Maxwell–Boltzmann statistics

It is applicable when the temperature is high enough or the particle density is low enough to render quantum effects negligible.

Maxwell–Boltzmann statistics grew out of the Maxwell–Boltzmann distribution, most likely as a distillation of the underlying technique.

[dubious – discuss] 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.

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.

Indeed, the Gibbs paradox is resolved if we treat all particles of a certain type (e.g., electrons, protons,photon etc.)

Maxwell–Boltzmann statistics can be derived in various statistical mechanical thermodynamic ensembles:[1] 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.

Suppose we have a container with a huge number of very small particles all with identical physical characteristics (such as mass, charge, etc.).

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.

The Maxwell–Boltzmann distribution is a mathematical function that describes about how many particles in the container have a certain energy.

However, because we can distinguish between which particles are occupying each energy level, the set of occupation numbers

To completely describe the state of the system, or the microstate, we must specify exactly which particles are in each energy level.

What follows next is a bit of combinatorial thinking which has little to do in accurately describing the reservoir of particles.

relates the thermodynamic entropy S to the number of microstates W, where k 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.

Under these conditions, we may use Stirling's approximation for the factorial: to write: Using the fact that

We constrain our solution using Lagrange multipliers forming the function: Finally In order to maximize the expression above we apply Fermat's theorem (stationary points), according to which local extrema, if exist, must be at critical points (partial derivatives vanish): By solving the equations above (

yields: or, rearranging: Boltzmann realized that this is just an expression of the Euler-integrated fundamental equation of thermodynamics.

Identifying E as the internal energy, the Euler-integrated fundamental equation states that : where T is the temperature, P is pressure, V is volume, and μ is the chemical potential.

Using the ideal gas equation of state (PV = NkT), It follows immediately that

Alternatively, we may use the fact that to obtain the population numbers as where Z is the partition function defined by: In an approximation where εi is considered to be a continuous variable, the Thomas–Fermi approximation yields a continuous degeneracy g proportional to

In the above discussion, the Boltzmann distribution function was obtained via directly analysing the multiplicities of a system.

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.

: which implies, for any state s of the system where Z is an appropriately chosen "constant" to make total probability 1.

Z is sometimes called the Boltzmann sum over states (or "Zustandssumme" in the original German).

If we index the summation via the energy eigenvalues instead of all possible states, degeneracy must be taken into account.

is simply the sum of the probabilities of all corresponding microstates: where, with obvious modification, this is the same result as before.

Maxwell–Boltzmann statistics can be used to derive the Maxwell–Boltzmann distribution of particle speeds in an ideal gas . Shown: distribution of speeds for 10 6 oxygen molecules at -100, 20, and 600 °C.
Equilibrium thermal distributions for particles with integer spin (bosons), half integer spin (fermions), and classical (spinless) particles. Average occupancy is shown versus energy relative to the system chemical potential , where is the system temperature, and is the Boltzmann constant.