8.14: Appendix V- Kramers-Krönig Relations
( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand\Dalpha
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[1], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dbeta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[2], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dgamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[3], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Ddelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[4], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Depsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[5], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dvarepsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[6], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dzeta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[7], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Deta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[8], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dtheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[9], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dvartheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[10], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Diota
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[11], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dkappa
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[12], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dlambda
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[13], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dvarpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[14], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\DGamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[15], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\DDelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\DTheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vmu
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[18], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vnu
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vxi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vom
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vvarpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vrho
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vvarrho
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vsigma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vvarsigma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vtau
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vupsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[29], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vphi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[30], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vvarphi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[31], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vchi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[32], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vpsi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[33], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vomega
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[34], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\VGamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[35], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\VDelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[36], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\newcommand\BI{\mib I}}
\)
\newcommand { M}
\newcommand { m}
}
\( \newcommand\tcb{\textcolor{blue}\)
\( \newcommand\tcr{\textcolor{red}\)
1$#1_$
\newcommand\SZ{\textsf Z}} \( \newcommand\kFd{k\ns_{\RF\dar}\)
\newcommand\mutB{\tilde\mu}\ns_\ssr{B}
\( \newcommand\xhihOZ
Callstack: at (Template:MathJaxArovas), /content/body/div/span[1], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
\( \newcommand\labar
Callstack: at (Template:MathJaxArovas), /content/body/div/span[2], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/08:_Nonequilibrium_Phenomena/8.14:_Appendix_V-_Kramers-Kronig_Relations), /content/body/p[1]/span, line 1, column 23
Suppose ˆχ(ω)≡ˆG(ω) is analytic in the UHP19. Then for all ν, we must have ∞∫−∞dν2πˆχ(ν)ν−ω+iϵ=0 , where ϵ is a positive infinitesimal. The reason is simple: just close the contour in the UHP, assuming ˆχ(ω) vanishes sufficiently rapidly that Jordan’s lemma can be applied. Clearly this is an extremely weak restriction on ˆχ(ω), given the fact that the denominator already causes the integrand to vanish as |ω|−1.
Let us examine the function 1ν−ω+iϵ=ν−ω(ν−ω)2+ϵ2−iϵ(ν−ω)2+ϵ2 . which we have separated into real and imaginary parts. Under an integral sign, the first term, in the limit ϵ→0, is equivalent to taking a principal part of the integral. That is, for any function F(ν) which is regular at ν=ω, limϵ→0∞∫−∞dν2πν−ω(ν−ω)2+ϵ2F(ν)≡℘∞∫−∞dν2πF(ν)ν−ω . The principal part symbol ℘ means that the singularity at ν=ω is elided, either by smoothing out the function 1/(ν−ϵ) as above, or by simply cutting out a region of integration of width ϵ on either side of ν=ω.
The imaginary part is more interesting. Let us write h(u)≡ϵu2+ϵ2 . For |u|≫ϵ, h(u)≃ϵ/u2, which vanishes as ϵ→0. For u=0, h(0)=1/ϵ which diverges as ϵ→0. Thus, h(u) has a huge peak at u=0 and rapidly decays to 0 as one moves off the peak in either direction a distance greater that ϵ. Finally, note that ∞∫−∞duh(u)=π , a result which itself is easy to show using contour integration. Putting it all together, this tells us that limϵ→0ϵu2+ϵ2=πδ(u) . Thus, for positive infinitesimal ϵ, 1u±iϵ=℘u∓iπδ(u) , a most useful result.
We now return to our initial result [kka], and we separate ˆχ(ω) into real and imaginary parts: ˆχ(ω)=ˆχ′(ω)+iˆχ″(ω) . (In this equation, the primes do not indicate differentiation with respect to argument.) We therefore have, for every real value of ω, 0=∞∫−∞dν2π[χ′(ν)+iχ″(ν)][℘ν−ω−iπδ(ν−ω)] . Taking the real and imaginary parts of this equation, we derive the Kramers-Krönig relations: χ′(ω)=+℘∞∫−∞dνπˆχ″(ν)ν−ωχ″(ω)=−℘∞∫−∞dνπˆχ′(ν)ν−ω .