Effective diffusivity in a periodic potential

How fast does a Langevin sampler actually explore a rough landscape? In one periodic dimension the answer is exact, 1/I₀(β)², which makes it a benchmark a sampler either reproduces or fails.

Effective diffusivity is the rate at which a diffusion actually covers ground in a landscape with barriers, as opposed to the rate it would cover ground with the barriers removed. It is the quantity that governs how quickly stochastic gradient Langevin dynamics explores a loss surface, how quickly an annealed sampler traverses a multimodal target at a given noise level, and how much of a sampler’s nominal step size survives the roughness of what it is stepping through. In general it is not computable.

In one periodic dimension it is, and the exception earns a page, because it is one of the few places where a sampler can be audited against an identity instead of against a longer run of itself.

Setting and assumptions

Let $V\in C^2(\mathbb{R})$ be $L$-periodic and let $\beta >0$. Consider the overdamped Langevin diffusion

$d{X}_t=-V^{\prime }(X_t)dt+\sqrt{2{\beta }^{-1}}d{W}_t,D_0:={\beta }^{-1},$

with $X_0$ deterministic or with any law having finite second moment. Write $\mathbb{T}=\mathbb{R}/L\mathbb{Z}$ and let $Y_t=X_t\mathrm{mod}L$ be the projected process, a diffusion on the torus with generator

$\mathcal{L}=-V^{\prime }(y){\partial }_y+D_0{\partial }_y^2.$

Because $\mathbb{T}$ is compact and $\mathcal{L}$ is uniformly elliptic with smooth coefficients, $Y$ has a unique invariant probability measure, and it is the Gibbs measure

$\pi (dy)=\frac{1}{Z}{e}^{-\beta V(y)}dy,Z={\int }_0^L{e}^{-\beta V}dy.$

The quantity of interest is the effective diffusivity

$D_{\mathrm{eff}}:={\lim }_{t\to \infty }\frac{\mathbb{E}[(X_t-X_0{)}^2]}{2t}.$

Note that $X$ itself is not projected: displacement accumulates, and that is what diffuses.

The corrector

The drift $b:=-V^{\prime }$ has mean zero under $\pi$:

${\int }_0^L{V}^{\prime }(y)e^{-\beta V(y)}dy=-\frac{1}{\beta }{\int }_0^L\frac{d}{dy}{e}^{-\beta V(y)}dy=0$

by periodicity. This is exactly the solvability condition for the cell problem, so by the Fredholm alternative for $\mathcal{L}$ on $\mathbb{T}$ there is a $\chi \in C^2(\mathbb{T})$, unique up to an additive constant, with

$\mathcal{L}\chi =V^{\prime }.$

Apply Itô’s formula to $\chi (Y_t)$ and use $\mathcal{L}\chi =V^{\prime }=-b$:

$\chi (Y_t)-\chi (Y_0)=-{\int }_0^{t}b(Y_s)ds+\sqrt{2D_0}{\int }_0^t{\chi }^{\prime }(Y_s)d{W}_s.$

Substituting ${\int }_0^{t}b(Y_s)ds$ back into $X_t-X_0={\int }_0^{t}b(Y_s)ds+\sqrt{2D_0}{W}_t$ gives the martingale decomposition

$X_t-X_0=\underset{\text{bounded}}{\underbrace{-[\chi (Y_t)-\chi (Y_0)]}}+\sqrt{2D_0}{\int }_0^t(1+{\chi }^{\prime }(Y_s))d{W}_s.$

The first term is bounded uniformly in $t$, because $\chi$ is continuous on a compact space; it contributes nothing at order $\sqrt{t}$. The second is a martingale whose quadratic variation is $2D_0{\int }_0^t(1+{\chi }^{\prime }{)}^2(Y_s)ds$, and the ergodic theorem for $Y$ gives $t^{-1}$ times that converging a.s. to $2D_0{\mathbb{E}}_{\pi }[(1+{\chi }^{\prime }{)}^2]$. Hence

$D_{\mathrm{eff}}=D_0{\int }_{\mathbb{T}}(1+{\chi }^{\prime }(y){)}^2\pi (dy).$

Two things follow before $\chi$ is even solved for. The formula is a variance, so $D_{\mathrm{eff}}\ge 0$ automatically, and the same decomposition gives a central limit theorem, $(X_t-X_0)/\sqrt{t}\Rightarrow \mathcal{N}(0,2D_{\mathrm{eff}})$, which says more than a statement about second moments would.

Solving the cell problem

In one dimension the cell problem integrates. Write $\mathcal{L}\chi =V^{\prime }$ as

$D_0{\chi }^{\prime \prime }-V^{\prime }{\chi }^{\prime }=V^{\prime },$

and multiply through by $\beta e^{-\beta V}$. Using $\frac{d}{dy}{e}^{-\beta V}=-\beta V^{\prime }{e}^{-\beta V}$, the left-hand side collapses to $(e^{-\beta V}{\chi }^{\prime }{)}^{\prime }$ and the right-hand side to $-(e^{-\beta V}{)}^{\prime }$, so

$\left(e^{-\beta V}\right(1+{\chi }^{\prime }){)}^{\prime }=0⟹1+{\chi }^{\prime }(y)=C{e}^{\beta V(y)}$

for a constant $C$. The constant is fixed by requiring $\chi$ to be periodic, i.e. ${\int }_0^L{\chi }^{\prime }=0$:

$L={\int }_0^L(1+{\chi }^{\prime })=C{\int }_0^L{e}^{\beta V}⟹C=\frac{1}{⟨e^{\beta V}⟩},$

writing $⟨f⟩:=L^{-1}{\int }_0^{L}f$. Substituting into the variance formula, with $Z=L⟨e^{-\beta V}⟩$,

$D_{\mathrm{eff}}=D_0\cdot \frac{1}{⟨e^{\beta V}{⟩}^2}\cdot \frac{1}{Z}{\int }_0^L{e}^{\beta V}=\boxed{\frac{D_0}{⟨e^{\beta V}⟩⟨e^{-\beta V}⟩}}$

which is the Lifson–Jackson formula. By Jensen’s inequality $⟨e^{\beta V}⟩⟨e^{-\beta V}⟩\ge 1$, with equality only for constant $V$: a periodic potential can only slow diffusion down, never speed it up. That is not obvious a priori, since the drift does not always oppose the motion, and it makes a useful sanity check on any implementation.

The cosine potential

Take $V(x)=\cos x$, so $L=2\pi$. Then

$⟨e^{\pm \beta \cos x}⟩=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{\pm \beta \cos x}dx=I_0(\beta ),$

the modified Bessel function of the first kind; the two signs agree because $x\mapsto x+\pi$ maps one integrand to the other. With $D_0={\beta }^{-1}$,

$D_{\mathrm{eff}}=\frac{1}{\beta I_0(\beta {)}^2}.$

At $\beta =1$, from the series $I_0(z)={\sum }_{k\ge 0}(z^2/4{)}^k/(k!{)}^2$,

$I_0(1)=1.2660658777520082\dots ,D_{\mathrm{eff}}=\frac{1}{I_0(1{)}^2}=0.6238603604320694\dots$

so a cosine barrier of height $2$ at unit temperature costs about $38\%$ of the free diffusivity.

The cold limit is Arrhenius, as it must be. From $I_0(\beta )\sim e^{\beta }/\sqrt{2\pi \beta }$ as $\beta \to \infty$,

$D_{\mathrm{eff}}\sim \frac{1}{\beta }\cdot 2\pi \beta e^{-2\beta }=2\pi e^{-\beta \Delta V},\Delta V=2,$

where $\Delta V=2$ is exactly the barrier height of $\cos x$ between its minimum and maximum. The formula reproduces Kramers’ law without being told about it.

Checking a simulation against it

This is the payoff. Discretize with Euler–Maruyama at step $h$, run $R$ independent replicas with $X_0$ drawn from the periodized Gibbs measure, and form the mean-squared-displacement estimator

$\hat{D}(t)=\frac{1}{R}{\sum }_{r=1}^R\frac{(X_t^{(r)}-X_0^{(r)}{)}^2}{2t}.$

Using the displacement rather than $\mathrm{Var}(X_t)$ matters at small $t$: the variance estimator carries an $O(1)/2t$ offset from the spread of $X_0$ and diverges as $t\to 0$, while the displacement estimator starts at $D_0=1$, which is the correct short-time answer.

With $R=4096$, $h=10^{-2}$, $\beta =1$, and standard errors from four disjoint replica groups:

$t$	$\hat{D}(t)$	$\pm$ s.e.	rel. dev.
1	0.87262	0.02012	$+39.9\%$
10	0.64524	0.02829	$+3.4\%$
100	0.60747	0.01687	$-2.6\%$
1000	0.61176	0.01861	$-1.9\%$
2000	0.62181	0.00984	$-0.3\%$

The convergence is slow, and it is slow for a structural reason rather than a statistical one: the systematic part of $\hat{D}(t)$ decays on the barrier-crossing timescale, not like $1/t$. At $\beta =1$ that timescale is short enough to reach in a browser tab. It would not be at $\beta =4$, and any plot that looked this clean at $\beta =4$ would be evidence of a bug.

The figure on the research page runs this estimator live against the dashed reference $1/I_0(1{)}^2$.

Why the exact value earns its keep

Two uses, both about sampling rather than about physics.

First, as a test. Any implementation of a Langevin sampler, including the inner loop of an annealed or score-based one, can be pointed at $V=\cos x$ and required to return $0.6239$. It is a stronger test than checking a marginal, because it depends on the sampler’s behavior over long trajectories instead of at stationarity, and it is where an integrator with a subtly wrong noise scaling shows up. A sampler that gets the invariant measure right and the diffusivity wrong has a bug in its time discretization.

Second, as a scale. The Arrhenius form D_{\mathrm{eff}} \sim 2\pi e^{-eta\Delta V} says the cost of roughness is exponential in the barrier and only linear in anything one might tune. That is the same arithmetic that makes tempering, annealing, and noise schedules necessary rather than optional: no step size buys back an exponential. The companion note on the Metropolis correction makes the same point from the other direction.

The sampler this note tests is in langevin-samplers.

References

• S. Lifson and J. L. Jackson (1962). On the self-diffusion of ions in a polyelectrolyte solution. Journal of Chemical Physics 36(9), 2410–2414.
• G. A. Pavliotis and A. M. Stuart (2008). Multiscale Methods: Averaging and Homogenization. Springer. Chapters 12–13 give the corrector argument in general dimension.
• R. Zwanzig (1988). Diffusion in a rough potential. PNAS 85(7), 2029–2030.