---
title: Eyring–Kramers Law in Metastable Dynamics
url: https://www.emergentmind.com/topics/eyring-kramers-law
type: topic
---

# Eyring–Kramers Law in Metastable Dynamics

The Eyring–Kramers law is the sharp low-noise or low-temperature asymptotic for rare transitions between metastable states. In its canonical form, it combines an Arrhenius exponential, determined by the relevant communication height or energy barrier, with a subexponential prefactor determined by local quadratic data near minima and transition states. Depending on normalization, it appears as a mean-time asymptotic proportional to $e^{\Delta V/\epsilon}$ or $e^{\beta \Delta V}$, and equivalently as a rate asymptotic proportional to $e^{-\Delta V/\epsilon}$ or $e^{-\beta \Delta V}$. Contemporary work treats not only reversible overdamped diffusions, but also hypoelliptic kinetic processes, non-reversible and non-Gibbsian diffusions, jump processes, and infinite-dimensional SPDEs, with the prefactor varying accordingly [1811.06786] [2503.12610] [1507.02104].

## 1. Classical reversible form

For overdamped Langevin dynamics
$$
dX_t=-\nabla V(X_t)\,dt+\sqrt{\frac{2}{\beta}}\,dW_t,
$$
with a nondegenerate local minimum $x_i$, an index-1 saddle $x_s$, and barrier $\Delta V=V(x_s)-V(x_i)$, the classical Eyring–Kramers rate is
$$
k_{i\to j}\approx \frac{|\lambda_-(x_s)|}{2\pi}\,
\sqrt{\frac{\det(\nabla^2 V(x_i))}{|\det(\nabla^2 V(x_s))|}}\,
e^{-\beta\Delta V},
$$
where $\lambda_-(x_s)<0$ is the unique unstable Hessian eigenvalue at the saddle. The reciprocal formulation gives the mean transition time
$$
\mathbb E[\tau_{i\to j}]
\sim
\frac{2\pi}{|\lambda_-(x_s)|}
\sqrt{\frac{|\det(\nabla^2 V(x_s))|}{\det(\nabla^2 V(x_i))}}\,
e^{\beta\Delta V}.
$$
In this reversible setting, the law describes not only metastable transition times but also the small spectrum of the generator, with the same communication heights governing the exponentially small eigenvalues [1811.06786].

The same exponential scale controls functional inequalities for the Gibbs measure. For smooth Morse potentials with a unique dominant critical depth $E_*$, the spectral gap and log-Sobolev constant satisfy
$$
\mathrm{Gap}(\beta)\asymp A_{EK}e^{-\beta E_*},
\qquad
\alpha(\beta)\asymp B_{EK}e^{-\beta E_*},
$$
which is the mechanism used to obtain quantitative convergence rates for simulated annealing schedules of the form $\beta(t)\asymp (\log t)/E$ when $E>E_*$ [2102.02339].

## 2. Exit problems, quasi-stationarity, and boundary saddles

A complementary formulation treats exit from a metastable domain $D$. For the Dirichlet realization of the overdamped generator on $D$, the principal eigenvalue $\lambda_D$ is simple, the associated quasi-stationary distribution (QSD) is explicit in terms of the principal eigenfunction, and if the process starts from the QSD then $\tau_D$ is exponential with parameter $\lambda_D$ and independent of the exit location. This gives an exact metastable Markov-jump reduction at the level of the QSD, with Eyring–Kramers asymptotics entering through the exit fluxes [1811.06786].

When the dominant escape channels are on the boundary rather than through interior saddles, the prefactor changes. If $x_\star$ is the unique minimum in $\Omega$ and $z_k\in\partial\Omega$ are the relevant local minima of $f|_{\partial\Omega}$ with $\partial_{n_\Omega}f(z_k)>0$, then the exit rates in the normalization
$$
dX_t=-\nabla f(X_t)\,dt+\sqrt h\,dB_t
$$
are
$$
k_{z_k}^{\mathrm{exit}}(h)
\sim
\frac{|\mu_{z_k}|}{\pi}
\sqrt{\frac{\det \mathrm{Hess}\,f(x_\star)}{|\det \mathrm{Hess}\,f(z_k)|}}
\exp\!\Big(-\frac{2}{h}(f(z_k)-f(x_\star))\Big),
$$
for higher saddles, with the obvious modification when several lowest boundary saddles have equal height. Here $\mu_{z_k}<0$ is the normal Hessian eigenvalue at the boundary saddle. The paper also emphasizes that exit rates and transition rates differ by a factor $1/2$ in the interior-saddle picture, because upon reaching the separatrix the process has asymptotically probability $1/2$ to return and $1/2$ to commit to the neighboring basin [2207.09284].

This boundary theory clarifies a frequent source of confusion: the Eyring–Kramers prefactor is not universally the inverse unstable Hessian eigenvalue. In domain-exit problems it can instead involve outward normal derivatives and tangential Hessians on $\partial\Omega$, with the exit distribution concentrating on generalized saddles of the boundary gradient flow [1811.06786].

## 3. Hypoelliptic and underdamped Langevin law

For the underdamped Langevin process in phase space,
$$
dq_t=p_t\,dt,\qquad
dp_t=-\nabla U(q_t)\,dt-\gamma p_t\,dt+\sqrt{2\gamma\epsilon}\,dB_t,
$$
with Hamiltonian
$$
V(q,p)=U(q)+\tfrac12|p|^2,
$$
Lee, Ramil, and Seo derive the sharp Eyring–Kramers asymptotic for the mean transition time between two wells connected by a unique index-1 saddle $\sigma$. If $m$ is the initial minimum, $H_A=\nabla^2U(m)$, $H_S=\nabla^2U(\sigma)$, $\lambda_S>0$ is the magnitude of the unique negative eigenvalue of $H_S$, and $\Delta V=U(\sigma)-U(m)$, then
$$
\mathbb E_{(m,0)}[\tau_{S_\epsilon}]
=
[1+o_\epsilon(1)]
\frac{2\pi}{\mu^\sigma}
\sqrt{\frac{| \det H_S|}{\det H_A}}
\exp\!\Big(\frac{\Delta V}{\epsilon}\Big),
$$
or, in inverse-temperature notation $\beta=1/\epsilon$,
$$
\mathbb E_{(m,0)}[\tau_{S_\epsilon}]
=
[1+o(\beta^{-1})]
\frac{2\pi}{\mu^\sigma}
\sqrt{\frac{| \det H_S|}{\det H_A}}
e^{\beta\Delta V},
$$
with
$$
\mu^\sigma=\frac{-\gamma+\sqrt{\gamma^2+4\lambda_S}}{2}>0.
$$
The factor $\mu^\sigma$ is the hypoelliptic unstable frequency at the saddle; friction and kinetic degrees of freedom enter only through this quantity [2503.12610].

This formula reduces to the overdamped prefactor in the high-friction scaling regime and to the low-friction Kramers regime as $\gamma\to0$. In particular, $\mu^\sigma\sim \lambda_S/\gamma$ as $\gamma\to\infty$, while $\mu^\sigma\to \sqrt{\lambda_S}$ as $\gamma\to0$ [2503.12610].

The derivation is structurally different from the elliptic overdamped case. Because diffusion acts only in momentum, the kinetic generator is hypoelliptic and the usual elliptic Dirichlet/Thomson capacity identities are not directly available. The argument instead introduces an equilibrium potential, a capacity-analog defined through a momentum-reversed test identity, and an elliptic regularization
$$
L_{\alpha,\epsilon}=L_\epsilon+\alpha(-\nabla U(q)\cdot \nabla_q+\epsilon\Delta_q),
$$
which yields a “magic formula” relating mean exit times, a harmonic surface measure, and the capacity-analog before passing to the limits $\alpha\to0$ and $N\to\infty$ [2503.12610].

## 4. Irreversibility, non-Gibbsianness, and symmetry corrections

For general irreversible elliptic diffusions
$$
dX_t^\epsilon=b(X_t^\epsilon)\,dt+\sqrt{2\epsilon}\,\sigma(X_t^\epsilon)\,dW_t,
$$
the barrier is no longer the potential difference but the Freidlin–Wentzell quasipotential $V(\bar x_1,x_\star)$. Under uniqueness and nondegeneracy of the instanton $\rho_t$ from the well $\bar x_1$ to the quasipotential saddle $x_\star$, the mean transition time satisfies
$$
\mathbb E[\tau_{\bar x_1\to\bar x_2}^\epsilon]
\sim
\frac{2\pi}{\lambda_+^\star}
\sqrt{\frac{|\det H_\star|}{\det(\mathrm{Hess}_xV(\bar x_1,\bar x_1))}}
\exp\!\Big(\int_{-\infty}^{+\infty}F(\rho_t)\,dt\Big)
\exp\!\Big(\frac{V(\bar x_1,x_\star)}{\epsilon}\Big).
$$
Here $\lambda_+^\star$ is the unstable eigenvalue of the Jacobian $Db(x_\star)$, $H_\star$ is the limiting Hessian of the quasipotential at the saddle, and the factor
$$
\exp\!\Big(\int_{-\infty}^{+\infty}F(\rho_t)\,dt\Big)
$$
measures non-Gibbsianness along the instanton. In the Gibbs-preserving case this correction is $1$ [1507.02104].

A more restrictive but important non-reversible class preserves the Gibbs invariant density while breaking reversibility. For
$$
d\boldsymbol{x}_\epsilon(t)=-(\nabla U+\boldsymbol{\ell})(\boldsymbol{x}_\epsilon(t))\,dt+\sqrt{2\epsilon}\,d\boldsymbol{w}_t,
$$
with $\nabla U\cdot \ell=0$ and $\nabla\cdot \ell=0$, the double-well prefactor is obtained by replacing the negative Hessian eigenvalue at the saddle by the unique negative eigenvalue of the non-selfadjoint linearization $\mathbb H^\sigma+\mathbb L^\sigma$:
$$
\mathbb E^\epsilon_{\boldsymbol m_1}[\tau_{\mathcal D_\epsilon(\boldsymbol m_2)}]
=
[1+o_\epsilon(1)]
\frac{2\pi}{\mu^\sigma}
\sqrt{\frac{-\det \mathbb H^\sigma}{\det \mathbb H^{\boldsymbol m_1}}}
\exp\!\Big(\frac{U(\sigma)-U(m_1)}{\epsilon}\Big).
$$
Since $\mu^\sigma\ge \lambda^\sigma$ in the reversible comparison, the non-reversible perturbation accelerates the metastable transition [2008.08291].

Analogous replacements occur for non-reversible random walks in a potential field. There the exponential barrier is unchanged, but the prefactor contains the unique negative eigenvalue of the Jacobian $\mathbb M_\sigma=\mathbb A\,\mathbb H_\sigma$ generated by the cycle structure rather than the negative Hessian eigenvalue itself, together with a non-Gibbs correction $e^{G(\sigma)-G(m_1)}$ [1605.01009].

Symmetry produces a different type of correction. For reversible metastable jump processes invariant under a finite group $G$, the trivial isotypic component has effective prefactors
$$
c_{ij}^\ast=c_{ab}|G_a\cap G_b|,
\qquad
m_i^\ast=\frac{m_a}{|G_a|},
$$
so stabilizers modify the standard Eyring–Kramers prefactor by explicit orbit-combinatorial factors. In nontrivial irreducible representations, additional Arrhenius exponents can appear through successor cycles and Schur complements of the symmetry-reduced generator [1312.0835].

The non-Gibbsian boundary-exit problem leads to yet another correction. For $dX_t=b(X_t)\,dt+\sqrt h\,dB_t$ with an orthogonal decomposition $b=-(\nabla f+\ell)$, $\ell\cdot \nabla f=0$, and non-characteristic boundary, the mean exit time from a bounded domain satisfies
$$
\mathbb E_x[\tau_\Omega]
=
\kappa(h)\sqrt h\,
\exp\!\Big\{\frac{2}{h}\big(\min_{\partial\Omega}f-f(x_0)\big)\Big\},
$$
with
$$
\frac{1}{\kappa_0}
=
\frac{\sqrt{\det \mathrm{Hess}\,f(x_0)}}{\sqrt\pi}
\sum_{z\in \mathscr P_{\mathrm{sp}}}
\frac{\partial_{n_\Omega}f(z)}
{\sqrt{\det \mathrm{Hess}(f|_{\partial\Omega})(z)}}
\exp\!\Big\{\int_0^\infty \mathrm{div}\,\ell(\psi_t(z))\,dt\Big\}.
$$
The new factor
$$
\exp\!\Big\{\int_0^\infty \mathrm{div}\,\ell(\psi_t(z))\,dt\Big\}
$$
is the explicit non-Gibbsian correction [2509.17678].

## 5. Infinite-dimensional, constrained, and field-theoretic extensions

In systems with conserved quantities, the naive determinant ratio may diverge because the mobility has zero modes. For reversible gradient diffusions constrained to a manifold $S$, the generalized Eyring–Kramers law replaces full determinants by determinants restricted to the tangent space:
$$
\tau
\approx
\frac{2\pi}{|\mu_-|}
\sqrt{\frac{| \det(H_s|_{T_{x_s}S})|}{\det(H_-|_{T_{x_-}S})}}
e^{\Delta U/\epsilon}.
$$
For a single conserved quantity with unit normal $\hat m$, this is equivalently
$$
\tau
\approx
\frac{2\pi}{|\mu_-|}
\sqrt{\frac{| \det H_s|}{\det H_-}}
\sqrt{\frac{\hat m\cdot H_s^{-1}\hat m}{\hat m\cdot H_-^{-1}\hat m}}
e^{\Delta U/\epsilon}.
$$
This framework is applied to fluctuating hydrodynamics, including stochastic thin-film rupture and a mass-conserving social-segregation SPDE, where ignoring the conservation law mispredicts the prefactor [2405.13490].

A rigorous infinite-dimensional spectral-gap version is now available for reversible stochastic gradient systems
$$
dX_t=-A\nabla F(X_t)\,dt+\sqrt{2\beta^{-1}A}\,dW_t
$$
on an abstract Wiener space. In the double-well case, with minima $x_\pm$, relevant saddles $x_j$, unstable eigenvalues $-\alpha_j<0$ of the linearized operator in the $A^{-1}$-metric, and Fredholm determinant factors
$$
C_\pm=\det[\nabla^2F(x_\pm)]^{-1/2},
\qquad
C_j=|\det[\nabla^2F(x_j)]|^{-1/2},
$$
the first nonzero eigenvalue satisfies, in the symmetric-well case,
$$
\lambda_\beta^{(1)}
=
\sum_{j=1}^{m_0}
\frac{C_-+C_+}{2\pi C_-C_+}\,
C_j\alpha_j\,e^{-\beta(h-F(x_+))}
\bigl(1+O(\beta^{-1/2})\bigr),
$$
which is an infinite-dimensional Eyring–Kramers law for the spectral gap [2606.16083].

Renormalized SPDEs require further determinant regularization. For the two-dimensional stochastic Allen–Cahn equation with Wick renormalization, the Arrhenius factor remains
$$
\exp\!\Big\{\frac{L^2}{4\epsilon}\Big\},
$$
but the finite-dimensional determinant ratio is replaced by the Carleman–Fredholm determinant
$$
\det_2\bigl(I+3P_\perp(-\Delta-1)^{-1}\bigr)^{-1/2},
$$
reflecting the ultraviolet renormalization of the quartic field theory [1604.05742].

Hyperbolic field theories admit a transition-state-theory variant. For the damped stochastic nonlinear wave equation with $\phi^4$ potential on $\mathbb T^d$, $d=2,3$, the transition frequency factorizes into a white-noise flux term and a renormalized Gibbs-measure ratio, producing
$$
\nu(\beta)
=
\kappa_{\mathrm{trans}}
\bigg[
\frac{1}{2\pi}
\sqrt{\prod_{|k|\le N}\frac{C_L|k|^2+2}{|C_L|k|^2-1|}
\exp\!\Big(-\frac{3C_NL^d}{2}\Big)}
\bigg]
\exp\!\Big(-\frac{\beta L^d}{4}\Big)
\bigl(1+O(\beta^{-1})\bigr).
$$
In the two-dimensional unforced hyperbolic model, the transmission coefficient satisfies $\kappa_{\mathrm{trans}}=1-\mathcal O(\beta^{-1/4+})$ [2410.03495].

Further SPDE variants show that the transition state itself can bifurcate. For the one-dimensional dynamical sine-Gordon stochastic heat equation, the transition state is constant when $\gamma\beta<1$ and becomes a kink–antikink pair when $\gamma\beta>1$, while the Arrhenius barrier remains $4\pi\gamma/(\epsilon\beta)$ [2509.13806].

## 6. Methods, applications, and conceptual boundaries

Three proof paradigms dominate the subject. The classical overdamped theory uses potential theory, Dirichlet spectral analysis, Witten Laplacians, and Agmon localization to identify the principal Dirichlet eigenfunction, its normal derivative at the boundary, and the corresponding capacity or flux. Non-selfadjoint Fokker–Planck operators with Gibbs stationary states can also be treated semiclassically by Gaussian quasimodes and graded Schur-complement analysis, without supersymmetry or PT symmetry, yielding prefactors involving the unique negative-real-part eigenvalue of
$$
\Lambda(s)=2\,\mathrm{Hess}\,f(s)\,A^0(s)+B^t(s)
$$
rather than a purely Hessian quantity [1811.06786] [2201.01660].

A distinct geometric viewpoint characterizes capacity by two weighted objects: a Gibbs-weighted geodesic distance
$$
d_\epsilon(A,B;\Omega)
=
\inf_{\gamma}
\int_0^1 e^{-F(\gamma(t))/\epsilon}|\gamma'(t)|\,dt
$$
and a Gibbs-weighted minimal separating hypersurface
$$
V_\epsilon(A,B;\Omega)
=
\inf_{S}
\int_S e^{-F(x)/\epsilon}\,dH^{n-1}(x).
$$
For a single saddle one obtains
$$
\mathrm{cap}(A,B)
\sim
\epsilon\,\frac{V_\epsilon}{d_\epsilon}\,e^{-F(z)/\epsilon},
$$
which recovers the classical Hessian-based Eyring–Kramers formula in the Morse case and extends it to flat minima, flat saddles, and multiple same-height saddles arranged in series or parallel [2206.13206].

Algorithmically, the law underpins metastable coarse-graining. In overdamped dynamics, the QSD makes the exit event exactly representable as a kinetic Monte Carlo step, with exponentially distributed holding time and independent exit channel. This is the basis for jump-process reductions and accelerated dynamics methods such as TAD. In Adaptive Kinetic Monte Carlo, Eyring–Kramers rates are used both to populate the low-temperature event catalog and, when reliable at the search temperature, to estimate the fraction of total reaction rate already discovered [1811.06786] [1506.05092].

The law also supplies quantitative convergence rates for optimization algorithms. In simulated annealing, the Eyring–Kramers asymptotics for the log-Sobolev constant and spectral gap yield polynomial decay bounds for
$$
\mathbb P(f(X_t)>\delta)
\quad\text{and}\quad
\mathbb P(f(x_k)>\delta)
$$
under cooling schedules with $E>E_*$ and suitable step-size conditions, directly linking metastable barrier structure to optimization complexity [2102.02339].

A recurring misconception is that the Eyring–Kramers law is a single formula with a universal prefactor. The data instead show a stable hierarchy of modifications. The Arrhenius exponent is robust and is usually determined by a communication height, a quasipotential barrier, or a domain-exit barrier. The prefactor, however, is sensitive to the operator class and geometry: unstable Hessian eigenvalues in the reversible overdamped case, $\mu^\sigma$ in the underdamped hypoelliptic case, normal derivatives and tangential Hessians for boundary exit, pseudo-determinants under conservation laws, Fredholm or Carleman–Fredholm determinants in SPDEs, stabilizer factors in symmetric jump processes, and non-Gibbsianness corrections along instantons or boundary characteristics [2503.12610] [2405.13490] [2509.17678].

Source: https://www.emergentmind.com/topics/eyring-kramers-law