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11.2: Following Many Systems- a “Gas” in Phase Space

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    We’ve looked at four paths in phase space, corresponding to four falling bodies, all beginning at \(\begin{equation}
    t=0
    \end{equation}\), but with different initial co-ordinates in \(\begin{equation}
    (p, x)
    \end{equation}\) Suppose now we have many falling bodies, so that at t=0 a region of phase space can be imagined as filled with a “gas” of points, each representing one falling body, initially at \(\begin{equation}
    \left(p_{i}, x_{i}\right), \quad i=1, \ldots, N
    \end{equation}\)

    The argument above about the phase space path of a point within the square at t=0 staying inside the square as time goes on and the square distorts to a parallelogram must also be true for any dynamical system, and any closed volume in phase space, since it depends on phase space paths never intersecting: that is,

    if at t = 0 some closed surface in phase space contains a number of points of the gas, those same points remain inside the surface as it develops in time -- none exit or enter.

    For the number of points N sufficiently large, the phase space time development looks like the flow of a fluid.

    This page titled 11.2: Following Many Systems- a “Gas” in Phase Space is shared under a not declared license and was authored, remixed, and/or curated by Michael Fowler.