Year: 1973

Authors: G.allavotti, C. Marchioro

Title On the calculation of an integral: Journal of Mathematical Analysis and Applications, 44, 661--675, 1973,

Abstract: The integral over $R^n$ $$I_n(g,\omega)=\int dq_1\ldots dq_n \exp\{-\frac12\sum_{i\ne j}^{1,n} \frac{g^2}{(q_i-q_j)^2}-\frac12\sum_{i=1}^n \omega^2 q_i^2\}$$ is exactly computed and shown to be $$I_n(g,\omega)=I_n(0,\omega) \exp\{-\omega g \frac{n(n-1)}2\}$$ In Sec. 5 we conjecture that the related Hamiltonian mechanical system is exactly integral and propose the form it would take in action-angle variables.

A Follow Up: The conjecture in Sec.5 has been proved partially (integrability) in "J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations Advances in Mathematics, 16:197--220, 1975", and completed (action-angle variables found) in "J.P. Francoise. Canonical partition functions of Hamiltonian systems and the stationary phase formula, Communications in Mathematical Physics, 117: 37--47, 1988".

Key words:  Statistical Mechanics, Path Integral, Brwonian Motion, FEynman-Kac Formula

Giovanni Gallavotti
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