To measure the viscosity of a complex fluid computationally, you could simulate it under an applied shear stress and measure the velocity gradient. The Green-Kubo formula offers an alternative. What is it?
AThere is no alternative — viscosity is inherently a non-equilibrium property requiring a non-equilibrium simulation
BCompute the viscosity from the static stress tensor at a single equilibrium snapshot
CCompute the viscosity from the time-integral of the stress autocorrelation function in an equilibrium simulation
DApply the Green-Kubo formula only to electrical conductivity; viscosity requires non-equilibrium methods
The Green-Kubo formula is η = (V/kT)∫₀^∞ ⟨σ_xy(t)σ_xy(0)⟩dt. You run an ordinary equilibrium simulation, record the stress tensor at each timestep, compute how it correlates with itself over time, and integrate. No shear flow, no applied force. Option A is the misconception this topic attacks: transport coefficients look non-equilibrium, but the fluctuation-dissipation theorem connects them to equilibrium dynamics. Option B misses the time correlation — a static snapshot doesn't capture memory or decay.
Question 2 Multiple Choice
In the Green-Kubo formula for viscosity, a longer autocorrelation decay time for the stress tensor corresponds to:
ALower viscosity — fast relaxation means the fluid resists shear less
BHigher viscosity — slow stress relaxation means the fluid retains shear stress memory longer
CHigher temperature — longer decay times indicate greater thermal fluctuation energy
DA shorter integration window needed, since the autocorrelation function decays quickly
Viscosity is the resistance to flow — how long a fluid 'remembers' an applied shear stress. A fluid whose stress autocorrelation decays slowly (long memory) requires large sustained force to flow: high viscosity. A fluid that rapidly loses stress memory flows easily: low viscosity. Honey has slow stress relaxation; water has fast stress relaxation. The integral of the autocorrelation function captures the total 'area under the memory curve,' which is the viscosity.
Question 3 True / False
The Green-Kubo formula follows from the fluctuation-dissipation theorem, which connects equilibrium fluctuations to the system's response to external perturbations.
TTrue
FFalse
Answer: True
True — the fluctuation-dissipation theorem is the physical principle that underlies the Green-Kubo formula. It states that the same microscopic dynamics governing how spontaneous thermal fluctuations relax also governs how the system responds to (and dissipates) an externally applied perturbation. This is why measuring equilibrium fluctuations tells you about non-equilibrium transport: the mechanism is the same in both cases.
Question 4 True / False
The Green-Kubo formula is specific to viscosity and cannot be extended to other transport coefficients like electrical conductivity or thermal conductivity.
TTrue
FFalse
Answer: False
False — the Green-Kubo framework applies uniformly across all linear transport coefficients. Electrical conductivity is the time-integral of the current-current autocorrelation function; thermal conductivity is the time-integral of the energy-flux autocorrelation function. Each transport coefficient is the time-integral of the relevant flux autocorrelation. This unification across different transport phenomena under a single mathematical framework is one of the deep results of non-equilibrium statistical mechanics.
Question 5 Short Answer
Why does measuring equilibrium fluctuations tell you about a system's response to an applied non-equilibrium perturbation? What principle connects them?
Think about your answer, then reveal below.
Model answer: The fluctuation-dissipation theorem connects them. At equilibrium, a system constantly undergoes thermal fluctuations — small, spontaneous deviations in stress, current, or energy flux. The rate at which these fluctuations decay is governed by the same microscopic dynamics that would dissipate an externally applied perturbation. In other words, the system does not 'know' whether a deviation from equilibrium was caused by a thermal fluctuation or an external force — it relaxes the same way in either case. The Green-Kubo formula turns this insight into a calculation: the decay rate of equilibrium fluctuations is the transport coefficient.
The key insight is that transport is not fundamentally about driving a system out of equilibrium — it is about how the system relaxes. Equilibrium fluctuations are constantly doing this relaxation work, and watching them reveals everything about transport without ever needing an external perturbation.