Appendix I
I.1 The Fluctuation-Dissipation Theorem
One way of formulating the FDT is by formally regarding the spontaneous
fluctuations of a quantity x as due to the action of some random force f , meaning
that the environment senses the system through the generalized susceptibility, α(ω),
and respond with a fluctuating force. The Fourier components x ω and f ω are related
by:
x ω = α(ω)f ω
(I.1)
The relation between the generalized impedance Z(ω) and α(ω) is:
Z(ω) =
i
ωα(ω)
(I.2)
As x ω = x 0ω e −iωt we can write:
f ω = Z(ω)
dx ω
dt
(I.3)
The spectral densities of the fluctuation are given by
(x
2 ) ω =| α(ω) |
2 (f
2 ) ω
(I.4)
The results of the FDT are:
(x
2 ) ω = ¯
hα
(ω) coth
¯
hω
2kT
(I.5)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9
187
I.1 The Fluctuation-Dissipation Theorem
One way of formulating the FDT is by formally regarding the spontaneous
fluctuations of a quantity x as due to the action of some random force f , meaning
that the environment senses the system through the generalized susceptibility, α(ω),
and respond with a fluctuating force. The Fourier components x ω and f ω are related
by:
x ω = α(ω)f ω
(I.1)
The relation between the generalized impedance Z(ω) and α(ω) is:
Z(ω) =
i
ωα(ω)
(I.2)
As x ω = x 0ω e −iωt we can write:
f ω = Z(ω)
dx ω
dt
(I.3)
The spectral densities of the fluctuation are given by
(x
2 ) ω =| α(ω) |
2 (f
2 ) ω
(I.4)
The results of the FDT are:
(x
2 ) ω = ¯
hα
(ω) coth
¯
hω
2kT
(I.5)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9
187