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fox26
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One version of Liouville’s Theorem for non-dissipative classical systems, governed by a conserved Hamiltonian, is that the volume in phase space (position-momentum space) of an ensemble of such systems (the volume is the Lebesgue measure of the set of points where the ensemble’s density is non-zero) does not vary with time. This is equivalent to the version of the Theorem which states that the density of systems along a flow line does not vary with time.
This so-called Theorem seems to be shown to be false by the following simple counterexample.
Let each system in the ensemble consist of one point or extended spherically symmetric non-rotating classical particle with mass 1 in an otherwise empty 3-dimensional Euclidean space. This system is non-dissipative and governed by a conserved Hamiltonian. The phase space is then a 6-dimensional space with 3 dimensions of space (position) and 3 of (linear) momentum. Let the ensemble in question at time t = 0 consist of such systems distributed with non-zero density d everywhere in a 6-dimensional hypercube with sides of length 1 (3 position, 3 momentum), centered on the origin and with sides parallel to the coordinate axes, and zero elsewhere. The ensemble’s phase space volume will then be 1, and its density d inside the cube and zero elsewhere. At time t = 1 the ensemble will have evolved to have non-zero density everywhere in a 6-dimensional rectilinear parallelepiped with edges still of length 1 in the 3 momentum coordinates, but of length 2 in the 3 position coordinates. The ensemble’s phase space volume will then be 8, and its average density d/8 inside the parallelepiped and zero elsewhere.
It seems that both the ensemble’s phase space volume and its phase space density have varied with time. Comments?
This so-called Theorem seems to be shown to be false by the following simple counterexample.
Let each system in the ensemble consist of one point or extended spherically symmetric non-rotating classical particle with mass 1 in an otherwise empty 3-dimensional Euclidean space. This system is non-dissipative and governed by a conserved Hamiltonian. The phase space is then a 6-dimensional space with 3 dimensions of space (position) and 3 of (linear) momentum. Let the ensemble in question at time t = 0 consist of such systems distributed with non-zero density d everywhere in a 6-dimensional hypercube with sides of length 1 (3 position, 3 momentum), centered on the origin and with sides parallel to the coordinate axes, and zero elsewhere. The ensemble’s phase space volume will then be 1, and its density d inside the cube and zero elsewhere. At time t = 1 the ensemble will have evolved to have non-zero density everywhere in a 6-dimensional rectilinear parallelepiped with edges still of length 1 in the 3 momentum coordinates, but of length 2 in the 3 position coordinates. The ensemble’s phase space volume will then be 8, and its average density d/8 inside the parallelepiped and zero elsewhere.
It seems that both the ensemble’s phase space volume and its phase space density have varied with time. Comments?
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