The quantum Boltzmann equation, also known as the Uehling–Uhlenbeck equation, is the quantum mechanical modification of the Boltzmann equation, which gives the nonequilibrium time evolution of a gas of quantum-mechanically interacting particles. Typically, the quantum Boltzmann equation is given as only the "collision term" of the full Boltzmann equation, giving the change of the momentum distribution of a locally homogeneous gas, but not the drift and diffusion in space. It was originally formulated by L.W. Nordheim (1928), and by and E. A. Uehling and George Uhlenbeck (1933). In full generality (including the p-space and x-space drift terms, which are often neglected) the equation is represented analogously to the Boltzmann equation.
[ ∂ ∂ t + v ⋅ ∇ x + F ⋅ ∇ p ] f ( x , p , t ) = Q [ f ] ( x , p ) {\displaystyle \left[{\frac {\partial }{\partial t}}+\mathbf {v} \cdot \nabla _{x}+\mathbf {F} \cdot \nabla _{p}\right]f(\mathbf {x} ,\mathbf {p} ,t)={\mathcal {Q}}[f](\mathbf {x} ,\mathbf {p} )}
where F {\displaystyle \mathbf {F} } represents an externally applied potential acting on the gas' p-space distribution and Q {\displaystyle {\mathcal {Q}}} is the collision operator, accounting for the interactions between the gas particles. The quantum mechanics must be represented in the exact form of Q {\displaystyle {\mathcal {Q}}} , which depends on the physics of the system to be modeled.
Quantum-statistical collision term For a dilute gas of identical particles undergoing binary elastic collisions, the defining difference from the classical Boltzmann equation lies in the collision operator. One common form of the Uehling–Uhlenbeck collision operator is
Q q ( f ) ( v ) = ∫ R d v ∫ S d v − 1 B ( v − v ∗ , ω ) [ f ′ f ∗ ′ ( 1 ± θ 0 f ) ( 1 ± θ 0 f ∗ ) − f f ∗ ( 1 ± θ 0 f ′ ) ( 1 ± θ 0 f ∗ ′ ) ] d ω d v ∗ , {\displaystyle {\mathcal {Q}}_{q}(f)(v)=\int _{\mathbb {R} ^{d_{v}}}\int _{S^{d_{v}-1}}B(v-v_{*},\omega )\left[f'f_{*}'(1\pm \theta _{0}f)(1\pm \theta _{0}f_{*})-ff_{*}(1\pm \theta _{0}f')(1\pm \theta _{0}f_{*}')\right]\,d\omega \,dv_{*},}
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