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advanced27 min studyLesson 358 in Physics

Relate thermal spreading to drift under a weak force

Derive the Einstein relation from equilibrium flux balance and a Langevin model.

Before you begin
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The question for today

What you’ll understand

  • Define mechanical mobility and diffusion with consistent units.
  • Derive D=μkBT from equilibrium zero flux.
  • Distinguish mechanical from electrical mobility and state the equilibrium assumptions.

A tracer jitters even with no mean force, yet a weak applied force produces a systematic drift. Diffusion quantifies the first effect and mobility the second. In thermal equilibrium their magnitudes are related by temperature: the Einstein relation. This connects fluctuation to response, but its validity depends on a bath and a near-equilibrium regime. It should not be applied automatically to an active particle, a strongly driven medium, or a coefficient defined with a different force convention.

Define the response before taking a ratio

Mechanical mobility is the steady drift velocity divided by a weak constant force. In an overdamped linear-drag model it is the reciprocal of the drag coefficient. Its units are seconds per kilogram, equivalently metres per newton second. Spatial diffusion is defined by the growth of centered displacement variance, not by the mean drift or by instantaneous speed. In one dimension a drift-free diffusive tracer has variance growing as two times the diffusivity times elapsed time.

A constant force adds a mean displacement proportional to time. The mean-square displacement then includes a squared-drift term as well as the diffusive variance. Subtracting the mean is therefore essential if diffusion is estimated from a driven ensemble. At very short times inertial motion can instead make the variance quadratic in time, so the long-time diffusive formula must not be used before velocity relaxation has occurred.

vdrift=μF,μ=1ζ,Var⁡[x(t)−x(0)]=2Dtv_{\rm drift}=\mu F,\qquad \mu=\frac1\zeta,\qquad \operatorname{Var}[x(t)-x(0)]=2Dt
μ is mechanical mobility in s/kg, ζ is linear drag in kg/s, F force in N, and D diffusivity in m²/s. The variance expression is for one-dimensional long-time Markov diffusion with constant coefficients.↔ Scroll wide equations horizontally.

Equilibrium requires zero probability current

A dilute tracer population in a weak external potential has a flux from both diffusion down density gradients and force-induced drift. At equilibrium its density follows the Boltzmann factor, and in a closed equilibrium setting the net current vanishes. Differentiate that factor and substitute the force as minus the potential gradient. The two contributions cancel for every weak potential only when diffusivity equals mechanical mobility times thermal energy.

The same relation follows from an inertial Langevin equation with white thermal noise: choose the noise covariance so that the stationary velocity variance obeys equipartition. Integrating the resulting exponential velocity correlation gives diffusion equal to thermal energy divided by drag. This route emphasizes the assumptions behind a constant friction model. Memory in a real fluid can change the short-time correlation shape while preserving an equilibrium fluctuation-response relation in the appropriate long-time limit.

J=−D∇n+μnF,neq∝e−U/(kBT),F=−∇U,D=μkBT\mathbf J=-D\nabla n+\mu n\mathbf F,\qquad n_{\rm eq}\propto e^{-U/(k_BT)},\quad\mathbf F=-\nabla U,\qquad D=\mu k_BT
J is tracer number flux in m⁻²s⁻¹, n density in m⁻³, U potential energy per tracer in J, and T bath temperature in K. The result assumes a dilute equilibrium bath, constant scalar coefficients, and linear response.↔ Scroll wide equations horizontally.

A changed convention changes the ratio

Electrical mobility is often defined as drift speed per electric-field magnitude, absorbing the magnitude of the carrier charge into the coefficient. The mechanical and electrical coefficients therefore differ by a charge factor. For a nondegenerate thermal carrier gas, diffusivity divided by positive electrical mobility equals thermal energy divided by charge magnitude. Signed carrier drift direction must still be handled separately. Mixing electrical mobility into the mechanical Einstein formula produces a dimensional error.

The heat lab visualizes diffusion of a deterministic spatial mode. Reset and set mode to 1 and diffusivity to 0.25, then Step twice; predict amplitude exp(−0.25). Reset and set mode to 1 and diffusivity to 1, then Step twice; predict exp(−1). This confirms the selected equation’s decay rate, but there are no random tracer trajectories or force controls. The display cannot measure mobility or verify the Einstein relation experimentally.

mv˙=−ζv+F+ξ(t),⟨ξ(t)ξ(t′)⟩=2ζkBTδ(t−t′),Dμelec=kBT∣q∣m\dot v=-\zeta v+F+\xi(t),\qquad \langle\xi(t)\xi(t')\rangle=2\zeta k_BT\delta(t-t'),\qquad \frac{D}{\mu_{\rm elec}}=\frac{k_BT}{|q|}
ξ is zero-mean thermal force in N, m is tracer mass in kg, and μelec=|q|μ has units m²/(V·s). The last formula uses positive electrical mobility magnitude and classical nondegenerate statistics.↔ Scroll wide equations horizontally.

Work it through

A tracer’s drift and spread

Take thermal energy kBT=4×10⁻²¹ J, drag ζ=2×10⁻⁸ kg/s, and weak constant force 6×10⁻¹⁵ N. At times well beyond inertial relaxation, find mechanical mobility, diffusion, mean displacement after 1 s, and one-dimensional standard deviation.

  1. The mechanical mobility is 1/ζ=5×10⁷ s/kg. Multiplying by the weak force gives drift speed 3×10⁻⁷ m/s, so the mean displacement after one second is 0.30 μm.

  2. The Einstein relation gives D=(5×10⁷)(4×10⁻²¹)=2×10⁻¹³ m²/s. Units work because thermal energy times seconds per kilogram equals square metres per second.

  3. The centered variance after one second is 2Dt=4×10⁻¹³ m². Its square root is about 6.32×10⁻⁷ m, or 0.632 μm. This spread coexists with the smaller directed mean; it is not the root of the total mean-square displacement including drift.

The result

The mobility is 5×10⁷ s/kg and D=2×10⁻¹³ m²/s; the one-second mean shift is 0.30 μm and centered standard deviation is 0.632 μm.

The living notebook

Explore the model

Interactive lab

Interactive lab · heat

Inspect a diffusion coefficient’s consequence

Reset and set n=1, κ=0.25, then Step twice; predict amplitude exp(−0.25). Reset and set n=1, κ=1, then Step twice; predict exp(−1).

Inspect a diffusion coefficient’s consequenceThe exact periodic heat-equation mode is u(x,t) = exp(−κn²t) sin(nx), on [0,2π]. Playback advances physical time. The initial sine is shown separately, and the unnormalized L² norm is √π exp(−κn²t). This analytic instrument does not run a finite-difference solver or test discretization stability. Current readings: Current amplitude: 1; Spatial L² norm: 1.772; Decay rate · κn²: 3.6 /s. All parameters are adjustable with the labeled controls below.Periodic heat equation mode 3 at time 0: the amplitude is 1.0π/2π3π/22π−1−0.500.51x · periodic intervalTemperature perturbation uExact mode · n = 3
Current temperature modeInitial sineCurrent amplitude
Current amplitude
1
Spatial L² norm
1.772
Decay rate · κn²
3.6 /s
t = 0 s

The exact periodic heat-equation mode is u(x,t) = exp(−κn²t) sin(nx), on [0,2π]. Playback advances physical time. The initial sine is shown separately, and the unnormalized L² norm is √π exp(−κn²t). This analytic instrument does not run a finite-difference solver or test discretization stability. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThis deterministic periodic heat mode demonstrates diffusion-rate scaling. It has no tracer samples, drift force, drag, or mobility measurement.

Pause & reason

Check your understanding

What relation uses mechanical mobility defined by drift velocity divided by force?

Your turn

Try it yourself

A thermal tracer has D=3×10⁻¹³ m²/s at kBT=4×10⁻²¹ J. Find mechanical mobility and the equilibrium density ratio across a region where potential energy rises by 2kBT.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Use the same equilibrium relation for transport coefficients and the Boltzmann factor for density.

Compare with the solution

Mobility is D/(kBT)=7.5×10⁷ s/kg. The equilibrium density ratio at higher versus lower potential is exp(−2)≈0.1353. A nonzero diffusive flux down this density gradient is exactly canceled by drift toward lower potential, so equilibrium does not require a uniform tracer density.

Keep the idea

Take it with you

  • Diffusion measures centered spreading while mobility measures weak-force drift.
  • Equilibrium zero current gives D=μkBT for mechanical mobility.
  • Different mobility conventions and nonequilibrium baths need separate checks.

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