In
statistical physics, the
fluctuation dissipation theorem is a powerful tool
for predicting the
non-equilibrium behavior of a
system — such as the
irreversible
dissipation of energy into
heat — from its
reversible fluctuations
in
thermal equilibrium.
The fluctuation dissipation theorem applies both to
classical and
quantum mechanical systems.Although
formulated originally by
Nyquist in
1928, the fluctuation-dissipation theorem was first proved by
Herbert B. Callen and Theodore A. Welton in 1951.
The fluctuation dissipation theorem relies on the assumption that
the response of a system in thermodynamic equilibrium to a small
applied force is the same as its response to a spontaneous
fluctuation. Therefore, there is a direct relation between the
fluctuation properties of the thermodynamic system and its linear
response properties. Often the linear response takes the form of
one or more exponential decays.
Example: Brownian motion
For example,
Einstein in his 1905
paper on
Brownian motion noted that
the same random forces which cause the erratic motion of a particle
in Brownian motion would also cause drag if the particle were
pulled through the fluid. In other words, the
fluctuation
of the particle at rest has the same origin as the
dissipative frictional force one must do work against, if
one tries to perturb the system in a particular direction.
From this observation he was able to use
statistical mechanics to derive a
previously unexpected connection, the
Einstein-Smoluchowski
relation:
- D = {\mu_p \, k_B T}
linking
D, the
diffusion constant, and
μ,
the
mobility of the particles.
(
μ is the ratio of the particle's terminal drift velocity
to an applied force,
μ = v_{d} / F).
k_{B} ≈ 1.38065 × 10
^{−23} m² kg
s
^{−2} K^{−1} is
Boltzmann's constant, and
T is
the
absolute temperature.
Example: Thermal noise in a resistor
In 1928,
John B. Johnson discovered and
Harry Nyquist explained
Johnson–Nyquist noise. With no
applied current, the mean-square voltage depends on the resistance
R, k_BT, and the bandwidth \Delta\nu over which the
voltage is measured:
- \langle V^2 \rangle = 4Rk_BT\,\Delta\nu.
General applicability
The examples above are consequences of the
fluctuation
dissipation theorem, a very general result of
statistical thermodynamics which
quantifies the relation between the fluctuations in a system at
thermal equilibrium and the
response of the system to applied perturbations. It thus allows,
for example, the use of molecular models to predict material
properties in the context of linear response theory. The theorem
assumes that applied perturbations (mechanical forces, electric
fields,
etc.) are weak enough that rates of
relaxation remain unchanged.
General form of the fluctuation dissipation theorem
The fluctuation-dissipation theorem can be formulated in many
ways;one particularly useful form is the following:
Let
x be an observable of a dynamical systemwith
Hamiltonian
H_{0}(
x) subject to thermal
fluctuations.The observable
x will fluctuate around its
mean value \langle x\rangle_0 with fluctuations characterized by a
power spectrum S_x(\omega) .Suppose
that we can switch on a (scalar) field
f which alters the
Hamiltonianto H(x)=H_0(x)+x f .The response of the observable
x to a field
f(
t) changing with time
ischaracterized (to first order) by the
susceptibility or
linear response function \chi(t) of
the system
- \langle x(t) \rangle = \langle x \rangle_0 + \int_{-\infty}^{t}
f(\tau) \chi(t-\tau) d\tau\,,
where the perturbation is adiabatically switched on at \tau
=-\infty.
Now the fluctuation dissipation theorem relates the power spectrum
to the imaginary part of the
Fourier
transform \tilde{\chi}(\omega)=\rm{Re}[{\tilde{\chi}}(\omega
)]+ i~\rm{Im}[{\tilde{\chi}}(\omega )] of the susceptibility
\chi(t)
- S_x(\omega) = \frac{2 k_B T}{\omega}
\rm{Im}[\tilde{\chi}(\omega)] .
The left-hand side describes fluctuations in x, the right-hand side
is closely related to the energy dissipated by the system when
pumped by an oscillatory field f(t) = f_0 sin(\omega t). .
(This is already the classical form of the theorem; quantum
fluctuations are taken into account byreplacing the prefactor by
{\hbar}\cdot {\rm cotanh}\frac{\beta\hbar \omega}{2}\,.. A proof
can be found by means of the
LSZ
reduction, an identity from quantum field theory.)
The fluctuation-dissipation theorem can be generalized in a
straight-forward way to the case of space-dependent fields,to the
case of several variables or to a quantum-mechanics setting.
Violations of FDT in glassy systems
While the FDT provides a general relation between the response of
equilibrium systems to small external perturbations and their
spontaneous fluctuations, no general relation is known for systems
out of equilibrium.
In the mid 1990s, in the study of non-equilibrium dynamics of
spin glass models it was discovered a
generalization of FDT valid for asymptotic non-stationary states,
where the temperature appearing in the equilibrium relation is
substituted by an effective temperature with a non-trivial
dependence on the time scales.This relation is proposed to hold in
glassy systems beyond the models for which it was initially
found.
Derivation I
We derive the fluctuation dissipation theorem in the form given
above, using the same notation.Consider the following test case:
The field
f has been on for infinite time and is switched
offat
t=0
- f(t)=f_0 \theta(-t) .
We can express the expectation value of
x by the
probability distribution
W(
x,0) and the transition
probability P(x',t | x,0)
- \langle x(t) \rangle = \int dx' \int dx \, x' P(x',t|x,0)
W(x,0) .
The probability distribution function
W(
x,0) is
an equilibrium distribution and hencegiven by the
Boltzmann distribution for the
Hamiltonian H(x)=H_0(x) + x f_0
- W(x,0)= \frac{\exp(-\beta H(x))}{\int dx' \, \exp(-\beta
H(x'))}.
For a weak field \beta f_0 \ll 1 , we can expand the right-hand
side
- W(x,0) \approx W_0(x) (1-\beta f_0 x),
here W_0(x) is the equilibrium distribution in the absence of a
field.Plugging this approximation in the formula for \langle x(t)
\rangle yields
- (*) \langle x(t) \rangle = \langle x \rangle_0 - \beta f_0
A(t),
where
A(
t) is the auto-correlation function of
x in the absence of a field.
- A(t)=\langle x(t) x(0) \rangle_0.
Note that in the absence of a field the system is invariant under
time-shifts.We can rewrite \langle x(t) \rangle - \langle x
\rangle_0 using the susceptibilityof the system and hence find with
the above equation (*)
- f_0 \int_0^{\infty} d\tau \, \chi(\tau) \theta(\tau-t) = \beta
f_0 A(t)
Consequently,
- (**) -\chi(t) = \beta \frac{d}{dt} ( A(t) \theta(t) ) .
For
stationary processes, the
Wiener-Khinchin theorem
states thatthe power spectrum equals twice the
Fourier transform of the
auto-correlationfunction
- S_x(\omega) = 2 \tilde{A}(\omega).
The last step is to Fourier transform equation (**) and to take
theimaginary part. For this it is useful to recall that the Fourier
transformof a real symmetric function is real, while the Fourier
transform of a realantisymmetric function is purely imaginary.We
can split \frac{d}{dt} ( A(t) \theta(t) ) into a symmetric and
ananti-symmetric part
- 2 \frac{d}{dt} ( A(t) \theta(t) ) = \frac{d}{dt} A(t) +
\frac{d}{dt}( A(t) {\rm sign}(t) ) .
Now the fluctuation dissipation theorem follows.
Derivation II
The following general derivation of the fluctuation-dissipation
theorem uses averaging in
phase space.
The derivation applies equally well to
classical as well as
quantum mechanical systems, although the
former uses a continuous integral over phase space, whereas the
latter uses a sum over quantum states. To represent both, we
introduce the
trace
notation, which applies both to classical and quantum
systems
\mathrm{Tr} X = \int d\Gamma X = \sum_{\mathrm{quantum\ states}\ n}
\langle n|X|n \rangle
where
dΓ represents an infinitesimal volume in
phase space. Thus, if a system is described by a
probability distribution
f(
q,
p) in
phase space, the average value of an arbitrary function
A
of the system's state is given by
\langle A \rangle = \mathrm{Tr} \left\{ A(q, p) \ f(q, p)
\right\}
where angular brackets are used to denote the averaging over the
ensemble. In particular, if the probability distribution is given
by the equilibrium
Boltzmann
distribution, the ensemble average equals
\langle A \rangle = \mathrm{Tr} \left\{ A(q, p) \ f(q, p) \right\}
=\frac{\mathrm{Tr} \left\{ A e^{-\beta H} \right\}}{\mathrm{Tr} \
e^{-\beta H}}
where β = 1/
k_{B}T,
k_{B} is the
Boltzmann constant and
T is the
temperature in
Kelvin.
Having defined our notation and basic variables, we now derive the
fluctuation-dissipation theorem. Consider a system that has reached
equilibrium under the Hamiltonian
H + h, where
h
is much smaller than the thermal energy
k_{B}T.
Being in thermal equilibrium, the probability of any state is
proportional to its Boltzmann factor
e^{−β(H +
h)}. At time
t = 0, let the perturbation
h be turned off; given
ergodicity, the system will gradually relax to a
new equilibrium, which has Boltzmann factors
e^{−βH}. The fluctuation-dissipation theorem
addresses the question of how quickly the system reaches its new
equilibrium.
Let the correlation function
c(
t) be
defined
c(t) = \langle \delta A (t) \ \delta A (0) \rangle =
\lim_{T\rightarrow\infty} \frac{1}{2T}\int_{-T}^{T} dt^{\prime}\
\delta A (t^{\prime} + t) \ \delta A (t^{\prime})
which may be written as
c(t) = \int d\Gamma\ \delta A(t; q, p) \ \delta A(0; q, p)\ f(q,
p)
where
δA(
t;
q,
p) is the
deviation from its mean at a time
t, given that the system
began at time
t = 0 at position (
q,
p)
in phase space. In other words, the integration is over all
initial positions of the system in phase space.
The mean value of
A as it evolves towards its new
equilibrium is given by
\bar{A}(t) =\frac{\mathrm{Tr} \left\{ A(t; p, q) e^{-\beta \left(H
+ h \right)} \right\}}{\mathrm{Tr} \ e^{-\beta \left(H + h
\right)}}.
Since
h is much smaller than the thermal energy
k_{B}T, we may expand the numerator
\mathrm{Tr} \left\{ A(t; p, q) e^{-\beta \left(H + h \right)}
\right\} =\mathrm{Tr} \left\{ A(t; p, q) e^{-\beta H} \left(1 -
\beta h + \cdots \right) \right\} \approx\mathrm{Tr} \left\{ A(t;
p, q) e^{-\beta H} \right\} -\mathrm{Tr} \left\{ A(t; p, q)
e^{-\beta H} \beta h \right\}.
We may likewise expand the denominator
\frac{1}{\mathrm{Tr} \ e^{-\beta \left(H + h \right)}}
\approx\frac{1}{\mathrm{Tr} \left\{ e^{-\beta H } \right\} -
\mathrm{Tr} \left\{ e^{-\beta H } \beta h \right\}}
=\frac{1}{\mathrm{Tr} e^{-\beta H }} \cdot \frac{1}{1 - \langle
\beta h \rangle} \approx\frac{1 + \langle \beta h
\rangle}{\mathrm{Tr} e^{-\beta H }}
where we have used
\langle \beta h \rangle = \frac{\mathrm{Tr} \left\{ \beta h
e^{-\beta H} \right\}}{\mathrm{Tr} \ e^{-\beta H}}
which is much less than one, by our assumption that
h is
much smaller than the thermal energy 1/β =
k_{B}T.
Combining the numerator and denominator, dropping quadratic and
high-order terms in <β
h>, and using the indifference
of equilibrium to time, we obtain
\bar{A}(t) - \langle A \rangle = - \beta \langle A h \rangle +
\beta \langle A \rangle \langle h \rangle.
Let the perturbation
h = −gA be proportional to the
variable
A with a constant
−g. Then this formula
becomes
\bar{A}(t) - \langle A \rangle =\beta g \left\{ \langle A(t) A(0)
\rangle - \langle A \rangle^{2} \right\} =\beta g \langle \delta
A(t) \delta A(0) \rangle = \beta g c(t).
Note that the system's relaxation is independent of
A and
linear in
g. These results imply that perturbations will
relax independently of one another; if two perturbations,
g_{1} and
g_{2} are applied, the
net relaxation will be the sum of the individual relaxations to
g_{1} and
g_{2} taken separately.
Such continuous linear systems have been well-studied, and many
methods developed for their solution, such as
Fourier transforms and
Laplace transforms.
See also
Notes
References
- H. B. Callen and T. A. Welton, Phys. Rev. 83,
34 (1951)
- L. D. Landau et E. M. Lifshitz, Cours de physique théorique t.5
Physique Statistique (Mir)
- "Fluctuation-Dissipation: Response Theory in Statistical
Physics" by Umberto Marini Bettolo Marconi, Andrea Puglisi,
Lamberto Rondoni, Angelo Vulpiani, [60767]
Further reading