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How to imagine a classical phase space for N particles?

  1. Oct 30, 2012 #1
    Classically a single particle will have 3 position coordinates and 3 momentum coordinates, and so it "exists" in a 6-dimensional phase space and moves around this space in relation to time (known as the phase trajectory). However I've read that when we have N classical particles, their position/momentum coordinates represent a 6N-dimensional space.

    Is this really a 6N-dimensional space? Or is it a 6 dimensional space (representing the possible momentum/position values possible) with N particles defined within that space each with their own trajectories around it?

    I guess what I mean, is it one of these two situations;

    1) Consider two particles and ignore their momentum for now, only focusing on their x-coordinates of position. Defined by x_1 and x_2 for particles 1 and 2 respectively. Is the phase space here one-dimensional but defined for two particles? As in, do I simply have a co-ordinate axis "x" defined from minus infinity to infinity (a line) and x_1 and x_2 are placed on that coordinate axis appropriately? So if x_1 = 2, x_2 = 100, I place x_1 on 2 and x_2 on 100 on the "x" axis?


    2) The phase space is actually 2 dimensional and with "x_1" and "x_2" coordinate axes. Like an x-y plane. I place x_1 at (2,0) and x_2 at (0,100)? The value for x_1 can only move along the x_1 coordinate axis, and the value for x_2 can only move along the x_2 coordinate axis.
    Last edited: Oct 30, 2012
  2. jcsd
  3. Oct 30, 2012 #2
    It's 2).

    1) is a possible representation of your system but not what is normally called phase space. In phase space every point corresponds to a state of the whole system, and each axis to a degree of freedom. In your example with two particles the system's state is (2,100). It's one point, not two or N.
  4. Oct 30, 2012 #3
    Thanks, your post made me realise the phase space is related to the system, not the individual particles of a system themselves. Hence, as you said, a point in a 6N-dimensional phase space describes the state of a system, and consequently then describes the property of each individual particle in the system. We don't "place" each individual particle in the phase space and at a frozen moment in time observe the positions of each particle (as I assumed in example (1)), and let their individual positions in the phase space be a representation of the state of the system.
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