A system at equilibrium shows large spontaneous fluctuations in its magnetization. According to linear response theory, what does this imply about the system's response to a small applied magnetic field?
ANothing — equilibrium fluctuations are unrelated to how the system responds to external fields
BThe system will show a large susceptibility, responding strongly to the applied field
CThe system will resist the applied field because the fluctuations saturate the response
DThe system needs to be driven out of equilibrium before its susceptibility can be measured
The fluctuation-dissipation theorem (FDT) directly connects equilibrium fluctuations to the response function. Large equilibrium fluctuations in magnetization indicate a large susceptibility — the system responds strongly to the same observable it fluctuates in. Option A is the common misconception: the Kubo formula shows that equilibrium fluctuations *are* the response, encoded in the time-correlation function. You don't need to apply a field to measure susceptibility — watching spontaneous fluctuations tells you everything.
Question 2 Multiple Choice
Why can the linear response of a system to an external perturbation be computed entirely from an unperturbed equilibrium molecular dynamics simulation?
ABecause linear systems don't change when perturbed, so equilibrium and driven dynamics are identical
BBecause the Kubo formula shows the response function equals an equilibrium time-correlation function
CBecause thermal fluctuations in equilibrium are suppressed to zero, leaving only the driven response
DBecause the external field is too weak to affect the trajectories of individual particles
The Kubo formula, χ_{AB}(t) = (i/ℏ)θ(t)⟨[A(t),B(0)]⟩₀, expresses the response function as an equilibrium commutator expectation value. This means every linear response property — susceptibility, conductivity, viscosity — is encoded in fluctuations that occur spontaneously in equilibrium. The simulation never needs to apply any perturbation. Option A misunderstands linearity: linear response means the *output* is proportional to the input, not that the system is unchanged.
Question 3 True / False
The imaginary part of the frequency-domain susceptibility χ''(ω) measures the energy stored (reactive response) in the system during periodic driving.
TTrue
FFalse
Answer: False
This is a common sign confusion. χ''(ω) — the imaginary part — measures *dissipation*: how much energy is absorbed from the driving field per cycle and converted to heat. The real part χ'(ω) measures the in-phase reactive response (energy stored and returned). The FDT connects χ''(ω) directly to the power spectrum of equilibrium fluctuations: χ''(ω) ∝ S(ω)/T. A system that dissipates strongly at a frequency also fluctuates strongly at that frequency.
Question 4 True / False
The response function χ(t − t') depends only on the time difference (t − t'), not on t and t' separately, because of time-translation invariance in equilibrium.
TTrue
FFalse
Answer: True
At thermal equilibrium, there is no preferred moment in time — the statistical properties of the system don't depend on when you start the clock. This symmetry means the response to a perturbation at time t' depends only on how long ago that perturbation occurred (t − t'), not on the absolute times. This is what allows the convolution ⟨δA(t)⟩ = ∫χ(t−t')h(t')dt' to be written as a simple product χ̃(ω)h̃(ω) in Fourier space — a major simplification for periodic driving analysis.
Question 5 Short Answer
Why is the fluctuation-dissipation theorem considered a profound result, rather than merely a convenient calculational shortcut? What does it reveal about the relationship between equilibrium and non-equilibrium physics?
Think about your answer, then reveal below.
Model answer: The FDT reveals that equilibrium and linear non-equilibrium physics are not separate regimes — they are two views of the same microscopic dynamics. The same thermal collisions that produce random fluctuations in equilibrium also produce the friction and damping seen when the system is driven. Dissipation is not an 'extra' phenomenon requiring a separate theory; it is the microscopic noise that becomes visible at the macroscopic level. Quantitatively: χ''(ω) = (ω/2kT)S(ω), so measuring noise at equilibrium gives you the dissipation at that frequency. This means you can, in principle, predict the resistance of a resistor by measuring its thermal (Johnson) noise — without connecting it to any circuit.
The depth of the FDT lies in its unification: it abolishes the distinction between 'passive equilibrium physics' and 'active response to driving' in the linear regime. It also has practical implications — Green-Kubo relations let you compute transport coefficients (thermal conductivity, shear viscosity, diffusion coefficients) from equilibrium MD simulations, avoiding the need to simulate driven systems that are harder to control and interpret.