Fluctuation–dissipation theorem

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:

$D={\mu \,k_{B}T}$

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, $k_{B}T$, and the bandwidth $\Delta \nu$ over which the voltage is measured:

$\langle V^{2}\rangle =4Rk_{B}T\,\Delta \nu .$

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 system with Hamiltonian H₀(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 Hamiltionian to $H(x)=H_{0}(x)+xf$. The response of the observable x to a field f(t) changing with time is characterized (to first order) by the susceptibility or linear response function $\chi (t)$ of the system

$\langle x(t)\rangle =\langle x\rangle _{0}+\int _{-\infty }^{t}\chi (\tau )f(t-\tau )d\tau \,,$

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 $\chi (t)$

$S_{x}(\omega )={\frac {2k_{B}T}{\omega }}{\rm {{Im}[{\tilde {\chi }}(\omega )]}}$.

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 by replacing 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 off at 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 hence given by the Boltzmann distribution for the Hamiltionian $H(x)=H_{0}(x)+xf_{0}$

$W(x,0)={\frac {\exp(-\beta H(x))}{\int dx\,\exp(-\beta H(x))}}.$

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.

$A(t)=\langle x(t)x(0)\rangle .$

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 (*)

$f_{0}\int _{0}^{\infty }d\tau \,\chi (\tau )\theta (\tau -t)=\beta f_{0}A(t)$

Consequently,

$(**)-\chi (t)=\beta {\frac {d}{dt}}(A(t)\theta (t)).$

For stationary processes, the Wiener-Khinchin theorem states, that the power spectrum equals twice the Fourier transform of the auto-correlation function

$S_{x}(\omega )=2{\tilde {A}}(\omega ).$

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 ${\frac {d}{dt}}(A(t)\theta (t))$ into a symmetric and an anti-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

$\langle A\rangle =\mathrm {Tr} \left\{A(q,p)\ f(q,p)\right\}={\frac {\mathrm {Tr} \left\{Ae^{-\beta H}\right\}}{\mathrm {Tr} \ e^{-\beta H}}}$

where β = 1/k_BT, 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_BT. 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_BT, 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

$\langle \beta h\rangle ={\frac {\mathrm {Tr} \left\{\beta he^{-\beta H}\right\}}{\mathrm {Tr} \ e^{-\beta H}}}$

which is much less than one, by our assumption that h is much smaller than the thermal energy 1/β = k_BT.

Combining the numerator and denominator, dropping quadratic and high-order terms in <βh>, and using the indifference of equilibrium to time, we obtain

${\bar {A}}(t)-\langle A\rangle =-\beta \langle Ah\rangle +\beta \langle A\rangle \langle h\rangle$

Let the perturbation h = −gA be proportional to the variable A with a constant −g. Then this formula becomes

${\bar {A}}(t)-\langle A\rangle =\beta g\left\{\langle A(t)A(0)\rangle -\langle A\rangle ^{2}\right\}=\beta g\langle \delta A(t)\delta A(0)\rangle =\beta gc(t).$

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, g₁ and g₂ are applied, the net relaxation will be the sum of the individual relaxations to g₁ and g₂ 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

• Non-equilibrium thermodynamics
• Green-Kubo relations
• Onsager reciprocal relations
• Equipartition theorem
• Boltzmann factor
• Dissipative system

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, [1]

Further resources

• Audio recording of a lecture by Prof. E. W. Carlson of Purdue University

• Weber J (1956). "Fluctuation Dissipation Theorem". Physical Review. 101: 1620–1626. doi:10.1103/PhysRev.101.1620.

• Felderhof BU (1978). "On the derivation of the fluctuation-dissipation theorem" (PDF). Journal of Mathematical Physics A. 11: 921–927. doi:10.1088/0305-4470/11/5/021.

• Chandler D (1987). Introduction to Modern Statistical Mechanics. Oxford University Press. pp. pp. 231–265. ISBN 978-0195042771. {{cite book}}: |pages= has extra text (help)

• Reichl LE (1980). A Modern Course in Statistical Physics. Austin TX: University of Texas Press. pp. pp. 545–595. ISBN 0-292-75080-3. {{cite book}}: |pages= has extra text (help)

• Plischke M, Bergersen B (1989). Equilibrium Statistical Physics. Englewood Cliffs, NJ: Prentice Hall. pp. pp. 251–296. ISBN 0-13-283276-3. {{cite book}}: |pages= has extra text (help)

• Pathria RK (1972). Statistical Mechanics. Oxford: Pergamon Press. pp. 443, 474–477. ISBN 0-08-018994-6.

• Huang K (1987). Statistical Mechanics. New York: John Wiley and Sons. pp. pp. 153, 394–396. ISBN 0-471-81518-7. {{cite book}}: |pages= has extra text (help)

• Callen HB (1985). Thermodynamics and an Introduction to Thermostatistics. New York: John Wiley and Sons. pp. pp. 307–325. ISBN 0-471-86256-8. {{cite book}}: |pages= has extra text (help)
• "Herbert B. Callen, 73, Theoretical Physicist", The New York Times obituary published on May 271993