Transport Coefficients

3.3. Transport Coefficients#

3.3.1. 1. Time Correlation Functions#

Let ( X(t) ) be a fluctuating variable sampled at time ( t ). When are two samples “independent”?

  • Autocorrelation function:

\[ C(t) = \langle X(0) X(t) \rangle \]

  • Normalized: ( C(0) = \langle X^2 \rangle ), and ( C(t) \to 0 ) as ( t \to \infty )

An “independent” sample can be taken after a time ( \tau ) has passed, where ( C(\tau) ) is sufficiently small.

To improve averaging, use multiple independent time origins:

\[ \langle X(t_0) X(t_0 + t) \rangle \]

3.3.1.1. Example#

A random variable ( X ) is chosen every time step from a uniform distribution over ([-1, 1]).

  • ( \langle X^2 \rangle = \int_{-1}^1 x^2 \cdot \frac{1}{2} dx = \frac{1}{3} )

  • For two independent samples: ( \langle X(t_0) X(t_0 + 1) \rangle = 0 )

  • ( \langle X(t_0) X(t_0 + 2) \rangle = 0 )

3.3.2. 2. Ways to Compute Autocorrelation#

3.3.2.1. Methods#

  • FFT (Fast Fourier Transform): Efficient for computing ( C(t) ) over long times.

  • Brute Force: Common for short-time correlations.

3.3.2.2. Brute Force Steps#

  1. Choose ( X_i ) in data series as ( t_0 )

  2. For ( j = i ) to ( i + \text{window} ):

    • Accumulate ( C_{j-i} += X_i X_j )

  3. Advance to next origin ( i ) and repeat

  4. Normalize: ( C_k = \frac{C_k}{N - k} )

3.3.3. 3. Green-Kubo and Einstein Relations#

Using linear response theory (small perturbations), we can show:

3.3.3.1. Green-Kubo Formula#

\[ X = \int_0^\infty dt \, \langle A(0) A(t) \rangle \]

  • ( X ): transport coefficient

  • ( A(t) ): dynamic observable

3.3.3.2. Einstein Relation#

\[ X = \lim_{t \to \infty} \frac{1}{2t} \langle [A(t) - A(0)]^2 \rangle \]

This is a fluctuation-dissipation relation.

3.3.4. 4. Diffusion#

3.3.4.1. Diffusion Coefficient ( D )#

\[ D = \lim_{t \to \infty} \frac{1}{2d} \frac{d}{dt} \langle [x(t) - x(0)]^2 \rangle \]

If isotropic:

\[ D = \frac{1}{3} (D_{xx} + D_{yy} + D_{zz}) \]

Or:

\[ D = \lim_{t \to \infty} \frac{1}{2d} \langle \Delta x(t)^2 \rangle \]

3.3.4.2. Random Walker in 1D#

  • Hop left or right with ( p = \frac{1}{2} )

  • ( \langle \Delta x \rangle = 0 )

  • ( \langle \Delta x^2 \rangle = n l^2 ) (binomial distribution)

Hence:

\[ D = \frac{l^2}{2 \Delta t} \]
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