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A concise derivation of the Wiener-Khinchin theorem

Introduction

While teaching a class on statistical physics, I found myself unhappy with textbook derivations of the Wiener-Khinchin theorem. I worked my way to a very short derivation that is free of integral bounds manipulations and of holes, or so I believe.

In either the books by Balakrishnan (Elements of Nonequilibrium Statistical Mechanics, Ane Books, 2008), Risken (The Fokker-Planck Equation, 2nd edition, Springer-Verlag, 1989), Coffey-Kalmykov-Waldron (The Langevin Equation, 2nd edition, World Scientific, 2004) or MathWorld, I could not find a short derivation that would cleanly take into account the averaging procedure or that would not resort to splittings of the domain of integration. I present here one derivation and a numerical illustration with Python.

{% notebook 2017/wiener_khinchine.ipynb %}

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