# Relativistic Boltzmann (including QM) equation from entropy?

• ohannuks
In summary, the conversation discusses the derivation of the relativistic Boltzmann equation from entropy in different physical theories, such as classical mechanics and quantum field theory. The assumption \frac{dN}{dt}= 0 is not applicable in the quantum field theory. Recommended readings for this topic include a paper by Hees and references listed therein, as well as a book by Cercignani. However, there is no known derivation for non-equilibrium QFT. A good introduction to this topic can be found in a paper by Cassing.
ohannuks
I'm interested in the derivation of relativistic Boltzmann equation from entropy after reading Scott Dodelson's wonderful cosmology book. Does anyone know of any good readings for this?

The usual way of doing things in classical mechanics is to assume $\frac{dN}{dt}= 0$ and go from there; but Boltzmann eq. can also be derived starting from maximizing entropy.

I wonder if there is any similar derivation for e.g. quantum field theory or just special/general relativistic theory. The assumption $\frac{dN}{dt}= 0$ is not applicable in the quantum field theory.

For the classical derivation of the (special-)relativistic Boltzmann equation, see

http://fias.uni-frankfurt.de/~hees/publ/kolkata.pdf

and the references listed therein (particularly I like Cercignani's book).

I've not manage to write up anything like this using non-equilibrium QFT. For a good introduction, see

W. Cassing From Kadanoff-Baym dynamics to off-shell parton transport, Eur. Phys. J. ST 168 (2009) 3-87
http://dx.doi.org/10.1140/epjst/e2009-00959-x
http://arxiv.org/abs/arXiv:0808.0715

ohannuks
Thank you a lot!

## 1. What is the Relativistic Boltzmann equation?

The Relativistic Boltzmann equation is a fundamental equation in physics that describes the behavior of a gas of particles in a relativistic framework. It is derived from the Boltzmann transport equation, which describes how a gas of particles evolves over time.

## 2. How does the Relativistic Boltzmann equation relate to entropy?

The Relativistic Boltzmann equation includes a term for entropy, which represents the randomness or disorder of a system. This term takes into account the fact that in a relativistic framework, particles can change their energy and momentum through interactions, leading to a change in the overall entropy of the system.

## 3. What is the significance of including quantum mechanics in the Relativistic Boltzmann equation?

Including quantum mechanics in the Relativistic Boltzmann equation allows for a more accurate description of the behavior of particles at small scales, where quantum effects become important. This is particularly relevant in high energy and high density systems, such as those found in particle colliders or in the early universe.

## 4. Are there any limitations to the Relativistic Boltzmann equation?

Like any scientific model, the Relativistic Boltzmann equation has its limitations. It assumes that particles behave as point-like objects and neglects quantum statistical effects. It also does not take into account interactions between particles, which can be important in certain systems.

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

The Relativistic Boltzmann equation is widely used in various fields of physics, including cosmology, astrophysics, and high energy physics. It can be used to study the evolution of the early universe, the behavior of particles in extreme conditions like neutron stars, and the production of particles in particle colliders. It also has applications in other fields, such as plasma physics and semiconductor devices.

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