A Energy hypersurface in a phase space (statistical physics)

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Energy hypersurfaces in phase space for systems with constant energy are closed to maintain a clear boundary between higher and lower energy regions. If these surfaces were not closed, it would result in the absence of energy contours, gradients, and forces, disrupting the fundamental principles of statistical physics. The closed nature of these hypersurfaces ensures that energy levels are well-defined and allows for the proper description of thermodynamic behavior. This closure is essential for the stability of systems and the predictability of their dynamics. Understanding this concept is crucial for analyzing energy distributions and transitions in statistical mechanics.
daniel444
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what is the reason for that the energy hypersurfaces in a phase space, which belong to systems with constant energy are closed? ( see picture )
237D2118-2CA1-4598-8FEF-7CF73AA8F22C.jpeg
 
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If not closed, there would be no border between higher energy region and lower energy region. No energy contour. No grad. No force.
 

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