Fluctuations, Response Functions & Fluctuation-Dissipation
Fluctuations, Response Functions & Fluctuation-Dissipation
Equilibrium fluctuations and linear response to external perturbations are linked by the fluctuation-dissipation theorem. This deep connection explains Brownian motion, Johnson noise, and the Kramers-Kronig relations.
Definition
Linear response: applying a small field \(h\) causes \(\langle m\rangle = \chi h\) where \(\chi\) is the susceptibility. Fluctuation-dissipation: \(\chi = \beta\langle(\delta m)^2\rangle\) — susceptibility equals variance divided by \(k_BT\).
Key Result
Johnson-Nyquist noise: a resistor \(R\) at temperature \(T\) generates voltage noise \(S_V = 4k_BTR\) (V²/Hz). Einstein relation for a Brownian particle: \(D = \mu k_BT\) where \(D\) is diffusivity and \(\mu\) is mobility.
Example 1
Heat capacity from fluctuations: \(C_V = \langle(\delta E)^2\rangle/(k_BT^2)\). For an ideal gas: \(\langle(\delta E)^2\rangle = Nk_BT^2c_v\), confirming \(C_V = Nc_v\) per particle.
Example 2
Kramers-Kronig relations link real and imaginary parts of the complex susceptibility \(\chi(\omega)\) via Hilbert transforms, allowing measurement of dissipation from known response and vice versa.
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Practice
- Derive the Einstein relation from the Langevin equation.
- How does the fluctuation-dissipation theorem apply to electrical circuits?
- What are the Onsager reciprocal relations?
- Explain why large systems have small relative fluctuations.
Show Answer Key
1. Langevin equation: $m\dot{v} = -\gamma v + \eta(t)$ with $\langle\eta(t)\eta(t')\rangle = 2\gamma k_BT\delta(t-t')$. At long times: $\langle x^2\rangle = 2Dt$ with $D = k_BT/\gamma$. This is the Einstein relation connecting diffusion coefficient $D$, friction $\gamma$, and temperature $T$.
2. Johnson-Nyquist noise: voltage fluctuations across a resistor $R$ at temperature $T$: $\langle V^2\rangle = 4k_BTR\Delta f$. The fluctuation (noise voltage) is related to the dissipation (resistance $R$) through $k_BT$. This is a direct application of the fluctuation-dissipation theorem to electrical circuits.
3. For coupled fluxes $J_i$ driven by forces $X_j$: $J_i = \sum_j L_{ij}X_j$. Onsager showed $L_{ij} = L_{ji}$ (from microscopic reversibility / time-reversal symmetry). Example: thermoelectric effects — Seebeck coefficient (heat current from voltage gradient) equals Peltier coefficient (divided by $T$).
4. Relative fluctuation: $\delta X/\langle X\rangle \sim 1/\sqrt{N}$ for an extensive variable $X$ in a system of $N$ particles. For $N \sim 10^{23}$: $\delta X/X \sim 10^{-12}$. This is why thermodynamic quantities (pressure, temperature, energy) have sharply defined values for macroscopic systems — fluctuations are completely negligible.