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  "body": "\n## Definition\n\nLet $B \\colon \\R^3 \\times S^2 \\to [0, \\infty)$ be a given function, the *collision kernel*, where $S^2$ is the unit sphere in $\\R^3$.\nFor $v, v_* \\in \\R^3$ and $\\sigma \\in S^2$ put\n$$v' = \\frac{v + v_*}{2} + \\frac{\\abs{v - v_*}}{2} \\sigma, \\qquad v'_* = \\frac{v + v_*}{2} - \\frac{\\abs{v - v_*}}{2} \\sigma.$$\nFor a function $g \\colon \\R^3 \\to \\R$ the *collision operator* is\n$$Q(g, g)(v) = \\int_{\\R^3} \\int_{S^2} B(v - v_*, \\sigma) \\big( g(v') g(v'_*) - g(v) g(v_*) \\big) \\, d\\sigma \\, dv_*.$$\nThe *Boltzmann equation* is the [[partial differential equation|GC5BDZ6QBC]]\n$$\\partial_t f + v \\cdot \\nabla_x f = Q(f, f)$$\nfor an unknown function $f \\colon [0, \\infty) \\times \\R^3 \\times \\R^3 \\to [0, \\infty)$, $(t, x, v) \\mapsto f(t, x, v)$.\nThe operator $Q$ acts on the variable $v$, and $\\nabla_x$ is the [[gradient|0ABK9V9Y64]] in $x$.\n\n## Remarks\n\nThe pair $(v', v'_*)$ runs over the velocities after an elastic collision of two particles with velocities $v$ and $v_*$: one has $v' + v'_* = v + v_*$ and $\\abs{v'}^2 + \\abs{v'_*}^2 = \\abs{v}^2 + \\abs{v_*}^2$.\n\nExample: for $\\rho, T \u003e 0$ and $w \\in \\R^3$, the Maxwellian $M(v) = \\rho (2\\pi T)^{-3/2} e^{-\\abs{v - w}^2/(2T)}$, constant in $t$ and $x$, is a solution.\nThe left side vanishes, and $M(v') M(v'_*) = M(v) M(v_*)$ by the two conservation laws above, so $Q(M, M) = 0$.\n\nNon-example: $f(t, x, v) = t M(v)$ is not a solution, because $\\partial_t f = M \\ne 0$ while $v \\cdot \\nabla_x f = 0$ and $Q(f, f) = t^2 Q(M, M) = 0$.\n\nFor hard spheres the kernel is $B(z, \\sigma) = c \\abs{z}$ for a constant $c \u003e 0$.\n",
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