EF-LD: Electronic Friction–Langevin Dynamics

• Electronic Friction–Langevin Dynamics (EF-LD) is a mixed quantum-classical model that employs a Langevin equation with friction and random forces to capture nonadiabatic nuclear motion near metal surfaces.
• It incorporates tensorial and coordinate-dependent friction to represent energy dissipation, mode coupling, and anisotropic effects beyond simple scalar damping.
• The framework extends to complex regimes including non-Markovian dynamics, quantum nuclear effects, and nonequilibrium conditions, aiding studies in surface scattering, adsorbate diffusion, and molecular electronics.

Electronic Friction–Langevin Dynamics (EF-LD) is a mixed quantum-classical framework for describing nonadiabatic nuclear motion near metal surfaces in which the fast electronic degrees of freedom are reduced to a mean force, a frictional back-action, and a stochastic force acting on classically propagated nuclei. In its standard Markovian form, EF-LD uses a Langevin equation in which dissipation and noise are linked by fluctuation-dissipation, while more elaborate variants incorporate tensorial friction, coordinate-dependent coupling, nonequilibrium driving, memory effects, and nuclear quantum effects. In the literature, EF-LD appears in surface scattering, adsorbate diffusion, electrochemistry, and molecular electronics, and is also discussed under closely related labels such as molecular dynamics with electronic friction (MDEF) (Askerka et al., 2016, Mäck et al., 2024, Liu et al., 25 May 2026).

1. Canonical formulation and basic physical content

The canonical EF-LD picture treats the nuclei as slow variables moving on an adiabatic potential energy surface or a potential of mean force, while metal electrons act as a thermal or steady-state bath. In tensorial notation this is written as

$$M\ddot{R_i}= - \frac{\partial V(\mathbf R)}{\partial R_i} - \sum_j \Lambda_{ij}\dot R_j + \mathscr R_i(t),$$

where $\Lambda_{ij}$ is the electronic friction tensor and $\mathscr R_i(t)$ is the random force. In a widely used scalar form for atom–metal scattering,

$$m\ddot{\mathbf r} = -\frac{\partial E}{\partial \mathbf r} - m\eta_{\mathrm{el}}\dot{\mathbf r} + \mathbf F_{\mathrm L}(t),$$

with

$$\left\langle \mathbf F_{\mathrm L}(t)\mathbf F_{\mathrm L}(t') \right\rangle = 2k_{\mathrm B}T_{\mathrm{el}}\,m\eta_{\mathrm{el}}\,\mathbf I\,\delta(t-t').$$

This makes explicit that the same coupling to substrate electrons that damps nuclear motion also injects thermal fluctuations into it (Hertl et al., 2021).

In this formulation, the friction term represents average energy dissipation into electron-hole-pair excitations, and the stochastic term represents the thermal fluctuations of those same electronic degrees of freedom. The standard reduction assumes slow nuclear motion relative to electronic relaxation, a low-velocity expansion of the electronic back-action, and, in the usual EF-LD form, a Markovian approximation in which the bath is represented by white noise and instantaneous friction. In equilibrium, the noise covariance is tied to the friction tensor by fluctuation-dissipation; in nonequilibrium, the same Langevin structure may still be used, but the equilibrium relation can be broken (Hertl et al., 2021, Mäck et al., 2024).

The formal scope of EF-LD is broader than a single scalar damping coefficient. It includes scalar isotropic friction, diagonal but anisotropic friction, and fully tensorial friction with off-diagonal couplings. It also includes models in which the friction depends on geometry, on the electronic state of the environment, or on external driving. This suggests a hierarchy of approximations ranging from local-density scalar friction to full coordinate-dependent Cartesian friction tensors (Askerka et al., 2016, Sachs et al., 2024).

2. Tensorial friction, anisotropy, and configuration dependence

A central development in EF-LD is the replacement of scalar damping by a full electronic friction tensor. Within first-order time-dependent perturbation theory evaluated on top of DFT/Kohn–Sham electronic structure, the tensor is written as

$$\Lambda_{ij} = \pi\hbar^2 \omega \sum_{\mathbf{k},\nu,\nu'} \langle \psi_{\mathbf{k}\nu}|\nabla_i \psi_{\mathbf{k}\nu'}\rangle \langle \psi_{\mathbf{k}\nu'}|\nabla_j \psi_{\mathbf{k}\nu}\rangle \delta(\epsilon_{\mathbf{k}\nu}-\epsilon_{\mathbf{k}\nu'}-\hbar \omega),$$

so that dissipative force along coordinate $i$ can be induced by motion along coordinate $j$. In the associated Langevin dynamics,

$-\sum_j \Lambda_{ij}\dot R_j$

need not be parallel to the velocity, and the friction tensor becomes a channel for friction-induced mode coupling and energy redistribution (Askerka et al., 2016).

The tensorial structure is not a small correction in representative surface systems. For H on Pd(100), the tensor is diagonal by symmetry at high-symmetry adsorption sites but strongly anisotropic; at the hollow site, friction along $x$ or $\Lambda_{ij}$0 is almost three times stronger than along $\Lambda_{ij}$1. Along the diffusion path, nonzero $\Lambda_{ij}$2 coupling appears and can reach about $\Lambda_{ij}$3 of the $\Lambda_{ij}$4 component. For CO on Cu(100), even at an upright atop geometry the full $\Lambda_{ij}$5 tensor has substantial off-diagonal atom–atom couplings, and at a tilted geometry it can be fully populated. Mode-resolved lifetimes extracted from the mass-weighted tensor are $\Lambda_{ij}$6 ps for the internal stretch, $\Lambda_{ij}$7 ps for the surface-adsorbate stretch, $\Lambda_{ij}$8 ps for the frustrated translation, and $\Lambda_{ij}$9 ps for the frustrated rotation. The friction eigenvectors are not identical to the vibrational normal modes, so a normal-mode-diagonal damping model is not generally justified (Askerka et al., 2016).

Configuration dependence introduces an additional layer. In a HEOM-based EF-LD formulation with coordinate-dependent metal–molecule coupling,

$\mathscr R_i(t)$0

the dependence of $\mathscr R_i(t)$1 on nuclear coordinates generates new contributions to the adiabatic force, friction tensor, and diffusion kernel. The resulting Langevin limit remains valid both in and out of equilibrium and for molecular models containing strong interactions. In the demonstrated models, these new terms are especially important near geometries where the level is near resonance with a lead chemical potential and where the coupling changes rapidly with position (Mäck et al., 2024).

A separate representation problem arises when one wants a friction tensor field for large-scale dynamics. An equivariant ACE-based construction addresses this by parameterizing a configuration-dependent diffusion tensor $\mathscr R_i(t)$2 and building the friction from it so that $\mathscr R_i(t)$3 is symmetric positive semidefinite at every configuration and satisfies fluctuation-dissipation by construction. In that framework,

$\mathscr R_i(t)$4

and, under the pairwise coupling used in the paper,

$\mathscr R_i(t)$5

Applied to a first-principles electronic-friction dataset for NO on Au(111), the final model with cutoff $\mathscr R_i(t)$6, correlation order $\mathscr R_i(t)$7, and polynomial degree $\mathscr R_i(t)$8 reached test errors

$\mathscr R_i(t)$9

while preserving the tensor’s anisotropy and internal-coordinate couplings along a scattering trajectory (Sachs et al., 2024).

3. Fluctuation-dissipation and the necessity of the random force

A defining feature of EF-LD is that the random force is not an optional add-on to friction. In the scalar Markovian formulation for atom–metal scattering, the stochastic force arises from thermal electron-hole pairs and satisfies

$$m\ddot{\mathbf r} = -\frac{\partial E}{\partial \mathbf r} - m\eta_{\mathrm{el}}\dot{\mathbf r} + \mathbf F_{\mathrm L}(t),$$0

Simulations of $$m\ddot{\mathbf r} = -\frac{\partial E}{\partial \mathbf r} - m\eta_{\mathrm{el}}\dot{\mathbf r} + \mathbf F_{\mathrm L}(t),$$1 eV H atoms scattering from $$m\ddot{\mathbf r} = -\frac{\partial E}{\partial \mathbf r} - m\eta_{\mathrm{el}}\dot{\mathbf r} + \mathbf F_{\mathrm L}(t),$$2 K Au and W surfaces show that omitting this random force by setting $$m\ddot{\mathbf r} = -\frac{\partial E}{\partial \mathbf r} - m\eta_{\mathrm{el}}\dot{\mathbf r} + \mathbf F_{\mathrm L}(t),$$3 causes the energy-loss spectra to “fail spectacularly”: instead of broad and smooth distributions, the results become structured and peaked, with peaks corresponding to different numbers of atom-surface collisions or “bounces” (Hertl et al., 2021).

The underlying reason is that the noise acts on velocity whereas experiments often resolve energy. If a projectile with initial velocity $$m\ddot{\mathbf r} = -\frac{\partial E}{\partial \mathbf r} - m\eta_{\mathrm{el}}\dot{\mathbf r} + \mathbf F_{\mathrm L}(t),$$4 receives a thermal fluctuation $$m\ddot{\mathbf r} = -\frac{\partial E}{\partial \mathbf r} - m\eta_{\mathrm{el}}\dot{\mathbf r} + \mathbf F_{\mathrm L}(t),$$5, then

$$m\ddot{\mathbf r} = -\frac{\partial E}{\partial \mathbf r} - m\eta_{\mathrm{el}}\dot{\mathbf r} + \mathbf F_{\mathrm L}(t),$$6

For fast projectiles, the cross term amplifies small thermal velocity fluctuations into much larger energy broadening. In the Ornstein–Uhlenbeck analysis used to rationalize the scattering calculations, the energy width reaches a maximum at

$$m\ddot{\mathbf r} = -\frac{\partial E}{\partial \mathbf r} - m\eta_{\mathrm{el}}\dot{\mathbf r} + \mathbf F_{\mathrm L}(t),$$7

with maximal width

$$m\ddot{\mathbf r} = -\frac{\partial E}{\partial \mathbf r} - m\eta_{\mathrm{el}}\dot{\mathbf r} + \mathbf F_{\mathrm L}(t),$$8

which for $$m\ddot{\mathbf r} = -\frac{\partial E}{\partial \mathbf r} - m\eta_{\mathrm{el}}\dot{\mathbf r} + \mathbf F_{\mathrm L}(t),$$9 behaves as

$$\left\langle \mathbf F_{\mathrm L}(t)\mathbf F_{\mathrm L}(t') \right\rangle = 2k_{\mathrm B}T_{\mathrm{el}}\,m\eta_{\mathrm{el}}\,\mathbf I\,\delta(t-t').$$0

The paper terms this an overshoot phenomenon: the width can temporarily exceed its equilibrium value during ballistic relaxation, even for collision times of about $$\left\langle \mathbf F_{\mathrm L}(t)\mathbf F_{\mathrm L}(t') \right\rangle = 2k_{\mathrm B}T_{\mathrm{el}}\,m\eta_{\mathrm{el}}\,\mathbf I\,\delta(t-t').$$1 fs (Hertl et al., 2021).

The same fluctuation-dissipation consistency also underlies discrete-time Langevin integrators. For the standard inertial Langevin equation

$$\left\langle \mathbf F_{\mathrm L}(t)\mathbf F_{\mathrm L}(t') \right\rangle = 2k_{\mathrm B}T_{\mathrm{el}}\,m\eta_{\mathrm{el}}\,\mathbf I\,\delta(t-t').$$2

with

$$\left\langle \mathbf F_{\mathrm L}(t)\mathbf F_{\mathrm L}(t') \right\rangle = 2k_{\mathrm B}T_{\mathrm{el}}\,m\eta_{\mathrm{el}}\,\mathbf I\,\delta(t-t').$$3

the integrated stochastic impulse over one step,

$$\left\langle \mathbf F_{\mathrm L}(t)\mathbf F_{\mathrm L}(t') \right\rangle = 2k_{\mathrm B}T_{\mathrm{el}}\,m\eta_{\mathrm{el}}\,\mathbf I\,\delta(t-t').$$4

satisfies

$$\left\langle \mathbf F_{\mathrm L}(t)\mathbf F_{\mathrm L}(t') \right\rangle = 2k_{\mathrm B}T_{\mathrm{el}}\,m\eta_{\mathrm{el}}\,\mathbf I\,\delta(t-t').$$5

In the Verlet-type discretization discussed below, this single Gaussian impulse per step preserves the discrete fluctuation-dissipation balance for the constant-friction Markovian problem and yields the exact Einstein diffusion coefficient

$$\left\langle \mathbf F_{\mathrm L}(t)\mathbf F_{\mathrm L}(t') \right\rangle = 2k_{\mathrm B}T_{\mathrm{el}}\,m\eta_{\mathrm{el}}\,\mathbf I\,\delta(t-t').$$6

for free diffusion (Grønbech-Jensen et al., 2012).

4. Numerical propagation and Verlet-type integration

Although EF-LD is a physical model before it is a numerical algorithm, practical work requires stable and statistically controlled propagation of second-order Langevin equations. A particularly relevant result is a simple Verlet revision for

$$\left\langle \mathbf F_{\mathrm L}(t)\mathbf F_{\mathrm L}(t') \right\rangle = 2k_{\mathrm B}T_{\mathrm{el}}\,m\eta_{\mathrm{el}}\,\mathbf I\,\delta(t-t').$$7

with constant scalar friction and white Gaussian noise. In velocity-Verlet-like form, the update is

$$\left\langle \mathbf F_{\mathrm L}(t)\mathbf F_{\mathrm L}(t') \right\rangle = 2k_{\mathrm B}T_{\mathrm{el}}\,m\eta_{\mathrm{el}}\,\mathbf I\,\delta(t-t').$$8

$$\left\langle \mathbf F_{\mathrm L}(t)\mathbf F_{\mathrm L}(t') \right\rangle = 2k_{\mathrm B}T_{\mathrm{el}}\,m\eta_{\mathrm{el}}\,\mathbf I\,\delta(t-t').$$9

where

$$\Lambda_{ij} = \pi\hbar^2 \omega \sum_{\mathbf{k},\nu,\nu'} \langle \psi_{\mathbf{k}\nu}|\nabla_i \psi_{\mathbf{k}\nu'}\rangle \langle \psi_{\mathbf{k}\nu'}|\nabla_j \psi_{\mathbf{k}\nu}\rangle \delta(\epsilon_{\mathbf{k}\nu}-\epsilon_{\mathbf{k}\nu'}-\hbar \omega),$$0

When $$\Lambda_{ij} = \pi\hbar^2 \omega \sum_{\mathbf{k},\nu,\nu'} \langle \psi_{\mathbf{k}\nu}|\nabla_i \psi_{\mathbf{k}\nu'}\rangle \langle \psi_{\mathbf{k}\nu'}|\nabla_j \psi_{\mathbf{k}\nu}\rangle \delta(\epsilon_{\mathbf{k}\nu}-\epsilon_{\mathbf{k}\nu'}-\hbar \omega),$$1, this reduces exactly to ordinary velocity-Verlet, so it is a true Verlet revision rather than a separate integrator family (Grønbech-Jensen et al., 2012).

The method has unusually strong analytical properties for two benchmark cases. For a flat potential, it reproduces

$$\Lambda_{ij} = \pi\hbar^2 \omega \sum_{\mathbf{k},\nu,\nu'} \langle \psi_{\mathbf{k}\nu}|\nabla_i \psi_{\mathbf{k}\nu'}\rangle \langle \psi_{\mathbf{k}\nu'}|\nabla_j \psi_{\mathbf{k}\nu}\rangle \delta(\epsilon_{\mathbf{k}\nu}-\epsilon_{\mathbf{k}\nu'}-\hbar \omega),$$2

independent of timestep $$\Lambda_{ij} = \pi\hbar^2 \omega \sum_{\mathbf{k},\nu,\nu'} \langle \psi_{\mathbf{k}\nu}|\nabla_i \psi_{\mathbf{k}\nu'}\rangle \langle \psi_{\mathbf{k}\nu'}|\nabla_j \psi_{\mathbf{k}\nu}\rangle \delta(\epsilon_{\mathbf{k}\nu}-\epsilon_{\mathbf{k}\nu'}-\hbar \omega),$$3, damping $$\Lambda_{ij} = \pi\hbar^2 \omega \sum_{\mathbf{k},\nu,\nu'} \langle \psi_{\mathbf{k}\nu}|\nabla_i \psi_{\mathbf{k}\nu'}\rangle \langle \psi_{\mathbf{k}\nu'}|\nabla_j \psi_{\mathbf{k}\nu}\rangle \delta(\epsilon_{\mathbf{k}\nu}-\epsilon_{\mathbf{k}\nu'}-\hbar \omega),$$4, and temperature $$\Lambda_{ij} = \pi\hbar^2 \omega \sum_{\mathbf{k},\nu,\nu'} \langle \psi_{\mathbf{k}\nu}|\nabla_i \psi_{\mathbf{k}\nu'}\rangle \langle \psi_{\mathbf{k}\nu'}|\nabla_j \psi_{\mathbf{k}\nu}\rangle \delta(\epsilon_{\mathbf{k}\nu}-\epsilon_{\mathbf{k}\nu'}-\hbar \omega),$$5, as long as $$\Lambda_{ij} = \pi\hbar^2 \omega \sum_{\mathbf{k},\nu,\nu'} \langle \psi_{\mathbf{k}\nu}|\nabla_i \psi_{\mathbf{k}\nu'}\rangle \langle \psi_{\mathbf{k}\nu'}|\nabla_j \psi_{\mathbf{k}\nu}\rangle \delta(\epsilon_{\mathbf{k}\nu}-\epsilon_{\mathbf{k}\nu'}-\hbar \omega),$$6. For a harmonic oscillator with $$\Lambda_{ij} = \pi\hbar^2 \omega \sum_{\mathbf{k},\nu,\nu'} \langle \psi_{\mathbf{k}\nu}|\nabla_i \psi_{\mathbf{k}\nu'}\rangle \langle \psi_{\mathbf{k}\nu'}|\nabla_j \psi_{\mathbf{k}\nu}\rangle \delta(\epsilon_{\mathbf{k}\nu}-\epsilon_{\mathbf{k}\nu'}-\hbar \omega),$$7 and $$\Lambda_{ij} = \pi\hbar^2 \omega \sum_{\mathbf{k},\nu,\nu'} \langle \psi_{\mathbf{k}\nu}|\nabla_i \psi_{\mathbf{k}\nu'}\rangle \langle \psi_{\mathbf{k}\nu'}|\nabla_j \psi_{\mathbf{k}\nu}\rangle \delta(\epsilon_{\mathbf{k}\nu}-\epsilon_{\mathbf{k}\nu'}-\hbar \omega),$$8, it gives the exact configurational variance

$$\Lambda_{ij} = \pi\hbar^2 \omega \sum_{\mathbf{k},\nu,\nu'} \langle \psi_{\mathbf{k}\nu}|\nabla_i \psi_{\mathbf{k}\nu'}\rangle \langle \psi_{\mathbf{k}\nu'}|\nabla_j \psi_{\mathbf{k}\nu}\rangle \delta(\epsilon_{\mathbf{k}\nu}-\epsilon_{\mathbf{k}\nu'}-\hbar \omega),$$9

and therefore the exact average potential energy

$i$0

for both underdamped and overdamped physical regimes within the usual Verlet stability range (Grønbech-Jensen et al., 2012).

The velocity variance, however, is not exact for the harmonic oscillator: $i$1 This means EF-LD propagated with this scheme is better interpreted as a good configurational sampler than as a method guaranteeing exact kinetic observables for bound modes. The formal stability limit remains

$i$2

and the paper also emphasizes the attenuation condition

$i$3

which corresponds to $i$4 and avoids unphysical sign-alternating attenuation (Grønbech-Jensen et al., 2012).

For true electronic-friction applications, this algorithm is directly applicable only when the EF model can be reduced to constant scalar Markovian friction with white noise. The paper itself does not address tensor friction, coordinate dependence, correlated multidimensional noise, multiplicative-noise drift corrections, or memory kernels. A plausible implication is that its main role in EF-LD is as a numerical template for the simplest Markovian limit rather than as a complete propagation strategy for general tensorial or non-Markovian EF models (Grønbech-Jensen et al., 2012).

5. Beyond the Markovian limit: quantum nuclei, memory, and frequency dependence

Several recent developments move EF-LD beyond the standard memoryless classical picture. One route starts from exact factorization of the electron-nuclear wavefunction,

$i$5

and derives a quantum nuclear equation of motion in which the electronic back-action appears as a retarded friction force

$i$6

with memory kernel

$i$7

In the Markov limit this reduces to a friction tensor, but the formulation also reinterprets Berry-curvature terms: electronic dynamics generally wash out the gauge fields appearing in adiabatic dynamics, whereas at $i$8 in the rapid-relaxation Markov limit the adiabatic pseudo-magnetic force is restored rather than double counted (Martinazzo et al., 2021).

A second route embeds electronic friction into ring-polymer molecular dynamics. In the path-integral generalized Langevin equation developed for H diffusion on Cu(111), the ring-polymer normal-mode momenta obey

$i$9

with colored noise satisfying

$j$0

For the Cu(111) application, the low-frequency friction spectrum is super-Ohmic, the tunnelling crossover temperature is about $j$1 K for H and about $j$2 K for D, and explicit harmonic-bath discretization requiring more than $j$3 bath modes can be replaced by an auxiliary-variable mapping converged with $j$4. The paper argues that previous apparent agreement of classical Markovian simulations with experiment can arise from cancellation between neglected nuclear quantum effects and overestimated dissipation efficiency (Trenins et al., 2024).

Nonequilibrium bias introduces an additional complication: the sign of the zero-frequency friction is not by itself a reliable guide to stability. In a donor–acceptor nanojunction with vibrationally assisted transport, the generalized Langevin equation is

$j$5

and the Markovian approximation replaces the kernel by $j$6. The paper shows that the same inelastic mechanism that generates negative Markovian friction also creates strong frequency structure in $j$7, with side peaks near $j$8. As a result, a Markovian instability can be spurious: for $j$9, Markovian EF-LD predicts a high-bias instability that does not appear in numerically exact HEOM simulations (Preston et al., 31 Mar 2026).

6. Scope, controversies, and advanced applications

EF-LD is both extensible and contested. One explicit warning comes from a two-dimensional scattering model based on an Anderson–Holstein impurity Hamiltonian, where electronic friction, a classical master equation (CME), and a broadened classical master equation (BCME) were compared. In that model, EF-LD is written as

$-\sum_j \Lambda_{ij}\dot R_j$0

with

$-\sum_j \Lambda_{ij}\dot R_j$1

The authors report strong evidence suggesting that EF results may be spurious for scattering problems with more than one nuclear dimension. Their concerns include exaggerated trapping, sensitivity to an artificial $-\sum_j \Lambda_{ij}\dot R_j$2, and the inability of continuous friction to represent sudden large-energy-transfer events that arise naturally in hopping pictures. In their assessment, BCME provides a more reliable interpolation between weak-hybridization hopping and strong-hybridization frictional behavior (Miao et al., 2017).

At the same time, EF-LD has been extended into regimes where simple mean-field or noninteracting pictures break down. In a two-dimensional Hubbard–Holstein model treated with DMFT and FDM-NRG, the one-dimensional Langevin dynamics

$-\sum_j \Lambda_{ij}\dot R_j$3

was supplied with a friction evaluated from many-body force-force correlations. DMFT yields two distinct friction peaks associated with electron attachment and detachment resonances with the solid Fermi level, whereas MFT gives only one broad central peak and misses the Fermi resonance. In the reported EF-LD trajectories, average kinetic energy relaxes similarly under DMFT- and MFT-based inputs, but the electronic population dynamics differ substantially, which the paper interprets as evidence that MFT is inadequate for nonadiabatic dynamics in strongly correlated systems (Liu et al., 25 May 2026). A closely related DMRG study of the Hubbard–Holstein model reaches the same qualitative conclusion: electron-electron interactions split a single noninteracting resonance into two conditions,

$-\sum_j \Lambda_{ij}\dot R_j$4

and MFT completely fails to recover the resulting Fermi resonance structure in the friction. The paper also reports that ED and DMRG agree at high temperature, while ED deviates strongly at low temperature because of finite-size limitations (Liu et al., 31 Aug 2025).

Periodic driving leads to a further generalization, Floquet electronic friction. Starting from a Floquet classical master equation, one can derive a Floquet Fokker–Planck equation

$-\sum_j \Lambda_{ij}\dot R_j$5

with

$-\sum_j \Lambda_{ij}\dot R_j$6

$-\sum_j \Lambda_{ij}\dot R_j$7

Because $-\sum_j \Lambda_{ij}\dot R_j$8 under strong driving, the second fluctuation-dissipation theorem is violated, and the nuclei can heat to kinetic energies above $-\sum_j \Lambda_{ij}\dot R_j$9. Benchmarking against Floquet quantum master equations and Floquet surface hopping shows that the friction picture works best in the intended regime of fast driving and fast electronic relaxation, but averaged Floquet friction misses low-frequency limit-cycle behavior (Wang et al., 2023).

Taken together, these developments define the present scope of EF-LD. In its simplest and most controlled form it is a Markovian, classical-nuclear reduction with friction and white noise. In more elaborate formulations it becomes tensorial, geometry dependent, strongly interacting, nonequilibrium, Floquet engineered, or non-Markovian. The literature also makes clear that each extension carries its own regime of validity, and that scalar, memoryless friction with equilibrium noise is not a universal surrogate for all nonadiabatic dynamics at metal surfaces (Miao et al., 2017, Trenins et al., 2024, Preston et al., 31 Mar 2026).