momentumdiffusion

Momentum diffusion refers to the stochastic changes in a particle’s momentum caused by random forces and collisions, leading to a spreading of the momentum distribution over time. It is the momentum-space counterpart to spatial diffusion and is an important ingredient in kinetic theory, plasmas, and condensed matter physics. The process is typically described statistically by a Fokker-Planck equation for the momentum distribution f(p,t).

The Fokker-Planck equation in momentum space is commonly written as

∂f/∂t = -∂/∂p [ A(p) f ] + ∂^2/∂p^2 [ D(p) f ].

Here A(p) is the drift coefficient, representing systematic forces that change momentum, and D(p) is the diffusion coefficient, quantifying random momentum fluctuations. In a Langevin representation, momentum evolves according to

dp/dt = A(p) + ξ(t),

where ξ(t) is a stochastic force with ⟨ξ(t)⟩ = 0 and ⟨ξ(t)ξ(t′)⟩ = 2D(p) δ(t−t′).

A common special case is linear drag with constant diffusion: A(p) = −γ p and D(p) = D. In many systems, a fluctuation-dissipation relation connects the diffusion and drag terms to temperature, so that at long times the momentum distribution approaches a thermal (Maxwell-Boltzmann) form.

Momentum diffusion appears in various contexts, including Brownian motion in a gas, transport of charged particles in turbulent or fluctuating fields, diffusion of cosmic-ray momenta, and momentum-space dynamics in cold-atom and plasma experiments. It is a foundational concept for understanding how random interactions shape momentum distributions in many-body systems.