Calculator Inputs
Formula Used
The fluctuation–dissipation relationship links equilibrium fluctuations to dissipation through the imaginary part of the response (susceptibility) χ″(ω).
| Case | Working relation used by this calculator |
|---|---|
| Classical limit | S(ω) = (2 kB T / ω) · χ″(ω) |
| Quantum form | S(ω) = (2 ħ / (1 − exp(−ħω/(kB T)))) · χ″(ω) |
| Unit conversion | S(f) = 2π · S(ω) , ω = 2πf |
Tip: χ″(ω) depends on your system definition (mechanical, electrical, magnetic). This tool keeps units symbolic and propagates them to S(ω) and S(f).
How to Use This Calculator
- Enter the system temperature in kelvin.
- Provide either frequency (Hz) or angular frequency (rad/s).
- Input the imaginary susceptibility χ″(ω) at that frequency.
- Select classical for high-temperature behavior, or quantum for general use.
- Optionally enter a bandwidth Δf to estimate integrated contribution.
- Press Calculate to view results above the form.
- Use the download buttons to export CSV or PDF.
Example Data Table
Sample inputs and computed spectra (illustrative only).
| T (K) | f (Hz) | χ″(ω) | Mode | S(ω) (per rad/s) | S(f) (per Hz) |
|---|---|---|---|---|---|
| 300 | 1,000 | 1.0×10−6 | Classical | 1.318×10−26 | 8.280×10−26 |
| 77 | 10,000 | 5.0×10−7 | Classical | 1.691×10−28 | 1.063×10−27 |
| 4 | 5,000,000 | 2.0×10−8 | Quantum | ≈ prefactor·χ″ | ≈ 2π·S(ω) |
For your system, use χ″(ω) from measurements or a model fit.
Fluctuation–Dissipation Guide
1) Why fluctuations matter
Thermal motion produces measurable noise in almost every physical system. Voltage noise in resistors, position jitter in micro-cantilevers, and magnetization noise in materials are all examples. The fluctuation–dissipation relationship connects these random fluctuations to how the same system absorbs energy when driven, making noise a quantitative diagnostic.
2) What this calculator computes
This tool estimates a fluctuation spectrum from the imaginary susceptibility χ″(ω) at a chosen frequency. In the classical limit it uses the prefactor 2kBT/ω; in the quantum form it uses 2ħ/(1−e−ħω/(kBT)). Results are shown as S(ω) per rad/s and S(f) per Hz.
3) Typical data sources for χ″(ω)
χ″(ω) can come from impedance spectroscopy, mechanical ringdown, dynamic light scattering fits, or susceptibility measurements in magnetic resonance. Many systems show peaks near resonances where χ″(ω) is largest, which often corresponds to higher predicted noise at that frequency.
4) Temperature scaling you can expect
In the classical regime, spectra scale linearly with temperature because kBT appears directly. For example, moving from 300 K to 77 K reduces the prefactor by about 3.9× at the same ω. At cryogenic temperatures, quantum corrections can prevent the spectrum from decreasing as fast.
5) Frequency and the ω dependence
With classical weighting, the prefactor includes 1/ω, so higher angular frequency reduces the contribution when χ″(ω) is constant. In real materials χ″(ω) is rarely flat; it may increase near relaxations or resonances, partially offsetting the 1/ω factor. Always interpret S with your measured χ″ curve.
6) When quantum effects become important
A practical crossover is ħω ≈ kBT. At 300 K this corresponds to f ≈ 6.2 THz. At 4 K it corresponds to f ≈ 83 GHz. If your operating frequency is comparable to these scales, choose the quantum option for more realistic weighting.
7) Bandwidth and integrated estimates
Many experiments measure noise in a finite bandwidth Δf. When Δf is supplied, the calculator reports an approximate integrated contribution S(f)·Δf, assuming the spectrum is nearly constant across the band. For wide bands or steep χ″(ω) changes, compute several points and integrate numerically.
8) Interpreting units and conventions
The calculator preserves your susceptibility convention. If χ″(ω) is dimensionless, S inherits the prefactor units; if χ″ has physical units, those propagate into S. Be consistent about one-sided versus two-sided spectra and about whether χ is defined per force, field, or generalized coordinate.
FAQs
1) What does χ″(ω) represent?
It is the dissipative part of the system response at angular frequency ω. Larger χ″ typically means stronger energy absorption and a larger predicted fluctuation spectrum at that frequency.
2) Should I enter f or ω?
Either is fine. If you enter ω, it overrides f. The calculator converts using ω = 2πf and reports both forms for easy cross-checking.
3) When should I use the quantum option?
Use it when ħω is not much smaller than kBT, such as cryogenic experiments or very high frequencies. In the small-ħω limit it smoothly approaches the classical result.
4) What are the units of S(ω) and S(f)?
They depend on how χ″ is defined for your observable. The tool reports “per rad/s” and “per Hz” forms and keeps the units symbolic so you can apply your convention consistently.
5) Why does the result change with frequency?
The prefactor contains 1/ω in the classical expression and a frequency-dependent quantum factor. In addition, χ″ itself usually varies strongly with frequency due to relaxations and resonances.
6) What does S(f)·Δf mean?
It is a quick estimate of the contribution integrated over a bandwidth Δf, assuming S(f) is nearly constant across that band. For wider bands, sample multiple frequencies and integrate.
7) Can this be used for electrical and mechanical systems?
Yes. The relationship is general for linear response. Use χ″ appropriate to your variable: electrical impedance/admittance, mechanical compliance, magnetic susceptibility, or another generalized response function.