Boltzmann equation and Hamiltonian

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SUMMARY

The discussion centers on the Boltzmann equation, specifically the relationship between the Liouville operator, denoted as $$L[f]$$, and the collision operator $$C[f]$$ in the context of Dark Matter (DM). The Liouville operator represents the time evolution of the phase-space distribution function $$f(x,v,t)$$, which incorporates the Hamiltonian, including potential energy and system interactions. The collision operator accounts for interactions not included in the Hamiltonian, such as self-interactions like DM annihilation, suggesting that not all interactions need to be represented in the Hamiltonian, particularly in gravitational contexts.

PREREQUISITES
  • Understanding of the Boltzmann equation and its components
  • Familiarity with the Liouville operator and its role in statistical mechanics
  • Knowledge of Hamiltonian mechanics and potential energy concepts
  • Basic concepts of Dark Matter and its interactions
NEXT STEPS
  • Study the derivation and applications of the Boltzmann equation in cosmology
  • Explore the role of the Liouville operator in phase-space dynamics
  • Research the implications of self-interactions in Dark Matter models
  • Investigate how different forces, such as gravity and weak force, are modeled in Hamiltonian systems
USEFUL FOR

Physicists, cosmologists, and researchers focusing on Dark Matter interactions and statistical mechanics, particularly those interested in the mathematical modeling of particle dynamics and phase-space distributions.

Malamala
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Hello! I read today, in the context of DM, about the Boltzmann equation: $$L[f]=C[f]$$ where ##L[f]## is the Liouville operator (basically ##\frac{df}{dt}##), with ##f(x,v,t)## being the phase-space distribution of the system and ##C[f]## being the collision operator. I am a bit confused about how should I think about this in general. When you calculate ##L[f]## you use the Hamiltonian, which includes the potential energy, hence all the interactions of the system. Then, what does ##C[f]## account for? Initially I thought that it is used when you have an external system interacting with the original one. But in the context of DM, it seems that it can be a self interaction (DM annihilation for example). So, now I assume that you don't really put all your interactions in the Hamiltonian, but I am not sure. Do you just put the gravity in the case of DM? And treat other interactions (weak force for example) as part of the collision operator? And in general, what you add to the Hamiltonian and what to the collision operator? Thank you!
 
Physics news on Phys.org
Malamala said:
in the context of DM
DM = ... ?
 
jtbell said:
DM = ... ?
Dark Matter
 

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