Maxwell-boltzmann Statistics

In Statistical Mechanics , Maxwell-Boltzmann statistics describes the statistical distribution of material particles over various energy states in Thermal Equilibrium , when the temperature is high enough and density is low enough to render quantum effects negligible. Maxwell-Boltzmann statistics are therefore applicable to almost any terrestrial phenomena for which the temperature is above a few tens of Kelvin s.

The expected number of particles with energy

\epsilon_i

for Maxwell-Boltzmann statistics is

N_i

where:

:

rac{N_i}{N} = rac {g_i} {e^{(\epsilon_i-\mu)/kT}} = rac{g_i e^{-\epsilon_i/kT}}{Z}

where:

$N_i$ is the number of particles in state ''i''

$\epsilon_i$ is the Energy of the ''i''-th state

$g_i$ is the Degeneracy of state ''i'', the number of microstates with energy $\epsilon_i$

μ is the Chemical Potential

''k'' is Boltzmann's Constant

''T'' is absolute Temperature

''N'' is the total number of particles

N=\sum_i N_i\,

''Z'' is the Partition Function

Z=\sum_i g_i e^{-\epsilon_i/kT}

e^(...) is the Exponential Function

Equivalently, the distribution is sometimes expressed as

:

rac{N_i}{N} = rac {1} {e^{(\epsilon_i-\mu)/kT}}= rac{e^{-\epsilon_i/kT}}{Z}

where the index ''i'' now specifies an individual microstate rather than the set of all states with energy

\epsilon_i

In this article, the Boltzmann Distribution will be derived using the assumption of distinguishable particles, as is usually done. The ''ad hoc'' correction for correct Boltzmann counting will be noted, as well as the relationship to the Fermi-Dirac and Bose-Einstein statistics.

A Derivation of the Maxwell-Boltzmann distribution

Suppose we have a number of energy levels, labelled by index ''i'' , each level having energy

\epsilon_i

and containing a total of

N_i

particles. To begin with, lets ignore the degeneracy problem. Assume that there is only one way to put

N_i

particles into energy level ''i''.

The number of different ways of selecting one object from

N

objects is obviously

N

. The number of different ways of selecting 2 objects from

N

objects, in a particular order, is thus

N(N-1)

and that of selecting

n

objects in a particular order is seen to be

N!/(N-n)!

. The number of ways of selecting 2 objects from

N

objects without regard to order is

N(N-1)

divided by the number of ways 2 objects can be ordered, which is 2!. It can be seen that the number of ways of selecting

n

objects from

N

objects without regard to order is the binomial coefficient:

N!/n!(N-n)!

. If we have a set of boxes numbered

1,2, \ldots, k

, the number of ways of selecting

N_1

objects from

N

objects and placing them in box 1, then selecting

N_2

objects from the remaining

N-N_1

objects and placing them in box 2 etc. is

:

W=\left(rac{N!}{N_1!(N-N_1)!}
ight)~\left(rac{(N-N_1)!}{N_2!(N-N_1-N_2)!}
ight)~\ldots
\left(rac{N_k!}{N_k!0!}
ight)

=N!\prod_{i=1}^k (1/N_i!)

where the extended product is over all boxes containing one or more objects. If the ''i''-th box has a "degeneracy" of

g_i

, that is, it has

g_i

sub-boxes, such that any way of filling the ''i''-th box where the number in the sub-boxes is changed is a distinct way of filling the box, then the number of ways of filling the ''i''-th box must be increased by the number of ways of distributing the

N_i

objects in the

g_i

boxes. The number of ways of placing

N_i

distinguishable objects in

g_i

boxes is

g_i^{N_i}

. Thus the number of ways (

W

) that

N

atoms can be arranged in energy levels each level

i

having

g_i

distinct states such that the ''i''-th level has

N_i

atoms is:

:

W=N!\prod rac{g_i^{N_i}}{N_i!}

For example, suppose we have three particles,

a

b

, and

c

, and we have three energy levels with degeneracies 1, 2, and 1 respectively. There are 6 ways to arrange the three particles

The six ways are calculated from the formula:

:

W=N!\prod rac{g_i^{N_i}}{N_i!}= 3!
\left(rac{1^2}{2!}
ight)
\left(rac{2^1}{1!}
ight)
\left(rac{1^0}{0!}
ight)=6

We wish to find the set of

N_i

for which

W

is maximised, subject to the constraint that there be a fixed number of particles, and a fixed energy. The maxima of

W

and

\ln(W)

are achieved by the same values of

N_i

and, since it is easier to accomplish mathematically, we will maximise the latter function instead. We constrain our solution using Lagrange Multipliers forming the function:

:

f(N_i)=\ln(W)+\alpha(N-\sum N_i)+\beta(E-\sum N_i \epsilon_i)

Using Stirling's Approximation for the factorials and taking the derivative with respect to

N_i

, and setting the result to zero and solving for

N_i

yields the Maxwell-Boltzmann population numbers:

:

N_i = rac{g_i}{e^{\alpha+\beta \epsilon_i}}

It can be shown thermodynamically that β = 1/''kT'' where

k

is Boltzmann's Constant and ''T'' is the Temperature , and that α = -μ/''kT'' where μ is the Chemical Potential , so that finally:

:

N_i = rac{g_i}{e^{(\epsilon_i-\mu)/kT}}

Note that the above formula is sometimes written:

:

N_i = rac{g_i}{e^{\epsilon_i/kT}/z}

where

z=exp(\mu/kT)

is the absolute Activity .

Alternatively, we may use the fact that

:

\sum_i N_i=N\,

to obtain the population numbers as

:

N_i = Nrac{g_i e^{-\epsilon_i/kT}}{Z}

where

Z

is the Partition Function defined by:

:

Z = \sum_i g_i e^{-\epsilon_i/kT}

CORRECT BOLTZMANN COUNTING

The entropy is equal to

k\ln(W)

where ''k'' is the Boltzmann Constant . When the entropy is calculated using the above expression for ''W'', it will not be extensive, that is, the sum of the entropies of two separate bodies of gas will not be equal to the entropy of both. This is the essence of Gibbs Paradox . The problem is generally solved by introducing "correct Boltzmann counting" in which the number of ways of distributing the energy is reduced by a factor of ''N!'' so that:

:

W=\prod_irac{g_i^{N_i}}{N_i!}

It can be shown that this redefinition of ''W'' will not change the expression for the Boltzmann distribution, but it will change the definition of entropy so that entropy will become an extensive property.

LIMITS OF APPLICABILITY

The Bose-Einstein and Fermi-Dirac distributions may be written:

:

N_i = rac{g_i}{e^{(\epsilon_i-\mu)/kT}\pm 1}

Assuming the minimum value of

\epsilon_i

is small, it can be seen that the conditions under which the Maxwell-Boltzmann distribution is valid is when

:

e^{-\mu/kT} \gg 1

For an Ideal Gas , we can calculate the chemical potential using the development in the Sackur-Tetrode article to show that:

:

\mu=\left(rac{\partial E}{\partial N}
ight)_{S,V}=-kT\ln\left(rac{V}{N\Lambda^3}
ight)

where

E

is the total internal energy,

S

is the Entropy ,

V

is the volume, and

\Lambda

is the Thermal De Broglie Wavelength . The condition for the applicability of the Maxwell Boltzmann distribution for an ideal gas is again shown to be:

:

rac{V}{N\Lambda^3}\gg 1

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Maxwell-boltzmann Statistics