Conservation laws and spin system modeling through principal component analysis

Abstract This paper examines several applications of principal component analysis (PCA) to physical systems. The central result demonstrates that the PCA can identify from the recorded system trajectories conserved quantities that take the form of polynomials in the system variables in an easily programmed and straightforward fashion. In particular, a data record composed of the positions and velocities generated from simulations of two-dimensional harmonic oscillator trajectories is recast into a feature record containing the values of the lowest order atomic polynomials formed from these quantities at each evaluation time. The combinations of the features that are conserved can then be obtained from the principal components of the feature record with the smallest explained variances. Additionally, two features of the application of the PCA to homogeneous periodic spin systems are identified and discussed. The first of these relates to certain characteristic behaviors of the explained variances associated with the principal components of the spin distribution that are found to be artifacts of the boundary geometry. The PCA is then employed to generate synthetic spin realizations with probability distributions in energy-magnetization space that resemble the corresponding input distributions although the associated statistical quantities are not sufficiently accurate.