Lorentz Atom Revisited by Solving the Abraham–Lorentz Equation of Motion

Appendix

Unique Solution of the AL Equation of Motion

The unique solution of (26), which is the general solution of the homogeneous equation plus a particular solution of the inhomogeneous equation, may be cast into the following form (t≥t₀),

where the compact form of the denominator results from applying the first of its identities (33). Here oscillator relaxation and response functions, ϕ(t) and χ(t), and frequency Ω˜ are defined in terms of (Ω, Γ) and are given, respectively, in (7) and (10). The oscillator parameters Ω=Ω(τ, ω₀) and Γ=Γ(τ, ω₀) are given in terms of the AL parameters (τ, ω₀) in (44).

As discussed in Section 3.3 above, (a2) implies that the unique solution of (26) for initial values (x₀, v₀, b₀) will diverge, if (t−t₀)→∞, because the characteristic polynomial of (26) has a positive root, z₂=Γ+1/τ>0. From limt0→−∞xAL(t, t0)=∞, I conclude that a steady-state solution of the AL equation does not exist. A steady-state solution would require that xALh(t, −∞)=0 for generic (x₀, v₀, b₀).

Fourier–Laplace Transform (FLT)

In (13), the Fourier–Laplace transform (FLT) of a bounded function f(t) (|f(t)|≤M<∞) has been introduced

which is an analytical function for all complex z outside the real axis. The FLT of f(t) has as a Cauchy integral representation

with f″(ω)=12i[f˜(ω+io)−f˜(ω−io)] denoting the spectral function, or dissipative part of f˜(ω+io), and f′(ω)=12[f˜(ω+io)+f˜(ω−io)] denoting the reactive part of f˜(ω+io). Dissipative and reactive parts obey dispersion relations

known as Kramers–Kronig relations in physics’ literature.

In general, f′(ω)=−∫−∞∞dω¯πf″(ω¯)ω¯−ω,   and f″(ω)=−∫−∞∞dω¯πf′(ω¯)ω¯−ω, will be complex functions of the real variable ω. Functions f(t), which vanish for large |t| (as is the case for response and relaxation functions discussed above), are related to their spectral function by conventional Fourier transform

and one easily verifies for the response function χ(t) (7), which is purely imaginary, odd in t, and vanishing for |t|→∞

a spectral function which is real, odd in ω, and 1/2 of the conventional Fourier transform of χ(t). Similarly, the relaxation function ϕ(t) (7), which is real, even in t, and vanishing for |t|→∞, will have a spectral function

which is real, even in ω, and just 1/2 of the conventional Fourier transform of ϕ(t). For response and relaxation spectrum, Kubo’s identity takes the simple form: χ″(ω)=ωϕ″(ω)/(mΩ²).