Fluctuations, Response, and Susceptibilities

Equilibrium fluctuations and linear response are two views of the same generating function. A distribution that spreads widely in an observable is sensitive to the field conjugate to that observable. The ordinary variance formula is exact for classical or commuting static response; noncommuting quantum response uses an imaginary-time correlation. Time-dependent dissipation adds causality and a frequency-resolved fluctuation–dissipation theorem whose normalization must be stated.

Required background. Calculus and Linear Algebra supplies multivariable differentiation and positive quadratic forms; covariance and the Fourier convention are defined below. Helpful background. Classical and Variational Mechanics supplies Hamiltonians; Quantum States, Observables, and Measurement supplies operator language, while Unitary Dynamics and Pictures of Motion supplies commutators and Heisenberg evolution for the optional quantum portions.

A source-dependent ensemble

Let a controllable field $h$ couple to an observable $X$ through the explicit sign convention

$$H(h)=H_0-hX.$$

At temperature $T\gt 0$, with $k_{\mathrm B}=1$ and $\beta=1/T$,

$$Z_{\mathrm{cl}}(h)=\sum_i e^{-\beta(H_{0,i}-hX_i)}$$

for a discrete classical system; a continuous classical system uses the corresponding phase-space integral with its specified measure. For a quantum system,

$$Z_{\mathrm q}(h)=\operatorname{Tr}e^{-\beta(H_0-hX)}.$$

Write either partition function as $Z(h)$ when the context is clear, and define $F(h)=-T\ln Z(h)$. Direct differentiation classically, or the Duhamel formula together with trace cyclicity quantum mechanically, gives without a commutativity assumption

$$\frac{\partial\ln Z}{\partial h}=\beta\langle X\rangle.$$

Differentiating again yields the ordinary variance formula classically, or quantum mechanically when $[H_0,X]=0$. In those cases the isothermal static susceptibility is

$$\boxed{ \chi_X \equiv \left(\frac{\partial\langle X\rangle}{\partial h}\right)_T =\beta\left(\langle X^2\rangle-\langle X\rangle^2\right) }.$$

Thus $\chi_X\geq0$ for the chosen $-hX$ coupling. Reversing the source sign reverses the first-derivative convention, which is why a susceptibility formula is incomplete unless the perturbing Hamiltonian is stated. Equivalently,

$$\frac{\partial^2\ln Z}{\partial h^2}=\beta^2\operatorname{Var}(X)\geq0, \qquad \frac{\partial^2F}{\partial h^2}=-\chi_X\leq0.$$

The convexity of $\ln Z$ and concavity of $F$ in the applied source are statistical stability checks.

With several sources, $H=H_0-\sum_a h_aX_a$, define $\Delta X_a=X_a-\langle X_a\rangle$ and

$$\operatorname{Cov}(X_a,X_b)=\langle\Delta X_a\Delta X_b\rangle.$$

The classical response matrix is

$$\chi_{ab} =\frac{\partial\langle X_a\rangle}{\partial h_b} =\beta\,\operatorname{Cov}(X_a,X_b).$$

The same formula holds quantum mechanically when $H_0$ and all source operators are jointly commuting: $[H_0,X_a]=0$ and $[X_a,X_b]=0$ for every $a,b$. The covariance matrix is positive semidefinite because, for every real vector $v_a$,

$$v_a\chi_{ab}v_b =\beta\operatorname{Var}\!\left(\sum_a v_aX_a\right)\geq0.$$

This supplies a compact consistency test for coupled susceptibilities and gives the bound $|\chi_{ab}|^2\leq\chi_{aa}\chi_{bb}$. Symmetry of the static matrix follows from equality of mixed derivatives when the free energy is smooth. Dynamical reciprocity needs additional time-reversal assumptions and should not be inferred from this static argument alone.

Quantum Statistical Mechanics and Density Operators develops density operators and thermal traces in detail. For noncommuting quantum observables, the derivative of an operator exponential cannot be treated as an ordinary scalar derivative. At zero source, with $\rho_0=Z_0^{-1}e^{-\beta H_0}$ and $\Delta X_a=X_a-\langle X_a\rangle_0$,

$$\chi_{ab} =\int_0^\beta d\tau\, \left\langle\Delta X_b(-i\tau)\,\Delta X_a(0)\right\rangle_0, \qquad X_b(-i\tau)=e^{\tau H_0}X_be^{-\tau H_0}.$$

This Kubo–Mori correlation reduces to $\beta\operatorname{Cov}(X_a,X_b)$ in the jointly commuting case; Kubo gives the corresponding canonical correlation in Kubo 1957, §5, equations (5.5)–(5.8), printed p. 580.

A two-level check of noncommuting response

Let $H(h)=\Delta\sigma_z-h\sigma_x$, with $\Delta\gt0$ an energy and $X=\sigma_x$ dimensionless. The Pauli matrices obey $\sigma_x^2=\sigma_z^2=I$ and $\sigma_x\sigma_z+\sigma_z\sigma_x=0$, so $H(h)^2=(\Delta^2+h^2)I$. Its energies are therefore $\pm\varepsilon_h$, where $\varepsilon_h=\sqrt{\Delta^2+h^2}$. Consequently,

$$Z(h)=2\cosh(\beta\varepsilon_h), \qquad \langle X\rangle_h =T\partial_h\ln Z =\frac{h}{\varepsilon_h}\tanh(\beta\varepsilon_h).$$

At zero source this gives

$$\chi_X=\frac{\tanh(\beta\Delta)}{\Delta}, \qquad \beta\operatorname{Var}_0(X)=\beta.$$

Both have units of inverse energy, but they differ because the source rotates the energy eigenvectors as well as changing their weights. At $\beta\Delta\ll1$ they agree to leading order. As $T\to0$, the actual susceptibility approaches $1/\Delta$, while the ordinary variance expression diverges. A nonzero quantum measurement variance in the ground state is therefore not a thermal fluctuation that may be inserted into the classical response formula.

Energy and particle-number fluctuations

For a $\beta$-independent Hamiltonian at fixed remaining controls, energy fluctuations follow by treating $\beta$ as the source-like parameter:

$$\operatorname{Var}(H)=\frac{\partial^2\ln Z}{\partial\beta^2} =T^2 C_V.$$

Tong derives this identity and its regular single-phase $N^{-1/2}$ relative-fluctuation scaling in Tong, current notes, §1.3.2, printed pp. 19–21. In a grand-canonical ensemble, $\operatorname{Var}(N)=T(\partial\langle N\rangle/\partial\mu)_{T,V}$ connects number fluctuations to particle-number response and, through the equation of state, to compressibility.

Exercise: independent paramagnetic spins

Take $N$ spins $s_i=\pm1$ with $X=\sum_i s_i$ and $H=-hX$. Compute the magnetization, susceptibility, and variance, then interpret the weak- and strong-field limits.

Solution

The partition function factorizes:

$$Z=(2\cosh\beta h)^N.$$

Therefore

$$\langle X\rangle =T\frac{\partial\ln Z}{\partial h} =N\tanh(\beta h),$$

and

$$\chi_X=N\beta\operatorname{sech}^2(\beta h).$$

Independence also gives $\operatorname{Var}(X)=N\operatorname{sech}^2(\beta h)$, so $\chi_X=\beta\operatorname{Var}(X)$ exactly. At $h=0$, $\chi_X=N/T$: lowering temperature increases response because thermal disorder less strongly opposes alignment. Saturation at $|\beta h|\gg1$ suppresses both variance and response.

From static response to causal response

Let the unperturbed equilibrium state be $\rho_0=Z_0^{-1}e^{-\beta H_0}$ and define Heisenberg operators by $A(t)=e^{iH_0t}Ae^{-iH_0t}$. For a time-dependent perturbation $H'(t)=-f(t)B$, linear response has the causal form

$$\delta\langle A(t)\rangle =\int_{-\infty}^{t}dt'\,\chi_{AB}(t-t')f(t').$$

With $\hbar=1$, quantum equilibrium, and this source sign, the retarded kernel is

$$\chi_{AB}(t)=i\,\theta(t)\langle[A(t),B(0)]\rangle_{\mathrm{eq}}.$$

Kubo’s original paper derives the response or “after-effect” function in Kubo 1957, §2, equations (2.8)–(2.19). Tokmakoff states the perturbation convention and commutator kernel in Tokmakoff 2009, printed pp. 7–9, equations (1.34) and (1.51), but writes the perturbation with the opposite source sign; translating his convention to $H'=-fB$ gives the displayed $+i$ kernel.

A normalized dynamical fluctuation–dissipation theorem

For a Hermitian observable $X$, define the connected ordered spectrum and the retarded transform by

$$\begin{aligned} S_{XX}(\omega) &=\int_{-\infty}^{\infty}dt\,e^{i\omega t} \langle\Delta X(t)\Delta X(0)\rangle_0,\\ \chi_{XX}(\omega) &=\int_{-\infty}^{\infty}dt\,e^{i\omega t}\chi_{XX}(t), \qquad \chi''_{XX}(\omega)=\operatorname{Im}\chi_{XX}(\omega). \end{aligned}$$

With these conventions and $\hbar=1$, the equilibrium spectral decomposition gives

$$S_{XX}(-\omega)=e^{-\beta\omega}S_{XX}(\omega), \qquad \chi''_{XX}(\omega) =\frac{1-e^{-\beta\omega}}{2}S_{XX}(\omega).$$

The first identity is detailed balance; the second is the quantum fluctuation–dissipation theorem. For the symmetrized spectrum

$$S^{\mathrm{sym}}_{XX}(\omega) =\frac12\int_{-\infty}^{\infty}dt\,e^{i\omega t} \left\langle\{\Delta X(t),\Delta X(0)\}\right\rangle_0,$$

the same result, for $\omega\neq0$ or for regular parts after separating zero-frequency elastic terms, becomes

$$S^{\mathrm{sym}}_{XX}(\omega) =\coth\!\left(\frac{\beta\omega}{2}\right)\chi''_{XX}(\omega).$$

Tong derives the ordered-spectrum relation with the same $e^{i\omega t}$ Fourier transform in Tong, Kinetic Theory, §4.3.2, printed pp. 96–98, equations (4.34)–(4.37). His source Hamiltonian is $H_{\mathrm{source}}=+\phi X$, so his response kernel and $\chi''$ carry the opposite sign. Setting $\phi=-f$ translates his convention to the $H'=-fX$ convention used here and gives the displayed positive relation.

For the regular low-frequency part in the classical regime $|\beta\omega|\ll1$, this reduces to $S^{\mathrm{sym}}_{XX}(\omega)\simeq2T\chi''_{XX}(\omega)/\omega$. The factors change if one changes the Fourier transform, source sign, operator ordering, or restores $\hbar$. This is why a static variance identity cannot be promoted to a frequency-domain statement without declaring conventions.

Limits and failure modes

The derivations assume equilibrium, a specified ensemble, differentiability, weak forcing, and convergent transforms. Near a continuous transition, a susceptibility may diverge in the thermodynamic limit while finite size rounds the divergence. Metastability can make measured response history-dependent. Far from equilibrium there is generally no universal relation between spontaneous fluctuations and response.

The order of limits can matter: taking frequency to zero before wave number may probe an equilibrated isothermal response, while reversing the order can retain a conserved-density constraint. A quoted “static susceptibility” must therefore identify which variables and conservation laws were allowed to relax.

If $X$ is exactly conserved, then $\Delta X(t)=\Delta X$ and

$$S^{\mathrm{sym}}_{XX}(\omega) =2\pi\operatorname{Var}(X)\,\delta(\omega), \qquad \chi''_{XX}(\omega)=0.$$

The zero-frequency elastic term must therefore be separated before using the displayed $\coth$ relation; it cannot be recovered by substituting $\omega=0$ into an identity for regular spectral parts.

Quantum Statistical Mechanics and Density Operators deepens the operator treatment and distinguishes thermal mixtures from reduced-state mixedness.

References

• Kubo, Ryogo. “Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems.” Journal of the Physical Society of Japan 12 (1957): 570–586, §§2–3 and §5. doi:10.1143/JPSJ.12.570.
• Tokmakoff, Andrei. “The Density Matrix,” MIT 5.74 Introductory Quantum Mechanics II (Spring 2009), pp. 7–9, equations (1.34)–(1.51). MIT OpenCourseWare PDF.
• Tong, David. Kinetic Theory, Chapter 4, “Linear Response,” University of Cambridge lecture notes, §§4.3–4.3.2, printed pp. 93–98. Official PDF.
• Tong, David. Statistical Physics. University of Cambridge Part II Mathematical Tripos lecture notes, current official undated-cover PDF, §1.3.2, printed pp. 19–21, and §1.4.1, printed pp. 27–29. Open PDF.