Response, Transport, and Inference

Transport coefficients connect microscopic response to macroscopic relaxation. Extracting them requires a specified source, the correct contact terms and limits, and data that actually resolve the relevant low-frequency behavior.

Enter this chapter

Begin with Sources, Linear Response, and Kubo Formulae to derive causal response to an applied source.

Helpful background. Sources, Linear Response, and Kubo Formulae and Spurions, Local Counterterms, and Symmetry Response develop the definitions used by the chapter’s opening pages.

Topics

1. Sources, Linear Response, and Kubo Formulae — How does a weak source produce retarded linear response, and which zero-frequency and zero-momentum limits define a transport coefficient?
2. Contact Terms, Magnetization Currents, and Order of Limits — Which contact, diamagnetic, polarization, and magnetization terms must be removed or retained, and why do static and transport limit orders differ?
3. Nonlinear Response and Higher-Order Kubo Relations — How are second- and higher-order response coefficients expressed through causal multipoint functions, and what new contact and frequency-order ambiguities arise?
4. Spectral Functions and Transport Peaks — How are transport peaks, Drude weights, diffusion poles, quasiparticle widths, continua, and sum-rule weight distinguished in a spectral function?
5. Transport Sum Rules and Ultraviolet Constraints — Which exact commutators, Ward identities, dispersion relations, and ultraviolet asymptotics constrain integrated transport spectral weight?
6. Diffusion, Conductivity, and Susceptibility — How are charge, heat, and thermoelectric diffusion matrices related to conductivities and susceptibilities, and which relation is frame- and momentum-relaxation dependent?
7. Shear and Bulk Viscosity — Which stress-tensor channels define shear and bulk viscosity, and how do pressure contacts, conserved modes, conformal symmetry, and criticality alter their extraction?
8. Memory Functions and Slow-Mode Projection — How do Mori–Zwanzig projection and memory matrices isolate slow operators and compute transport when quasiparticles are absent or momentum is weakly relaxed?
9. Transport Extraction, Inverse Problems, and Error Budgets — Which transport combinations are identifiable from a given Euclidean, real-time, kinetic, experimental, or simulation dataset once priors and model discrepancy are included?

How the ideas fit together

Source differentiation fixes the response kernel, including local contacts. Conservation laws then relate its long-wavelength poles to thermodynamic susceptibility and dissipative transport; for one isolated diffusive density, conductivity and susceptibility determine the diffusion constant Kubo 1957, §§ 2–3.

That relationship does not identify a coefficient from arbitrary data. Euclidean kernels can smooth away the width of a transport peak, while exactly conserved overlaps can produce delta functions. Sum rules constrain integrated weight, and slow-mode projection identifies mechanisms, but neither replaces a resolution analysis of the inverse problem.

Where to go next

Continue to Thermalization, Integrability Breaking, and Quantum Chaos to distinguish local equilibration from thermalization and dephasing.

References

• Kubo, Ryogo. 1957. “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 (6): 570–586. DOI.