Order that appears when energy flows

Pour a thin layer of oil or water into a flat pan and heat it gently from below. For a while nothing visible happens: heat simply passes through the liquid by conduction, molecule to molecule. Then, as the difference in temperature between the bottom and the top passes a certain value, the liquid organises itself. It breaks into regular cells, rolls or hexagons, in which warm liquid rises in the middle and cool liquid sinks at the edges. Millions of molecules move together in a pattern no one designed.

These are convection cells, first studied carefully by Henri Bénard around 1900. Lord Rayleigh worked out the theory for a layer heated from below in 1916; in Bénard’s own shallow dishes, with a free surface, the cells turned out to be driven mainly by surface tension, but the principle is the same. They are the standard example of what Ilya Prigogine called a dissipative structure: an ordered state that exists only because energy flows through the system and is dissipated as heat (Nicolis and Prigogine 1977). Turn off the burner and the cells vanish within seconds. The order is not stored in the liquid. It is the shape of the flow.

The cells do not appear gradually. Below a threshold, there are none; above it, they are there. For a layer of liquid between two plates, the threshold is set by a dimensionless number, the Rayleigh number, which compares the push of buoyancy with the brakes of viscosity and heat diffusion. Convection begins when it exceeds about 1708.

g = 9.81           # m/s^2
beta = 2.07e-4     # thermal expansion of water at 20 °C, 1/K
nu = 1.0e-6        # kinematic viscosity, m^2/s
kappa = 1.43e-7    # thermal diffusivity, m^2/s
Ra_c = 1708        # critical Rayleigh number, layer between two rigid plates

for d_mm in (2, 5, 10):
    d = d_mm / 1000
    dT = Ra_c * nu * kappa / (g * beta * d ** 3)
    print(f"water layer {d_mm:2d} mm: convection cells appear above a difference of {dT:6.3f} K")
# water layer  2 mm: convection cells appear above a difference of 15.035 K
# water layer  5 mm: convection cells appear above a difference of  0.962 K
# water layer 10 mm: convection cells appear above a difference of  0.120 K

Two lessons from the pan carry over to everything that follows. First, ordered structure in an open system needs a driving flow above a threshold. Second, how much flow is needed depends on the structure: a deeper layer organises with a far smaller push, because the threshold falls with the cube of the depth.

Thresholds, collapse and hysteresis

Physicists and mathematicians describe this kind of change with an order parameter, a single number $x$ that is zero when the system is disordered and grows as order appears. Near a threshold, many very different systems obey the same simple equations. One of the most instructive is

$$\frac{dx}{dt} = \mu x + x^3 – x^5 ,$$

where $\mu$ measures how far the driving flow is above or below its threshold (Strogatz 2015). The $x^3$ term means that order, once it exists, helps to sustain itself; the $x^5$ term keeps it from growing without limit.

def settle(mu, x, dt=0.01, steps=20000):
    """dx/dt = mu*x + x**3 - x**5: x is how much shared order the system holds."""
    for _ in range(steps):
        x += dt * (mu * x + x ** 3 - x ** 5)
    return x

mus = [round(0.30 - 0.05 * i, 2) + 0.0 for i in range(14)]     # input falls from 0.30 to -0.35 ...
x, down = 0.9, {}
for mu in mus:
    x = settle(mu, max(x, 1e-3))
    down[mu] = x
x, up = 1e-3, {}
for mu in reversed(mus):                                  # ... and then rises again
    x = settle(mu, max(x, 1e-3))
    up[mu] = x

print("  mu    order (input falling)   order (input rising)")
for mu in mus:
    print(f"{mu:+.2f}   {down[mu]:21.2f}   {up[mu]:20.2f}")
#   mu    order (input falling)   order (input rising)
# +0.30                    1.11                   1.11
# +0.25                    1.10                   1.10
# +0.20                    1.08                   1.08
# +0.15                    1.06                   1.06
# +0.10                    1.04                   1.04
# +0.05                    1.02                   1.02
# +0.00                    1.00                   0.00
# -0.05                    0.97                   0.00
# -0.10                    0.94                   0.00
# -0.15                    0.90                   0.00
# -0.20                    0.85                   0.00
# -0.25                    0.71                   0.00
# -0.30                    0.00                   0.00
# -0.35                    0.00                   0.00

The table holds three surprises.

Order resists decline. As the drive falls below zero, the ordered state does not disappear. It sags only a little, from 1.00 to 0.85, while the drive falls by a fifth of its range.

Then it collapses. Between $\mu = -0.25$ and $-0.30$, a small further decline wipes out the order completely. Nothing dramatic happened to the drive; it fell by the same small step as before.

Coming back costs more than staying. For every value of $\mu$ between $-0.25$ and $0$, the system can be ordered or disordered, depending only on where it came from. A system that has collapsed stays collapsed until the drive is raised all the way to zero, well above the level that was enough to keep it going. This dependence on history is called hysteresis.

A relationship as a dissipative structure

Now return to the couple of the first two chapters. Their shared life is a kind of order: an improbable arrangement in which two people’s attention, time and plans are coordinated. Chapter 1 argued that, left alone, this arrangement relaxes, because shared attention is statistically unlikely. Chapter 2 argued that the natural source of renewal in the early months, the prediction errors of novelty, dries up as the partners learn each other.

The model is deliberately crude: two people are not a pan of water, and $J$, $L$ and $c$ are not yet quantities anyone can read off a meter. Its value is in the shape it predicts. It says that the time to act is long before anything feels wrong, because the ordered state hides its decline until it fails; and that it is cheaper to keep a fire going than to relight it.

Leaking: an old word for a new idea

The idea that a living order loses its energy through outflows is not new. It is one of the oldest metaphors in Asian contemplative traditions.

Both texts describe what this chapter’s model calls $L$. The Daodejing’s “openings” are usually read as the senses and desires; to “block” them is not to stop seeing and hearing but to stop being drained by everything seen and heard. The Buddhist outflows are more specific: not attention as such, but craving, the compulsive pull towards more.

A preview: the mind that predicts

One more idea completes the frame of the book. Karl Friston’s free-energy principle proposes that any self-organising system that persists, from a cell to a brain, must keep itself within a limited range of states, and does so by minimising a quantity called variational free energy, an upper bound on how surprised it is by its sensations (Friston 2010).

This raises an obvious objection, which Friston and his colleagues called the dark-room problem: if minds minimise surprise, why do they not sit in a dark, quiet room forever? Their answer is that a creature’s model of the world includes expectations about the states it should be in, and that reducing uncertainty about the world is itself valuable, so the same principle that seeks predictability also drives exploration and curiosity (Friston, Thornton, and Clark 2012).

For a couple, this is the central tension in one sentence. Each partner needs the other to be predictable enough to feel safe, and unpredictable enough to remain interesting. Chapter 4 takes up that tension. Chapter 5 returns to the leak, and asks how much of a person’s attention is still theirs to give.