Questions: Linear Response Theory and Susceptibilities

Question 1

A physicist wants to calculate how a material's magnetization responds to a weak oscillating magnetic field. According to the Kubo formula, what is the most direct way to compute this response?

• A. Calculate the equilibrium magnetization-magnetization correlation function in the unperturbed system
• B. Solve the Schrödinger equation for the perturbed Hamiltonian H − hA
• C. Measure the magnetization in the presence of the field and divide by field strength
• D. Expand the partition function in powers of h and keep the first-order term

Answer: (A)

The Kubo formula χ_BA(t) = iθ(t)⟨[B(t), A(0)]⟩₀/ℏ says the response function is an equilibrium correlator in the unperturbed system. You never need to solve the perturbed problem — the fluctuations of the system at equilibrium fully determine how it will respond to a weak external perturbation. This is the profound insight of linear response theory.

Question 2

For a system's frequency-dependent susceptibility χ(ω), what does the imaginary part Im[χ(ω)] physically represent?

• A. The energy dissipated (absorbed) from an oscillating field at frequency ω
• B. The reactive (dispersive) response — the in-phase part of the system's reaction
• C. The equilibrium fluctuation amplitude at frequency ω
• D. The phase shift between the applied field and the system's response

Answer: (A)

Im[χ(ω)] measures dissipation — how much energy is absorbed per cycle from a field oscillating at frequency ω. The real part Re[χ(ω)] gives the reactive (dispersive) response, which is in phase with the field. This split between dissipative and reactive response is analogous to the resistance and reactance in an electrical circuit.

Question 3

A system with large spontaneous fluctuations in magnetization (even without any applied field) will tend to have a large magnetic susceptibility.

• T. True
• F. False

Answer: True

True. This is the core message of the fluctuation-dissipation connection embedded in the Kubo formula. A system that fluctuates easily in equilibrium is precisely one that can be easily pushed into a new state by a small external perturbation. Mathematically, susceptibility is proportional to the integral of the equilibrium correlation function, so large fluctuations mean large susceptibility.

Question 4

To use the Kubo formula to calculate a system's linear response, you is expected to first solve the dynamics of the perturbed Hamiltonian H + H'.

• T. True
• F. False

Answer: False

False. This is the wrong approach that linear response theory supersedes. The Kubo formula expresses the response function entirely in terms of equilibrium expectation values in the unperturbed system — ⟨[B(t), A(0)]⟩₀ uses the unperturbed equilibrium state. The entire point is that you do not need to solve the perturbed problem; equilibrium correlators contain all the information about the linear response.

Question 5

Why is the Kubo formula considered a unifying result in statistical mechanics? What does it reveal about the relationship between equilibrium and non-equilibrium behavior?

Model answer: The Kubo formula shows that a system's response to any weak external perturbation is completely determined by its equilibrium fluctuations. This means apparently non-equilibrium quantities — electrical conductivity, magnetic susceptibility, viscosity — are all accessible from equilibrium calculations alone. Equilibrium and linear non-equilibrium behavior are not separate regimes requiring different theories; they are two faces of the same underlying physics, connected by the spontaneous fluctuations of the system.

Before Kubo's formula, computing transport and response coefficients seemed to require solving non-equilibrium problems. The formula reveals this is unnecessary for the linear regime: the retarded Green's function (response) equals the equilibrium commutator correlator, connecting dynamics to thermodynamics. This unifies conductivity, susceptibility, viscosity, and compressibility as instances of the same idea: a system that fluctuates easily at equilibrium responds easily to perturbations.