Nyquist formula

At high temperatures, E(ui,T) takes its equipartition value of E(u,T) kT, (kT >> Hu) to yield the so-called Nyquist formula that describes voltage fluctuations (Nyquist noise) generated in a linear electrical system, i.e. 

Application of the F-D theorem produced [122] several significant results. Apart from the Nyquist formula these include the correct formulation of Brownian motion, electric dipole and acoustic radiation resistance, and a rationalization of spontaneous transition probabilities for an isolated excited atom. 

Summing up, when the particle environment is a thermal bath, the two fluctuation-dissipation theorems are valid. In both theorems the bath temperature T plays an essential role. In its form (157) or (159) (Einstein relation), the first FDT involves the spectral density of a dynamical variable linked to the particle (namely its velocity), while, in its form (161) (Nyquist formula), the second FDT involves the spectral density of the random force, which is a dynamical variable of the bath. 

Such a quantity, denoted as 7 eff(co), and parameterized by the age of the system, has been defined, for real to, via an extension of both the Einstein relation and the Nyquist formula. It has been argued in Refs. 5 and 6 that the effective temperature defined in this way plays in out-of-equilibrium systems the same role as does the thermodynamic temperature in systems at equilibrium (namely, the effective temperature controls the direction of heat flow and acts as a criterion for thermalization). 

Interestingly, due to the linearity of the generalized Langevin equation (22), the same effective temperature T,eff(( ) can consistently be used in the modified Nyquist formula linking the noise spectral density C/ /- ([Pg.313]

Let us emphasize that the function (co) is not the effective temperature involved in both the modified Einstein relation (184) and the modified Nyquist formula (185). Instead it is the quantity which appears in the modified Kubo formula for the mobility [Eq. (187)]. Let us add that one could also have... 

Nonequilibrium noise generated by carrier-mediated ion transport was studied in lipid bilayers modified by tetranactin (41). As expected, deviations of measured spectral density from the values calculated from the Nyquist formula 1 were found. The instantaneous membrane current was described as the superposition of a steady-state current and a fluctuating current, and for the complex admittance in the Nyquist formula only a small-signal part of the total admittance was taken. The justification of this procedure is occasionally discussed in the literature (see, for example, Tyagai (42) and references cited therein), but is unclear. 

A generalization of the Nyquist formula, eq 1, was proposed by Grafov and Levich (43) to describe fluctuations in a nonlinear steady state. This approach is based on the fluctuation-dissipation thermodynamics of irreversible nonlinear systems and introduces the so-called dissipative resistance (42), which differs from small-signal resistance in a general case. This result indicates that separation of equilibrium and transport noise is not a well-defined procedure. 

A noise that has a clearly distinct origin from noise discussed in previous sections is the electric noise that originates in modulation of ion transport by fluctuations in system conductance. These temporal fluctuations can be measured, at least in principle, even in systems at equilibrium. Such a measurement was conducted by Voss and Clark in continuous metal films (44). The idea of the Voss and Clark experiment was to measure low-frequency fluctuations of the mean-square Johnson noise of the object. In accordance with the Nyquist formula, fluctuations in the system conductance result in fluctuations in the spectral density of its equilibrium noise. Measurement of these fluctuations (that is, measurement of the noise of noise) yields information on conductance fluctuations of the system without the application of any external perturbations. The samples used in these experiments require rather large amplitude conductance fluctuations to be distinguished from Johnson noise fluctuations because of the intrinsic limitation of statistics. Voss and... 

Thermal noise denotes electrical fluctuations spontaneously generated in a sample resulting from thermal agitation. The noise level is given by the Nyquist formula (2) ... 

The sample resistance fac) calculated from the observed thermal noise voltage spectra using the Nyquist formula was compared with the values measured with a conventional resistance bridge (General Radio, model 1620). Similar measuring equipment has been described in previous publications (S,4). 

Within the range of frequency, temperature, shear rate etc. covered by the experiment, all the measured thermal noise levels agreed well with the predictions based on the Nyquist formula. This implies that the thermal noise level could have been calculated from resistivity measurements and also that the noise peaks in the vicinity of Tg and Tm would have appeared in the corresponding resistivity-temperature diagrams. This was actually verified in numerous experimental runs. On the other hand, the measurement of thermal noise has the advantage that no external voltage has to be applied across the sample. This eliminates the possibility that the observed peaks arise from polarization effects (6,7). 

A recent study by DeFelice and Firth (1971) has shown that the electrical noise present in glass microelectrodes is in excess of the Johnson noise predicted by the Nyquist formula given in Section 4.4.3. If the microelectrode lumen is filled with an electrolyte of concentration ri2 and the external electrolyte in which the electrode is immersed has the concentration when = ri2 2i noise voltage is observed as predicted by the Nyquist formula. When Ml offset voltage is noted (tip potential) because of the... 

The Johnson noise of the load resistor R at the temperature T which gives, according to the Nyquist formula, an rms-noise current... 

Figure 18.29 shows the spectral content of the input signal after sampling. Frequencies below 50 Hz, the Nyquist frequency (/s/2), appear correctly. However, frequencies above the Nyquist appear as aliases below the Nyquist frequency. For example, FI appears correctly however, both F2, F3, and F4 have aliases at 30, 40, and 10 Hz, respectively The resulting frequency of aliased signals can be calculated with the following formula ... 

Furthermore, the impedance of a depressed Nyquist plot due to surface roughness, dielectric inhomogeneities and diffusion is defined as the Cole-Cole impedance formula [33-36]. Hence,... 

The amount of thermal noise contributed by the electrode is given by Nyquist s formula ... 

Figure 13. The typical electrochemical response of graphite and hard cattxm electrodes (a, b, respectively). Slow scan CV, the variation of the diffusion coefficient vs. potential (log D kk E), and selected impedance spectra (Nyquist plots at several potentials, as indicated) are presented. The various phases of the lithiated graphite that correspond to the various CV peaks (13a) are also presented. The dashed line that connects the points of the lowest frequency in the Nyquist plots of the graphite electrodes correspond to the CV peak around 0.12 V (Li/Li ). The intercalaUon capacity corresponding to the phase transition between stages III and II can be calculated from this dashed line according to the formula Cw = l/Z"-m (as m 0). See text and references 96,105-107.

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