Author Archives: proteneer

One of the major issues with the stability of the timemachine is whether or not your Hessians are PSD. A non-PSD Hessian induces a transition matrix with a spectral radius , causing a divergent trajectory. Recall Brownian dynamics: Assuming a zero mixed partial, we can differentiate with respect to G and H are the Gradients […]

Brownian Dynamics has the following equations of motion, where we assume that the geometries is dependent on the forcefield parameters We can differentiate the above w.r.t. For the purposes of stability analysis and without loss of generality we can assume: So we can proceed to simplify into: It can be shown that this converges if […]

I encountered a really counter-intuitive result this past week regarding the Hessians of a simple 2D Harmonic Oscillator: It is clear that the energies for and are the same, and the forces are identical in magnitude except opposite in direction. However, while the internal Hessian and therefore positive semi-definite (PSD) every where w.r.t. , this […]

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