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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

epsilon_i

for Maxwell-Boltzmann statistics is

N_i

where:

rac = rac = rac

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

$Z=sum_i g_i e^$

e^(...) is the exponential function

Equivalently, the distribution is sometimes expressed as

rac = rac = rac

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

epsilon_i

    Maxwell–Boltzmann statistics
        A derivation of the Maxwell-Boltzmann distribution
        Another derivation
                Comments
        Limits of applicability
        See also

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A derivation of the Maxwell-Boltzmann distribution

In this particular derivation, the Boltzmann distribution will be derived using the assumption of distinguishable particles, even though the ad hoc correction for Boltzmann counting is ignored, the results remain valid.

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, let's 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 performing an ordered selection of 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

$W=left(rac$

ight)~left(rac ight)~ldots left(rac ight)

$=N!prod_^k (1/N_i!)$

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^

. 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:

$W=N!prod rac$

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 3 particles so that

N_1

= 2,

N_2

= 1 and

N_3

= 0.

The six ways are calculated from the formula:

$W=N!prod rac= 3!$

left(rac ight) left(rac ight) left(rac 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)+eta(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

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

Note that the above formula is sometimes written:

N_i = rac

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 = Nrac

where

Z

is the partition function defined by:

Z = sum_i g_i e^

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Another derivation

In the above discussion, the Boltzmann distribution function was obtained via directly analysing the multiplicities of a system. Alternatively, one can make use of the canonical ensemble. In a canonical ensemble, a system is in thermal contact with a reservoir. While energy is free to flow between the system and the reservoir, the reservoir is thought to have infinitely large heat capacity as to maintain constant temperature, T, for the combined system.

In the present context, our system is assumed to be have energy levels

epsilon _i

with degeneracies

g_i

. As before, we would like to calculate the probability that our system has energy

epsilon_i

.

If our system is in state

; s_1

, then there would be a corresponding number of microstates available to the reservoir. Call this number

; Omega _ R (s_1)

. By assumption, the combined system (of the system we are interested in and the reservoir) is isolated, so all microstates are equally probable. Therefore, for instance, if

Omega _ R (s_1) = 2
Omega _ R (s_2) , we can conclude that our system is twice as likely to be in state $; s_1$ than $; s_2$ . In general, if $; P(s_i)$ is the probability that our system is in state $; s_i$ ,

$rac = rac.$

Since the entropy of the reservoir

; S_R = k ln Omega _R

, the above becomes

$rac = rac = e^.$

Next we recall the thermodynamic identity:

$d S_R = rac (d U_R + P d V_R - mu d N_R)$ .

In a canonical ensemble, there is no exchange of particles, so the

d N_R

term is zero. Similarly,

d V_R = 0

. This gives

$(S_R (s_1) - S_R (s_2) = rac (U_R (s_1) - U_R (s_2)) = - rac (E(s_1) - E(s_2))$

, where

; U_R (s_i)

and

; E(s_i)

denote the energies of the reservoir and the system at

s_i

, respectively. For the second equality we have used the conservation of energy. Substituting into the first equation relating

P(s_1),

P(s_2):

rac = rac

, which implies, for any state s of the system

P(s) = rac e^

, where Z is an appropriately chosen "constant" to make total probability 1. (Z is constant provided that the temperature T is invariant.) It is obvious that

$; Z = sum _s e^$

where the index s run through all microstates of the system. (Z is sometimes called the Boltzmann sum over states.) If we index the summation via the energy eigenvalues instead of all possible states, degeneracy must be taken into account. The probability of our system having energy

epsilon _i

is simply the sum of the probabilities of all corresponding microstates:

$P (epsilon _i) = rac g_i e^$

where, with obvious modification,

Z = sum _j g_j  e^.

This is the same result as before.

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Comments

Notice that in this formulation, the initial assumption "... suppose the system has total N particles..." is dispensed with. Indeed, the number of particles possessed by the system plays no role in arriving at the distribution. Rather, how many particles would occupy states with energy

epsilon _i

follows as an easy consequence.

What has been presented above is essentially a derivation of the canonical partition function. As one can tell by comparing the definitions, the Boltzman sum over states is really no different from the canonical partition function.

Exactly the same approach can be used to derive Fermi-Dirac and Bose-Einstein statistics. However, there one would replace the canonical ensemble with the grand canonical ensemble, since there is exchange of particles between the system and the reservoir. Also, the system one considers in those cases is a single particle state, not a particle. (In the above discussion, we could have assumed our system to be a single atom.)

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Limits of applicability

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

N_i = rac

Assuming the minimum value of

epsilon_i

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

$e^ gg 1$

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

$mu=left(rac$

ight)_=-kTlnleft(rac 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

$racgg 1.$

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