## A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo

Print publication date: 2014

Print ISBN-13: 9780199581931

Published to Oxford Scholarship Online: October 2014

DOI: 10.1093/acprof:oso/9780199581931.001.0001

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# (p.382) Appendix B Evaluation of the Laplace Integral Outside the Analyticity Strip

Source:
Physics of Long-Range Interacting Systems
Publisher:
Oxford University Press

The microcanonical partition function in formula (2.76) can be expressed using the Laplace representation of the Dirac $δ$ function as

(B.1)
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with $β>0$; this is the expression given in (2.78), with $E=Nε$. As explained in Section 2.5, we divide the integral in (B.1) into three intervals, defined by $λI<−δ,−δ<λI<δ$ and $λI>δ$, respectively, with $0<δ<Δ$. Here we show that the contribution to the integral in λ‎ coming from values of $λI$ outside the strip, i.e. for values of $λI$ with $|λI|>Δ$, is exponentially small in N.

Let us then consider first the value of $Z(λ,N)$ in the two external intervals, i.e. for $|λI|>Δ$. We have

(B.2)
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We see that this expression is proportional to $⟨exp−iλIH⟩$, the canonical expectation value of $exp−iλIH$, which we expect to be exponentially small for large N. We confirm this expectation rewriting the last expression as

(B.3)
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where we have used the expression of the canonical partition in terms of the microcanonical entropy as in Eq. (1.64). This is an integral with a large phase. For N going to infinity, its value will be determined by the value of the integrand for $ε$ equal to the integration extremes and to the values of the possible nonanalyticities of $s(ε)$, all denoted by $εk$ (see, e.g. (Bender and Orszag 1978)). We then have

(B.4)
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(p.383) where ck are coefficients that could in principle be determined. It is clear that the successive integration over $λI$ in any one of the two external intervals of integration will then give a vanishing contribution, due to the very large oscillations. We are then left with

(B.5)
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which is the first equality given in Eq. (2.79).