Local-Density Friction Approximation

• Local-Density Friction Approximation is a framework that models friction by replacing complex many-body interactions with a local, isotropic friction coefficient derived from material properties.
• It is applied in electronic friction for molecular adsorbates, Casimir friction in dense media, and dislocation friction in crystalline solids to predict energy-dissipative behaviors.
• The method employs both independent-atom and atoms-in-molecules approaches to accurately incorporate local electron densities and material parameters into scalable simulations.

The Local-Density Friction Approximation (LDFA) is a theoretical framework that replaces the full non-adiabatic, non-local friction response by a strictly local, isotropic description, where friction coefficients are obtained from the local properties of the medium, such as local electron (or dislocation) density or local material parameters. LDFA has been rigorously developed in the contexts of electronic non-adiabatic energy loss for molecular adsorbates on surfaces, Casimir friction between dense media, and frictional/dissipative stress in dislocation networks. Its appeal lies in the reduction of an inherently complex, many-body phenomenon to a numerically tractable, local quantity that can be effectively implemented in large-scale simulations while retaining quantitative accuracy under appropriate physical conditions (Rittmeyer et al., 2015, Høye et al., 2013, Zaiser, 2015).

1. LDFA in Electronic Friction for Molecular Adsorbates

In the modeling of non-adiabatic energy dissipation (electronic friction) for molecules adsorbed on metal surfaces, LDFA provides scalar, atomic friction coefficients $\eta_i$, each computed as a function of a local "embedding" electron density $\rho_{\rm emb}$ evaluated at the nuclear positions. The explicit expression, known as the Persson–Hellsing formula, is

$$\eta_i^{\rm LDFA}(\rho_{\rm emb,i}) = 4\pi\rho_{\rm emb,i}k_F\sum_{l=0}^\infty(l+1)\sin^2[\delta_l^F-\delta_{l+1}^F]$$

where $k_F=(3\pi^2\rho_{\rm emb})^{1/3}$ is the Fermi momentum and $\delta_l^F$ are phase shifts for each angular momentum channel at the Fermi energy. In the prevalent Independent-Atom Approximation (IAA), the embedding density for atom $i$ is chosen as the electron density of the clean metal substrate sampled at $R_i$,

$\rho_{\rm emb,i}^{\rm IAA} = \rho_{\rm surf}(R_i)$

and thus

$$\eta_i^{\rm IAA} = \eta_i^{\rm LDFA}(\rho_{\rm surf}(R_i))$$

This construction neglects intra-molecular density contributions. As a consequence, LDFA–IAA systematically underestimates energy dissipation and overestimates vibrational lifetimes of diatomics, missing bond-length dependence and density modulation effects (Rittmeyer et al., 2015).

To rectify this, the Atoms-in-Molecules (AIM) extension introduces a Hirshfeld-type partitioning of the total self-consistent electron density of the full adsorbate+surface system, yielding

$$\rho_{\rm emb,i}^{\rm AIM} = \rho_{\rm SCF}(R_i) - \rho^{\rm H}_i(R_i) = [1 - w^H_i(R_i)]\,\rho_{\rm SCF}(R_i)$$

with $w^H_i(r)$ the Hirshfeld sharing function. This inclusion restores interatomic density contributions and allows friction coefficients to reflect both substrate and adsorbate neighbors, achieving near-quantitative agreement with both advanced orbital-dependent theories and experimental results for vibrational lifetimes. The tensors generated in both IAA and AIM remain isotropic and diagonal in Cartesian coordinates.

2. LDFA in Casimir Friction of Dense Media

In the context of Casimir friction, LDFA emerges as the continuum limit of a statistical-mechanical treatment of dipolar, polarizable "particles" in dense media. Two semi-infinite dielectric half-spaces are considered, separated by a vacuum gap and sliding at small velocity. The discrete lattice of interacting dipoles is replaced by continuous media described solely via local permittivities $\varepsilon(\omega)$.

By summing all internal (intraplane and interplane) dipolar correlations using ring–chain resummation techniques, the friction force per unit area at finite temperature for, e.g., Drude metals is

$$F = -\frac{\hbar v}{4 d^4} \frac{(k_BT)^2(\hbar\nu)^2}{(\hbar\omega_p)^4}$$

where $d$ is the gap, $v$ the sliding velocity, $T$ the temperature, $\omega_p$ the plasma frequency, and $\nu$ the damping rate. This result encodes the finite-temperature and local-density character: all nonlocality is incorporated in the exponential transfer factor $e^{-2qd}$, while the material-specific response is described by $\varepsilon(\omega)$ at each spatial point (Høye et al., 2013). Comparison with earlier approaches (Pendry, Volokitin-Persson) confirms that LDFA yields quantitatively consistent results in the appropriate parameter regimes.

3. LDFA for Dislocation Friction in Crystalline Solids

For three-dimensional dislocation systems, LDFA is derived by coarse-graining the discrete elastic interaction energy of curved dislocation lines into functionals of local dislocation density variables. When dislocation segments on different slip systems form "junctions," energy is dissipated as these are repeatedly formed and broken during plastic deformation.

The junction (friction) energy is approximated in the local-density sense as

$$E_{\rm jun} = \int d^3 r \sum_{\beta,\beta'} E_{\beta\beta'}\, h_{\beta\beta'}\,\rho_\beta(\mathbf r)\,\rho_{\beta'}(\mathbf r)$$

where $\rho_\beta(\mathbf r)$ is the orientation-averaged segment density for slip system $\beta$, $E_{\beta\beta'}$ is the per-unit-length junction energy, and $h_{\beta\beta'}$ an overlap factor. The friction (flow) stress on slip system $\beta$ becomes

$$\tau^{\rm fr}_\beta(\mathbf r) = b \sum_{\beta'} E_{\beta\beta'} h_{\beta\beta'} \rho_{\beta'}(\mathbf r) = \frac{\mu b^2}{4\pi(1-\nu)}\sum_{\beta'} h_{\beta\beta'} \rho_{\beta'}(\mathbf r)$$

where $\mu$ is the shear modulus, $b$ the Burgers vector magnitude, and $\nu$ Poisson's ratio. Assumptions underlying this LDFA include the locality of the junction density, separation of scales, and weak spatial gradients relative to the dislocation spacing. The LDFA friction stress is strictly local and dissipative, distinct from nonlocal Hartree or back-stress terms (Zaiser, 2015).

4. Theoretical Basis and Connection to Linear Response

The equivalence between LDFA-derived friction and results from rigorous linear-response theory is established across applications. For electronic friction, the connection to the Fermi Golden Rule for vibrational damping is made by equating the exponential decay rate of vibrational energy in the friction law with that calculated from the transition matrix elements (but without requiring explicit evaluation of $\partial V/\partial Q$ terms). In statistical treatments of dipolar friction, Kubo's fluctuation–dissipation theorem connects the friction kernel to imaginary-time correlation functions, and in field-theoretical formulations, the photon propagator defines the dissipative response. In dislocation systems, LDFA friction stress emerges via energy-dissipation balance when breaking junctions, providing a clear dissipative mechanism distinct from variational derivative (conservative) stresses (Rittmeyer et al., 2015, Høye et al., 2013, Zaiser, 2015).

5. Numerical Implementation and Computational Considerations

The practical application of LDFA proceeds straightforwardly in each domain:

• For electronic friction, the primary computational cost is the self-consistent field (SCF) density calculation for the full system, while Hirshfeld partitioning and density interpolation (AIM) add negligible overhead (<5%). Phase-shift sums needed for friction coefficients can be precomputed and tabulated (Rittmeyer et al., 2015).
• In Casimir friction, the main technical requirement is an accurate characterization of local permittivities $\varepsilon(\omega)$ as a function of position and frequency, with all nonlocality handled analytically via integration over wavevectors (Høye et al., 2013).
• For dislocations, LDFA requires only the local densities on each slip system and readily incorporates standard elasticity parameters, allowing efficient incorporation into continuum-scale simulations (Zaiser, 2015).

In all settings, the strictly local nature of LDFA enables efficient computation and seamless coupling with atomistic or field-theoretic approaches.

6. Scope of Applicability and Limitations

LDFA is quantitatively accurate under several key assumptions:

• The medium exhibits scale separation such that the relevant variation in density or material parameter is slow on the length scale of the core interaction (e.g., mean free path, Fermi wavelength, or dislocation spacing).
• Dissipative processes are dominated by local, quasi-homogeneous physics (e.g., chemisorption with electron–hole pair damping, junction-breaking in statistically homogeneous dislocation tangles).
• Non-resonant, nonlocal, and anisotropic effects (e.g., strong molecular-state crossing, field-enhanced resonant friction, long-range GND interactions) are negligible.

LDFA is not suitable for describing phenomena where such nonlocal or memory effects control dissipation, or in materials or regimes with strong spatial inhomogeneity at the core scale. In the context of electronic friction, it treats the friction tensor as diagonal and isotropic and cannot describe off-diagonal or tensorial memory effects (Rittmeyer et al., 2015, Høye et al., 2013, Zaiser, 2015).

7. Comparative Analysis Across Physical Domains

While LDFA originated from distinct fields, its conceptual basis and consequences are highly analogous across electronic friction, Casimir friction, and dislocation friction:

Application Domain	Local Density	Friction Functional
Electronic friction	$\rho_{\rm emb}(R_i)$	$\eta_i \propto$ FEG scattering
Casimir friction	$\varepsilon(\omega, \mathbf r)$	$F \propto$ local permittivity
Dislocation friction	$\rho_\beta(\mathbf r)$	$\tau^{\rm fr}_\beta \propto \rho_{\beta'}$

This universal structure—replacing full many-body interactions by local-density-derived dissipative kernels—accounts for the theoretical appeal and broad applicability of the LDFA framework.

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References:

(Rittmeyer et al., 2015, Høye et al., 2013, Zaiser, 2015)