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Title: A Deal with the Devil: From Divergent Perturbation Theory to an Exponentially-Convergent Self-Consistent Expansion

Abstract: For many nonlinear physical systems, approximate solutions are pursued by conventional perturbation theory in powers of the non-linear terms. Unfortunately, this often produces divergent asymptotic series, collectively dismissed by Abel as "an invention of the devil." An alternative method, the self-consistent expansion, has been introduced by Schwartz and Edwards. Its basic idea is a rescaling of the zeroth-order system around which the solution is expanded, to achieve optimal results. While low-order self-consistent calculations have been remarkably successful in describing the dynamics of non-equilibrium many-body systems (e.g., the Kardar-Parisi-Zhang equation), its convergence properties have not been elucidated before. To address this issue we apply this technique to the canonical partition function of the classical harmonic oscillator with a quartic $gx^{4}$ anharmonicity, for which perturbation theory's divergence is well-known. We explicitly obtain the $N^{\text{th}}$ order self-consistent expansion for the partition function, which is rigorously found to converge exponentially fast in $N$, and uniformly in $g$, for any coupling $g>0$. Comparing the self-consistent expansion with other methods that improve upon perturbation theory (Borel resummation, hyperasymptotics, Pad\'e approximants, and the Lanczos $\tau$-method), it compares favorably with all of them for small $g$ and dominates over them for large $g$. Remarkably, the self-consistent expansion is shown to successfully capture the correct partition function for the double-well potential case, where no perturbative expansion exists. Our treatment is generalized to the case of many oscillators, as well as to a more general nonlinearity of the form $g|x|^{q}$ with $q\ge0$ and complex $g$. These results allow us to treat the Airy function, and to see the fingerprints of Stokes lines in the self-consistent expansion.
 Comments: 37 pages, 11 figures Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech) Cite as: arXiv:1803.06631 [math-ph] (or arXiv:1803.06631v1 [math-ph] for this version)

Submission history

From: Benjamin Remez [view email]
[v1] Sun, 18 Mar 2018 09:39:27 GMT (2390kb,D)