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Wikipedia / Maxwell–Boltzmann statistics

In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the particle density is low enough to render quantum effects negligible.

The expected number of particles where:

where the index i.

History

Maxwell–Boltzmann statistics grew out of the Maxwell–Boltzmann distribution, most likely as a distillation of the underlying technique. The distribution was first derived by Maxwell in 1860 on heuristic grounds. Boltzmann later, in the 1870s, carried out significant investigations into the physical origins of this distribution. The distribution can be derived on the ground that it maximizes the entropy of the system.

Relation with Maxwell–Boltzmann Distribution

Maxwell–Boltzmann distribution where:

  • ,

  • ,

  • is the Boltzmann constant,

  • is the absolute temperature.

where:

  • is the probability density function of particle speeds,

  • is the mass of a particle,

  • is the Boltzmann constant,

  • is the absolute temperature,

  • is the speed of the particle.

Derivation

Applicability

Maxwell–Boltzmann statistics is used to derive the Maxwell–Boltzmann distribution of an ideal gas. However, it can also be used to extend that distribution to particles with a different energy–momentum relation, such as relativistic particles (resulting in Maxwell–Jüttner distribution), and to other than three-dimensional spaces.

Maxwell–Boltzmann statistics is often described as the statistics of "distinguishable" classical particles. In other words, the configuration of particle A in state 1 and particle B in state 2 is different from the case in which particle B is in state 1 and particle A is in state 2. This assumption leads to the proper (Boltzmann) statistics of particles in the energy states, but yields non-physical results for the entropy, as embodied in the Gibbs paradox.

At the same time, there are no real particles that have the characteristics required by Maxwell–Boltzmann statistics. Indeed, the Gibbs paradox is resolved if we treat all particles of a certain type (e.g., electrons, protons, etc.) as principally indistinguishable. Once this assumption is made, the particle statistics change. The change in entropy in the entropy of mixing example may be viewed as an example of a non-extensive entropy resulting from the distinguishability of the two types of particles being mixed.

Quantum particles are either bosons (following Bose–Einstein statistics) or fermions (subject to the Pauli exclusion principle, following instead Fermi–Dirac statistics). Both of these quantum statistics approach the Maxwell–Boltzmann statistics in the limit of high temperature and low particle density.

Derivations

Maxwell–Boltzmann statistics can be derived in various statistical mechanical thermodynamic ensembles:

In each case it is necessary to assume that the particles are non-interacting, and that multiple particles can occupy the same state and do so independently.

Derivation from microcanonical ensemble

Suppose we have a container with a huge number of very small particles all with identical physical characteristics (such as mass, charge, etc.). Let's refer to this as the system. Assume that though the particles have identical properties, they are distinguishable. For example, we might identify each particle by continually observing their trajectories, or by placing a marking on each one, e.g., drawing a different number on each one as is done with lottery balls.

The particles are moving inside that container in all directions with great speed. Because the particles are speeding around, they possess some energy. The Maxwell–Boltzmann distribution is a mathematical function that describes about how many particles in the container have a certain energy. More precisely, the Maxwell–Boltzmann distribution gives the non-normalized probability (this means that the probabilities do not add up to 1) that the state corresponding to a particular energy is occupied.

is the occupation number of the energy level, then we know the total energy of the system. However, because we can distinguish between which does not completely describe the state of the system. To completely describe the state of the system, or the microstate, we must specify exactly which particles are in each energy level. Thus when we count the number of possible states of the system, we must count each and every microstate, and not just the possible sets of occupation numbers.

. With the concept of combination

This is just the multinomial coefficient, the number of ways of arranging N items into k boxes, the lth box holding Nl items, ignoring the permutation of items in each box.

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 distinct states such that the i

This is the form for W first derived by Boltzmann relates the thermodynamic entropy S to the number of microstates W, where kB is the Boltzmann constant. It was pointed out by Gibbs however, that the above expression for W does not yield an extensive entropy, and is therefore faulty. This problem is known as the Gibbs paradox. The problem is that the particles considered by the above equation are not indistinguishable. In other words, for two particles (A and B) in two energy sublevels the population represented by [A, B] is considered distinct from the population [B, A] while for indistinguishable particles, they are not. If we carry out the argument for indistinguishable particles, we are led to the Bose–Einstein expression for W

. Under these conditions, we may use Stirling's approximation

This is essentially a division by N! of Boltzmann's original expression for W, and this correction is referred to as .

and, since it is easier to accomplish mathematically, we will maximize the latter function instead. We constrain our solution using Lagrange multipliers

In order to maximize the expression above we apply Fermat's theorem (stationary points)

Boltzmann realized that this is just an expression of the Euler-integrated fundamental equation of thermodynamics. Identifying E as the internal energy where T is the temperature, P is pressure, V is volume, and μ is the chemical potential with the constant of proportionality being the Boltzmann constant. Using the ideal gas equation of state (PV=NkBT

is the absolute activity.

where Z is the partition function

In an approximation where εi is considered to be a continuous variable, the Thomas–Fermi approximation yields a continuous degeneracy g which is just the Maxwell–Boltzmann distribution for the energy.

Derivation from canonical ensemble

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.

.

Since the entropy

Next we recall the thermodynamic identity (from the first law of thermodynamics and second law of thermodynamics

which implies, for any state s where Z is an appropriately chosen "constant" to make total probability 1. (Z is constant provided that the temperature T where the index s runs through all microstates of the system. Z is sometimes called the Boltzmann sum over states this is the same result as before.

Comments on this derivation:

  • Notice that in this formulation, the initial assumption "...suppose the system has total N particles follows as an easy consequence.

  • What has been presented above is essentially a derivation of the canonical partition function. As one can see by comparing the definitions, the Boltzmann sum over states is equal to 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.)

Derivation from canonical ensemble

The Maxwell-Boltzmann distribution describes the probability of a particle occupying an energy state E

For a system of indistinguishable particles, we start with the canonical ensemble formalism.

be the number of particles in state i

accounts for the number of ways to distribute N indistinguishable particles among the states.

.

for small x

This is the derivation of the Maxwell–Boltzmann distribution in both energy regimes.

See also

Notes

References

Bibliography

  • Carter, Ashley H., "Classical and Statistical Thermodynamics", Prentice–Hall, Inc., 2001, New Jersey.

  • Raj Pathria, "Statistical Mechanics", Butterworth–Heinemann, 1996.