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  "body": "\n## Definition\n\nLet $d \\ge 1$, let $T \u003e 0$, and let $F$ assign to each function $f \\colon [0, T) \\times \\R^d \\times \\R^d \\to [0, \\infty)$ a force field $F[f] \\colon [0, T) \\times \\R^d \\to \\R^d$.\nThe *Vlasov equation* with force $F$ is the [[partial differential equation|GC5BDZ6QBC]]\n$$\\partial_t f + v \\cdot \\nabla_x f + F[f](t, x) \\cdot \\nabla_v f = 0.$$\nThe unknown is a density $f(t, x, v) \\ge 0$, and $\\nabla_x$ and $\\nabla_v$ are the [[gradients|0ABK9V9Y64]] in $x$ and in $v$.\nThe main case is the *Vlasov–Poisson system* in $d = 3$, where $\\gamma \\in \\set{1, -1}$ and\n$$F[f](t, x) = \\frac{\\gamma}{4\\pi} \\int_{\\R^3} \\frac{x - y}{\\abs{x - y}^3} \\rho(t, y) \\, dy, \\qquad \\rho(t, y) = \\int_{\\R^3} f(t, y, v) \\, dv.$$\nThe sign $\\gamma = 1$ gives a repulsive force, as for a plasma of one species, and $\\gamma = -1$ gives an attractive force, as for a self-gravitating system.\n\n## Remarks\n\nThe force of the Vlasov–Poisson system is $F = -\\gamma \\nabla_x \\phi$, where $-\\Delta \\phi = \\rho$.\nWith a force that does not depend on $f$, the equation is a linear [[transport equation|PVD7VZJZ9B]] along the trajectories of $\\dot x = v$, $\\dot v = F$.\n\nExample: for $d = 1$ and the external force $F(t, x) = -x$, every function $f(t, x, v) = h(x^2 + v^2)$ with $h$ differentiable is a stationary solution.\nIndeed $v \\, \\partial_x f - x \\, \\partial_v f = 2xv \\, h' - 2xv \\, h' = 0$.\n\nNon-example: with the same force, $f = h(x^2 + 2v^2)$ is not a solution when $h' \\ne 0$, because $v \\, \\partial_x f - x \\, \\partial_v f = -2xv \\, h'(x^2 + 2v^2)$.\n",
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