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Physics of Long-Range Interacting Systems$

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

(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)
Ω(ε,N)=12πiβiβ+idλeNλεZ(λ,N),

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)
Z(β+iλI,N)={S1,,SN}expβH({Si})expiλIH({Si}).

We see that this expression is proportional to expiλIH, the canonical expectation value of expiλIH, which we expect to be exponentially small for large N. We confirm this expectation rewriting the last expression as

(B.3)
Z(β+iλI,N)=NdεexpNβε+iλIεs(ε),

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)
Z(β+iλI,N)NkckexpNβεk+iλIεks(εk),

(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)
Ω(ε,N)N+12πiβiδβ+iδdλeNλεZ(λ,N),

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