Liouville Flow Volume and Density

Worked Example

Consider the "Triangular expansion" flow field. Its map is given by (e^tx₀, e^ty₀ + x₀²(e^2t - e^t)). Let us verify the properties for Initial x = 1, Initial y = 0, and Time t = 1.

Flow parameters:
x0 = 1, y0 = 0, t = 1

1. Flow coordinates (x_t, y_t):
x_t = e^1 * 1 = e
    ≈ 2.718281828459045
y_t = e^1 * 0 + 1^2 * (e^2 - e^1) = e^2 - e
    ≈ 4.670774270471606

2. Jacobian determinant (det J):
For Triangular expansion, det J = e^(2t)
det J = e^2
      ≈ 7.38905609893065

3. Jacobian lower-left (J_21):
J_21 = 2 * x0 * (e^(2t) - e^t)
J_21 = 2 * 1 * (e^2 - e)
     ≈ 9.341548540943212

4. Log volume ratio:
ln(det J) = 2t = 2

5. Divergence:
The divergence is constant: div = 2.

6. Material density:
Initial density = 1
Density transports as exp(-div * t)
Density = exp(-2 * 1) = e^-2
        ≈ 0.1353352832366127

Experiment

Use the interactive module at https://riano.app/module/liouville-flow-volume-density to observe the local area change.

1. Set the Flow to Linear sink. Observe how the divergence is -3.
2. Set Initial x to 1 and Initial y to 0.
3. Change the Time to 1. Notice the parallelogram shrinks, matching the Jacobian determinant e^-3 and a negative log volume ratio. The material density increases to compensate.
4. Set the Flow to Hamiltonian shear or Hyperbolic. Notice the divergence is 0, the Jacobian determinant is 1, and the volume ratio is 0. The area is preserved but the lengths are not necessarily preserved.

Limitations

The Jacobian determinant describes the local volume scaling (for an infinitesimal material parallelogram), but not necessarily the image of an arbitrary finite nonlinear rectangle. The plotted visualization uses a finite square to approximate the infinitesimal area, so distortions on its boundary may not perfectly match the true flow map of its corners. The initial material density is assumed to be locally 1, but this represents local material density, not a globally normalized uniform law on the whole plane.

References

For further reading and a deeper understanding of these concepts, refer to the following source:

• Continuum Mechanics and Dynamical Systems (CMDS) - Explains the relationship between flow maps, Liouville's equation, and material density transport.