Transport Coefficients and Green–Kubo Relations

We take from SM10 the two theorems this article stands on and does not re-derive: the Green–Kubo relations (aq-sm-green-kubo-relations) already prove that every transport coefficient equals a time-integrated equilibrium flux autocorrelation, $L=\tfrac{1}{Vk_BT}\int_0^\infty\langle\hat J(0)\hat J(t)\rangle\,\mathrm dt$ , and the Kubo formula (aq-sm-kubo-formula) already supplies the microscopic current–current response. Our job is the SM11 angle: how you actually compute a transport coefficient from a stochastic trajectory, and the exact equivalence of the two routes to it — the Green–Kubo (autocorrelation-integral) route and the Einstein (mean-square-displacement-slope) route. Slightly out of equilibrium, a system relaxes by carrying conserved quantities down their gradients: momentum (Newton's viscosity law), energy (Fourier's law), charge (Ohm's law), particles (Fick's law). The remarkable content of nonequilibrium statistical mechanics is that these dissipative coefficients are fixed entirely by equilibrium fluctuations. The practical payload is that you never have to drive the system: you run a long equilibrium molecular-dynamics trajectory, evaluate the relevant microscopic flux, accumulate its autocorrelation, and integrate — or, equivalently and often more robustly, measure the long-time slope of a mean-square displacement. We prove the equivalence $D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm dt=\lim_{t\to\infty}\langle\Delta x^2\rangle/2t$ exactly, work it out closed-form for the Ornstein–Uhlenbeck velocity (giving the Einstein relation $D=k_BT/m\gamma$ ), and confirm both routes agree in a real numeric simulation. The one genuine subtlety — hydrodynamic long-time tails $\langle J(0)J(t)\rangle\sim t^{-d/2}$ that make the integral converge only marginally, and diverge in two dimensions — is the frontier the estimator has to respect.

Key takeaways

Transport coefficients (viscosity $\eta$ , thermal conductivity $\kappa$ , conductivity $\sigma$ , diffusion $D$ ) are set by equilibrium current fluctuations — recalled from SM10's Green–Kubo relations, which we do not re-derive.
The two computational routes are exactly equivalent: the Green–Kubo integral $D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm dt$ equals the Einstein slope $\lim_{t\to\infty}\langle\Delta x^2\rangle/2t$ ; differentiating the MSD twice returns the velocity autocorrelation.
For the Ornstein–Uhlenbeck velocity $\langle v(0)v(t)\rangle=(k_BT/m)e^{-\gamma t}$ both routes give the Einstein relation $D=k_BT/m\gamma$ in closed form.
In practice you simulate an equilibrium trajectory, accumulate the flux autocorrelation (or the displacement), and integrate — no nonequilibrium driving needed; the MSD form is numerically more robust to noise and boundaries.
Hydrodynamic long-time tails $\langle J(0)J(t)\rangle\sim t^{-d/2}$ are integrable in $d=3$ ( $t^{-3/2}$ ) but log-divergent in $d=2$ ( $t^{-1}$ ), so 2D transport coefficients can fail to exist.
Key idea: A transport coefficient is the equilibrium time-integral of a microscopic flux autocorrelation, and it can be read off in two mathematically identical ways — as that integral (Green–Kubo) or as the long-time slope of the integrated flux's mean-square displacement (Einstein) — because the MSD is the double time-integral of the autocorrelation.

The core insight

A transport coefficient is the equilibrium time-integral of a microscopic flux autocorrelation. Because the mean-square displacement of the integrated flux is the double time-integral of that autocorrelation, its long-time slope equals the same integral. So the Green–Kubo (autocorrelation) and Einstein (mean-square-displacement) routes are two readings of one identity — and the Einstein relation $D=k_BT/m\gamma$ is the closed-form result for an exponentially-decaying velocity autocorrelation.

Where this lives today

Green–Kubo is the standard route to transport in molecular simulation. Liquid viscosities come from long equilibrium trajectories with the off-diagonal virial stress $P_{xy}$ accumulated and integrated (or the Einstein "momentum-displacement" form for stability). Thermal conductivity of solids and liquids uses the heat-current autocorrelation, with the famous subtlety that per-atom energy partitions are gauge-dependent while the integrated $\kappa$ is not — a live issue for machine-learned interatomic potentials. Electrical conductivity follows from the charge-current autocorrelation, whose Fourier transform is the AC (optical) conductivity, connecting directly to the quantum Kubo–Greenwood formula in electronic-structure codes. And when fluxes couple — heat and charge in thermoelectrics, heat and particles in thermodiffusion — the coefficients become a matrix $L_{ij}$ of cross-correlations constrained by the Onsager reciprocal relations (aq-sm-onsager-reciprocal-relations-sm11).

The physical picture

Watch the random jiggling of momentum, energy or charge in an equilibrium fluid with no applied force. These fluctuations are constantly born and forgotten; the time a fluctuation takes to forget itself — its correlation time — measures how readily the same microscopic processes will carry a steady current when you do impose a gradient. A liquid whose momentum fluctuations persist is one that resists shear strongly: high viscosity. That is the whole content of Green–Kubo, and Onsager's regression hypothesis is the hinge: a released macroscopic disturbance decays by the same law as a spontaneous fluctuation of the same observable — the system cannot tell whether a fluctuation was placed by an experimenter or arose by chance.

Figure

The velocity autocorrelation C(t)=⟨v(0)v(t)⟩ decays; its shaded area is the diffusion coefficient D (Green–Kubo route).

Think of the Einstein form. The time-integral of a current is a displacement: for a tagged particle $\int_0^t v\,\mathrm dt'=x(t)-x(0)$ , so $\langle\Delta x^2\rangle$ grows linearly with slope $2D$ at late times. The autocorrelation route and the MSD route are the same identity read from two ends — differentiate the MSD twice and you recover the velocity autocorrelation. Early on the motion is ballistic ( $\langle\Delta x^2\rangle\propto t^2$ , the particle hasn't collided yet); late on it is diffusive ( $\langle\Delta x^2\rangle\to2Dt$ ). The crossover happens on the correlation time $\tau=1/\gamma$ .

Figure

Punchline — the two routes to D must agree: the shaded area under C(t) (Green–Kubo) equals the slope 2D of the diffusive MSD asymptote (Einstein). Ballistic t² bends over into diffusive 2Dt.

The long-time tail is the subtle physics. Naively correlations decay exponentially, giving a finite coefficient; but conservation laws and hydrodynamic backflow let a fluctuation re-correlate through collective modes, producing algebraic tails $\langle J(0)J(t)\rangle\sim t^{-d/2}$ . In three dimensions $t^{-3/2}$ is integrable (the coefficient exists but converges slowly); in two dimensions $t^{-1}$ gives a logarithmically divergent diffusion constant — 2D hydrodynamic transport is genuinely subtle.

The core equations

The generic statement is recalled from aq-sm-green-kubo-relations: a transport coefficient is a time-integrated equilibrium flux autocorrelation. We quote it because the whole article is about evaluating it.

L=\frac{1}{V k_B T}\int_0^\infty \langle \hat J(0)\cdot \hat J(t)\rangle_{eq}\,\mathrm{d}t \qquad (1)

where: $L$ — transport coefficient ( $\sigma$ , $\kappa$ , $\eta$ ); $V$ — volume; $\hat J(t)$ — microscopic flux of the conserved quantity, evolved by the unperturbed dynamics; $\langle\cdot\rangle_{eq}$ — equilibrium canonical average. The dissipation lives entirely in how fast the equilibrium current forgets itself. For self-diffusion the two routes appear side by side — this identity is the article's owned result.

D=\frac{1}{3}\int_0^\infty \langle \mathbf{v}(0)\cdot\mathbf{v}(t)\rangle\,\mathrm{d}t=\lim_{t\to\infty}\frac{\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle}{6t} \qquad (2)

where: $D$ — self-diffusion coefficient; $\mathbf v(t)$ — tagged-particle velocity; $\mathbf r(t)$ — its position; the factors $3,6$ are the three Cartesian dimensions (in 1D they are $1,2$ ). The left form is Green–Kubo, the right is Einstein; they are exactly equal. The Einstein relation is the closed-form value when the velocity autocorrelation is a single exponential — the Ornstein–Uhlenbeck (Langevin) case.

\langle v(0)v(t)\rangle=\frac{k_BT}{m}\,e^{-\gamma|t|}\ \Longrightarrow\ D=\frac{k_BT}{m}\int_0^\infty e^{-\gamma t}\,\mathrm dt=\frac{k_BT}{m\gamma} \qquad (3)

where: $\gamma$ — friction rate; the amplitude $k_BT/m$ is fixed by equipartition $\tfrac12m\langle v^2\rangle=\tfrac12k_BT$ . With Stokes friction $m\gamma=6\pi\eta a$ this becomes the Stokes–Einstein relation $D=k_BT/6\pi\eta a$ . The concrete fluxes for the mechanical coefficients carry different temperature powers because their conjugate thermodynamic forces differ.

\eta=\frac{V}{k_B T}\!\int_0^\infty\!\langle P_{xy}(0)P_{xy}(t)\rangle\,\mathrm dt,\qquad \kappa=\frac{1}{V k_B T^2}\!\int_0^\infty\!\langle J^q_x(0)J^q_x(t)\rangle\,\mathrm dt,\qquad \sigma=\frac{1}{V k_B T}\!\int_0^\infty\!\langle J_x(0)J_x(t)\rangle\,\mathrm dt \qquad (4)

where: $P_{xy}$ — off-diagonal microscopic stress (virial) tensor; $J^q$ — heat current; $J$ — charge current. Thermal conductivity carries $1/T^2$ (not $1/T$ ) because the force conjugate to heat flux is $\nabla(1/T)=-\nabla T/T^2$ — the extra $1/T$ is not optional.

Derivations

Derivation AThe Green–Kubo integral equals the Einstein mean-square-displacement slope

This is the structural result the article owns: the two ways of computing a transport coefficient from an equilibrium trajectory are not merely both correct — they are the same number, because the mean-square displacement of the integrated flux is the double time-integral of the flux autocorrelation. We do not re-derive Green–Kubo itself (that lives in aq-sm-green-kubo-relations and aq-sm-kubo-formula); we prove the estimator-level identity that every molecular-dynamics practitioner relies on, and that lets you cross-check one route against the other. The proof needs only stationarity of the equilibrium correlation and fast-enough decay.

Step 1. Write the displacement as the time-integral of the velocity: $\Delta x(t)=x(t)-x(0)=\int_0^t v(t')\,\mathrm dt'$ . This holds for any conserved-quantity "displacement" — a tagged-particle position, a momentum moment, an energy moment.

Step 2. Square and average: $\langle\Delta x^2(t)\rangle=\int_0^t\!\!\int_0^t\langle v(t')v(t'')\rangle\,\mathrm dt'\,\mathrm dt''$ . Because the equilibrium correlation is stationary, $\langle v(t')v(t'')\rangle=C(t''-t')$ depends only on the time difference.

Step 3. Change variables to the difference $s=t''-t'$ ; the double integral collapses to a single integral with a triangular weight: $\langle\Delta x^2(t)\rangle=2\int_0^t (t-s)\,C(s)\,\mathrm ds$ . Why the triangular $(t-s)$ weight: pairs of times separated by $s$ occur with multiplicity proportional to $(t-s)$ inside the square window $[0,t]^2$ — short separations are over-counted relative to long ones.

Step 4. Differentiate once: $\dfrac{\mathrm d}{\mathrm dt}\langle\Delta x^2\rangle=2\int_0^t C(s)\,\mathrm ds$ . As $t\to\infty$ this tends to $2\int_0^\infty C(s)\,\mathrm ds$ , so the long-time slope of the MSD is $2\int_0^\infty C=2D$ — the Einstein route equals the Green–Kubo integral.

Step 5. Differentiate a second time: $\dfrac{\mathrm d^2}{\mathrm dt^2}\langle\Delta x^2\rangle=2C(t)$ — the consistency check that recovers the velocity autocorrelation from the MSD. The two routes are the same identity read from the two ends. Why the decay condition matters: the limit exists only if $\int_0^\infty s\,C(s)\,\mathrm ds<\infty$ , which fails for the $t^{-1}$ long-time tail in two dimensions — exactly why 2D diffusion can be ill-defined. Verified: the symbolic equivalence (GK integral = Einstein slope), $\mathrm d^2\langle\Delta x^2\rangle/\mathrm dt^2=2C(t)$ , and a numeric OU simulation where both routes give $D=k_BT/m\gamma$ within a few percent are in verify_aq-sm-transport-coefficients-green-kubo.py.

Go deeper · Derivation B · 1 minEinstein relation from the Ornstein–Uhlenbeck velocity

Derivation BEinstein relation from the Ornstein–Uhlenbeck velocity

We compute the simplest, fully closed-form Green–Kubo example: the diffusion coefficient of a Brownian particle whose velocity obeys a Langevin equation. This exhibits the whole machinery on one line and delivers the Einstein–Sutherland relation $D=k_BT/m\gamma$ , the historical bridge between diffusion and friction. We recall the Langevin equation (aq-sm-langevin-equation) rather than re-deriving it, and use the fluctuation–dissipation fixing of the noise strength.

Step 1. The velocity Langevin equation is $m\dot v=-m\gamma v+\xi(t)$ with white noise $\langle\xi(t)\xi(t')\rangle=2m\gamma k_BT\,\delta(t-t')$ . The noise amplitude is not free: it is fixed by the requirement that the stationary state satisfy equipartition. Why the noise strength is $2m\gamma k_BT$ : solving for the stationary variance gives $\langle v^2\rangle=\text{(noise)}/(2m^2\gamma)$ ; demanding $\langle v^2\rangle=k_BT/m$ forces exactly this value — this is the fluctuation–dissipation theorem in miniature.

Step 2. The stationary velocity autocorrelation of a linear SDE decays as a single exponential with the friction rate: $\langle v(0)v(t)\rangle=(k_BT/m)\,e^{-\gamma|t|}$ , amplitude set by equipartition at $t=0$ .

Step 3. Insert into the Green–Kubo formula (1D): $D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm dt=\dfrac{k_BT}{m}\int_0^\infty e^{-\gamma t}\,\mathrm dt=\dfrac{k_BT}{m\gamma}$ . The single exponential integrates to $1/\gamma$ .

Step 4. With Stokes drag $m\gamma=6\pi\eta a$ for a sphere of radius $a$ in a fluid of viscosity $\eta$ , this becomes $D=k_BT/6\pi\eta a$ — the Stokes–Einstein relation, letting you read off molecular sizes from measured diffusion. Verified: $D=k_BT/m\gamma$ from the exponential integral (symbolic), the equipartition-fixed noise strength, and the Stokes–Einstein form are checked in verify_aq-sm-transport-coefficients-green-kubo.py.

Limiting cases

Case	Result	Meaning
Markovian / single exponential	$C(t)=\langle J^2\rangle e^{-t/\tau}\Rightarrow L=\langle J^2\rangle\tau/Vk_BT$	Static fluctuation × correlation time; the Drude / single-relaxation-time picture.
Dilute gas (Boltzmann)	$\eta\sim\tfrac13 nm\bar v\ell\sim\sqrt{mk_BT}/\sigma_{\rm coll}$	Collision-dominated decay; $\eta$ density-independent (recalled from aq-sm-kinetic-theory-boltzmann-equation).
Hydrodynamic long-time tail	$C(t)\sim (k_BT/\rho)[4\pi(D+\nu)t]^{-d/2}$	Backflow gives $t^{-d/2}$ ; integrable in $d=3$ , log-divergent in $d=2$ .
Zero-frequency spectrum	$L=\tilde C(\omega{=}0)/Vk_BT$	DC transport is the zero-frequency flux power spectrum; finite $\omega$ gives the AC response.

Worked examples

Example 1 — Ornstein–Uhlenbeck: both routes give the same D.

Step 1. Take a Brownian particle with $k_BT/m=1$ and friction $\gamma=2$ (natural units), so the exact answer is $D=k_BT/m\gamma=\mathbf{0.5}$ .

Step 2. Green–Kubo route: the velocity autocorrelation is $C(t)=e^{-2t}$ , and $\int_0^\infty e^{-2t}\,\mathrm dt=\tfrac12=\mathbf{0.5}$ .

Step 3. Einstein route: simulate $4000$ trajectories, integrate $v$ to get $x$ , and take the late-time MSD slope $\langle\Delta x^2\rangle/2t$ . The numeric estimate matches $0.5$ to within a few percent, and the Green–Kubo integral of the measured autocorrelation agrees with the Einstein slope to the same tolerance — the two routes are one number, confirmed in code.

Example 2 — Water self-diffusion → friction and correlation time.

Step 1. Water self-diffusion at room temperature is $D\approx2.3\times10^{-9}\ \mathrm{m^2/s}$ . Using the Einstein relation $D=k_BT/m\gamma$ with $m\approx3.0\times10^{-26}$ kg and $T=298$ K, the implied friction rate is $\gamma=k_BT/(mD)=\mathbf{6.0\times10^{13}\ s^{-1}}$ .

Step 2. The velocity correlation time is therefore $\tau=1/\gamma\approx\mathbf{17\ fs}$ — a molecule "forgets" its velocity in about seventeen femtoseconds, comparable to a single collision time in a dense liquid. This is the physical timescale a molecular-dynamics simulation must resolve to converge the diffusion coefficient by either route.

Deep context

Origins · 1 mintransport coefficients and how this was discovered

transport coefficients

framework 1928–1957 · The equilibrium-fluctuation theory of viscosity, conductivity and diffusion, completed by Green (1954) and Kubo (1957).

Transport coefficients are the constants of proportionality in the phenomenological laws of irreversible flow — Newton's, Fourier's, Ohm's and Fick's laws — and the deep result of nonequilibrium statistical mechanics is that they are determined entirely by equilibrium fluctuations. Lars Onsager's regression hypothesis (1931) supplied the physical principle; Melville Green (1952, 1954) derived the autocorrelation formulas for a classical fluid; and Ryogo Kubo (1957) gave the general linear-response derivation that now bears their joint name. The framework unified Einstein's 1905 relation $D=k_BT/m\gamma$ , the Drude conductivity, and the modern molecular-dynamics computation of viscosity and thermal conductivity into one statement: dissipation is the time-integral of a fluctuation.

History

1905

— Einstein and Sutherland relate diffusion to friction, D = kT/mγ

1928

— Drude-type single-relaxation-time transport theory in metals

1931

— Onsager states the regression hypothesis and reciprocal relations

1952

— Green derives autocorrelation expressions for transport coefficients

1954

— Green's Markov-random-process formulation of the transport formulas

1957

— Kubo gives the general linear-response derivation (Kubo formula)

1970

— Alder & Wainwright discover the t^(-d/2) long-time tail in simulation

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 15 (linear response, fluctuation–dissipation, Kubo formula) · Kardar, Statistical Physics of Particles (kinetic theory and transport); Statistical Physics of Fields (hydrodynamics) · Reif, Fundamentals of Statistical and Thermal Physics, Ch. 15 · Sethna, Statistical Mechanics, Ch. 10 (correlations, response, dissipation) · Landau & Lifshitz, Statistical Physics Part 1 §118–124 and Physical Kinetics (Vol. 10) · Kubo, Toda & Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics · Hansen & McDonald, Theory of Simple Liquids, Ch. 7–8 · Allen & Tildesley, Computer Simulation of Liquids

Provenance. Equations and derivations cross-checked against Pathria & Beale Ch. 15 (Kubo formula, FDT), Kubo–Toda–Hashitsume Statistical Physics II, Hansen & McDonald Theory of Simple Liquids Ch. 7–8 (microscopic stress, heat current, GK/Einstein equivalence), Reif Ch. 15, and Landau–Lifshitz Physical Kinetics. A verifier should: (1) confirm $D=k_BT/m\gamma$ by integrating $\langle v(0)v(t)\rangle=(k_BT/m)e^{-\gamma t}$ ; (2) verify the GK–Einstein equivalence by the triangular-weight integral and $\mathrm d^2\langle\Delta x^2\rangle/\mathrm dt^2=2C(t)$ ; (3) confirm the $1/T^2$ vs $1/T$ prefactors from the conjugate-force argument; (4) check the $t^{-d/2}$ long-time-tail integrability by dimension (Alder–Wainwright 1970). Standard textbook values: Drude $\sigma_0=ne^2\tau/m$ , Stokes–Einstein $D=k_BT/6\pi\eta a$ .

Builds on: Green–Kubo Relations · Kubo Formula · Fluctuation–Dissipation Theorem · Langevin Equation and Stochastic Differential Equations · Fokker–Planck Equation Leads to: Molecular Dynamics Simulation · Thermostats and Ensemble Control in Molecular Dynamics · Onsager Reciprocal Relations

Why are the Green–Kubo and Einstein formulas exactly the same and not just similar?

Because the mean-square displacement is literally the double time-integral of the velocity autocorrelation. Writing $\Delta x=\int_0^t v\,\mathrm dt'$ , squaring and averaging gives $\langle\Delta x^2\rangle=2\int_0^t(t-s)C(s)\,\mathrm ds$ . Differentiate once and the long-time slope is $2\int_0^\infty C=2D$ (Einstein equals Green–Kubo); differentiate twice and you get $2C(t)$ back. There is no approximation — only the requirement that $C$ decay fast enough for the integral to converge. They differ only in numerical character: the MSD form is robust to noise and periodic boundaries, the autocorrelation form exposes the microscopic decay and long-time tails.

Do I need to drive the system out of equilibrium to measure a transport coefficient?

No — that is the whole point. Green–Kubo expresses the coefficient as an equilibrium fluctuation, so you run an ordinary equilibrium trajectory (NVE or a gently thermostatted NVT), evaluate the microscopic flux at each step, and integrate its autocorrelation. Nonequilibrium methods (imposing a real gradient and measuring the steady flux) also work and can be more efficient for large signals, but they require careful control of the driving strength to stay in the linear regime. The equilibrium route has no such worry and is the default for viscosity, conductivity and diffusion.

What is Onsager's regression hypothesis and why does it matter here?

It states that a macroscopic disturbance, once released, relaxes by the same law as a spontaneous equilibrium fluctuation of the same quantity. Physically the system cannot distinguish a fluctuation an experimenter placed from one that arose by chance, so the dissipative response and the equilibrium autocorrelation must coincide. Linear-response theory (the Kubo formula, aq-sm-kubo-formula) proves the hypothesis to first order in the perturbation, and it is exactly what licenses replacing the driven-flux definition of a transport coefficient by the equilibrium-autocorrelation definition — the conceptual heart of Green–Kubo.

Why can the diffusion coefficient fail to exist in two dimensions?

Hydrodynamic backflow gives the velocity autocorrelation an algebraic long-time tail $\langle v(0)v(t)\rangle\sim t^{-d/2}$ . In $d=2$ this is $t^{-1}$ , whose time integral diverges logarithmically, so the Green–Kubo integral does not converge and $D$ depends on system size and observation time rather than being an intrinsic constant. In $d=3$ the tail $t^{-3/2}$ is integrable, so $D$ exists — but the slow decay still forces long simulations and careful tail extrapolation. Alder and Wainwright discovered these tails numerically in 1970, overturning the naive expectation of exponential decay.

Why does the microscopic stress tensor have both a kinetic and a virial part?

Momentum is transported across a plane in two ways: particles physically carrying their momentum through the plane (the kinetic term $\sum m_i v_{i\alpha}v_{i\beta}$ ), and forces acting across the plane between particles on opposite sides (the virial term $\sum_{i<j}r_{ij\alpha}F_{ij\beta}$ ). In a dilute gas the kinetic term dominates; in a dense liquid or solid the virial (interaction) term dominates. The off-diagonal component $P_{xy}$ is the shear stress whose autocorrelation gives the viscosity via Eq. (4). Only the sum is the correct conserved-momentum flux.

How does this connect to the Drude formula for a metal?

If the charge-current autocorrelation decays by a single relaxation time $\tau$ , $\langle J(0)J(t)\rangle\propto e^{-t/\tau}$ , then the Green–Kubo integral gives $\sigma_0=ne^2\tau/m$ — exactly the Drude DC conductivity. Its Fourier transform gives the AC conductivity $\sigma(\omega)=\sigma_0/(1-i\omega\tau)$ . So the phenomenological Drude picture is the single-exponential (Markovian) limit of the general Green–Kubo formula, and the quantum generalization is the Kubo–Greenwood expression used in first-principles electronic-transport calculations.