Fluctuation dissipation theorem
Encyclopedia
The fluctuation-dissipation theorem (FDT) is a powerful tool in statistical physics
for predicting the behavior of non-equilibrium thermodynamical
systems. These systems involve the irreversible
dissipation
of energy into heat
from their reversible
thermal fluctuations
at thermodynamic equilibrium
. The fluctuation-dissipation theorem applies both to classical
and quantum mechanical
systems.
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, the theorem connects the linear response relaxation of a system from a prepared non-equilibrium state to its statistical fluctuation properties in equilibrium. Often the linear response takes the form of one or more exponential decays.
The fluctuation-dissipation theorem was originally formulated by Harry Nyquist
in 1928, and later proven by Herbert Callen
and Theodore A. Welton in 1951.
and the response of the system to applied perturbations.
The model 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, e.g., mechanical forces or electric fields, are weak enough that rates of relaxation
remain unchanged.
noted in his 1905 paper on Brownian motion
that the same random forces that 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 Einstein was able to use statistical mechanics
to derive a previously unexpected connection, the Einstein-Smoluchowski relation
:
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 = vd / F). kB ≈ 1.38065 × 10−23 m2 kg s−2 K−1 is the Boltzmann constant, and T is the absolute temperature.
discovered and Harry Nyquist
explained Johnson–Nyquist noise
. With no applied current, the mean-square voltage depends on the resistance R,
, and the bandwidth 
over which the voltage is measured:
Let
be an observable of a dynamical system with Hamiltonian
subject to thermal fluctuations.
The observable
will fluctuate around its mean value 
with fluctuations characterized by a power spectrum
.
Suppose that we can switch on a scalar field
which alters the Hamiltonian
to
.
The response of the observable
to a time-dependent field 
is
characterized to first order by the susceptibility
or linear response function
of the system
where the perturbation is adiabatically switched on at
.
Now the fluctuation-dissipation theorem relates the power spectrum of
to the imaginary part of the Fourier transform
of the susceptibility 
,
The left-hand side describes fluctuations in
, the right-hand side is closely related to the energy dissipated by the system when pumped by an oscillatory field 
.
This is the classical form of the theorem; quantum fluctuations are taken into account by
replacing
with 
(whose limit for 
is 
). 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.
Consider the following test case: The field f has been on for infinite time and is switched off at t=0
We can express the expectation value of x by the probability distribution W(x,0) and the transition probability
The probability distribution function W(x,0) is an equilibrium distribution and hence
given by the Boltzmann distribution
for the Hamiltonian
For a weak field
, we can expand the right-hand side
here
is the equilibrium distribution in the absence of a field.
Plugging this approximation in the formula for
yields
where A(t) is the auto-correlation function of x in the absence of a field.
Note that in the absence of a field the system is invariant under time-shifts.
We can rewrite
using the susceptibility
of the system and hence find with the above equation (*)
Consequently,
Statistical physics
Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...
for predicting the behavior of non-equilibrium thermodynamical
Non-equilibrium thermodynamics
Non-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium; for they are changing or can be triggered to change over time, and are continuously and discontinuously...
systems. These systems involve the irreversible
Irreversibility
In science, a process that is not reversible is called irreversible. This concept arises most frequently in thermodynamics, as applied to processes....
dissipation
Dissipation
In physics, dissipation embodies the concept of a dynamical system where important mechanical models, such as waves or oscillations, lose energy over time, typically from friction or turbulence. The lost energy converts into heat, which raises the temperature of the system. Such systems are called...
of energy into heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
from their reversible
Reversible process (thermodynamics)
In thermodynamics, a reversible process, or reversible cycle if the process is cyclic, is a process that can be "reversed" by means of infinitesimal changes in some property of the system without loss or dissipation of energy. Due to these infinitesimal changes, the system is in thermodynamic...
thermal fluctuations
Thermal fluctuations
In statistical mechanics, thermal fluctuations are random deviations of a system from its equilibrium. All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they disappear altogether as temperature approaches absolute zero.Thermal fluctuations are a...
at thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...
. The fluctuation-dissipation theorem applies both to classical
Classical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...
and quantum mechanical
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
systems.
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, the theorem connects the linear response relaxation of a system from a prepared non-equilibrium state to its statistical fluctuation properties in equilibrium. Often the linear response takes the form of one or more exponential decays.
The fluctuation-dissipation theorem was originally formulated by Harry Nyquist
Harry Nyquist
Harry Nyquist was an important contributor to information theory.-Personal life:...
in 1928, and later proven by Herbert Callen
Herbert Callen
Herbert B. Callen was an American physicist best known as the author of the textbook Thermodynamics and an Introduction to Thermostatistics, the most frequently cited thermodynamic reference in physics research literature...
and Theodore A. Welton in 1951.
General applicability
The fluctuation-dissipation theorem is a general result of statistical thermodynamics that quantifies the relation between the fluctuations in a system at thermal equilibriumThermal equilibrium
Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space...
and the response of the system to applied perturbations.
The model 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, e.g., mechanical forces or electric fields, are weak enough that rates of relaxation
Relaxation time
In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium.Each relaxation process can be characterized by a relaxation time τ...
remain unchanged.
Brownian motion
For example, Albert EinsteinAlbert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
noted in his 1905 paper on Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
that the same random forces that 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 Einstein was able to use statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
to derive a previously unexpected connection, the Einstein-Smoluchowski relation
Einstein relation (kinetic theory)
In physics the Einstein relation is a previously unexpected connection revealed independently by Albert Einstein in 1905 and by Marian Smoluchowski in their papers on Brownian motion...
:
linking D, the diffusion constant
Fick's law of diffusion
Fick's laws of diffusion describe diffusion and can be used to solve for the diffusion coefficient, D. They were derived by Adolf Fick in the year 1855.- Fick's first law :...
, and μ, the mobility of the particles. (μ is the ratio of the particle's terminal drift velocity to an applied force, μ = vd / F). kB ≈ 1.38065 × 10−23 m2 kg s−2 K−1 is the Boltzmann constant, and T is the absolute temperature.
Thermal noise in a resistor
In 1928, John B. JohnsonJohn B. Johnson
John Bertrand "Bert" Johnson was a Swedish-born American electrical engineer and physicist...
discovered and Harry Nyquist
Harry Nyquist
Harry Nyquist was an important contributor to information theory.-Personal life:...
explained Johnson–Nyquist noise
Johnson–Nyquist noise
Johnson–Nyquist noise is the electronic noise generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage...
. With no applied current, the mean-square voltage depends on the resistance R,
, and the bandwidth over which the voltage is measured:
General formulation
The fluctuation-dissipation theorem can be formulated in many ways; one particularly useful form is the following:Let
be an observable of a dynamical system with HamiltonianHamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
subject to thermal fluctuations.The observable
will fluctuate around its mean value with fluctuations characterized by a power spectrum
.Suppose that we can switch on a scalar field
which alters the Hamiltonianto
.The response of the observable
to a time-dependent field ischaracterized to first order by the susceptibility
Susceptibility
*In physics, the susceptibility of a material or substance describes its response to an applied field. There are many kinds of susceptibilities, for example:These two susceptibilities are particular examples of a linear response function;...
or linear response function
Linear response function
A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response...
of the system
where the perturbation is adiabatically switched on at
.Now the fluctuation-dissipation theorem relates the power spectrum of
to the imaginary part of the Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the susceptibility ,-
.
The left-hand side describes fluctuations in
, the right-hand side is closely related to the energy dissipated by the system when pumped by an oscillatory field .This is the classical form of the theorem; quantum fluctuations are taken into account by
replacing
with (whose limit for is ). 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.
Derivation
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 off at t=0
We can express the expectation value of x by the probability distribution W(x,0) and the transition probability
The probability distribution function W(x,0) is an equilibrium distribution and hence
given by the Boltzmann distribution
Boltzmann distribution
In chemistry, physics, and mathematics, the Boltzmann distribution is a certain distribution function or probability measure for the distribution of the states of a system. It underpins the concept of the canonical ensemble, providing its underlying distribution...
for the Hamiltonian
For a weak field
, we can expand the right-hand side
here
is the equilibrium distribution in the absence of a field.Plugging this approximation in the formula for
yields
where A(t) is the auto-correlation function of x in the absence of a field.
Note that in the absence of a field the system is invariant under time-shifts.
We can rewrite
using the susceptibilityof the system and hence find with the above equation (*)
Consequently,
-
For stationary processStationary processIn the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...
es, the Wiener-Khinchin theorem states that
the power spectrum equals twice the Fourier transformFourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the auto-correlation
function
The last step is to Fourier transform equation (**) and to take the
imaginary part. For this it is useful to recall that the Fourier transform
of a real symmetric function is real, while the Fourier transform of a real
antisymmetric function is purely imaginary.
We can split
into a symmetric and an
anti-symmetric part
-
Now the fluctuation-dissipation theorem follows.
Violations in glassy systems
While the fluctuation-dissipation theorem 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. Glassy systems at low temperatures, as well as real glasses, are characterized by slow approaches to equilibrium states. Thus these systems require large time-scales to be studied while they remain in disequilibrium.
In the mid 1990s, in the study of non-equilibrium dynamics of spin glassSpin glassA spin glass is a magnet with frustrated interactions, augmented by stochastic disorder, where usually ferromagnetic and antiferromagnetic bonds are randomly distributed...
models, a generalization of the fluctuation-dissipation theorem was discovered that holds 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.
See also
- Non-equilibrium thermodynamicsNon-equilibrium thermodynamicsNon-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium; for they are changing or can be triggered to change over time, and are continuously and discontinuously...
- Green-Kubo relationsGreen-Kubo relationsThe Green–Kubo relations give the exact mathematical expression for transport coefficients in terms of integrals of time correlation functions.-Thermal and mechanical transport processes:...
- Onsager reciprocal relationsOnsager reciprocal relationsIn thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists....
- Equipartition theoremEquipartition theoremIn classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition...
- Boltzmann factorBoltzmann factorIn physics, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state i in a multi-state system in thermodynamic equilibrium at temperature T...
- Dissipative systemDissipative systemA dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter....
Further reading
- Audio recording of a lecture by Prof. E. W. Carlson of Purdue UniversityPurdue UniversityPurdue University, located in West Lafayette, Indiana, U.S., is the flagship university of the six-campus Purdue University system. Purdue was founded on May 6, 1869, as a land-grant university when the Indiana General Assembly, taking advantage of the Morrill Act, accepted a donation of land and...
- Kubo's famous text: Fluctuation-dissipation theorem
- Non-equilibrium thermodynamics
-
