Questions: Time-Correlation Functions and Relaxation

Question 1

A liquid has a large initial velocity autocorrelation function C(0) but it decays extremely rapidly — faster than any other liquid you've measured. A student claims this means the liquid must have a large diffusion coefficient. Is the student correct?

• A. Yes — C(0) sets the scale of the diffusion coefficient directly
• B. No — the diffusion coefficient is the time integral ∫C(t)dt, so a large but rapidly decaying C(t) integrates to a small area and yields a small D
• C. Yes — C(0) = kT/m by equipartition, and this ratio equals D
• D. No — because C(t) decays to zero in ergodic systems, D must always be zero

Answer: (B)

The Green-Kubo relation D = (1/3) ∫₀^∞ ⟨v(t)·v(0)⟩ dt makes clear that what matters is the area under the time-correlation function, not its initial value. A large C(0) with a short relaxation time τ integrates to a small D. Transport is determined by both the fluctuation magnitude and the fluctuation lifetime together — neither alone is sufficient.

Question 2

Which statement best describes the behavior of a time-correlation function C(t) = ⟨A(t)A(0)⟩ for an ergodic equilibrium system?

• A. C(t) oscillates indefinitely because the system is in thermal equilibrium
• B. C(t) grows over time as the system explores more of phase space
• C. C(t) decays to zero as t → ∞ because the system explores all accessible phase space, erasing memory of the initial value
• D. C(t) is constant for equilibrium systems because statistical properties don't change in time

Answer: (C)

Ergodicity means the system explores all regions of phase space over long times. Once many collisions have occurred, the current value of A(t) is statistically independent of the initial value A(0), so ⟨A(t)A(0)⟩ → ⟨A⟩² = 0 for zero-mean observables. The decay timescale encodes how long memory persists — the relaxation time τ. This is precisely what makes the time integral well-defined and finite.

Question 3

The diffusion coefficient D can be computed entirely from equilibrium dynamics by integrating the equilibrium velocity autocorrelation function — a property known as a Green-Kubo relation.

• T. True
• F. False

Answer: True

This is the central result connecting equilibrium fluctuations to non-equilibrium transport. D = (1/3) ∫₀^∞ ⟨v(t)·v(0)⟩ dt expresses a non-equilibrium property (how fast a diffusing particle spreads) purely in terms of equilibrium fluctuations. The fluctuation-dissipation theorem guarantees this connection: the same thermal fluctuations that randomize velocities at equilibrium also determine how energy is dissipated when the system is perturbed.

Question 4

A time-correlation function C(t) that starts at a larger initial value C(0) generally corresponds to a larger transport coefficient.

• T. True
• F. False

Answer: False

C(0) and the transport coefficient are not directly proportional. The transport coefficient is the time integral of C(t), which depends on both C(0) and the relaxation timescale τ. A system with a small C(0) but a very long τ (slow decay) can have a much larger diffusion coefficient than one with a large C(0) that decays almost instantly. The interplay between fluctuation magnitude and fluctuation lifetime determines transport, not either quantity alone.

Question 5

Why can equilibrium time-correlation functions reveal non-equilibrium transport properties like diffusion coefficients and viscosity?

Model answer: The fluctuation-dissipation theorem establishes that the same microscopic dynamics that govern equilibrium thermal fluctuations also determine how the system responds to and dissipates external perturbations. At equilibrium, molecules undergo constant thermal motion — velocities fluctuate and correlate over timescales set by collisions. These equilibrium fluctuations are statistically equivalent to the relaxation dynamics seen when the system is driven slightly out of equilibrium. Green-Kubo relations formalize this: they express each transport coefficient as the time integral of the relevant equilibrium correlation function, so measuring equilibrium dynamics gives complete information about non-equilibrium transport.

This is one of the deepest results in statistical mechanics. Non-equilibrium coefficients like D and η describe how the system relaxes when perturbed, but that relaxation is governed by the same molecular interactions that produce equilibrium fluctuations. You don't need to actually perturb the system — the equilibrium dynamics already contain all the information, accessible through time-correlation functions.