2026-10-01 · chapter Yang–Mills and the mass gapPart 2 Physical preparationrelaxation and the dynamical exponent

The relaxation time and the dynamical exponent z — by what power of the correlation length does the approach to equilibrium slow down

What 2-08 and 2-09 treated was the equilibrium distribution, that is, "which configurations appear with what probability". There is no time in it. But when a computer draws samples from an equilibrium distribution, or when a physical system in contact with a heat bath moves towards equilibrium, the configuration moves in time. This chapter measures the speed of that movement.

As a metaphor, the equilibrium distribution is a single photograph, and the dynamics is the machine that makes it. Among machines that make the same photograph, some are fast and some slow. Precisely, a Markov process (a random motion whose next move depends only on the current configuration) leaving a given probability measure $\mu$ invariant is not unique, and the way it approaches equilibrium depends on the choice.

The time $s$ of this chapter is a different thing from the imaginary time of 2-09 (for the contrast see 8-03 §05).

The subject is the relation $\tau\approx\xi^{z}$, and $z$ is called the dynamical exponent. The assumptions of the later parts (A-06 Index of assumptions) have the form of placing $z=2$ at the level of a power. Here we first line up examples where that form holds and examples where it breaks.

Suppose we want a single real number $X$ to follow the probability density $e^{-V(x)}/Z$. If we move it in the direction down the slope $V$ and at the same time shake it with random numbers, the distribution stops where the two effects balance. This is Langevin dynamics.

$W$ is Brownian motion (1-09 §06), and $dW$ is a Gaussian increment with mean 0 and variance $ds$ over time $ds$. The coefficient $\sqrt2$ is chosen so that the distribution at which it stops is exactly $e^{-V}$.

The operator producing the time change of the expectation of a function $f$ is called the generator, written $\mathcal L$. Expanding $f(X+dX)$ to second order, the first-order term in $dX$ leaves $-V'f'\,ds$ in the direction of the slope, and the second-order term leaves $\tfrac12f''\cdot2\,ds$ since the mean of $dW^2$ is $ds$ (orders higher than $ds$ are dropped). Thus for (1),

($\mu=e^{-V}dx/Z$; the last equality is integration by parts). By the last equality $-\mathcal L$ is symmetric in $L^2(\mu)$ and non-negative. It is 0 only for constants, and we write the next eigenvalue (the gap) as $\mathrm{gap}(\mathcal L)$.

The inverse of the gap is the Poincaré constant (1-10 §07).

Running from equilibrium, the autocorrelation of every observable decreases faster than $e^{-s/C_P}$, and there is an observable that decreases at exactly this rate. That is, the slowest relaxation time $\tau_{\exp}$ equals $C_P$ Known. This equality is a property of reversible dynamics in continuous time; for chains in discrete time it splits into three quantities (8-03 §06).

Example. For $V=x^2/(2\sigma^2)$, (1) is the Ornstein–Uhlenbeck process, $f=x$ is an eigenfunction, and $\mathcal Lx=-x/\sigma^2$. Hence $C_P=\sigma^2$: the larger the variance (the wider the tails), the slower the relaxation. With many variables, we set up (1) for the field $\phi_x$ at each point,

($W_x$ independent at each point), and replace the $|f'|^2$ of (3) by the squared length of the gradient $|\nabla f|^2$. The distribution at which it stops is $e^{-S}/Z$. In field theory this is called stochastic quantisation (6-09).

The spins of the Ising model are $\pm1$, so they cannot move continuously down a slope. Instead, each spin carries an independent clock, and when the clock rings the spin is redrawn from its conditional distribution. This is called the heat-bath rule, and the dynamics doing this in continuous time is called Glauber dynamics (Glauber 1963 analysed a process of this type in detail on a closed one-dimensional chain) Known.

With $K$ including the inverse temperature in the coupling (as in 2-09), let $\mu(\sigma)\propto\exp(K\sum_{\langle xy\rangle}\sigma_x\sigma_y)$. Writing the configuration with spin $j$ flipped as $\sigma^j$ and the sum of neighbours as $h_j=\sum_{y\sim j}\sigma_y$, the heat-bath flip rate is

The right-hand formula is detailed balance: it says that "the probability flowing from $\sigma$ to $\sigma^j$" equals the flow in the opposite direction. From this, $\mu$ is the distribution at which it stops, and the dynamics is reversible. The more aligned a spin ($\sigma_jh_j\gt0$), the less readily it flips. The generator is $(\mathcal Lf)(\sigma)=\sum_jw_j(\sigma)\bigl(f(\sigma^j)-f(\sigma)\bigr)$, the $\int|f'|^2$ on the right side of (3) is replaced by $\tfrac12\sum_\sigma\mu(\sigma)\sum_jw_j(\sigma)\bigl(f(\sigma^j)-f(\sigma)\bigr)^2$, and $C_P=1/\mathrm{gap}(\mathcal L)$ is the same.

The one-dimensional chain can be solved

In one dimension $h_j\in\{-2,0,2\}$, so we can write $w_j=\tfrac12\bigl(1-\tfrac g2\sigma_j(\sigma_{j-1}+\sigma_{j+1})\bigr)$, $g=\tanh2K$. The expectation of a spin changes by $2$ in one flip, so $\frac{d}{ds}\langle\sigma_j\rangle=-2\langle\sigma_jw_j\rangle$, and

closes. The mean of the total magnetisation $M=\sum_j\sigma_j$ decreases like $e^{-(1-g)s}$. On the other hand, the static correlation is $\langle\sigma_0\sigma_r\rangle=(\tanh K)^r$ (2-09 §03), and

The limits follow from $1-\tanh2K\approx2e^{-4K}$, $-\log\tanh K\approx2e^{-2K}$. The colder it is, the longer the correlations, and relaxation slows by its square: for this dynamics $z=2$.

ComputedTable 1 One-dimensional Glauber dynamics. We diagonalised the generator of the ring of $N=8,\ 10$ points on all states ($2^N$ of them) and checked that the gap agrees with $1-\tanh2K$ to 10 digits for all of $K=0.25,\ 0.5,\ 1,\ 1.5$. On these two rings the decay of magnetisation is the slowest relaxation. For general $N$, the decay rate of magnetisation $1-\tanh2K$ gives an upper bound on the gap (since magnetisation is an eigenfunction of $\mathcal L$).

We apply (4) to the Gaussian chain of (7) of 2-08 (the $H(\phi)$ there; here written as the action $S$) $S=\tfrac12\sum_x[(\phi_{x+1}-\phi_x)^2+m^2\phi_x^2]$. Since $\partial S/\partial\phi_x=(2+m^2)\phi_x-\phi_{x+1}-\phi_{x-1}$ is linear, the equations separate for each Fourier component $\hat\phi_q=\sum_xe^{-iqx}\phi_x$ of wave number $q$ (1-11).

Every component is the Ornstein–Uhlenbeck process of (1), decreasing at rate $\omega_q$. The slowest is the uniform component $q=0$, with rate $m^2$. Therefore $C_P=1/m^2$ (the same number as "the $C_P$ of a Gaussian measure is the largest eigenvalue of the covariance" of 1-10 §09) Known.

The correlation length was $\cosh(1/\xi)=1+m^2/2$ by (8) of 2-08. Using $\cosh u-1=2\sinh^2(u/2)$, the two are connected by a single formula.

This is an exact equality; on the lattice $C_P$ is slightly smaller than $\xi^2$, and becomes $C_P/\xi^2\to1$ in the continuum limit ($\xi\to\infty$). For the continuous free field $C_P=\xi^2$ holds exactly Known (an elementary derivation; the numbers in Table 2 are Computed).

Why the square

Why the square. The mean motion of (8), with the noise removed, is the heat equation with mass $\frac{d}{ds}\phi=\Delta_{\rm lat}\phi-m^2\phi$. Diffusion takes a time of order $\ell^2$ to smooth out bumps of length $\ell$ (as for a random walk), and the mass term removes waves longer than $\xi$ at a uniform rate $m^2$. So the time is $\xi^2$. Precisely, the content of this explanation is that the $\omega_q$ of (8) approaches $m^2+q^2$ for small $q$, and bottoms out at $m^2$ for $q\lesssim1/\xi$.

ComputedThe curves in the figure are the exact solution $\hat\phi_q(s)=e^{-\omega_qs}\hat\phi_q(0)$ of (8), computed on the spot (the initial shape was made with random numbers from a fixed seed). This is the mean motion with the noise removed, not a sample of the fluctuations.

ComputedTable 2 $\xi$ and $C_P$ of the Gaussian chain. The last two columns are the heat-bath lower bound of section 08 (determined by the decrease on linear functions)

A table set beside the gap of the transfer matrix is in Table 1 of 8-03, and a table measured with a checkerboard heat bath is in 5-07 §07.

With interactions, the formulas do not close. So on $L\times L$ boxes (periodic boundary) of the two-dimensional Ising model we diagonalise the generator of Glauber dynamics (5) numerically and find $C_P$. There are $2^{L^2}$ states, about 34 million for $L=5$. The slowest relaxation is expected to lie in the direction of magnetisation (the subspace invariant under translations and changing sign under the global flip). For $L=3,\ 4$ the gap of the whole space and the gap of this subspace agreed to 8 digits at all seven temperatures below Computed. For $L=5$ the whole space cannot be handled, so we found the gap restricted to this subspace (671,104 dimensions). That value is at least the gap of the whole space, so the $C_P$ for $L=5$ is a lower bound on the true $C_P$.

At the critical point $K_c=\tfrac12\log(1+\sqrt2)$ (2-08 §04) the $\xi$ of the infinite system is infinite, and the length over which correlations reach inside a box levels off at the box size $L$ (2-09 §07). So instead of $\xi$ we use $L$ as the ruler. If $\tau\approx\xi^z$ is right, then

$z_{\rm eff}$ is an effective exponent read from neighbouring sizes, and is expected to approach $z$ as $L\to\infty$.

ComputedTable 3 Two-dimensional Ising, continuous-time heat bath (each spin redrawn at rate 1), $C_P=1/\mathrm{gap}$ with $L\times L$ periodic boundary. The unit of time is "one attempt per spin"

Three behaviours

Three behaviours can be read. On the high-temperature side ($K/K_c=0.6$), $C_P$ levels off around 6, and $z_{\rm eff}$ tends to 0: if $\xi$ is finite, so is the relaxation time. On the low-temperature side ($K/K_c=1.5$), $\log C_P$ increases with $L$ by an almost constant difference (the differences are 2.07, 1.97), growing more like an exponential than a power. This is thought to be because moving from "almost all $+$" to "almost all $-$" requires making a wall across the box, whose weight becomes small like an exponential $e^{-cL}$ proportional to the length of the wall Physics. In a box with periodic boundary two walls are needed, so $c$ is expected to be about twice the interface tension $\sigma$. However, the $2\sigma\approx1.55$ at $K/K_c=1.5$ and the difference of about 2 for $L\le5$ do not yet agree, and what this box can say goes only as far as "a tendency to grow exponentially". A similar picture is the closing gap of 2-09 §08.

At the critical point $C_P/L^2$ keeps increasing as $4.77,\ 5.18,\ 5.50$, and $z_{\rm eff}$ is $2.29,\ 2.27$. It lies on the side greater than 2. The value known from numbers on larger boxes (up to $15\times15$) is $z=2.1665(12)$ (Nightingale–Blöte 1996, heat bath) Physics; the values from the small boxes on this machine can be read as approaching it from above, but two points cannot decide this. What can be said goes only as far as "in the range $L\le5$, $C_P/L^2$ does not settle to a constant".

ComputedThe numbers in this figure were computed on this machine (the values of Table 3 embedded). L = 2 is excluded, as it is a special box in which neighbours coincide.

In physics one puts forward the hypothesis that as $\xi$ grows the relaxation time grows as a power of $\xi$. It is called the dynamic scaling hypothesis, and is held to be the extension of static scaling (2-08 §06) to correlations involving time (Ferrell et al. 1967, Halperin–Hohenberg 1967, with a detailed development in 1969) Physics. Put in mathematical language for reversible continuous-time dynamics,

$A\asymp B$ means that the ratio $A/B$ is bounded above and below by positive constants. In this book $\eta_{\rm dyn}$ is defined by this formula. The name comes from its being the quantity that measures, as a power, the departure from the $C_P\approx\xi^2$ of the free field.

It is a different quantity from the $\eta$ of (13) of 2-08. That one is the departure in the power of the static correlation $G(x)\sim|x|^{-(d-2+\eta)}$ at the critical point, and is written $\eta_s$ when it must be distinguished from the dynamic quantity. The two coincide only within the conventional theory (called by the name of van Hove) that holds "the relaxation time is proportional to the susceptibility"; for two-dimensional Ising, $\eta_s=1/4\gt0$ while $\eta_{\rm dyn}\approx-0.17\lt0$, so they differ even in sign (Table 2 in 8-03 §07).

Assumption H of the later parts (10-01 The line of construction and Assumption H) is the statement $\lim\ln C_P/\ln\xi=2$ ($C_P=\xi^{2+o(1)}$, $\eta_{\rm dyn}=0$) that places this left-hand form at the level of a power, for the Langevin dynamics of lattice Yang–Mills. The band statement $C_P\asymp\xi^2$ is called $\text{H}_{\rm band}$ in 10-01 and is treated separately from H (A-06 Index of assumptions). Whether it holds is an unproved assumption, and the physical counterpart is set out in the form of questions in 8-05 §03. What this chapter wants to confirm is that the form itself is not trivial: change the dynamics once and it breaks. The example is the next section.

Table 4 lists known examples in which the pattern $C_P\asymp\xi^2$ breaks. The ways of breaking divide into three.

Interaction shifts the power. For Glauber dynamics of two- and three-dimensional Ising, numerically $z$ slightly exceeds 2. Even with the rule kept local, interaction moves the power. In mathematics it has been shown that for Glauber dynamics of two-dimensional critical Ising the inverse of the gap is bounded by a polynomial in $L$ (Lubetzky–Sly) Known. From below, it has been shown that at the critical point with periodic boundary (and on the low-temperature side) the heat-bath gap is at most $C(\log L/L)^2$, that is, $z\ge2$ (Masaoka–Soejima–Watanabe 2025; extending to local reversible single-spin-flip dynamics) Known. At the two-dimensional critical point $(L/\log L)^2\lesssim C_P(L)\le\mathrm{poly}(L)$ is a theorem, and the "too fast" side of §06 does not occur for local Ising dynamics. No theorem determining the value of the power $z$ itself has been found.

Conserved quantities slow it down. In dynamics that do not change the total magnetisation (exchanging neighbouring spins), the uniform component cannot move, and long waves are carried by diffusion, so it is slower still. The prediction of physics is held to be $z=4-\eta_s$ (model B in the classification of Hohenberg–Halperin) Physics.

The way of updating speeds it up. The Swendsen–Wang cluster method is non-local dynamics that flips whole clusters, and its abstract says it breaks dynamic universality, giving 'unusually small values of the dynamical critical exponent' Physics. Even for the same equilibrium distribution, changing the machine changes $z$.

PhysicsTable 4 Example values of $z$ (numerical and predicted values from the abstracts of primary sources; $\eta_{\rm dyn}=2-z$ computed on this machine)

The last row is a slowness that actually appears in computations of lattice gauge theory. HMC (hybrid Monte Carlo; an update method that adds fictitious momenta and moves deterministically; 5-07 §08) is dynamics different from Langevin (next section), and whether the same slowness occurs for Langevin is treated in 10-02 Roads tried on the four-dimensional lattice, and walls.

Detour: $z=4$ for conserving dynamics

"The relaxation time" is not determined until we say whose dynamics' time it is. The heat bath and Langevin have different generators for the same distribution.

We compare them on the Gaussian chain. For the continuous-time heat bath (each point, at rate 1, redrawn from the conditional distribution $\mathcal N\bigl((\phi_{x-1}+\phi_{x+1})/(2+m^2),\,1/(2+m^2)\bigr)$ with its neighbours fixed), the mean of the field values moves by

Linear functions are eigenfunctions of this generator, and the decay rate of the uniform component is $m^2/(2+m^2)$. Therefore the $C_P$ of the heat bath is at least $(2+m^2)/m^2$ Computed (checked with the matrix of a ring of 64 points). This lower bound is about twice $\xi^2$, so for the heat bath $z\ge2$. As the right two columns of Table 2 show, the ratio of the lower bound to the $C_P$ of Langevin approaches 2 as $\xi\to\infty$. Even if the powers are the same, the constants differ.

Differences other than the constant

The difference is not only in the constant.

• The choice of variables. The heat bath is the same process even if the variable at each point is relabelled ($\psi_x=c\,\phi_x$ and so on), because the operation of drawing from a conditional distribution does not depend on coordinates. The $C_P$ of Langevin is measured by the length of the gradient, so measured in $\psi=c\,\phi$ it becomes $c^2$ times as large. This is why in lattice gauge theory $C_P$ changes by a factor $\beta$ depending on whether one measures by "the angle of the link $\vartheta$ or the field $A=\sqrt\beta\,\vartheta$". Since $\beta\sim\ln\xi$, the power statement H does not change, but the band statement does (1-10 §09, 10-01 The line of construction and Assumption H).
• The unit of time. The time of the heat bath is "the number of attempts per point", and the time of Langevin is the $s$ of (1). The heat bath that advances one sweep at a time in discrete time (5-07 §07) is yet another quantity, and $C_P$, the relaxation time and $\tau_{\exp}$ split apart (8-03 §06).
• The exponent. In physics, two local reversible dynamics with the same conserved quantities (both of model A type: not conserving magnetisation) are expected to have the same $z$ (the same dynamic universality class) Physics. The conserving dynamics of §07 has different conserved quantities (model B), so it is not a counterexample to this expectation. For the Gaussian chain, what can be said by formulas is $z=2$ for Langevin, and, for the heat bath, $z\ge2$ from the lower bound from linear functions. That the gap of the heat bath is determined by linear functions, with equality, is not shown in this chapter. In general, this chapter has no theorem showing that the $z$ of two dynamics are equal.

When this book writes $C_P$, unless stated otherwise it is the inverse of the gap of the generator of Langevin (4), with the variable whose gradient it is measured by stated alongside. Reading numbers from the heat bath or HMC as the $C_P$ of Langevin is an assumption.

Langevin dynamics (4) can be set up for lattice Yang–Mills too. Since the variables are group elements on the links, the gradient is taken as a derivative on the group (1-03). Doing the same for a continuous field is stochastic quantisation (6-09), and drawing samples from the same distribution on a computer is the Monte Carlo method (5-07).

At strong coupling (the side where $\beta$ is small), the Poincaré inequality and the log-Sobolev inequality (an inequality stronger than (3), controlling the decrease of entropy) have been shown for lattice Yang–Mills (Shen–Zhu–Zhu 2023) Known. This is a result on the side where $\xi$ is finite. In four dimensions there is no theorem determining the power of $C_P$ as $\xi\to\infty$.

What this chapter can say

Sources and reproduction

Next chapter: 3-01 The gauge principle. This chapter continues in 5-07 §07 (critical slowing down of the Monte Carlo method), 8-03 §05–§07 (the dictionary of the two times) and 10-01 (Assumption H).

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