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Collision Theory

The theory explaining how particles interact during collisions, influencing momentum and energy transfer within a system.
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The statement of the theorem

Let f(r,v,t)f(\mathbf{r}, \mathbf{v}, t) be the single-particle distribution function in phase space Γ=R3×R3\Gamma = \mathbb{R}^3 \times \mathbb{R}^3. The evolution of ff is governed by the Boltzmann equation: ∂f∂t+v⋅∇rf+Fm⋅∇vf=(∂f∂t)coll \frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{r}} f + \frac{\mathbf{F}}{m} \cdot \nabla_{\mathbf{v}} f = \left(\frac{\partial f}{\partial t}\right)_{coll} where F\mathbf{F} is the external force. The collision integral, (∂f∂t)coll\left(\frac{\partial f}{\partial t}\right)_{coll}, quantifies the rate of change due to particle interactions and is defined as: (∂f∂t)coll=∫v′∫Ωσ(g,g′,v,v′)[f′f′−ff′]⋅g⋅dgdv′ \left(\frac{\partial f}{\partial t}\right)_{coll} = \int_{\mathbf{v}'} \int_{\Omega} \sigma(\mathbf{g}, \mathbf{g}', \mathbf{v}, \mathbf{v}') \left[ f' f' - f f' \right] \cdot \mathbf{g} \cdot d\mathbf{g} d\mathbf{v}' Here, g=v−v′\mathbf{g} = \mathbf{v} - \mathbf{v}' is the relative velocity, σ(g,g′,v,v′)\sigma(\mathbf{g}, \mathbf{g}', \mathbf{v}, \mathbf{v}') is the differential cross-section for collision between particles with relative velocities g\mathbf{g} and g′\mathbf{g}', and dgdv′d\mathbf{g} d\mathbf{v}' represents the integration over all possible outgoing velocities and incoming velocities, respectively. The theory postulates that the collision frequency ν\nu is proportional to the integral of the cross-section over the relative velocity distribution.