Haoran Sun

Haoran Sun

Chemistry PhD student @UCSD

Posts by Tags

CUDA

Classical mechanics

An interesting property of Hamiltonian system

1 minute read

Published:

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

  1. Arnol’d, V. I. (2013). Mathematical methods of classical mechanics (Vol. 60). Springer Science & Business Media. ↩

Computational chemistry

Differential equations

Liouville's formula

1 minute read

Published:

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

1 minute read

Published:

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

  1. Arnol’d, V. I. (2013). Mathematical methods of classical mechanics (Vol. 60). Springer Science & Business Media. ↩

Jupyter

Mathemacis

Mathematics

Monte Carlo

PDE

Physics

Energy distribution under NVT ensemble

1 minute read

Published:

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.

An interesting property of Hamiltonian system

1 minute read

Published:

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

  1. Arnol’d, V. I. (2013). Mathematical methods of classical mechanics (Vol. 60). Springer Science & Business Media. ↩

Python

Statistical mechanics

Energy distribution under NVT ensemble

1 minute read

Published:

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.