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Quantum Boltzmann equation, also known as the Uehling–Uhlenbeck equation, is the quantum-mechanical generalization of the classical Boltzmann equation. It describes the nonequilibrium time evolution of the distribution function of a gas of quantum particles, incorporating Fermi–Dirac or Bose–Einstein statistics.^[1]^[2]

It was originally formulated by L. W. Nordheim (1928)^[3] and later developed by E. A. Uehling and G. E. Uhlenbeck (1933).^[4]

General form

In full generality, including drift and external forces, the equation takes the form

$[\frac{\partial}{\partial t} + 𝐯 \cdot \nabla_{x} + 𝐅 \cdot \nabla_{p}] f (𝐱, 𝐩, t) = 𝒬 [f] (𝐱, 𝐩),$

where:

$f (𝐱, 𝐩, t)$ is the distribution function
$𝐅$ is an external force
$𝒬 [f]$ is the collision operator

The quantum nature of the system is entirely encoded in the structure of the collision term $𝒬$ , which incorporates quantum statistical effects such as Pauli blocking and Bose enhancement.^[1]

In many applications, only the collision term is retained, corresponding to a spatially homogeneous system.

Irreversibility and entropy

The quantum Boltzmann equation exhibits irreversible behavior and defines an arrow of time. Despite the underlying time-reversibility of quantum mechanics, irreversibility emerges because phase information is discarded and only occupation numbers are retained.^[5]

As a result, the system evolves toward an equilibrium distribution. The equation is valid on time scales short compared to the Poincaré recurrence time, which is typically extremely large.

Applications

The quantum Boltzmann equation is widely used in semiconductor physics and quantum kinetic theory.^[6]

Experimental studies have verified its predictions, for example in the relaxation of exciton gases toward equilibrium distributions measured with time-resolved techniques.^[7]

Semiconductor model

In semiconductor applications, the equation is often simplified under the assumptions:

spatial homogeneity
negligible external forces
dilute particle gas

Under these conditions, the collision operator can be written explicitly. For electron–electron scattering with momentum transfer $𝐪$ , one obtains

$\begin{aligned} 𝒬 [f] (𝐤) & = \frac{- 2}{ℏ (2 π)^{5}} \int d 𝐪 \int d k_{𝟏} | \hat{v} (𝐪) |^{2} \\ \times δ (\frac{ℏ^{2}}{2 m} (| 𝐤 - 𝐪 |^{2} + | k_{𝟏} + 𝐪 |^{2} - {k_{𝟏}}^{2} - 𝐤^{2})) \\ \times [f_{𝐤} f_{k_{𝟏}} (1 - f_{𝐤 - 𝐪}) (1 - f_{k_{𝟏} + 𝐪}) - f_{𝐤 - 𝐪} f_{k_{𝟏} + 𝐪} (1 - f_{𝐤}) (1 - f_{k_{𝟏}})] \end{aligned}$

which explicitly shows the role of quantum statistics via the factors $(1 - f)$ .

Table of contents (138 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

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Physics:Quantum Boltzmann equation

General form

Irreversibility and entropy

Applications

Semiconductor model

See also

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