Abstract
In this chapter, we lay the foundation of the kinetic theory approach to study of interacting particle systems. We start with defining microstates, macrostates, and the phase space (the \(\Gamma \)-space) of interacting particle systems involving \(N \gg 1\) particles, and then discuss two broad classes of inter-particle interactions: Short and long-range interactions. In a suitably-defined thermodynamic limit, we then introduce the notion of N-particle distribution function in the \(\Gamma \) space and the single-particle distribution function in the single-particle phase space (the \(\mu \)-space). For the evolution of the N-particle distribution function, we discuss in turn the Liouville equation and the BBGKY hierarchy, and obtain the evolution of the single-particle distribution function by suitably closing the hierarchy. For short-range systems, we obtain the corresponding Boltzmann equation. The derivation of the corresponding equation for long-range systems, which is known as the Vlasov equation, is then discussed in detail. As an illustration of the techniques developed, we then obtain for a model of interacting classical Heisenberg spins the corresponding Vlasov equation, via the aforementioned route involving the BBGKY hierarchy and also via an alternative Klimontovich approach.
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Notes
- 1.
This is in analogy with the convention adopted in studying fluid flows, whereby the Lagrangian characterization of the flow field corresponds to monitoring of fluid motion in terms of motion of an individual fluid ‘packet’ as it moves through space and time. The Eulerian characterization instead corresponds to monitoring as a function of time the motion at specific locations of the space through which the fluid flows. A parallel may be drawn between the Lagrangian viewpoint and the viewpoint of an observer on a boat that is drifting down a river, while the Eulerian viewpoint corresponds to an observer stationed on the bank of the river and monitoring the river flowing by.
- 2.
The hierarchy was proposed independently by Nikolay Bogolyubov, Max Born, Herbert S. Green, John Gamble Kirkwood, and Jacques Yvon. Therefore, it is named as BBGKY hierarchy.
- 3.
For a system of N particles, each with mass \(m_i\) and position vector \(\boldsymbol{q}_i\);Â \(1\le i\le N\), the position vector of the center of mass is given by \(\boldsymbol{Q} = (\,\sum ^{N}_{i=1} m_i{\boldsymbol{q}_i} \,)/ (\,\sum ^{N}_{i=1} m_i\,)\) . Here, we have two particles with the same mass m, and hence, \(\boldsymbol{Q} = (\boldsymbol{q}_1+\boldsymbol{q}_2 )/2 \,\).
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Das, D., Gupta, S. (2023). Kinetic Theory. In: Facets of Noise. Fundamental Theories of Physics, vol 214. Springer, Cham. https://doi.org/10.1007/978-3-031-45312-0_3
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