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Energy distribution under NVT ensemble
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Under the NVT ensemble, the probability density of a particular state is given by the famous Boltzmann distribution. Note that \(Q(N,V,T)\) is the partition function, $\mathbf{x}$ denotes the phase space coordinate, $\beta=1/kT$, and $\mathcal{H}(\mathbf{x})$ is the Hamiltonian of the system.
Liouville's formula
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Problem. Prove Liouville's formula \(W=W_0 e^{\int \mathrm{tr}\ A dt}\) for the Wronskian determinant of the linear system \(\dot{\mathbf{x}} = A(t)\mathbf{x}\).
An interesting property of Hamiltonian system
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Problem. Show that in an hamiltonian system it is impossible to have asymptotically stable equilibrium positions and asymptotically stable limit cycles in the phase space.1
Arnol’d, V. I. (2013). Mathematical methods of classical mechanics (Vol. 60). Springer Science & Business Media. ↩
Trajectory click-to-view scheme
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