deviation from this distribution of particle velocities
is extremely improbable (see above: multinomial distribution.)
This makes for the great practical importance of the MB distribution:
it is simply **the** distribution of velocities in a many particle
system which we may assume to hold, neglecting all other possible but
improbable distributions.

As we can see from the figure, is a skewed distribution;
its **maximum** is located at

(2.32) |

(2.33) |

(2.34) |

EXAMPLE:The mass of an molecule is ; at room temperature (appr. ) we have ; therefore the most probable speed of such a molecule under equilibrium conditions is .

2005-01-25