Commit 9ea6982b authored by Tianbai Xiao's avatar Tianbai Xiao
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Update physics.rst


Former-commit-id: b5b4ac61
parent ab99b436
================
Transport Theory
Kinetic Theory
================
The kinetic theory is dedicated to describe the dynamical behavior of a many-particle system through ensemble averaging.
......@@ -27,13 +27,13 @@ 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
.. math::
\mathbf{F} = m \mathbf{a}
As a first order system it reads
.. code:: math
.. math::
\frac{d \mathbf x}{dt} = \mathbf v, \ \frac{d \mathbf v}{dt} = \frac{\mathbf F}{m}
......@@ -57,7 +57,7 @@ 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
.. math::
\frac{\partial f}{\partial t}+ \mathbf v \cdot \nabla_\mathbf x f + \mathbf a \cdot \nabla_\mathbf v f = Q(f)
......@@ -66,7 +66,7 @@ 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
.. math::
Q(f)=\int_{\mathbb R^3} \mathcal B(\mathbf v_*, \mathbf v) \left[ f(\mathbf v_*)-f(\mathbf v)\right] d\mathbf v_*
......@@ -75,7 +75,7 @@ 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
.. 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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