BBGKY QBE

- BBGKY(Bogoliubov-Born-Green-Kirkwood-Yvon) hierarchy

The BBGKY hierarchy is a set of equations that describe the statistical properties of a system of particles. These equations are derived from the Liouville equation, which governs the time evolution of the phase space distribution function for a many-particle system. It provides a way to deal with the complexity of many-body interactions by breaking down the problem into a series of coupled equations for reduced distribution functions.

Let f_n be the n-particle reduced distribution function. The BBGKY hierarchy consists of a series of equations for f_n, each of which is coupled to f_n+1. The first few equations of the hierarchy are as follows:

1. For the single-particle distribution function f_1:

2. For the two-particle distribution function f_2: 

And so on. 

- Quantum Boltzmann Equation

The Quantum Boltzmann Equation(QBE) is a quantum mechanical version of the classical Boltzmann equation. It describes the statistical behavior of a dilute gas of quantum particles. The QBE takes into account quantum mechanical effects such as quantum statistics (Fermi-Dirac or Bose-Einstein statistics) and quantum scattering.

The Quantum Boltzmann Equation can be written as:

where f is the distribution function in phase space, v is the particle velocity, F is the force acting on the particles, and the right-hand side represents the collision term, which is significantly more complex in the quantum case than in the classical case. This term accounts for quantum mechanical scattering processes.

- Relationship

1. The BBGKY Hierarchy is more general and can be applied to dense systems with strong interactions, whereas the Quantum Boltzmann Equation is typically used for dilute gases where interactions can be approximated as binary collisions.

2. Under certain conditions, the first equation of the BBGKY Hierarchy can be reduced to a form resembling the (quantum) Boltzmann equation. This happens in the limit of weak correlations between particles, allowing the system to be described predominantly by the one-particle distribution function.

3. Both frameworks can incorporate quantum effects, but the Quantum Boltzmann Equation inherently includes these from the start, especially in terms of particle statistics and quantum scattering.

저번에 수업에 어떤 수학과(아마 PDE 하시는 분?) 사람이 발표 평가에 재미난거 들고와서 조만간 좀 가볍게 공부해볼듯

Stat mech 솔직히 대충 들어서 복습도 하는 겸에

오랜만에 좀 빡세게 놀아서 내일 출근할 엄두가 안 난다

하지만 난 아직 학부생이니까.. 걍 연구실 가지말까 ㅋㅋ