Boltzmann Equation

The Boltzmann equation shows the statistical behaviour of a thermodynamic system. The Boltzmann equation was given by Ludwig Boltzmann in 1872. In modern literature, the term Boltzmann equation is often in use in a more general sense, referring to any kinetic equation that shows the change of a macroscopic quantity in a thermodynamic system. These quantities can be energy, charge or particle number.

Introduction to Boltzmann Equation

The Boltzmann equation comes up not by analyzing the individual positions and momenta of each particle in the fluid. It is rather considering a probability distribution for the position and momentum of a particle. This means the probability that the particle occupies a given very small region of space. Mathematically the volume of the element \(d^3r\) centred at the position r and has momentum nearly equal to a given momentum vector, at an instant of time.

The Boltzmann equation can be in use to determine how physical quantities change i.e., heat energy and the momentum when a fluid is in transport. The Boltzmann equation is a nonlinear integrodifferential equation, and the unknown function in the equation is a probability density function in the six-dimensional space of a particle position and momentum.

Boltzmann equation is:

\(\frac{P_{S b}}{P_{S a}}=\frac{N_{b}}{N_{a}}=\frac{g b e^{-\frac{E B}{k T}}}{g a^{-\frac{E a}{k T}}}=\frac{g b}{g a} e^{-\frac{(E b-E a)}{k T}}\)

Ludwig Boltzmann

Ludwig Boltzmann was a Physicist and a Philosopher. He made the statistical explanation of the second law of thermodynamics. He contributed to the development of statistical mechanics. Boltzmann Kinetic theory of gas was also one of the significant contributions by him.

The Boltzmann equation is used to study the statistical behaviour of the thermodynamic system which is not in a state of equilibrium.

Applications of Boltzmann Equation

• Conservation equations: Boltzmann equation is in use in the derivation of conservation laws of mass, momentum, charge, and energy that are part of fluid dynamics.
• Hamiltonian mechanics: Classical mechanics was made as Hamiltonian mechanics with the help of different mathematical formulation.
• Quantum theory and violation of particle number: Quantum Boltzmann equation is in use to find out the number of particles that are not conserved in the collisions. These are broadly in use in physical cosmology.
• General relativity and astronomy: Boltzmann equation finds application in galactic dynamics.

FAQ on Boltzmann Equation

Question: Find the temperature at which the number of hydrogen atoms is equal in the ground state when n = 1 and the second excited state is n = 3. Also, the required energy is \(E_3\) = -1.5 eV.

Answer: Given,

Ground state, n = 1

Excited-state, n = 3

The required energy, E₃ = -1.5 eV

Using the Boltzmann equation,

\(N_3/N_1=1=18/2e^{-12.1/kT}\)

\(kT = -(12.1/ln 0.111) = 5.51 eV\)

\(T = 5.51 \times  40 \times  300 = 66,000 K\)

However, we know that for \(N_1 = N_2\), the required temperature is 85,400 K

For this temperature, kT = 7.12 eV

Therefore, \(N_3 = N_1 18/2 e^{-12.1/7.12}\)

\(N_3 = 1.64 N_1\)