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The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in
statistical physics Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approxim ...
for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the theorem is a proof that thermodynamic fluctuations in a physical variable predict the response quantified by the
admittance In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance & resistance are defined. The SI unit of admittanc ...
or impedance (to be intended in their general sense, not only in electromagnetic terms) of the same physical variable (like voltage, temperature difference, etc.), and vice versa. The fluctuation–dissipation theorem applies both to classical and
quantum mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
systems. The fluctuation–dissipation theorem was proven by Herbert Callen and Theodore Welton in 1951 and expanded by
Ryogo Kubo was a Japanese mathematical physicist, best known for his works in statistical physics and non-equilibrium statistical mechanics. Work In the early 1950s, Kubo transformed research into the linear response A linear response function describ ...
. There are antecedents to the general theorem, including
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
's explanation of
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
during his ''
annus mirabilis ''Annus mirabilis'' (pl. ''anni mirabiles'') is a Latin phrase that means "marvelous year", "wonderful year", "miraculous year", or "amazing year". This term has been used to refer to several years during which events of major importance are re ...
'' and
Harry Nyquist Harry Nyquist (, ; February 7, 1889 – April 4, 1976) was a Swedish-American physicist and electronic engineer who made important contributions to communication theory. Personal life Nyquist was born in the village Nilsby of the parish Stora ...
's explanation in 1928 of
Johnson noise Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a ...
in electrical resistors.


Qualitative overview and examples

The fluctuation–dissipation theorem says that when there is a process that dissipates energy, turning it into heat (e.g., friction), there is a reverse process related to
thermal fluctuation In statistical mechanics, thermal fluctuations are random deviations of a system from its average state, that occur in a system at equilibrium.In statistical mechanics they are often simply referred to as fluctuations. All thermal fluctuations b ...
s. This is best understood by considering some examples: * '' Drag and
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
'' *:If an object is moving through a fluid, it experiences drag (air resistance or fluid resistance). Drag dissipates kinetic energy, turning it into heat. The corresponding fluctuation is
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
. An object in a fluid does not sit still, but rather moves around with a small and rapidly-changing velocity, as molecules in the fluid bump into it. Brownian motion converts heat energy into kinetic energy—the reverse of drag. * '' Resistance and
Johnson noise Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a ...
'' *:If electric current is running through a wire loop with a
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...
in it, the current will rapidly go to zero because of the resistance. Resistance dissipates electrical energy, turning it into heat (
Joule heating Joule heating, also known as resistive, resistance, or Ohmic heating, is the process by which the passage of an electric current through a conductor produces heat. Joule's first law (also just Joule's law), also known in countries of former US ...
). The corresponding fluctuation is
Johnson noise Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a ...
. A wire loop with a resistor in it does not actually have zero current, it has a small and rapidly-fluctuating current caused by the thermal fluctuations of the electrons and atoms in the resistor. Johnson noise converts heat energy into electrical energy—the reverse of resistance. * '' Light absorption and
thermal radiation Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) i ...
'' *:When light impinges on an object, some fraction of the light is absorbed, making the object hotter. In this way, light absorption turns light energy into heat. The corresponding fluctuation is
thermal radiation Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) i ...
(e.g., the glow of a "red hot" object). Thermal radiation turns heat energy into light energy—the reverse of light absorption. Indeed,
Kirchhoff's law of thermal radiation In heat transfer, Kirchhoff's law of thermal radiation refers to wavelength-specific radiative emission and absorption by a material body in thermodynamic equilibrium, including radiative exchange equilibrium. It is a special case of Onsage ...
confirms that the more effectively an object absorbs light, the more thermal radiation it emits.


Examples in detail

The fluctuation–dissipation theorem is a general result of statistical thermodynamics that quantifies the relation between the fluctuations in a system that obeys detailed balance and the response of the system to applied perturbations.


Brownian motion

For example,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
noted in his 1905 paper on
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
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 In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
to derive the
Einstein–Smoluchowski relation In physics (specifically, the kinetic theory of gases), the Einstein relation is a previously unexpected connection revealed independently by William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchowski in 1906 in their works on ...
: D = which connects the
diffusion constant Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equ ...
''D'' and the particle mobility ''μ'', the ratio of the particle's
terminal Terminal may refer to: Computing Hardware * Terminal (electronics), a device for joining electrical circuits together * Terminal (telecommunication), a device communicating over a line * Computer terminal, a set of primary input and output dev ...
drift velocity In physics, a drift velocity is the average velocity attained by charged particles, such as electrons, in a material due to an electric field. In general, an electron in a conductor will propagate randomly at the Fermi velocity, resulting in an a ...
to an applied force. ''k''B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, and ''T'' is the
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
.


Thermal noise in a resistor

In 1928, John B. Johnson discovered and
Harry Nyquist Harry Nyquist (, ; February 7, 1889 – April 4, 1976) was a Swedish-American physicist and electronic engineer who made important contributions to communication theory. Personal life Nyquist was born in the village Nilsby of the parish Stora ...
explained
Johnson–Nyquist noise Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the electronic noise generated by the thermal agitation of the charge carriers (usually the electrons) inside an electrical conductor at equilibrium, which happens rega ...
. With no applied current, the mean-square voltage depends on the resistance R, k_T, and the bandwidth \Delta\nu over which the voltage is measured: : \langle V^2 \rangle \approx 4Rk_T\,\Delta\nu. This observation can be understood through the lens of the fluctuation-dissipation theorem. Take, for example, a simple circuit consisting of a
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...
with a resistance R and a
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
with a small capacitance C. Kirchhoff's law yields :V=-R\frac+\frac and so the
response function In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
for this circuit is :\chi(\omega)\equiv\frac=\frac In the low-frequency limit \omega\ll (RC)^, its imaginary part is simply :\text\left chi(\omega)\rightapprox \omega RC^2 which then can be linked to the power spectral density function S_V(\omega) of the voltage via the fluctuation-dissipation theorem :S_V(\omega)=\frac\approx \frac\text\left chi(\omega)\right2Rk_T The Johnson–Nyquist voltage noise \langle V^2 \rangle was observed within a small frequency
bandwidth Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...
\Delta \nu=\Delta\omega/(2\pi) centered around \omega=\pm \omega_0. Hence :\langle V^2 \rangle\approx S_V(\omega)\times 2\Delta \nu\approx 4Rk_T\Delta \nu


General formulation

The fluctuation–dissipation theorem can be formulated in many ways; one particularly useful form is the following:. Let x(t) be an
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
of a
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
with
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
H_0(x) subject to thermal fluctuations. The observable x(t) will fluctuate around its mean value \langle x\rangle_0 with fluctuations characterized by a
power spectrum The power spectrum S_(f) of a time series x(t) describes the distribution of Power (physics), power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discre ...
S_x(\omega) = \langle \hat(\omega)\hat^*(\omega) \rangle. Suppose that we can switch on a time-varying, spatially constant field f(t) which alters the Hamiltonian to H(x)=H_0(x)-f(t)x. The response of the observable x(t) to a time-dependent field f(t) is characterized to first order by the susceptibility or
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. Because of its many applications in information t ...
\chi(t) of the system : \langle x(t) \rangle = \langle x \rangle_0 + \int_^ \! f(\tau) \chi(t-\tau)\,d\tau, where the perturbation is adiabatically (very slowly) switched on at \tau =-\infty. The fluctuation–dissipation theorem relates the two-sided power spectrum (i.e. both positive and negative frequencies) of x to the imaginary part of the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
\hat(\omega) of the susceptibility \chi(t): S_x(\omega) = -\frac \operatorname\hat(\omega). Which holds under the Fourier transform convention f(\omega)=\int_^\infty f(t) e^\, dt. 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 \sin(\omega t + \phi). This is the classical form of the theorem; quantum fluctuations are taken into account by replacing 2 k_\mathrm T / \omega with \hbar \, \coth(\hbar\omega / 2k_\mathrmT) (whose limit for \hbar\to 0 is 2 k_\mathrm T/\omega). 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 straightforward way to the case of space-dependent fields, to the case of several variables or to a quantum-mechanics setting. A special case in which the fluctuating quantity is the energy itself is the fluctuation-dissipation theorem for the frequency-dependent specific heat.


Derivation


Classical version

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 : f(t)=f_0 \theta(-t) , where \theta(t) is the
Heaviside function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
. 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 hence given by the
Boltzmann distribution In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability t ...
for the Hamiltonian H(x) = H_0(x) - x f_0 : W(x,0)= \frac \,, where \beta^ = k_T. For a weak field \beta x f_0 \ll 1 , we can expand the right-hand side : W(x,0) \approx W_0(x) +\beta f_0 (x(0)-\langle x \rangle_0) 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 where ''A''(''t'') is the auto-correlation function of ''x'' in the absence of a field: : A(t)=\langle (t)-\langle x \rangle_0 x(0)-\langle x \rangle_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 susceptibility of the system and hence find with the above equation (*) : f_0 \int_0^ d\tau \, \chi(\tau) \theta(\tau-t) = \beta f_0 A(t) Consequently, To make a statement about frequency dependence, it is necessary to take the Fourier transform of equation (**). By integrating by parts, it is possible to show that : -\hat\chi(\omega) = i\omega\beta \int_0^\infty e^ A(t)\, dt -\beta A(0). Since A(t) is real and symmetric, it follows that : 2 \operatorname hat\chi(\omega)= -\omega\beta \hat A(\omega). Finally, for
stationary process In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...
es, the
Wiener–Khinchin theorem In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary ...
states that the two-sided
spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, o ...
is equal to the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the auto-correlation function: : S_x(\omega) = \hat(\omega). Therefore, it follows that : S_x(\omega) = -\frac \operatorname hat\chi(\omega)


Quantum version

The fluctuation-dissipation theorem relates the
correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables rep ...
of the observable of interest \langle \hat(t)\hat(0)\rangle (a measure of fluctuation) to the imaginary part of the
response function In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
\text\left chi(\omega)\right\left chi(\omega)-\chi^*(\omega)\right2i in the frequency domain (a measure of dissipation). A link between these quantities can be found through the so-called Kubo formula :\chi(t-t')=\frac\theta(t-t')\langle hat(t),\hat(t')\rangle which follows, under the assumptions of the
linear response 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. Because of its many applications in information ...
theory, from the time evolution of the
ensemble average In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...
of the observable \langle\hat(t)\rangle in the presence of a perturbing source. Once Fourier transformed, the Kubo formula allows writing the imaginary part of the response function as :\text\left chi(\omega)\right\frac\int_^\langle \hat(t)\hat(0)-\hat(0)\hat(t)\rangle e^dt. In the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat b ...
, the second term can be re-expressed as :\langle \hat(0) \hat(t)\rangle=\text e^\hat(0)\hat(t)=\text \hat(t) e^\hat(0)=\text e^\underbrace_\hat(0)=\langle \hat(t-i\hbar\beta) \hat(0)\rangle where in the second equality we re-positioned \hat(t) using the cyclic property of trace. Next, in the third equality, we inserted e^e^ next to the trace and interpreted e^ as a time evolution operator e^ with
imaginary time Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories. M ...
interval \Delta t=-i\hbar\beta. The imaginary time shift turns into a e^ factor after Fourier transform :\int_^\langle \hat(t-i\hbar\beta)\hat(0)\rangle e^dt=e^\int_^\langle \hat(t)\hat(0)\rangle e^dt and thus the expression for \text\left chi(\omega)\right/math> can be easily rewritten as the quantum fluctuation-dissipation relation :S_(\omega)=2\hbar\left _(\omega)+1\righttext\left chi(\omega)\right/math> where the power spectral density S_(\omega) is the Fourier transform of the auto-correlation \langle \hat(t) \hat(0)\rangle and n_(\omega)=\left(e^-1\right)^ is the Bose-Einstein distribution function. The same calculation also yields :S_(-\omega)=e^S_(\omega) = 2\hbar\left _(\omega)\righttext\left chi(\omega)\rightneq S_(+\omega) thus, differently from what obtained in the classical case, the power spectral density is not exactly frequency-symmetric in the quantum limit. Consistently, \langle \hat(t)\hat(0)\rangle has an imaginary part originating from the commutation rules of operators. The additional "+1" term in the expression of S_x(\omega) at positive frequencies can also be thought of as linked to
spontaneous emission Spontaneous emission is the process in which a quantum mechanical system (such as a molecule, an atom or a subatomic particle) transits from an excited energy state to a lower energy state (e.g., its ground state) and emits a quantized amount of ...
. An often cited result is also the symmetrized power spectral density :\frac=2\hbar\left _(\omega)+\frac\righttext\left chi(\omega)\right\hbar\coth\left(\frac\right)\text\left chi(\omega)\right The "+1/2" can be thought of as linked to
quantum fluctuation In quantum physics, a quantum fluctuation (also known as a vacuum state fluctuation or vacuum fluctuation) is the temporary random change in the amount of energy in a point in space, as prescribed by Werner Heisenberg's uncertainty principle. ...
s, or to zero-point motion of the observable \hat. At high enough temperatures, n_\approx (\beta\hbar\omega)^\gg 1, i.e. the quantum contribution is negligible, and we recover the classical version.


Violations in glassy systems

While the fluctuation–dissipation theorem provides a general relation between the response of systems obeying detailed balance, when detailed balance is violated comparison of fluctuations to dissipation is more complex. Below the so called glass temperature T_, glassy systems are not equilibrated, and slowly approach their equilibrium state. This slow approach to equilibrium is synonymous with the violation of detailed balance. Thus these systems require large time-scales to be studied while they slowly move toward equilibrium. To study the violation of the fluctuation-dissipation relation in glassy systems, particularly
spin glasses In condensed matter physics, a spin glass is a magnetic state characterized by randomness, besides cooperative behavior in freezing of spins at a temperature called 'freezing temperature' ''Tf''. In ferromagnetic solids, component atoms' mag ...
, Ref. performed numerical simulations of macroscopic systems (i.e. large compared to their correlation lengths) described by the three-dimensional Edwards-Anderson model using supercomputers. In their simulations, the system is initially prepared at a high temperature, rapidly cooled to a temperature T=0.64 T_ below the glass temperature T_g, and left to equilibrate for a very long time t_ under a magnetic field H. Then, at a later time t + t_, two dynamical observables are probed, namely the
response function In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
\chi(t+t_,t_)\equiv\left.\frac\_ and the spin-temporal
correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables rep ...
C(t+t_,t_)\equiv \frac\left.\sum_\langle S_x(t_) S_x(t+t_)\rangle\_ where S_x=\pm 1 is the spin living on the node x of the cubic lattice of volume V, and m(t)\equiv \frac \sum_ \langle S_(t) \rangle is the magnetization density. The fluctuation-dissipation relation in this system can be written in terms of these observables as T\chi(t+t_, t_)=1-C(t+t_, t_) Their results confirm the expectation that as the system is left to equilibrate for longer times, the fluctuation-dissipation relation is closer to be satisfied. In the mid-1990s, in the study of dynamics of
spin glass In condensed matter physics, a spin glass is a magnetic state characterized by randomness, besides cooperative behavior in freezing of spins at a temperature called 'freezing temperature' ''Tf''. In ferromagnetic solids, component atoms' magne ...
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 thermodynamics Non-equilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium but can be described in terms of macroscopic quantities (non-equilibrium state variables) that represent an ext ...
*
Green–Kubo relations The Green–Kubo relations ( Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients \gamma in terms of integrals of time correlation functions: :\gamma = \int_0^\infty \left\langle \dot(t) \dot ...
*
Onsager reciprocal relations In 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. "Reciprocal relations" occur betwe ...
*
Equipartition theorem In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
*
Boltzmann distribution In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability t ...
*
Dissipative system A 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. A tornado may be thought of as a dissipative system. Dis ...


Notes


References

* * *


Further reading


Audio recording
of a lecture by Prof. E. W. Carlson of
Purdue University Purdue University is a public land-grant research university in West Lafayette, Indiana, and the flagship campus of the Purdue University system. The university was founded in 1869 after Lafayette businessman John Purdue donated land and money ...

Kubo's famous text: Fluctuation-dissipation theorem
* * * * * * * * * * {{DEFAULTSORT:Fluctuation-Dissipation Theorem Statistical mechanics Non-equilibrium thermodynamics Physics theorems Statistical mechanics theorems