Maxwell-Boltzmann statistics
From Free net encyclopedia
Template:Statistics (stat. mech.)
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 kelvins.
The expected number of particles with energy <math>\epsilon_i</math> for Maxwell-Boltzmann statistics is <math>N_i</math> where:
- <math>
\frac{N_i}{N} = \frac {g_i} {e^{(\epsilon_i-\mu)/kT}} = \frac{g_i e^{-\epsilon_i/kT}}{Z} </math>
where:
- <math>N_i</math> is the number of particles in state i
- <math>\epsilon_i</math> is the energy of the i-th state
- <math>g_i</math> is the degeneracy of state i, the number of microstates with energy <math>\epsilon_i</math>
- μ is the chemical potential
- k is Boltzmann's constant
- T is absolute temperature
- N is the total number of particles
- <math>N=\sum_i N_i\,</math>
- Z is the partition function
- <math>Z=\sum_i g_i e^{-\epsilon_i/kT}</math>
- e(...) is the exponential function
Equivalently, the distribution is sometimes expressed as
- <math>
\frac{N_i}{N} = \frac {1} {e^{(\epsilon_i-\mu)/kT}}= \frac{e^{-\epsilon_i/kT}}{Z} </math>
where the index i now specifies an individual microstate rather than the set of all states with energy <math>\epsilon_i</math>
Template:Physics/ParticleDistributions
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.
Contents |
Derivation of the Maxwell-Boltzmann distribution
Suppose we have a number of energy levels, labelled by index i , each level having energy <math>\epsilon_i</math> and containing a total of <math>N_i</math> particles. To begin with, lets ignore the degeneracy problem. Assume that there is only one way to put <math>N_i</math> particles into energy level i.
The number of different ways of selecting one object from <math>N</math> objects is obviously <math>N</math>. The number of different ways of selecting 2 objects from <math>N</math> objects, in a particular order, is thus <math>N(N-1)</math> and that of selecting <math>n</math> objects in a particular order is seen to be <math>N!/(N-n)!</math>. The number of ways of selecting 2 objects from <math>N</math> objects without regard to order is <math>N(N-1)</math> 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 <math>n</math> objects from <math>N</math> objects without regard to order is the binomial coefficient: <math>N!/n!(N-n)!</math>. If we have a set of boxes numbered <math>1,2, \ldots, k</math>, the number of ways of selecting <math>N_1</math> objects from <math>N</math> objects and placing them in box 1, then selecting <math>N_2</math> objects from the remaining <math>N-N_1</math> objects and placing them in box 2 etc. is
- <math>W=\left(\frac{N!}{N_1!(N-N_1)!}\right)~\left(\frac{(N-N_1)!}{N_2!(N-N_1-N_2)!}\right)~\ldots
\left(\frac{N_k!}{N_k!0!}\right)</math>
- <math>=N!\prod_{i=1}^k (1/N_i!)</math>
where the extended product is over all boxes containing one or more objects. If the i-th box has a "degeneracy" of <math>g_i</math>, that is, it has <math>g_i</math> 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 <math>N_i</math> objects in the <math>g_i</math> boxes. The number of ways of placing <math>N_i</math> distinguishable objects in <math>g_i</math> boxes is <math>g_i^{N_i}</math>. Thus the number of ways (<math>W</math>) that <math>N</math> atoms can be arranged in energy levels each level <math>i</math> having <math>g_i</math> distinct states such that the i-th level has <math>N_i</math> atoms is:
- <math>W=N!\prod \frac{g_i^{N_i}}{N_i!}</math>
For example, suppose we have three particles, <math>a</math>, <math>b</math>, and <math>c</math>, and we have three energy levels with degeneracies 1, 2, and 1 respectively. There are 6 ways to arrange the three particles
| . | . | . | . | . | . | |||||||||||
| c | . | . | c | b | . | . | b | a | . | . | a | |||||
| ab | ab | ac | ac | bc | bc | |||||||||||
The six ways are calculated from the formula:
- <math>W=N!\prod \frac{g_i^{N_i}}{N_i!}= 3!
\left(\frac{1^2}{2!}\right) \left(\frac{2^1}{1!}\right) \left(\frac{1^0}{0!}\right)=6 </math>
We wish to find the set of <math>N_i</math> for which <math>W</math> is maximised, subject to the constraint that there be a fixed number of particles, and a fixed energy. The maxima of <math>W</math> and <math>\ln(W)</math> are achieved by the same values of <math>N_i</math> 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:
- <math>
f(N_i)=\ln(W)+\alpha(N-\sum N_i)+\beta(E-\sum N_i \epsilon_i) </math>
Using Stirling's approximation for the factorials and taking the derivative with respect to <math>N_i</math>, and setting the result to zero and solving for <math>N_i</math> yields the Maxwell-Boltzmann population numbers:
- <math>
N_i = \frac{g_i}{e^{\alpha+\beta \epsilon_i}} </math>
It can be shown thermodynamically that β = 1/kT where <math>k</math> is Boltzmann's constant and T is the temperature, and that α = -μ/kT where μ is the chemical potential, so that finally:
- <math>
N_i = \frac{g_i}{e^{(\epsilon_i-\mu)/kT}} </math>
Note that the above formula is sometimes written:
- <math>
N_i = \frac{g_i}{e^{\epsilon_i/kT}/z} </math>
where <math>z=exp(\mu/kT)</math> is the absolute activity.
Alternatively, we may use the fact that
- <math>\sum_i N_i=N\,</math>
to obtain the population numbers as
- <math>
N_i = N\frac{g_i e^{-\epsilon_i/kT}}{Z} </math>
where <math>Z</math> is the partition function defined by:
- <math>
Z = \sum_i g_i e^{-\epsilon_i/kT} </math>
Correct Boltzmann counting
The entropy is equal to <math>k\ln(W)</math> 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:
- <math>W=\prod_i\frac{g_i^{N_i}}{N_i!}</math>
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:
- <math>
N_i = \frac{g_i}{e^{(\epsilon_i-\mu)/kT}\pm 1} </math>
Assuming the minimum value of <math>\epsilon_i</math> is small, it can be seen that the conditions under which the Maxwell-Boltzmann distribution is valid is when
- <math>e^{-\mu/kT} \gg 1</math>
For an ideal gas, we can calculate the chemical potential using the development in the Sackur-Tetrode article to show that:
- <math>\mu=\left(\frac{\partial E}{\partial N}\right)_{S,V}=-kT\ln\left(\frac{V}{N\Lambda^3}\right)</math>
where <math>E</math> is the total internal energy, <math>S</math> is the entropy, <math>V</math> is the volume, and <math>\Lambda</math> 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:
- <math>\frac{V}{N\Lambda^3}\gg 1</math>