Discrete distribution

Maxwell—Boltzmann distribution can be written as a discrete distribution of a single particle’s discrete energy spectrum. Maxwell-Boltzmann statistics gives the average number of particles found in a given single-particle microstate.

Notation:

1. \(k_\text{B}\) (k_B) is boltzmann_constant.

Conditions:

1. Particles do not interact and are classical.

2. The system is in thermal equilibrium.

Links:

1. Wikipedia.

occupancy_of_state

Occupancy of, or expected number of particles in, the single-particle microstate \(i\).

Symbol:

N_i

Latex:

\(N_{i}\)

Dimension:

dimensionless

particle_count

Total particle_count of the system.

Symbol:

N

Latex:

\(N\)

Dimension:

dimensionless

energy_of_state

energy of single-particle microstate \(i\).

Symbol:

E_i

Latex:

\(E_{i}\)

Dimension:

energy

equilibrium_temperature

temperature of the system.

Symbol:

T

Latex:

\(T\)

Dimension:

temperature

single_particle_partition_function

Single-particle partition_function, which acts as a normalizing factor of the distribution.

Symbol:

Z

Latex:

\(Z\)

Dimension:

dimensionless

law

N_i = N / Z * exp(-E_i / (k_B * T))

Latex:

\[N_{i} = \frac{N}{Z} \exp{\left(- \frac{E_{i}}{k_\text{B} T} \right)}\]