Chapter 3 The Double Well and Kramers’ Escape Rate

Two valleys

Imagine a rolling landscape with two valleys separated by a ridge. A ball rests at the bottom of the left valley and is shaken continually at random. Most shakes only jiggle it near the bottom; now and then a run of shakes that happen to point the same way pushes it up over the ridge, and it rolls into the right valley. Chemical reactions, protein folding, the flipping of magnetic domains and the switching of populations between two states can all be described this way.

This chapter borrows the picture to describe habits: the left valley is a familiar, effortless old habit; the right valley is a better new one; the ridge is the “activation energy” that must be paid to get from old to new. This is still a simplified model: a person’s behaviour is not a one-dimensional coordinate and has no measurable “potential energy”. The model’s value is that it gives a quantitative relation that can be discussed: how the waiting time depends on the height of the ridge and the strength of the shaking.

The double well and the overdamped Langevin equation

Calculation

The simplest double well is

$U(x) = a x^4 – b x^2 \quad (a, b > 0),$

with minima at $x = \pm\sqrt{b/2a}$ , the maximum (the top of the barrier) at $x = 0$ , and barrier height $\Delta U = b^2/4a$ . If motion is dominated by strong friction (overdamped), the state changes over time according to the overdamped Langevin equation

$\mathrm{d}x = -U'(x)\,\mathrm{d}t + \sqrt{2D}\,\mathrm{d}W,$

where the first term is the “downhill force” of the potential and the second is random disturbance of strength $D$ (in physics, $D$ corresponds to the temperature $k_B T$ ).

Evidence

In 1889 Arrhenius summarised experiments showing that reaction rates change exponentially with temperature (Arrhenius 1889). In 1940 Kramers derived the rate of crossing a barrier from the theory of Brownian motion; in the overdamped limit

$r = \frac{\sqrt{U''(x_{\min})\,|U''(x_{\max})|}}{2\pi}\, e^{-\Delta U / D},$

so the mean waiting time is about $\tau = 1/r \propto e^{\Delta U/D}$ (Kramers 1940). The result holds when the barrier is much higher than the noise ( $\Delta U/D \gg 1$ ), and over fifty years it was tested and extended in a great many experiments and numerical studies (Hänggi et al. 1990).

Simulation against formula

Calculation

Take $a = 1/4$ and $b = 1/2$ , so the valley bottoms are at $x = \pm 1$ and the barrier height is 0.25. The program starts the ball at the bottom of the left valley, records how long it takes to reach the bottom of the right valley, repeats this 100 times for each noise strength, and compares the mean with Kramers’ formula.

# Overdamped Langevin motion in the double well U(x) = a x^4 - b x^2  (a = 1/4, b = 1/2):
#   dx = -U'(x) dt + sqrt(2 D) dW
# Wells at x = -1 and x = +1, barrier at x = 0 of height dU = b^2 / (4a) = 0.25.
# Kramers (1940), overdamped limit: escape rate r = sqrt(U''(well) |U''(barrier)|) / (2 pi) * exp(-dU / D).
# Simplified model; "noise" D stands for the everyday fluctuations that occasionally push a state over.
import math, random
random.seed(7)
A, B = 0.25, 0.5
DU = B * B / (4 * A)
U2_WELL, U2_TOP = 4 * B, 2 * B                  # U'' at the wells and |U''| at the barrier: 2 and 1
DT = 0.01

def force(x):
    return -(4 * A * x ** 3 - 2 * B * x)

def mean_escape_time(D, n=100):
    s = math.sqrt(2 * D * DT)
    total = 0.0
    for _ in range(n):
        x, t = -1.0, 0.0
        while x < 1.0:                    # from the bottom of one well to the bottom of the other
            x += force(x) * DT + s * random.gauss(0, 1)
            t += DT
        total += t
    return total / n

print("noise D   dU/D   simulated mean time   Kramers 1/r   ratio")
for D in (1 / 16, 1 / 20, 1 / 24):
    sim = mean_escape_time(D)
    kr = 2 * math.pi / math.sqrt(U2_WELL * U2_TOP) * math.exp(DU / D)
    print(f"{D:7.4f}   {DU / D:4.1f}   {sim:19.0f}   {kr:11.0f}   {sim / kr:5.2f}")
print("Kramers is the leading term for large dU/D; the simulation also includes the slide into the far well")
print("each +1 in dU/D multiplies the waiting time by e = 2.7")
# noise D   dU/D   simulated mean time   Kramers 1/r   ratio
#  0.0625    4.0                   274           243    1.13
#  0.0500    5.0                   752           659    1.14
#  0.0417    6.0                  2080          1792    1.16
# Kramers is the leading term for large dU/D; the simulation also includes the slide into the far well
# each +1 in dU/D multiplies the waiting time by e = 2.7

The simulated means are about 15 % above Kramers’ formula, as expected: the formula is the leading term for large $\Delta U/D$ , while the simulation also includes the time to slide into the far valley after crossing the ridge. What matters is the scaling: each increase of 1 in $\Delta U/D$ multiplies the waiting time by about 2.7 (the number $e$ ). From 4 to 6, the waiting time rises more than sevenfold. The waiting time is extremely sensitive to the ratio of barrier height to noise, and this is the starting point for the rest of the chapter.

Two ways to shorten the wait

The waiting time $\tau \propto e^{\Delta U/D}$ offers two routes: lower the barrier $\Delta U$ , or tilt the whole landscape.

Lowering the barrier corresponds to breaking the change into very small steps, each needing little “activation energy”.
Tilting corresponds to a steady outside pull, $U(x) – Fx$ . A partner’s expectations, an agreement with others, a fixed date for showing one’s work can all be seen as forces that tilt the landscape towards the new habit.

Calculation

Make the old habit (the left valley) slightly deeper, and take noise $D = 0.025$ , so that the baseline $\Delta U/D$ is about 11. The program computes the waiting time to leave the old valley under each intervention, and the waiting time to “relapse” from the new valley back to the old.

# Two ways to shorten the Kramers waiting time tau ~ exp(dU / D) for leaving the old well:
#   (1) lower the barrier (break the change into small steps that each need little activation);
#   (2) tilt the landscape with a steady outside pull F (a partner, a commitment), U(x) - F x.
# Old habit = left well (made slightly deeper by c), new habit = right well. Simplified model.
import math
A, B, C, D = 0.25, 0.5, 0.03, 0.025

def landscape(b, F):
    U = lambda x: A * x ** 4 - b * x ** 2 + (C - F) * x
    dU = lambda x: 4 * A * x ** 3 - 2 * b * x + (C - F)
    d2U = lambda x: 12 * A * x ** 2 - 2 * b
    roots = []
    for x0 in (-1.5, 0.0, 1.5):                 # Newton from three starting points
        x = x0
        for _ in range(60):
            x -= dU(x) / d2U(x)
        roots.append(x)
    left, top, right = sorted(roots)
    if abs(top - left) < 1e-6 or abs(right - top) < 1e-6:
        return None                              # the barrier has vanished: no waiting at all
    pref = lambda w: 2 * math.pi / math.sqrt(d2U(w) * abs(d2U(top)))
    tau_go = pref(left) * math.exp((U(top) - U(left)) / D)
    tau_back = pref(right) * math.exp((U(top) - U(right)) / D)
    return U(top) - U(left), tau_go, tau_back

base = landscape(B, 0.0)
print(f"baseline: barrier {base[0]:.3f}, dU/D = {base[0] / D:.1f}, waiting time {base[1]:.2e}, relapse time {base[2]:.1e}")
cases = [("barrier lowered by a quarter", B * math.sqrt(0.75), 0.0),
         ("barrier halved", B * math.sqrt(0.5), 0.0),
         ("steady pull F = 0.05", B, 0.05),
         ("steady pull F = 0.10", B, 0.10),
         ("barrier halved + pull F = 0.05", B * math.sqrt(0.5), 0.05)]
for label, b, F in cases:
    r = landscape(b, F)
    print(f"{label:31s} barrier {r[0]:.3f}  waiting time {r[1]:.2e}  ({base[1] / r[1]:8.0f}x shorter)"
          f"  relapse time {r[2]:.1e}")
print(f"pull removed after the switch: relapse time back to {base[2]:.1e} (the tilt must persist to hold)")
# baseline: barrier 0.281, dU/D = 11.2, waiting time 3.27e+05, relapse time 3.1e+04
# barrier lowered by a quarter    barrier 0.216  waiting time 2.85e+04  (      11x shorter)  relapse time 3.2e+03
# barrier halved                  barrier 0.151  waiting time 2.57e+03  (     127x shorter)  relapse time 3.7e+02
# steady pull F = 0.05            barrier 0.230  waiting time 4.52e+04  (       7x shorter)  relapse time 2.2e+05
# steady pull F = 0.10            barrier 0.184  waiting time 7.37e+03  (      44x shorter)  relapse time 1.8e+06
# barrier halved + pull F = 0.05  barrier 0.109  waiting time 4.98e+02  (     657x shorter)  relapse time 1.8e+03
# pull removed after the switch: relapse time back to 3.1e+04 (the tilt must persist to hold)

Three landscapes. Blue: baseline, with the old habit on the left and slightly deeper. Gold: barrier height roughly halved (the change broken into small steps). Red: a steady outside force F = 0.10 tilts the landscape to the right.

Three points stand out.

First, with these parameters, halving the barrier shortens the waiting time about 127-fold, and halving it while adding a small steady pull shortens it about 657-fold — two to three orders of magnitude. The original idea that interventions can shorten the escape time “by several orders of magnitude” holds in the model, but only when $\Delta U/D$ is large; in a person we cannot measure $\Delta U/D$ , so “orders of magnitude” is a statement about the model, not a prediction for anyone.

Second, lowering the barrier alone makes relapse just as much easier. A barrier works in both directions: lower the ridge, and the way back from the new valley to the old one is shorter too. In the “barrier halved” row, the relapse time drops from about 30,000 to a few hundred. Small steps help a person start; they do not help a person stay.

Third, tilting shortens the time to leave and lengthens the time to relapse at once: a steady pull of F = 0.10 raises the relapse time to about 1.8 million. But the last line shows that once the pull is removed, the relapse time returns to its baseline. In this model the outside pull must persist for the new state to hold, unless the landscape itself has changed (Chapter 4).

What the evidence says

Evidence

Interventions of the “lower the threshold to start” kind have fairly good evidence. A meta-analysis of implementation intentions (“when situation X arises, I will do Y”) found a medium-to-large effect on goal attainment, d of about 0.65 (Gollwitzer and Sheeran 2006). Habit research shows that when a simple behaviour is repeated in a stable context, its automaticity rises with repetition and levels off; in a study that followed 96 volunteers in daily life, the 39 whose curves fitted well took a median of 66 days to reach 95 % of their plateau, with individuals ranging from 18 to 254 days (Lally et al. 2010; Wood and Rünger 2016).

Evidence

The evidence on “outside pulls” is more mixed. A large study comparing 54 interventions to raise gym attendance found that many had small to moderate effects while they ran, but the effects were mostly modest and the evidence that they lasted after the programme ended was weak (Milkman et al. 2021). This fits the model’s third point: once the pull is removed, relapse returns to its old pace.

Note

Popular practices such as “tiny habits” or the “two-minute rule” currently lack randomised trials that directly compare step sizes. This book treats them as one way of lowering the barrier and states the idea below as a hypothesis.

Hypothesis · not established

Not established. The model predicts that “lower the barrier only” and “lower the barrier plus a steady outside pull” differ little at the start, but clearly differ in relapse after support is withdrawn: the barrier-only group relapses faster. This can be tested in a three-arm randomised trial: group A sets goals as usual; group B breaks the goal into steps of two minutes or less; group C does what B does and also reports weekly to a partner. The primary outcome is the share still doing the behaviour in weeks 4 to 8 after the intervention ends. If C and B do not differ in post-intervention maintenance, or B’s maintenance is no lower than A’s, this prediction of the model is refuted.

Limits of the analogy

Noise in physics is directionless thermal motion; the “disturbances” of a life — a trip, an illness, a new friend — often have a direction and also change the landscape itself. A physical barrier has a definite height; a person’s “barrier” shifts with mood, fatigue and surroundings, and differs from day to day. Most important, a ball does not decide which valley to go to, and a person does: intentions, values and plans are variables this one-dimensional model leaves out. The model is useful as a reminder of one relation — the waiting time is exponentially sensitive to the ratio of barrier to disturbance — so small structural changes may do more than doubled willpower. It cannot tell anyone how high their barrier is.

References

Arrhenius, Svante. 1889. “Über Die Reaktionsgeschwindigkeit Bei Der Inversion von Rohrzucker Durch säuren.” Zeitschrift für Physikalische Chemie 4: 226–48.

Gollwitzer, Peter M., and Paschal Sheeran. 2006. “Implementation Intentions and Goal Achievement: A Meta-Analysis of Effects and Processes.” Advances in Experimental Social Psychology 38: 69–119.

Hänggi, Peter, Peter Talkner, and Michal Borkovec. 1990. “Reaction-Rate Theory: Fifty Years After Kramers.” Reviews of Modern Physics 62 (2): 251–341.

Kramers, H. A. 1940. “Brownian Motion in a Field of Force and the Diffusion Model of Chemical Reactions.” Physica 7 (4): 284–304.

Lally, Phillippa, Cornelia H. M. van Jaarsveld, Henry W. W. Potts, and Jane Wardle. 2010. “How Are Habits Formed: Modelling Habit Formation in the Real World.” European Journal of Social Psychology 40 (6): 998–1009.

Milkman, Katherine L., Dena Gromet, Hung Ho, et al. 2021. “Megastudies Improve the Impact of Applied Behavioural Science.” Nature 600: 478–83.

Wood, Wendy, and Dennis Rünger. 2016. “Psychology of Habit.” Annual Review of Psychology 67: 289–314.

Fan, Y. (2026). Classical and Statistical Mechanics of Human Action: Psychological Inertia, Static Friction Thresholds, Kramers Escape Rate in Double-Well Potentials and Habit Phase Transitions, Read Beside East Asian Philosophy of Practice [行动的经典与统计力学] (Version 1.0.0). 5D Library. https://doi.org/10.5281/zenodo.23738774 · Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0).