Commit ab99b436 by Tianbai Xiao

Update physics.rst


Former-commit-id: b038b0f0
parent 8eaec665
 ... ... @@ -21,3 +21,61 @@ In the Kit-RT solver, we consider the liner Boltzmann equation where the particles don't interact with one another but scatter with the background material. For convenience, we reformulate the particle velocity into polar coordinates. The physical world shows a diverse set of behaviors on different characteristic scales. Consider the molecular motion of gases as an example. Down to the finest scale of a many-particle system, the Newton’s second law depicts particle motions via .. code:: math \mathbf{F} = m \mathbf{a} As a first order system it reads .. code:: math \frac{d \mathbf x}{dt} = \mathbf v, \ \frac{d \mathbf v}{dt} = \frac{\mathbf F}{m} An intuitive numerical algorithm is to get the numerous particles on board and track the trajectories of them. A typical example is the Molecular Dynamics_. This is not going to be efficient since there are more than 2e25 molecules per cubic meter in normal atmosphere, and things get even more complicated when you count on the N-body interactions all the time. Some methods have been proposed to simplify the computation. As an example, the Direct simulation Monte Carlo_ employs certain molecular models and conduct the intermolecular collisions in a stochastic manner. It significantly reduces the computational cost, while the trade-off is the artificial fluctuations. Many realizations must be simulated successively to average the solutions and reduce the errors. An alternative strategy is made from ensemble averaging, where the coarse-grained modeling is used to provide a bottom-up view. At the mean free path and collision time scale of molecules, particles travel freely during most of time with mild intermolecular collisions. Such dynamics can be described with an operator splitting approach, i.e. the kinetic transport equation .. code:: math \frac{\partial f}{\partial t}+ \mathbf v \cdot \nabla_\mathbf x f + \mathbf a \cdot \nabla_\mathbf v f = Q(f) where the left and right hand sides model particle transports and collisions correspondingly. Different collision models can be inserted into such equation. If the particles only collide with a background material one obtains linear Boltzmann collision operator .. code:: math Q(f)=\int_{\mathbb R^3} \mathcal B(\mathbf v_*, \mathbf v) \left[ f(\mathbf v_*)-f(\mathbf v)\right] d\mathbf v_* where the collision kernel \mathcal B models the strength of collisions at different velocities. If the interactions among particles are considered, the collision operator becomes nonlinear. For example, the two-body collision results in nonlinear Boltzmann equation .. code:: math Q(f)=\int_{\mathbb R^3} \int_{\mathcal S^2} \mathcal B(\cos \beta, |\mathbf{v}-\mathbf{v_*}|) \left[ f(\mathbf v')f(\mathbf v_*')-f(\mathbf v)f(\mathbf v_*)\right] d\mathbf \Omega d\mathbf v_*
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