Boltzmann equation and Hamiltonian

DM) is a term used in physics to refer to hypothetical matter that is thought to account for a large part of the total mass in the universe. In the context of DM, the Boltzmann equation is used to describe the behavior of a system using the Liouville operator ##L[f]## and the collision operator ##C[f]##. While ##L[f]## represents the Liouville operator and takes into account all interactions of the system, including potential energy, ##C[f]## accounts for additional interactions such as self-interactions or interactions with external systems. In the case of DM, the Hamiltonian may only include gravity, with other interactions such as weak force being considered part of the collision operator. The specific breakdown
  • #1
Malamala
299
27
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!
 
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  • #2
Malamala said:
in the context of DM
DM = ... ?
 
  • #3
jtbell said:
DM = ... ?
Dark Matter
 

1. What is the Boltzmann equation?

The Boltzmann equation is a fundamental equation in statistical mechanics that describes the behavior of a gas of particles. It relates the time evolution of a system to the interactions between particles and their surroundings, taking into account factors such as temperature, pressure, and energy.

2. What is the Hamiltonian in physics?

The Hamiltonian is a mathematical function that represents the total energy of a system in classical mechanics. It takes into account the kinetic and potential energy of all the particles in the system and is used to calculate the equations of motion for those particles.

3. How are the Boltzmann equation and Hamiltonian related?

The Boltzmann equation and Hamiltonian are related through a process known as coarse-graining. This involves taking into account the interactions between particles at a macroscopic level, rather than considering each individual particle. The Hamiltonian is used to calculate the macroscopic properties of a system, which are then used in the Boltzmann equation to describe the behavior of the system over time.

4. What is the significance of the Boltzmann equation and Hamiltonian in physics?

The Boltzmann equation and Hamiltonian are important in understanding the behavior of physical systems, particularly in thermodynamics and statistical mechanics. They provide a mathematical framework for predicting the behavior of gases, liquids, and other systems at a macroscopic level, based on the interactions of their individual particles.

5. How is the Boltzmann equation used in practical applications?

The Boltzmann equation is used in a wide range of practical applications, including in the study of atmospheric and oceanic flows, plasma physics, and the behavior of materials. It is also used in engineering and technology, such as in the development of advanced materials and in the design of spacecraft and aircraft. Additionally, the Boltzmann equation has applications in fields such as economics and social sciences, where it can be used to model the behavior of large systems of individuals.

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