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The bandwidth is the width of a specified frequency interval, $\Delta \nu ={\nu }_h-{\nu }_l$. Its meaning depends on how the interval is selected: a signal bandwidth and an effective noise bandwidth can differ even for the same receiving system.

For a linear filter with chosen nonzero reference voltage gain $H_0$, define its effective noise bandwidth byThis is the width of an ideal rectangular linear filter with gain magnitude $|H_0|$ passing the same mean square of input white noise. ConsequentlyUse the full voltage transfer from the open-circuit voltage source to the readout, including any electrical load, when applying this formula. A nominal signal bandwidth stated between two cutoff frequencies does not specify the integral: gradual tails and a nonuniform passband matter. For an ideal rectangular unit-gain passband $[f_1,f_2]$, $B_{\mathrm{n}}=f_2-f_1$. For an unloaded RC low-pass filter with $H_0=1$ and cutoff frequency $f_c=1/(2\pi RC)$,The last equality follows by substituting $x=f/f_c$ and integrating $1/(1+x^2)$ as $\arctan x$. Using the nominal $f_c$ as the noise bandwidth underestimates the variance by a factor $\pi /2$, or the root mean square voltage by $\sqrt{\pi /2}$.

Let $K(f)$ be the complete current-to-voltage frequency response from detector input to readout, in $V/A$. For a real stable linear time-invariant system, the shot contribution to the output voltage variance isIf $K(f)=R_{F}GH(f)$ with feedback electrical resistance $R_F$, a reference voltage gain $G$, and dimensionless normalized frequency response $H$, thenThis is the effective noise bandwidth for unit reference magnitude. The transimpedance amplifier factor $R_F$ converts electric current to voltage; $G$ multiplies voltage. Neither is an additional noise source merely because it amplifies an existing fluctuation. In contrast, a real amplifier adds input voltage noise and input current noise, and its feedback resistor contributes thermal electrical noise. Those terms need their own transfers to the output. In particular, input voltage noise generally has a different transfer from an injected detector electric current. Simply multiplying every input specification by $R_{F}G$ is incorrect.

For a unit-gain exponential low-pass filter, with time constant $\tau >0$,The cutoff frequency $f_c$ is not $B_{\mathrm{n}}$. An independent time-domain derivation sums independent event contributions to giveThe same reasoning for a rectangular averaging gate has $h=1/T$ on $[0,T]$ and zero elsewhere, hence effective noise bandwidth $1/(2T)$ and variance $qI/T$, agreeing with the shot noise counting model. Equal numerical durations $T=\tau$ do not make these two different linear filters equivalent.

Johnson-Nyquist noise is equilibrium electrical fluctuation caused by thermal agitation of charge carriers. A resistor can generate a fluctuating terminal voltage even when its mean current is zero. In the classical frequency range $hf\ll k_{B}T$, the Fluctuation-dissipation theorem relates this noise to dissipative resistance. The one-sided voltage power spectral density of an unloaded resistor isEquivalently its short-circuit current spectrum is $4k_{B}T/R$. Over a flat bandwidth $B$ the rms voltage is $\sqrt{4k_{B}TRB}$. A real measurement instead integrates $S_V(f)|H(f){|}^2$ over positive frequency, with loading and the filter's effective noise bandwidth included. A flat spectrum does not imply an infinite physically measurable variance; the circuit and the quantum high-frequency regime limit the ideal white-noise model.

Cooling or narrowing the accepted bandwidth reduces thermal noise. Reducing resistance lowers its open-circuit voltage noise but raises its short-circuit current noise, so the appropriate choice depends on source impedance and amplifier loading. Thermal agitation is not a fluctuating externally applied voltage that can be removed by perfect shielding alone.

Shot noise arises from discrete transfer events. In the simplest shot noise counting model, equal charges $q$ arrive independently at constant rate $r$. A gate of duration $t$ contains a Poisson number $K$, with mean and variance $rt$. Therefore mean current is $I=qr$, and the averaged-current variance is $qI/t$. The corresponding one-sided low-frequency spectrum isfor a flat accepted bandwidth. Electron current has $q=e$. Vacuum emission or photodetection can approximate independent arrivals; correlations in a conductor can suppress or enhance the fluctuations. Discreteness by itself does not guarantee Poisson statistics.

For a fixed mean current the ideal shot-noise formula is not proportional to temperature, whereas Johnson noise is. Increasing current raises absolute shot noise but lowers its fractional size, since $\sqrt{2qIB}/I\propto I^{-1/2}$. Both can appear approximately white within a useful frequency band, and independent noise powers add after referring them to a common measurement point. Their spectra must use the same convention: two-sided densities are half these one-sided values. A real low-current experiment can also contain amplifier noise, low-frequency drift and correlated transfer noise, so a mean current or resistance alone does not specify its complete noise budget.

Temperature changes affect measurements through several routes. Thermal expansion changes lengths and alignments; temperature-dependent resistance and refractive index change electrical and optical calibrations; temperature differences between dissimilar junctions generate Seebeck effect voltages. These slow environmental changes are distinct from unavoidable equilibrium thermal fluctuations and Johnson-Nyquist noise. A useful strategy identifies the coupling to the measured signal before choosing a remedy.

Reduce environmental coupling by enclosing the apparatus against drafts, using thermal insulation and radiation shields, isolating heat sources, and minimizing dissipated power. A vacuum enclosure suppresses convection but leaves thermal conduction and thermal radiation paths to control. A large heat capacity reduces rapid temperature excursions. In a simple thermal model with heat capacity $C_T$ and thermal conductance $G_T$ to a reservoir, the relaxation time is $C_T/G_T$, so adequate settling time is necessary after changing the apparatus. A temperature sensor close to the sensitive element and stable negative feedback heating or cooling can regulate its temperature; poorly placed sensors or excessive feedback gain can leave gradients or cause oscillations.

Electrical thermal noise has the classical low-frequency one-sided voltage power spectral densityfor a resistor measured over a flat bandwidth $B$ without significant loading. More generally replace $B$ by the effective noise bandwidth. Lower temperature or narrower bandwidth reduces the noise, and resistance choices must account for source loading and the amplifier's own voltage and current noise. A lock-in amplifier uses modulation and narrowband detection to reject much low-frequency drift and out-of-band noise, while thermal noise inside the accepted band remains. Temperature stabilization does not remove equilibrium fluctuations.

Use a matched reference or differential bridge to reject common environmental changes. Keep junctions isothermal and choose matched materials to reduce thermoelectric offsets. For a resistance measurement, reversing a small test current separates a stable offset from the wanted voltage:Reversal must be fast enough that the offset stays approximately fixed, with enough settling time and sufficiently little self-heating. Recording the sample temperature permits a measured calibration slope $dS/dT$ to correct a signal to a reference temperature; the correction's calibration and thermometer uncertainties must be propagated. Independent white noise fluctuations decrease under averaging, whereas correlated temperature drift may not decrease as the inverse square root of the number of readings. Temperature records and repeated reference measurements help distinguish the two.

Temperature as an equilibrium relation. Classical thermodynamics begins with the zeroth law of thermodynamics: thermal equilibrium is transitive. A thermometer can therefore assign a common empirical temperature to systems in mutual thermal equilibrium. Different thermometer properties initially give different monotone empirical scales; a resistance reading by itself does not define the absolute scale.

The Second law of thermodynamics provides a substance-independent thermodynamic scale. A reversible engine operating between reservoirs has a heat ratio depending only on their temperatures. One chooses absolute temperature so that $Q_h/Q_c=T_h/T_c$, with $Q_h,Q_c$ the positive heat magnitudes. Its reversible efficiency is $1-T_c/T_h$. Equivalently, reversible heat defines entropy through $dS=\delta Q_{\mathrm{rev}}/T$. For an equilibrium simple compressible system,Classical thermodynamics uses this relation without needing a microscopic account of entropy. A dilute-gas thermometer realizes an empirical scale approaching the thermodynamic scale after corrections for nonideal behavior.

Statistical meaning of the same derivative. In the microcanonical description, an equilibrium energy shell containing $\Omega (U,V,N)$ accessible states has $S=k_B\ln \Omega$. Its statistical temperature is consequentlyTo see why this is an equilibrium temperature, let two weakly coupled systems exchange a small energy while their total energy stays fixed. The combined multiplicity is ${\Omega }_A(U_A){\Omega }_B(U_{\mathrm{tot}}-U_A)$. Maximizing its logarithm giveswhich is equality of the two inverse temperatures. Stability requires the total entropy maximum, not only stationarity. The overwhelmingly dominant macrostate in a large system therefore realizes the thermodynamic equilibrium condition.

A large reservoir gives a small subsystem the canonical probabilities $p_i\propto e^{-\beta E_i}$, with $\beta =1/(k_{B}T)$. In that ensemble, $F=-k_{B}T\ln Z$ and $S=-k_B{\sum }_i{p}_i\ln p_i$ agree with macroscopic thermodynamic relations in their usual equilibrium regime. The equipartition identity for a classical dilute monatomic gas, $⟨p^2/(2m)⟩=3k_{B}T/2$, is a useful operational connection between microscopic motion and temperature, but is not a universal definition: quantum modes need not obey equipartition, and a system away from equilibrium need not have one temperature at all.

For finite systems one must specify the entropy convention and energy resolution; surface and fluctuation corrections can distinguish definitions that agree in the ordinary thermodynamic limit. Bounded-energy population-inverted systems can also have a negative entropy derivative and hence a negative statistical temperature. That possibility lies outside an ordinary positive-temperature ideal-gas thermometer; it does not mean temperature is merely an arbitrary average kinetic energy.

Obtaining the energy–temperature conversion. The Boltzmann constant connects entropy measured in $\mathrm{J}{\mathrm{K}}^{-1}$ with the dimensionless logarithm of a state count. Historically it could be determined from the ideal gas law $pV=N{k}_{B}T$: measure the molar gas constant $R$ and independently determine the Avogadro constant $N_A$, then obtain $k_B=R/N_A$. The independent particle-number determination matters; using $R$ alone would not fix energy per particle.

An electrical method uses Johnson-Nyquist noise. A resistor of resistance $R_e$ at known temperature has, in the classical range $hf\ll k_{B}T$, a one-sided open-circuit voltage-noise spectrum $S_V(f)=4k_{B}T{R}_e$. Over a calibrated flat bandwidth $\Delta f$,A practical measurement uses an equilibrium resistor, a calibrated low-noise amplifier and a known transfer function, subtracts or independently characterizes amplifier noise, and replaces a nominal bandwidth by the actual effective noise bandwidth. Varying temperature and fitting the mean-square voltage is more informative than treating one amplified voltage sample as the thermal variance. Other primary methods relate temperature to sound speed in a dilute gas or to calibrated particle diffusion and mobility.

There is a historically relevant unit-definition qualification: by the date of this exam, the revised SI had already taken effect on 20 May 2019. The kelvin is now defined by fixingThe same experiments now realize or compare thermodynamic temperatures using this fixed constant, whereas historically they measured the conversion constant on an independently defined temperature scale. The physical distinction between equilibrium temperature and microscopic state counting is unchanged.^[ref]

Connect one physical resistor $R$ across an ideal lossless capacitor $C$, with no applied source and an ideal readout that draws no electric current. Represent the resistor by its noisy Thévenin equivalent. The resulting RC low-pass frequency response gives the stationary capacitor voltage power spectral densityIts mean square follows by the effective noise bandwidth calculation:This is also the equipartition theorem: the mean capacitor energy is $\frac{1}{2}C⟨v_C^2⟩=\frac{1}{2}{k}_{\mathrm{B}}T$. Raising $R$ makes the low-frequency spectrum taller and its width narrower in exactly compensating proportions. It changes the equilibration time $RC$, not the final variance.

The integral to infinity belongs to the ideal classical model. For a physical electrical circuit, assume thermal equilibrium, linear components with negligible extra dissipation, and a corner far below the frequencies where parasitic effects, frequency-dependent electrical resistance, or $hf\sim k_{\mathrm{B}}T$ invalidate the model; the omitted high-frequency tail must be negligible. A finite-bandwidth instrument measures only part of this variance. Its input capacitance contributes to the total $C$, and its finite input electrical resistance introduces both loading and its own thermal electrical noise.

For $T=300\mathrm{K}$ and $C=1\mathrm{nF}$, the predicted equilibrium variance is $4.141947\times 10^{-12}{\mathrm{V}}^2$ and the root mean square voltage is $2.0352\mu \mathrm{V}$. At $R=10\mathrm{k}\Omega$, $RC=10\mu \mathrm{s}$ and $B_{\mathrm{n}}=25\mathrm{kHz}$. At $R=100\mathrm{k}\Omega$, $RC=100\mu \mathrm{s}$ and $B_{\mathrm{n}}=2.5\mathrm{kHz}$; the equilibrium root mean square voltage is identical.

To obtain the growth curve, prepare $v_C(0)=0$ exactly and connect the resistor at $t=0$ while maintaining its temperature. The RC low-pass filter's causal impulse response is $h(t)=e^{-t/(RC)}/(RC)$ for $t\ge 0$. Classical ideal white noise with two-sided power spectral density $2k_{\mathrm{B}}TR$ givesThe factor $2k_{\mathrm{B}}TR$ here is the two-sided time-domain noise intensity; the equilibrium spectral integral above used the one-sided $4k_{\mathrm{B}}TR$. They give the same long-time limit. The variance approaches equilibrium with time constant $RC/2$, although a deterministic initial voltage decays with time constant $RC$. If the capacitor already starts in thermal equilibrium, its initial variance supplies the missing exponentially decaying term and the total variance stays constant.