Fluctuations, Linear Response and the Fluctuation-Dissipation Relation
At fixed T,V,N in the canonical ensemble, Var(E)=k_B T^2 C_V. At fixed T and density, extensive short-range systems with finite correlation length usually have relative energy fluctuations proportional to 1/sqrt(N). The fluctuation-dissipation relation links equilibrium fluctuations to dissipative linear response through temperature and frequency factors. Resistor noise also depends on measurement bandwidth.
What a learner can do afterwards
- Relates the variance of energy in the canonical ensemble to heat capacity
- Explains relative-fluctuation scaling for extensive systems with finite correlation length and describes its limits
- States what the fluctuation-dissipation relation connects and names one measurable case
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At fixed V,N in the canonical ensemble, Var(E)=k_B T^2 C_V. T is in kelvin. Boltzmann's constant k_B and heat capacity C_V, the temperature derivative of mean energy at fixed V,N, both have units energy/kelvin. Their product needs T^2 for energy-squared units. Variance is the mean squared deviation from mean energy. Its square root is the root-mean-square (RMS) energy fluctuation.
Voltage is energy transferred per unit charge. For an ideal resistor, resistance R relates applied voltage to driven current by voltage=R times current. In thermal equilibrium, charge-carrier motion produces Johnson-Nyquist voltage fluctuations even with no battery attached or driven current. A measurement records these fluctuations; no audible sound is required.
At equilibrium T, the classical positive-frequency voltage-noise spectrum is S_V=4 k_B T R in V^2/Hz. Assume constant R and photon energies much smaller than k_B T in the band. An ideal open-circuit measurement draws no current. With unit gain in the band and zero outside, mean-square voltage = 4 k_B T R delta f. Bandwidth delta f is the band's width in Hz. To infer R, divide mean-square voltage by the known factors multiplying R.
At the same positive T, Var(E) is proportional to C_V. If sample B has four times sample A's C_V, B has four times the energy variance but only twice the root-mean-square fluctuation. Compare variance with variance, and fluctuation size with its square root.
The bars show relative variances, not root-mean-square fluctuation sizes.
Linear response means an observable's change is proportional to a sufficiently weak perturbation. Dissipative response converts drive energy to heat. The fluctuation-dissipation relation links equilibrium correlations or spectra to that response through temperature and frequency factors. Johnson-Nyquist noise is a measurable case: T and R set the voltage-noise spectrum; the same R dissipates energy when driven. Noise and response are distinct; predictions require the relevant conditions.
For extensive short-range systems at fixed T and density, away from critical regimes and with finite correlation length, mean energy and energy variance usually grow in proportion to N. The root-mean-square fluctuation grows as sqrt(N). Dividing it by an extensive, nonzero mean energy gives relative size proportional to 1/sqrt(N). Equal growth of variance and mean does not imply constant relative fluctuation size.
The usual size scaling assumes correlations do not grow across the whole sample. Near a critical point, correlations can extend far and fluctuation scaling needs further analysis. Long-range interactions can also change the result. Smaller N often means larger relative fluctuations in the regular regime, but size alone does not justify the rule in every system.
Fixed T,V,N: Var(E)=k_B T^2 C_V. In extensive short-range systems with finite correlations, relative RMS size usually falls as 1/sqrt(N). Noise relates to dissipative response through T and frequency.
Original lesson, questions and flashcards written for this learning dataset.
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