1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
35
I d (t) = G I s (t − t d ), with t d =
π
2ω r
and G 1
(1.136)
It is now assumed that each oscillator is driven by a stochastic voltage of the form
V T (t) =
√
2k B T 0 R (t) , where is a Gaussian noise. Moreover, the very high
quality factor of the oscillators imply that the currents I s , I d and I have frequency
quite close to ω r = 1/
√
LC, while their amplitudes and phases change very slowly.
Therefore, the following approximation
I (t) = ˆ
I (t) sin[ω r t + ˆ
φ(t)]
and the analogous ones for the other currents result quite accurate. Now, although
Eq. (1.136) implies memory effects, due to contributions from times t − t d , the above
quasi-harmonic approximation implies I s (t − t d ) ω r q s (t), and I s (t) = dq s (t)/dt
obeys:
L
d I s (t)
dt
+ I s (t) [R + R d ] +
q s (t)
C
=
2k B T 0 R (t)
(1.137)
with R d = Gω r L in playing the role of the viscous damping. Introducing g = R d /R,
as the efficiency of the feedback mechanism, Eq. (1.137) turns identical to the
Langevin equation with damping R + R d , in an equilibrium state at the fictitious
temperature T eff = T 0 /(1 + g).
While the fact that T eff differs from the bath temperature T 0 reveals the nonequilibrium nature of the phenomenon, one may formally solve Eq. (1.137) as usual in
the equilibrium case. This has been done in Ref. [11], and the analytical as well as the
numerical simulations results have been compared with experimental data collected
in five years time, obtaining perfect agreement, within the experimental paramenters
uncertainty.
Clearly, the legacy of the Brownian motion, which describes the most diverse
equilibrium phenomena, such as the random motion of pollen in water and the miroscopic current fluctuations of shot noise, extends to the case of nonequilibrium systems. Much research continues to be performed within this realm, the framework of
stochastic thermodynamics being one instance of that [15].
1.5 Fluctuations Dissipation Relations
The study of the Brownian motion is important because its range of applicability
includes an incredibly wide set of fluctuating phenomena. At the same time it is
limited to equilibrium phenomena characterized by equipartition and by fast decay
of correlations, which excludes, for instance, the gravitational antenna described
above. To treat different phenomena, one may begin relaxing the condition of white
spectrum on the stochastic term , introducing retarded frictions. In particular one
may consider
35
I d (t) = G I s (t − t d ), with t d =
π
2ω r
and G 1
(1.136)
It is now assumed that each oscillator is driven by a stochastic voltage of the form
V T (t) =
√
2k B T 0 R (t) , where is a Gaussian noise. Moreover, the very high
quality factor of the oscillators imply that the currents I s , I d and I have frequency
quite close to ω r = 1/
√
LC, while their amplitudes and phases change very slowly.
Therefore, the following approximation
I (t) = ˆ
I (t) sin[ω r t + ˆ
φ(t)]
and the analogous ones for the other currents result quite accurate. Now, although
Eq. (1.136) implies memory effects, due to contributions from times t − t d , the above
quasi-harmonic approximation implies I s (t − t d ) ω r q s (t), and I s (t) = dq s (t)/dt
obeys:
L
d I s (t)
dt
+ I s (t) [R + R d ] +
q s (t)
C
=
2k B T 0 R (t)
(1.137)
with R d = Gω r L in playing the role of the viscous damping. Introducing g = R d /R,
as the efficiency of the feedback mechanism, Eq. (1.137) turns identical to the
Langevin equation with damping R + R d , in an equilibrium state at the fictitious
temperature T eff = T 0 /(1 + g).
While the fact that T eff differs from the bath temperature T 0 reveals the nonequilibrium nature of the phenomenon, one may formally solve Eq. (1.137) as usual in
the equilibrium case. This has been done in Ref. [11], and the analytical as well as the
numerical simulations results have been compared with experimental data collected
in five years time, obtaining perfect agreement, within the experimental paramenters
uncertainty.
Clearly, the legacy of the Brownian motion, which describes the most diverse
equilibrium phenomena, such as the random motion of pollen in water and the miroscopic current fluctuations of shot noise, extends to the case of nonequilibrium systems. Much research continues to be performed within this realm, the framework of
stochastic thermodynamics being one instance of that [15].
1.5 Fluctuations Dissipation Relations
The study of the Brownian motion is important because its range of applicability
includes an incredibly wide set of fluctuating phenomena. At the same time it is
limited to equilibrium phenomena characterized by equipartition and by fast decay
of correlations, which excludes, for instance, the gravitational antenna described
above. To treat different phenomena, one may begin relaxing the condition of white
spectrum on the stochastic term , introducing retarded frictions. In particular one
may consider