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Wyatt's Notes — University

Fluctuation-Dissipation Theorem

13.1 Linear Response Theory

The fluctuation-dissipation theorem (FDT) connects the response of a system to a small perturbation with the spontaneous fluctuations of the system at equilibrium.

Consider a Hamiltonian H0\mathcal{H}_0 perturbed by a time-dependent field:

H(t)=H0−f(t)A\mathcal{H}(t) = \mathcal{H}_0 - f(t)A

Where AA is an observable conjugate to the field f(t)f(t). The change in ⟨A(t)⟩\langle A(t) \rangle to first order in ff is:

⟨A(t)⟩−⟨A⟩0=∫−∞tχAA(t−t") f(t′) dt′\langle A(t) \rangle - \langle A \rangle_0 = \int_{-\infty}^{t} \chi_{AA}(t - t")\, f(t')\, dt'

Where the response function is:

χAA(t)=iℏθ(t)⟨[A(t),A(0)]⟩0\chi_{AA}(t) = \frac{i}{\hbar}\theta(t)\langle[A(t), A(0)]\rangle_0

13.2 Classical FDT

In the classical limit, the FDT takes a simpler form. The dynamic susceptibility χ(ω)=χ′(ω)+iχ′′(ω)\chi(\omega) = \chi'(\omega) + i\chi''(\omega) relates to the power spectrum S(ω)S(\omega) of fluctuations:

S(ω)=2kBTω χ′′(ω)S(\omega) = \frac{2k_B T}{\omega}\,\chi''(\omega)

For a harmonic oscillator with damping γ\gamma and natural frequency ω0\omega_0:

χ′′(ω)=γω(ω02−ω2)2+γ2ω2\chi''(\omega) = \frac{\gamma\omega}{(\omega_0^2 - \omega^2)^2 + \gamma^2\omega^2}

The fluctuation spectrum is Lorentzian, peaked at ω0\omega_0.

13.3 Johnson—Nyquist Noise

The FDT predicts thermal (Johnson—Nyquist) noise in a resistor:

⟨V2⟩=4kBTRΔf\langle V^2 \rangle = 4k_B T R \Delta f

Where RR is the resistance and Δf\Delta f is the bandwidth. This noise is fundamental — it arises from thermal fluctuations of charge carriers and cannot be eliminated.

Worked Example 13.1: Johnson--Nyquist Noise Calculation

A 1010 kΩ\Omega resistor at room temperature (T=300T = 300 K) measured with bandwidth Δf=1\Delta f = 1 MHz:

⟨V2⟩=4×1.38×10−23×300×104×106\langle V^2 \rangle = 4 \times 1.38 \times 10^{-23} \times 300 \times 10^4 \times 10^6

=4×1.38×10−23×3×1012= 4 \times 1.38 \times 10^{-23} \times 3 \times 10^{12}

=1.66×10−10 V2= 1.66 \times 10^{-10} \text{ V}^2

Vrms=1.66×10−10≈1.29×10−5 V=12.9 \muVV_{\text{rms} = \sqrt{1.66 \times 10^{-10}} \approx 1.29 \times 10^{-5} \text{ V} = 12.9 \text{ \mu\text{V}}}

This sets a fundamental limit on the sensitivity of electrical measurements.

Worked Example 13.2: Brownian Motion and Einstein Relation

The Einstein relation is a special case of the FDT for Brownian motion. The diffusion constant DD relates to the mobility μ\mu:

D=μkBTD = \mu k_B T

For a spherical particle of radius rr in a fluid with viscosity η\eta:

μ=16πηr(Stokes drag)\mu = \frac{1}{6\pi\eta r} \quad \text{(Stokes drag)}

So D=kBT/(6πηr)D = k_B T/(6\pi\eta r).

For a 11 μ\muM diameter sphere in water (η=10−3\eta = 10^{-3} Pa⋅\cdotS) at T=300T = 300 K:

D=1.38×10−23×3006π×10−3×0.5×10−6=4.14×10−219.42×10−9≈4.39×10−13 m2/sD = \frac{1.38 \times 10^{-23} \times 300}{6\pi \times 10^{-3} \times 0.5 \times 10^{-6}} = \frac{4.14 \times 10^{-21}}{9.42 \times 10^{-9}} \approx 4.39 \times 10^{-13} \text{ m}^2/\text{s}

The mean squared displacement in time tt is ⟨x2⟩=2Dt\langle x^2 \rangle = 2Dt. In 1 second: ⟨x2⟩≈0.94\sqrt{\langle x^2 \rangle} \approx 0.94 μ\muM.