Problem Statement
State the fluctuation-dissipation theorem and explain its physical content.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: Statement: The spontaneous fluctuations of a system in thermal equilibrium are related to the system’s response (dissipation) to an external perturbation by the same mechanism.
Step 2 — Apply the relevant physical law or equation: For a linear response function $\chi”(\omega)$ (imaginary part, characterising dissipation), the power spectral density of fluctuations of the conjugate variable is:
Step 3 — Solve algebraically for the unknown: $$S(\omega) = \frac{2k_BT}{\pi\omega}\chi”(\omega) \quad \text{(classical limit)}$$
Step 4 — Substitute numerical values with units: Examples:
Step 5 — Compute and check the result:
- Johnson-Nyquist noise: electrical resistance $R$ (dissipation) ↔ voltage fluctuations $S_V = 4k_BTR$.
- Brownian motion: viscous drag coefficient $\gamma$ ↔ random force: $\langle F^2\rangle = 2\gamma k_BT$ (Einstein relation).
- Nyquist theorem for any impedance: $S_I = 4k_BT\,\text{Re}(1/Z(\omega))$.
Step 6: Physical content: Thermal fluctuations drive and dissipation damps — they are two sides of the same coin. Any mechanism that dissipates energy must also produce fluctuations at equilibrium.
Worked Calculation
$$S(\omega) = \frac{2k_BT}{\pi\omega}\chi”(\omega) \quad \text{(classical limit)}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
$$\boxed{S(\omega) = \frac{2k_BT}{\pi\omega}\chi”(\omega) \quad \text{(classical limit)}}$$
Statement: The spontaneous fluctuations of a system in thermal equilibrium are related to the system’s response (dissipation) to an external perturbation by the same mechanism.
For a linear response function $\chi”(\omega)$ (imaginary part, characterising dissipation), the power spectral density of fluctuations of the conjugate variable is:
$$S(\omega) = \frac{2k_BT}{\pi\omega}\chi”(\omega) \quad \text{(classical limit)}$$
Examples:
- Johnson-Nyquist noise: electrical resistance $R$ (dissipation) ↔ voltage fluctuations $S_V = 4k_BTR$.
- Brownian motion: viscous drag coefficient $\gamma$ ↔ random force: $\langle F^2\rangle = 2\gamma k_BT$ (Einstein relation).
- Nyquist theorem for any impedance: $S_I = 4k_BT\,\text{Re}(1/Z(\omega))$.
Physical content: Thermal fluctuations drive and dissipation damps — they are two sides of the same coin. Any mechanism that dissipates energy must also produce fluctuations at equilibrium.
Answer
$$\boxed{S(\omega) = \frac{2k_BT}{\pi\omega}\chi”(\omega) \quad \text{(classical limit)}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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