2.2 The Maxwell-Boltzmann distribution

We want to apply statistical procedures to the swarm of points in Boltzmann's $\mu$ space. To do this we first divide that space in $6$-dimensional cells of size $(\Delta x)^{3} (\Delta v)^{3}$, labelling them by $j$ ($=1, \dots, m$). There is a characteristic energy $E_{j} \equiv E(\vec{r}_{j},\vec{v}_{j})$ pertaining to each such cell. For instance, in the ideal gas case this energy is simply $E_{j}=mv_{j}^{2}/2$, where $\vec{v}_{j}$ is the velocity of the particles in cell $j$.

Now we distribute the $N$ particles over the $m$ cells, such that $n_{j}$ particles are allotted to cell no. $j$. In a closed system with total energy $E$ the population numbers $n_{j}$ must fulfill the condition $\sum_{j} n_{j}E_{j}=E$. The other condition is, of course, the conservation of the number of particles, $\sum_{j} n_{j} =N$. 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

$\parbox{360pt}{ \mbox{}\\ {\bf Playing Boltzmann's game \\ \vspace{2ex} {\small [Code: LBRoulette]} } }$

Instead of playing the game we may calculate its outcome by probability theory. For good statistics we require that $N » m » 1$. A specific $m$-tuple of population numbers $\vec{n} \equiv \{ n_{j}; j=1,\dots , m \}$ 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 $\vec{n} \equiv \{ n_{j}; j=1,\dots , m \}$ is

$$P(\vec{n})= \frac{N!}{n_{1}! n_{2}! \dots n_{m}!}$$ (2.20)

(In combinatorics this is called permutations of $m$ elements - i. e. cell numbers - with repetition).

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 $\vec{n}^{*}$ which renders the expression 2.20 a maximum, given the additional constraints

$$\sum_{j=1}^{m} n_{j} - N = 0 \;\;\;{\rm and} \;\;\; \sum_{j=1}^{m} n_{j}E_{j} - E =0$$ (2.21)

Since the logarithm is a monotonically increasing function we may scout for the maximum of $\ln P$ instead of $P$ - 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

$$\delta \ln P -\delta \left( \alpha \sum_{j} n_{j} + \beta \sum_{j} E_{j} n_{j}\right) = 0$$ (2.22)

with the undetermined multipliers $\alpha$ and $\beta$ leads us, using the Stirling approximation for $\ln P$, to

$$- \ln n_{j}^{*} - \alpha - \beta E_{j} = 0$$ (2.23)

Thus the optimal partitioning $n_{j}^{*}$ is given by

$$n_{j}^{*} = e^{ - \alpha - \beta E_{j}} \; {\rm ;} \; \; j=1,\dots m$$ (2.24)

Consequently

$$f(\vec{r}_{j},\vec{v}_{j}) \propto n_{j}^{*} = A \exp\{-\beta E(\vec{r}_{j},\vec{v}_{j})\}$$ (2.25)

In particular, we find for a dilute gas, which in the absence of external forces will be homogeneous with respect to $\vec{r}$,

$$f(\vec{v}_{j})= B \exp \{ -\beta (m v_{j}^{2}/2) \}$$ (2.26)

Using the normalizing condition $\int f(\vec{v}) d\vec{v}=1$ or

$$B \int 4 \pi v^{2} e^{-\beta m v^{2}/2} dv = 1$$ (2.27)

we find $B = \left( \beta m / 2 \pi \right)^{3/2}$ and therefore

$$f(\vec{v}) = \left[ \frac{\beta m}{2 \pi } \right]^{3/2} e^{-\beta mv^{2}/2 }$$ (2.28)

Now we take a closer look at the quantity $\beta$ which we introduced at first just for mathematical convenience. The mean kinetic energy of a particle is given by

$$\langle E \rangle \equiv \int d\vec{v} (mv^{2}/2)f(\vec{v}) = \frac{3}{2 \beta}$$ (2.29)

But we will learn in Section 2.3 that the average kinetic energy of a molecule is related to the macroscopic observable quantity $T$ (temperature) according to $\langle E \rangle = 3 kT/2$; therefore we have $\beta \equiv 1/kT$. Thus we may write the distribution density of the velocity in the customary format

$$f(\vec{v}) = \left[ \frac{m}{2 \pi kT} \right]^{3/2} e^{-mv^{2}/2 kT}$$ (2.30)

This density in velocity space is commonly called Maxwell-Boltzmann distribution density. The same name is also used for a slightly different object, namely the distribution density of the modulus of the particle velocity (the ``speed'') which may easily be derived as (see equ. 1.66).

$$f(\vert \vec{v} \vert) = 4 \pi v^{2} f(\vec{v}) = 4 \pi \left[ \frac{m}{2 \pi k T} \right]^{3/2} v^{2}e^{-mv^{2}/2kT}$$ (2.31)

So we have determined the population numbers $n_{j}^{*}$ of the cells in $\mu$ 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.

Figure 2.3: Maxwell-Boltzmann distribution

As we can see from the figure, $f(\vert\vec{v}\vert)$ is a skewed distribution; its maximum is located at

$$\tilde{v} = \sqrt{\frac{2kT}{m}}$$ (2.32)

This most probable speed is not the same as the mean speed,

$$\langle v \rangle \equiv \int_{0}^{\infty} dv v f(\vert\vec{v}\vert) = \sqrt{\frac{8}{\pi}} \sqrt{\frac{kT}{m}}$$ (2.33)

or the root mean squared velocity or r.m.s. velocity),

$$\overline{v}_{rms} \equiv \sqrt{\langle v^{2}\rangle} = \sqrt{\frac{3kT}{m}}$$ (2.34)

EXAMPLE: The mass of an $H_{2}$ molecule is $m=3.346 \cdot 10^{-27} kg$; at room temperature (appr. $300 K$) we have $kT = 1.380 \cdot 10^{-23} . 300 = 4.141 \cdot 10^{-21}J$; therefore the most probable speed of such a molecule under equilibrium conditions is $\tilde{v}= 1.573 \cdot 10^{3}m/s$.