KMS Relations and Fluctuation–Dissipation

The fluctuation–dissipation relation is the frequency-space consequence of the KMS condition. Stationarity alone is insufficient: a driven steady state may depend only on time differences while its Wightman functions fail the imaginary-time KMS relation and therefore admit no universal thermal factor.

Required background. Use thermal density operators and KMS and the chapter’s causal/statistical conventions.

Helpful background. Detailed balance and stochastic fluctuation–dissipation gives the classical effective counterpart.

From KMS to Wightman functions

For a neutral bosonic operator in $\rho_\beta=e^{-\beta H}/Z$, the trace argument underlying Kubo 1957, §§ 2–3 gives

$$\langle O(t)O(0)\rangle_\beta =\langle O(0)O(t+i\beta)\rangle_\beta.$$

With $G^>(t)=-i\langle O(t)O(0)\rangle$ and $G^<(t)=-i\langle O(0)O(t)\rangle$, and Fourier transform $G(\omega)=\int dt\,e^{i\omega t}G(t)$,

$$G^>(\omega)=e^{\beta\omega}G^<(\omega).$$

Since $G^K=G^>+G^<$ and $G^R-G^A=G^>-G^<$,

$$G^K(\omega)=\coth\!\left(\frac{\beta\omega}{2}\right) [G^R(\omega)-G^A(\omega)]$$

for bosons. For fermions the antiperiodic KMS sign yields

$$G^K(\omega)=\tanh\!\left(\frac{\beta\omega}{2}\right) [G^R(\omega)-G^A(\omega)]$$

under the corresponding fermionic definitions. These compact formulas are convention sensitive; the Wightman ratio is the safest invariant starting point.

If $O$ carries charge $q$ and the state is $e^{-\beta(H-\mu Q)}$, the thermal factor becomes $\omega-\mu q$, provided $[Q,O]=-qO$ under the declared sign convention. Pairing charged conjugate operators matters: applying a neutral formula to $OO$ when the nonzero correlator is $OO^\dagger$ is an error.

KMS appears only in the final, explicitly conditional box of the contour schematic. The contour and $r/a$ basis exist for arbitrary normalized initial states; thermal analyticity is the extra hypothesis that closes spectral and statistical information into a fluctuation–dissipation relation.

Unitarity and largest-time identities constrain every normalized closed-time-path theory, while KMS additionally requires a thermal state and time-translation-invariant equilibrium dynamics. The diagram’s quadratic-inversion step applies to the two-point propagators; thermal vertex relations require the separately rotated interaction and KMS symmetry. Under the equilibrium hypotheses, KMS relates Wightman functions and hence the spectral and statistical correlators. Stationarity alone does not justify the last relation. The diagram is schematic and not to scale.

The sections From KMS to Wightman functions and Why stationarity is not KMS give the text and equation equivalent of the equilibrium hypothesis, derived relation, and failure boundary.

Classical and zero-frequency limits

For $|\beta\omega|\ll1$,

$$\coth(\beta\omega/2)=\frac{2T}{\omega}+O(\omega/T).$$

Thus low-frequency fluctuations are enhanced relative to dissipation. The apparent $1/\omega$ singularity is interpreted only together with the small-$\omega$ behavior of $G^R-G^A$. A conserved density can contain a delta function or hydrodynamic pole, so setting $\omega=0$ before taking the regulated limit is unsafe.

For a free oscillator, $G^R-G^A$ has peaks at $\omega=\pm\omega_0$. Multiplication by $\coth(\beta\omega/2)$ gives weights $2n_B(\omega_0)+1$, exactly reproducing $G^K$. This checks the sign at both positive and negative frequency.

Why stationarity is not KMS

A density matrix diagonal in the energy basis is stationary, but unless its weights have the Gibbs form—or generalized chemical potentials associated with the operator algebra—its Wightman ratio is not $e^{\beta\omega}$ with a single $\beta$. A periodically driven steady state, generalized Gibbs ensemble, or mode-dependent Gaussian state can likewise be stationary without ordinary KMS.

Defining

$$\coth\!\left(\frac{\omega}{2T_{\mathrm{eff}}(\omega)}\right) =\frac{G^K}{G^R-G^A}$$

is only a parametrization when the ratio is well defined. A physical effective temperature requires a frequency window, operator independence, positive thermodynamic interpretation, and consistency with independent observables. One fitted ratio does not establish thermalization.

Failure tests

• Test the Wightman KMS ratio directly, including the operator charge and chemical-potential shift.
• State the Fourier sign; reversing it changes which exponential appears.
• Vary the frequency window and operator before quoting an effective temperature.
• Resolve zero-frequency distributions and contact terms before taking limits.
• Do not impose KMS on a driven or aging state merely because a late-time correlator looks stationary over a short interval.

Exercise

Starting from $G^>=e^{\beta\omega}G^<$, derive the bosonic $\coth$ factor.

Solution

Write $G^K=(e^{\beta\omega}+1)G^<$ and $G^R-G^A=(e^{\beta\omega}-1)G^<$. Their ratio is $(e^{\beta\omega}+1)/(e^{\beta\omega}-1)=\coth(\beta\omega/2)$.

Continue

Use nonlinear response for higher source derivatives. In a general nonequilibrium state, evolve $F$ and $\rho$ independently rather than imposing this relation.

References

• Callen, H. B., and Welton, T. A. (1951). “Irreversibility and Generalized Noise.” Physical Review 83, 34–40. DOI.
• Kubo, R. (1957). “Statistical-Mechanical Theory of Irreversible Processes. I.” Journal of the Physical Society of Japan 12, 570–586. DOI.
• Martin, P. C., and Schwinger, J. (1959). “Theory of Many-Particle Systems. I.” Physical Review 115, 1342–1373. DOI.