← Back to Statistical mechanics and kinetic theory

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 content (89 articles)

Full contents

↑ Back to index

1. Foundations (7)

↑ Back to index

2. Conceptual and interpretations (4)

↑ Back to index

3. Mathematical structure and systems (9)

↑ Back to index

4. Atomic and spectroscopy (5)

↑ Back to index

5. Wavefunctions and modes (8)

↑ Back to index

6. Quantum information and computing (6)

↑ Back to index

7. Quantum optics and experiments (4)

Template:Quantum optics operators

↑ Back to index

8. Open quantum systems (7)

↑ Back to index

9. Quantum field theory (18)

↑ Back to index

10. Statistical mechanics and kinetic theory (9)

↑ Back to index

11. Plasma and fusion physics (3)

Physics:Plasma physics (fusion context)
Physics:Tokamak physics
Physics:Tokamak edge physics and recycling asymmetries

Hierarchy of modern physics models showing the progression from quantum statistical mechanics to kinetic theory and plasma physics, culminating in tokamak edge transport and recycling asymmetries.

↑ Back to index

12. Timeline (2)

↑ Back to index

13. Advanced and frontier topics (6)

References

↑ ^1.0 ^1.1 Filbet, F.; Hu, J.; Jin, S. (2012). "A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime". ESAIM: M2AN 46 (2): 443–463. doi:10.1051/m2an/2011051.
↑ Bao, W.; Markowich, P.; Pareschi, L. (2004). "Quantum kinetic theory: Modelling and numerics for Bose–Einstein condensation". Modeling and Computational Methods for Kinetic Equations. pp. 287–320. doi:10.1007/978-0-8176-8200-2_10. ISBN 9781461264873.
↑ Nordheim, L. W. (1928). "On the kinetic method in the new statistics and application in the electron theory of conductivity". Proceedings of the Royal Society A 119 (783): 689–698. doi:10.1098/rspa.1928.0126.
↑ Uehling, E. A.; Uhlenbeck, G. E. (1933). "Transport Phenomena in Einstein–Bose and Fermi–Dirac Gases". Physical Review 43 (7): 552–561. doi:10.1103/PhysRev.43.552.
↑ Snoke, D. W.; Liu, G.; Girvin, S. M. (2012). "The basis of the Second Law of thermodynamics in quantum field theory". Annals of Physics 327 (7): 1825–1851. doi:10.1016/j.aop.2011.12.016.
↑ Snoke, D. W. (2011). "The quantum Boltzmann equation in semiconductor physics". Annalen der Physik 523 (1–2): 87–100. doi:10.1002/andp.201000102.
↑ Snoke, D. W.; Braun, D.; Cardona, M. (1991). "Carrier thermalization in Cu₂O". Physical Review B 44 (7): 2991–3000. doi:10.1103/PhysRevB.44.2991.

Author: Harold Foppele

0.00

(0 votes)

[Filbet-1] 1.0 ^1.1 Filbet, F.; Hu, J.; Jin, S. (2012). "A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime". ESAIM: M2AN 46 (2): 443–463. doi:10.1051/m2an/2011051.

[Bao-2] Bao, W.; Markowich, P.; Pareschi, L. (2004). "Quantum kinetic theory: Modelling and numerics for Bose–Einstein condensation". Modeling and Computational Methods for Kinetic Equations. pp. 287–320. doi:10.1007/978-0-8176-8200-2_10. ISBN 9781461264873.

[Nordheim-3] Nordheim, L. W. (1928). "On the kinetic method in the new statistics and application in the electron theory of conductivity". Proceedings of the Royal Society A 119 (783): 689–698. doi:10.1098/rspa.1928.0126.

[UehlingUhlenbeck-4] Uehling, E. A.; Uhlenbeck, G. E. (1933). "Transport Phenomena in Einstein–Bose and Fermi–Dirac Gases". Physical Review 43 (7): 552–561. doi:10.1103/PhysRev.43.552.

[SnokeArrow-5] Snoke, D. W.; Liu, G.; Girvin, S. M. (2012). "The basis of the Second Law of thermodynamics in quantum field theory". Annals of Physics 327 (7): 1825–1851. doi:10.1016/j.aop.2011.12.016.

[SnokeSemi-6] Snoke, D. W. (2011). "The quantum Boltzmann equation in semiconductor physics". Annalen der Physik 523 (1–2): 87–100. doi:10.1002/andp.201000102.

[SnokeExp-7] Snoke, D. W.; Braun, D.; Cardona, M. (1991). "Carrier thermalization in Cu₂O". Physical Review B 44 (7): 2991–3000. doi:10.1103/PhysRevB.44.2991.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

Anonymous

Search

Physics:Quantum Boltzmann equation

Namespaces

More

Page actions

Contents

General form

Irreversibility and entropy

Applications

Semiconductor model

See also

Table of content (89 articles)

Index

Full contents

References

Navigation

Navigation

Resources

Help

googletranslator

Navigation

Wiki tools

Wiki tools

Anonymous

Search

Physics:Quantum Boltzmann equation

General form

Irreversibility and entropy

Applications

Semiconductor model

See also

Table of content (89 articles)

Index

Full contents

References

Navigation

Wiki tools

Page tools

Other projects

Categories