Boltzmann equation (kinetic theory)

Nonlinear integro-differential equation for the one-particle distribution in a dilute gas; collision operator encodes binary interactions and yields the H-theorem.

Boltzmann equation (kinetic theory)

The Boltzmann equation describes the evolution of a dilute gas at the level of the one-particle distribution, bridging microscopic dynamics and macroscopic thermodynamics.

Prerequisites: thermodynamic entropy , temperature , canonical ensemble , Boltzmann H-theorem , H-functional .

Unknown and physical meaning

The unknown is the one-particle distribution $f(t,x,v)\ge 0$ on time $t\ge 0$ , position $x\in\mathbb{R}^3$ (or a box), and velocity $v\in\mathbb{R}^3$ . Macroscopic fields are velocity moments:

\rho(t,x)=\int_{\mathbb{R}^3} f(t,x,v)\,dv, \qquad \rho u(t,x)=\int_{\mathbb{R}^3} v f(t,x,v)\,dv,

and (kinetic) energy density

\mathcal{E}(t,x)=\int_{\mathbb{R}^3} \frac{|v|^2}{2}\,f(t,x,v)\,dv.

Boltzmann equation (with external force)

The kinetic equation is

\partial_t f + v\cdot\nabla_x f + \frac{F}{m}\cdot\nabla_v f = Q(f,f),

where $F$ is an external force, $m$ the particle mass, and $Q(f,f)$ is the collision operator.

Collision operator (standard binary-collision form)

A common form is

Q(f,f)(v)=\int_{\mathbb{R}^3}\int_{\mathbb{S}^2} B(|v-v_*|,\cos\theta)\,\Big(f(v')f(v_*')-f(v)f(v_*)\Big)\,d\omega\,dv_*,

where $v_*$ is the collision partner velocity, $\omega\in\mathbb{S}^2$ parametrizes scattering, and $(v',v_*')$ are the post-collisional velocities determined by conservation of momentum and energy in a binary collision. The kernel $B$ encodes the interaction model (hard spheres, Maxwell molecules, etc.).

Collision invariants and conservation laws

For collision invariants $\phi(v)\in\{1, v, |v|^2\}$ one has

\int_{\mathbb{R}^3} Q(f,f)(v)\,\phi(v)\,dv = 0,

which yields local balance laws for mass, momentum, and energy (up to transport terms).

Maxwellian equilibria

Spatially homogeneous equilibria are Maxwellians:

M(v)=\rho\left(\frac{m}{2\pi k T}\right)^{3/2}\exp\!\left(-\frac{m|v-u|^2}{2kT}\right),

parameterized by density $\rho$ , bulk velocity $u$ , and temperature $T$ . This connects to equilibrium ideas such as the canonical ensemble .

H-theorem (entropy monotonicity)

Define the Boltzmann H-functional

H[f]=\int f\log f\,dv\,dx.

Along sufficiently regular solutions (with appropriate boundary/decay conditions),

\frac{d}{dt}H[f(t)]\le 0,

with equality only at (local) Maxwellians. This is the kinetic-theory route to entropy increase and is developed in Boltzmann H-theorem and H-functional .

Modeling assumptions (context)

The equation is derived for a dilute gas under a molecular-chaos assumption (factorization of pre-collisional correlations). It is therefore an effective description rather than an exact rewriting of microscopic Hamiltonian dynamics.