Linear response theory

Linear response theory approximates a small change in an observable by a linear function of an applied perturbation around a specified reference state. A weak force can produce a proportional displacement, and a weak applied magnetic field can produce proportional magnetization. The response coefficient depends on the material and state; linearity is an approximation to test by changing the drive size.

For a causal time-dependent response, writeThe kernel $\chi$ describes the delayed effect of the perturbation $f$. A sinusoidal input probes the frequency-dependent response: its in-phase part and out-of-phase part determine the response magnitude and phase lag. Causality links the real and imaginary parts through Kramers-Kronig relations under the appropriate convergence assumptions. Comparing the response at several small drive sizes tests linearity; a linear-looking response at one size alone cannot establish the range of validity.

The Kubo formula expresses a linear response through an equilibrium correlation function. For a perturbation $H^{\prime }(t)=-f(t)B$ and observable $A$, one convention gives $\delta ⟨A(t)⟩={\int }_{-\infty }^t{\chi }_{AB}(t-t^{\prime })f(t^{\prime })d{t}^{\prime }$, withThe step function enforces causality, and the commutator records the quantum response.

The sign depends on the declared perturbing-Hamiltonian convention. Applying the framework to currents or polarization relates transport coefficients to fluctuations in an unperturbed system, forming a foundation of linear response theory and the fluctuation-dissipation theorem.^[ref]

The random phase approximation treats the independent-particle response to the self-consistent total field, including the induced mean Coulomb potential. With a positive static number-response coefficient ${\Pi }_0=-\delta n/\delta U$, a potential $U=-e\varphi$ gives $\delta n=e{\Pi }_0{\varphi }_{\mathrm{tot}}$. Combining induced charge $-e\delta n$ with Poisson equation gives $ϵ=1+e^2{\Pi }_0/({ϵ}_0{q}^2)$. Exchange-correlation vertex corrections beyond this self-consistent response are omitted. It is a response approximation, not a literal assumption that every microscopic electron phase is independently randomized.

For an integrable linear response kernel $K$ and input $V$, the convolution theorem gives $\hat{I}(\omega )=\hat{K}(\omega )\hat{V}(\omega )$. The integrated output is therefore zero for every integrable input if and only if $\hat{K}(0)=\int K(t)dt=0$. For a sum of causal damped sines, $K(t)=\Theta (t){\sum }_j{a}_j{e}^{-{\gamma }_{j}t}\sin (p_{j}t)$, the condition is ${\sum }_j{a}_j{p}_j/({\gamma }_j^2+p_j^2)=0$. If all $a_j$ are positive, nonzero terms of both signs are needed; a term with $p_j=0$ vanishes identically.

Causality constrains the real and imaginary parts of a frequency-dependent linear response. For a dielectric with ${ϵ}_r(\omega )\to 1$ at large frequency, the $e^{-i\omega t}$ convention givesThe integral is a Cauchy principal value. Absorption and optical dispersion therefore cannot be specified independently.

An Onsager reciprocal relation connects cross-couplings between near-equilibrium transport processes. Choose thermodynamic forces $X_j$ and fluxes $J_i$ so that their products give the entropy production, and write $J_i={\sum }_j{L}_{ij}{X}_j$. With microscopic reversibility, no imposed time-reversal-breaking field and variables with the appropriate matching time-reversal parity, the coefficients obey $L_{ij}=L_{ji}$. The choice of conjugate force and flux matters: arbitrary rescaling of just one coefficient need not preserve its numerical symmetry.

For coupled thermal and electrical transport, a thermal driving force can induce charge flow while an electrical driving force induces heat flow. Reciprocity relates those cross effects, reducing the number of independent transport measurements. Lars Onsager's 1931 paper develops this reasoning from microscopic reversibility and distinguishes it from thermodynamic positivity alone.^[ref] External magnetic fields and odd time-reversal variables require a generalized parity-sensitive relation rather than an unconditional symmetric matrix.