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Drude model

The Drude model is a classical theory of electrical conduction in metals, proposed by Paul Drude in 1900 to explain the transport properties of electrons in materials, particularly how an applied electric field induces a current through the motion of free electrons.^[1] In this model, metals are envisioned as a lattice of stationary positive ions surrounded by a gas of conduction electrons that behave like classical particles, accelerating under the influence of an electric field but undergoing frequent collisions with the ions that randomize their velocities and establish a steady-state drift.^[2] These collisions are characterized by a relaxation time τ, typically on the order of 10⁻¹⁴ seconds at room temperature, during which electrons travel a mean free path before scattering.^[1] The model's key achievement is deriving the electrical conductivity σ as σ = n e² τ / m, where n is the electron density, e is the elementary charge, and m is the electron mass, which quantitatively matches experimental values for metals (1–10 μΩ·cm resistivity) and underpins Ohm's law (J = σ E) in the linear response regime.^[1] It also successfully predicts the Hall effect, cyclotron resonance, and the form of the Wiedemann-Franz law relating electrical and thermal conductivities (κ / σ T = (3/2) (k_B / e)², where κ is thermal conductivity and k_B is Boltzmann's constant), though the numerical prefactor differs from the experimental value.^[1] Drude's framework built on the recent discovery of the electron by J.J. Thomson in 1897 and drew analogies from kinetic theory of gases, assuming no electron-electron interactions and treating scattering as abrupt events independent of the field.^[1] Despite its successes, the Drude model has notable limitations, as it fails to account for quantum mechanical effects, incorrectly attributing scattering solely to ion cores (whereas perfect lattices show no scattering in quantum theory), and predicting an erroneous temperature dependence for resistivity (proportional to √T rather than the observed linear T).^[1] These shortcomings were later addressed by refinements, such as Hendrik Lorentz's 1905 correction of a factor-of-two error in Drude's original derivation and the quantum-based Sommerfeld model in 1927, yet the Drude approach remains foundational for understanding classical transport and serves as a starting point for more advanced theories in solid-state physics.^[2]

Historical Development

Origins in Classical Physics

The foundations of the Drude model emerged from 19th-century advancements in kinetic theory, initially developed for gases but later adapted to conceptualize electrical conduction in solids. James Clerk Maxwell laid key groundwork in his 1860 work on the dynamical theory of gases, where he modeled gas particles as colliding spheres to derive the velocity distribution and transport properties, providing a framework for treating charge carriers similarly. Ludwig Boltzmann extended this in 1872 with his transport equation, describing how particle distributions evolve under collisions and external forces, which proved essential for analyzing drift motion in conductors. Building on these ideas, Wilhelm Weber proposed in 1871 that electrical phenomena in matter arise from interactions between charged particles within atoms, envisioning positive charges surrounded by orbiting negative ones, an early atomic model influencing later views of conduction.^[3] Eduard Riecke advanced this in the 1890s by modeling metals as lattices of neutral atoms containing free electrons behaving like a gas, introducing concepts of electron drift velocity under an electric field to explain current flow. Riecke's 1898 theory explicitly applied kinetic principles to estimate electron numbers and mobilities in metals, treating conduction as the collective motion of these particles scattered by lattice vibrations.^[3] The discovery of the electron by J.J. Thomson in 1897 provided the crucial empirical basis for such free-electron pictures, as his cathode-ray experiments identified negatively charged corpuscles far lighter than atoms, confirming the existence of mobile subatomic particles capable of carrying charge in metals.^[4] However, pre-Drude models like those of Weber and Riecke struggled with key observations, notably failing to explain the positive temperature dependence of resistivity, as their classical scattering assumptions predicted either temperature-independent or weakly varying resistance, contrary to experimental increases with thermal agitation.^[3] These classical precursors set the stage for Paul Drude's 1900 synthesis, which integrated kinetic theory with the confirmed electron to form a cohesive model of metallic conduction.^[3]

Key Formulations by Drude and Lorentz

In 1900, Paul Drude published a seminal paper introducing a classical model for electrical conduction in metals, positing that conduction arises from a gas of free electrons undergoing random thermal motion akin to particles in an ideal gas. Drawing from classical kinetic theory, Drude envisioned these electrons as mobile charge carriers drifting through the metallic lattice under an applied electric field while colliding with ions. Drude's formulation included an initial estimate of electrical conductivity derived from the mean free path of electrons and their density, which he approximated using data from electrolysis experiments, leading to the prediction of approximately one conduction electron per atom in metals. This electron density assumption, grounded in Faraday's laws of electrolysis, marked a key conceptual advance, though it overestimated the number for polyvalent metals and was later refined in subsequent theories. In 1905, Hendrik Lorentz refined Drude's model through a series of papers on the motion of electrons in metallic bodies, introducing the relaxation time \tau as a parameter to describe the average time between collisions more realistically.^[5] Lorentz's approach, employing the Boltzmann transport equation under a relaxation-time approximation, corrected inconsistencies in Drude's treatment of scattering, such as the use of separate relaxation times for electrical and thermal processes. Lorentz also corrected a factor-of-two error in Drude's original derivation by showing that the relaxation time is the same for electrical and thermal conductivity, improving the prediction for the Wiedemann-Franz law.^[6] These refinements established a unified framework that better aligned with experimental observations like the Wiedemann-Franz law and solidified the model's foundational equations for conductivity.^[5]

Core Assumptions

Electron Behavior in Metals

In the Drude model, metals are conceptualized as a regular lattice of positively charged ions that remain fixed in position, with the valence electrons detached from their parent atoms and forming a gas of free charge carriers capable of moving throughout the material.^[7] This "free electron gas" arises because the valence electrons are loosely bound and can wander freely, neutralizing the positive background charge of the ion lattice to maintain overall electrical neutrality.^[8] The model, introduced by Paul Drude in 1900, draws an analogy to the kinetic theory of gases, treating these electrons as classical particles in a container defined by the metal's boundaries.^[9] These free electrons are assumed to be in thermal equilibrium with the lattice, exhibiting random thermal velocities due to their kinetic energy. According to the equipartition theorem, each electron has an average kinetic energy of \frac{3}{2} kT, where k is Boltzmann's constant and T is the temperature, leading to a root-mean-square speed of v_{\rms} = \sqrt{\frac{3kT}{m}}, with m the electron mass.^[7] At room temperature, this speed is on the order of $10^7 cm/s, far exceeding typical drift velocities under applied fields.^[10] A key simplification in the model is the neglect of interactions between electrons themselves; instead, electrons are treated as non-interacting particles that only experience scattering from the fixed ions or impurities in the lattice.^[11] This independent electron approximation allows the application of dilute gas kinetics to the dense electron gas in metals.^[12] The density of these free electrons, n, is estimated from the number of valence electrons per atom and the atomic density of the metal, typically yielding n \approx 10^{22} cm^{-3} for common metals like sodium or copper, where one or more valence electrons per atom contribute to the gas.^[13] For instance, in copper with one free electron per atom, n \approx 8.5 \times 10^{22} cm^{-3}.^[7] This high density underscores the model's innovation in bridging classical gas dynamics with the phenomenon of conduction in solids, providing a foundational classical picture despite its simplifications.^[7]

Scattering and Relaxation Processes

In the Drude model, conduction electrons in metals are treated as a free electron gas that undergoes frequent collisions primarily with lattice ions, which are modeled as fixed scattering centers. These collisions are assumed to be elastic, preserving the electron's kinetic energy on average, but randomizing its momentum direction, leading to a loss of directional drift and thus electrical resistance.^[14]^[15] The average time between such collisions is denoted by the relaxation time τ, which characterizes the momentum relaxation process. The mean free path λ, representing the average distance an electron travels between collisions, is given by λ = v_rms τ, where v_rms = √(3k_B T / m) is the root-mean-square speed of the electrons, with k_B the Boltzmann constant, T the temperature, and m the electron mass. Typical values for τ at room temperature are on the order of 10^{-14} to 10^{-15} seconds, resulting in λ ≈ 10–100 Å for common metals like copper.^[16]^[14] The relaxation time τ exhibits a strong temperature dependence due to scattering by thermal vibrations of the lattice ions. In the classical Drude model, as temperature increases, the vibration amplitudes (∝ √T) and electron speeds (∝ √T) enhance the scattering rate, yielding τ ∝ T^{-1/2} for this contribution.^[17] In addition to scattering from lattice vibrations, impurities and defects introduce a temperature-independent scattering rate 1/τ_imp, which adds a fixed contribution to the total scattering rate via Matthiessen's rule: 1/τ = 1/τ_vib + 1/τ_imp. This impurity term accounts for the residual resistivity observed at low temperatures, where scattering from vibrations diminishes, leaving a finite resistivity even as T → 0.^[14]^[16] The model assumes a Markovian scattering process, wherein each collision completely randomizes the electron's velocity according to the equilibrium Maxwell-Boltzmann distribution, with no memory of its pre-collision state, justifying the use of a constant average τ independent of prior history.^[16]^[15]

Electrical Conductivity Derivation

Direct Current (DC) Response

In the Drude model, the direct current (DC) response of a metal to a constant electric field \mathbf{E} is analyzed through the classical equation of motion for a conduction electron, which balances the accelerating force from the field against a frictional drag due to collisions. The equation is

m \frac{d\mathbf{v}}{dt} = -e \mathbf{E} - \frac{m \mathbf{v}}{\tau},

where m is the electron mass, e > 0 is the elementary charge, \mathbf{v} is the electron velocity, and \tau is the average relaxation time between collisions.^[1] This form, introduced by Paul Drude in 1900, models collisions as randomizing the electron momentum exponentially with timescale \tau.^[18] For steady-state DC conditions, the acceleration term vanishes (d\mathbf{v}/dt = 0), yielding the drift velocity

\mathbf{v}_d = -\frac{e \tau}{m} \mathbf{E}.

This represents the average velocity superimposed on the random thermal motion of electrons, arising from the balance between field-induced acceleration and collision-induced deceleration.^[1] The resulting current density is then \mathbf{J} = -n e \mathbf{v}_d, where n is the electron number density, giving

\mathbf{J} = \frac{n e^2 \tau}{m} \mathbf{E}.

Thus, the DC electrical conductivity is \sigma = n e^2 \tau / m, which linearly relates current density to the applied field via Ohm's law in the form \mathbf{J} = \sigma \mathbf{E}.^[1] The corresponding resistivity is \rho = 1/\sigma = m / (n e^2 \tau).^[10] The temperature dependence of conductivity enters primarily through \tau(T); in the classical Drude model, phonon scattering leads to \tau \propto 1/\sqrt{T} and thus \sigma \propto 1/\sqrt{T}, though experiments show linear T at higher temperatures.^[10] This derivation of \sigma provides the electrical foundation for the Wiedemann-Franz law, where the shared \tau links electrical conductivity to thermal transport, predicting \kappa / (\sigma T) = constant (with the classical Lorentz number L = (3/2) (k_B / e)^2).

Alternating Current (AC) Response

The Drude model extends to alternating current (AC) by considering time-varying electric fields, where the oscillatory nature of the field introduces frequency dependence into the electron dynamics. In this framework, the electric field is assumed to take the form \mathbf{E}(t) = \mathbf{E}_0 e^{-i \omega t}, with the real part representing the physical field. The equation of motion for an electron under this field, incorporating the damping due to scattering, is given by

m \frac{d \mathbf{v}}{dt} + \frac{m}{\tau} \mathbf{v} = -e \mathbf{E}_0 e^{-i \omega t},

where m is the electron mass, \mathbf{v} is the velocity, e is the electron charge, and \tau is the relaxation time.^[19] Assuming a steady-state solution of the form \mathbf{v}(t) = \mathbf{v}(\omega) e^{-i \omega t}, the velocity in frequency space becomes

\mathbf{v}(\omega) = -\frac{e \mathbf{E}_0 / m}{-i \omega + 1/\tau}.

This solution captures the balance between acceleration by the field, inertial effects from the frequency term, and frictional damping. The current density \mathbf{j}(\omega) = -n e \mathbf{v}(\omega), where n is the electron density, leads to the complex conductivity

\sigma(\omega) = \frac{n e^2 \tau / m}{1 - i \omega \tau} = \frac{\sigma_0}{1 - i \omega \tau},

with \sigma_0 = n e^2 \tau / m being the DC conductivity.^[20]^[19] The frequency-dependent conductivity relates to the dielectric function via Maxwell's equations in frequency space, yielding

\varepsilon(\omega) = 1 + \frac{i \sigma(\omega)}{\varepsilon_0 \omega},

where \varepsilon_0 is the vacuum permittivity. Substituting \sigma(\omega) gives

\varepsilon(\omega) = 1 + \frac{i \omega_p^2}{\omega (1/\tau - i \omega)},

revealing the plasma frequency \omega_p = \sqrt{n e^2 / \varepsilon_0 m}, which characterizes collective electron oscillations in metals.^[20]^[19] In the low-frequency limit (\omega \tau \ll 1), the conductivity approaches the DC value \sigma(\omega) \approx \sigma_0, and the response resembles steady-state conduction with minimal inertial effects. At high frequencies (\omega \tau \gg 1), the term -i \omega \tau dominates, making \sigma(\omega) \approx i n e^2 / (m \omega); here, electrons cannot follow the rapid field oscillations due to inertia, leading to reduced conductivity and high reflectivity in metals as the dielectric function becomes negative for \omega < \omega_p.^[19] These features explain key optical properties of metals, such as the high reflectivity at infrared and visible frequencies, where the Drude model predicts a characteristic "Drude tail" in reflectivity spectra—a gradual decrease in reflectivity at higher frequencies approaching the plasma edge, beyond which the material becomes more transparent. This tail arises from the frequency dependence of \varepsilon(\omega) and is observed in spectra of simple metals like sodium and silver.^[21]

Thermal and Thermoelectric Properties

Thermal Conductivity Mechanism

In the Drude model, thermal conductivity arises from the transport of kinetic energy by free conduction electrons in response to a temperature gradient. Electrons in the hotter region possess higher average kinetic energies due to the local temperature and diffuse toward the cooler region between scattering events, establishing a net heat flux from hot to cold. This process mirrors the drift of electrons under an electric field for electrical conduction, but here the driving force is the spatial variation in thermal energy rather than charge separation. The model assumes that electrons behave as a classical gas, with collisions randomizing their velocities and the relaxation time τ governing both momentum and energy transfer.^[18] The heat current density \mathbf{J}_Q is derived from the flux of electron kinetic energy across a plane perpendicular to the temperature gradient \nabla T. Considering electrons crossing a unit area with mean free path λ and root-mean-square speed v_{\rms} = \sqrt{3 k_B T / m}, where k_B is Boltzmann's constant and m is the electron mass, the net energy transport yields Fourier's law \mathbf{J}_Q = -\kappa \nabla T. The thermal conductivity κ is then given by \kappa = \frac{1}{3} n v_{\rms} \lambda c_v, where n is the electron density, λ = v_{\rms} τ is the mean free path, and c_v = \frac{3}{2} k_B is the specific heat per electron following the Dulong-Petit law for a classical monatomic gas. This expression captures the electron contribution to heat conduction, neglecting lattice vibrations (phonons) as the dominant mechanism in metals at room temperature.^[22] Substituting the expressions for v_{\rms} and λ into the formula for κ, and assuming classical equipartition of energy, simplifies to \kappa = \frac{3}{2} \frac{n k_B^2 T \tau}{m}. However, to align with experimental observations in metals, where the classical specific heat overestimates the effective electron contribution at low temperatures, the model is often refined using Fermi-Dirac statistics in a semiclassical approximation, yielding \kappa = \frac{\pi^2}{3} \frac{n k_B^2 T \tau}{m}. This form treats the electrons as degenerate but retains the Drude scattering dynamics. The assumption of a single relaxation time τ for both energy and momentum transport is crucial here, linking thermal and electrical properties.^[11] A key prediction of the Drude framework is the Wiedemann-Franz law, which relates thermal and electrical conductivities through the ratio \frac{\kappa}{\sigma T} = L_0, where σ = \frac{n e^2 \tau}{m} is the electrical conductivity and e is the electron charge. Using the classical form, L_0 = \frac{3}{2} \left( \frac{k_B}{e} \right)^2 \approx 1.11 \times 10^{-8} W Ω K⁻², but the Fermi-adjusted expression gives L_0 = \frac{\pi^2}{3} \left( \frac{k_B}{e} \right)^2 \approx 2.45 \times 10^{-8} W Ω K⁻², known as the Lorenz number, which matches measurements for many metals. This law emerges directly from the shared τ and electron parameters, assuming isothermal electrical conditions and no phonon heat transport. Lorentz refined Drude's original derivation in 1905 by applying the Boltzmann transport equation, correcting a factor-of-two error in the mean free path and confirming the classical limit.^[22]^[11]

Thermopower and Seebeck Effect

The thermopower, also known as the Seebeck coefficient S, quantifies the thermoelectric voltage generated across a material due to a temperature gradient and is defined as S = -\frac{[\Delta V](/page/Delta-v)}{\Delta T}, where \Delta V is the open-circuit voltage difference between hot and cold junctions separated by \Delta T. In the Drude model, this effect originates from the diffusive transport of charge carriers: electrons at the hotter end possess higher kinetic energy and thus greater average speeds, leading to a net flux toward the colder end. This diffusion creates a charge imbalance, with excess electrons accumulating at the cold junction, which in turn induces an electric field that opposes further net carrier flow in the steady state.^[11]^[23] Within the classical Drude framework, the Seebeck coefficient is derived by balancing the average electron drift velocity induced by the temperature gradient against that from the resulting electric field. The temperature-gradient contribution to the mean velocity is \mathbf{v}_Q = -\frac{\tau}{6} \frac{d \langle v^2 \rangle}{dT} \nabla T, where \tau is the relaxation time and \langle v^2 \rangle is the mean-square speed, while the field-induced velocity is \mathbf{v}_E = -\frac{e \tau}{m} \mathbf{E}. Setting the total velocity to zero for zero net current yields S = \frac{\mathbf{E}}{\nabla T} = -\frac{k_B}{2e}, assuming the classical specific heat per electron c_v = \frac{3}{2} k_B and deriving from the energy flux associated with carrier transport. This classical approximation can also be interpreted through entropy transport, where S \approx -\frac{k_B}{e} \frac{k_B T}{E_F}, reflecting the reduced effective entropy carried by carriers in a degenerate electron gas, though the Drude model originally treats electrons as a classical gas.^[23]^[11] A more detailed treatment in kinetic theory, which the Drude model simplifies, expresses the Seebeck coefficient as

S = -\frac{1}{eT} \frac{\int (\varepsilon - \mu) \sigma(\varepsilon) \left( -\frac{\partial f}{\partial \varepsilon} \right) d\varepsilon}{\int \sigma(\varepsilon) \left( -\frac{\partial f}{\partial \varepsilon} \right) d\varepsilon},

where \varepsilon is the electron energy, \mu is the chemical potential, \sigma(\varepsilon) is the energy-dependent conductivity, and f is the distribution function. In the Drude approximation, assuming a Maxwellian distribution and energy-independent scattering time \tau, this integral reduces to the classical form S = -\frac{k_B}{2e} \approx -43 \, \mu\mathrm{V/K}, independent of temperature. For metals, incorporating partial degeneracy leads to a refined Drude-like expression S = -\frac{\pi^2 k_B^2 T}{3 e E_F}, where E_F is the Fermi energy, emphasizing the role of states near the Fermi level.^[23]^[24] The Drude model predicts a negative Seebeck coefficient for electron conductors, consistent with the negative charge of carriers, with a magnitude on the order of k_B / e \approx 86 \, \mu\mathrm{V/K} scaled by factors near unity in the classical limit, or reduced to 10--100 \mu\mathrm{V/K} when accounting for degeneracy. However, the classical prediction overestimates the value by about two orders of magnitude at room temperature compared to typical metallic values of a few \mu\mathrm{V/K}, due to the neglect of Fermi-Dirac statistics and energy-dependent scattering.^[11]^[23] The Seebeck effect is thermodynamically linked to the Peltier effect through Kelvin relations, which state that the Peltier coefficient \Pi, representing heat transported per unit charge current, satisfies \Pi = S T, ensuring consistency in the coupled transport of charge and heat.^[25]

Model Validity and Limitations

Experimental Agreements and Discrepancies

The Drude model achieves notable success in predicting the order-of-magnitude value of electrical conductivity \sigma in metals, with experimental measurements for copper yielding \sigma \approx 6 \times 10^7 S/m at room temperature, aligning closely with the model's estimate using electron density n \approx 8.5 \times 10^{28} m^{-3} and relaxation time \tau \approx 2.5 \times 10^{-14} s.^[26] However, the classical Drude model predicts an incorrect \rho \propto \sqrt{T} temperature dependence for resistivity from phonon scattering, rather than the observed linear \rho \propto T at high temperatures. At low temperatures, the model fails to predict the observed \rho \propto T^5 dependence due to reduced phonon scattering, instead assuming classical behavior.^[26] Furthermore, the model explains the Hall effect through the coefficient R_H = -1/(n e), which matches experimental data for simple metals like alkali metals, confirming the negative charge of carriers and providing n values consistent with one conduction electron per atom; for instance, in sodium, theoretical and measured R_H \approx -2.5 \times 10^{-10} m³/C show strong agreement at low fields and room temperature.^[27] Despite these strengths, the Drude model encounters significant discrepancies in thermal properties, particularly specific heat. The model predicts an electronic contribution C_v = \frac{3}{2} n k_B, comparable to the lattice term and violating the Dulong-Petit law, which experiments confirm applies only to ionic vibrations with total C_v \approx 3 N k_B per mole of atoms, while the observed electronic share is much smaller (\ll n k_B) at room temperature due to quantum degeneracy effects.^[14] The assumption of temperature-independent electron density n in the Drude model overlooks band structure influences, leading to overpredictions of carrier mobility at low temperatures, where experiments reveal saturation or decreases due to reduced scattering not captured by classical statistics.^[26] In optical properties, the model agrees well with the Drude tail in the infrared regime, reproducing high reflectivity via the Hagen-Rubens relation R \approx 1 - 2 \sqrt{\frac{m \omega}{2 \pi n e^2 \tau}} for metals like silver, where experimental spectra show near-unity reflectance below the plasma frequency \omega_p \approx 10^{15} rad/s.^[21] However, it fails in the ultraviolet, underestimating absorption because it neglects interband transitions between valence and conduction bands, which experiments attribute to quantum band structure effects peaking at energies around 3-5 eV in noble metals.^[21] Quantitatively, the electron density n derived from conductivity roughly matches atomic valency (e.g., one electron per atom in monovalent metals), supporting the free-electron picture.^[26] Yet, the classical Drude model predicts a Seebeck coefficient S \approx 0, which is much smaller than experimental values (several \muV/K in metals like copper) and often the wrong sign, as non-zero S requires energy-dependent scattering not captured classically.^[26]

Relation to Modern Theories

The Drude model served as a foundational classical framework that was significantly refined by quantum mechanics in the late 1920s, particularly through Arnold Sommerfeld's incorporation of Fermi-Dirac statistics to describe the free electron gas in metals.^[28] In this 1927 extension, Sommerfeld replaced the classical Maxwell-Boltzmann distribution with quantum statistics, which corrected the model's erroneous prediction of the electronic specific heat capacity—classically constant at (3/2) n k_B but observed to vanish at low temperatures—yielding instead C_v \propto T for T \ll T_F, where T_F is the Fermi temperature, while preserving the electrical conductivity \sigma \approx n e^2 \tau / m near room temperature due to the dominance of states near the Fermi level.^[28] This quantum refinement established the Drude approach as the high-temperature limit of the free electron Fermi gas model, where classical equipartition holds above T_F.^[29] In modern solid-state physics, the Drude model's assumption of free electrons is extended by introducing the effective mass m^* to account for interactions with the crystal lattice, allowing the conductivity formula \sigma = n e^2 \tau / m^* to describe band structure effects in semiconductors and metals without fully resolving the periodic potential.^[29] Felix Bloch's 1928 quantum mechanical treatment further advanced this by demonstrating that electrons in a periodic lattice potential form Bloch waves, leading to energy bands rather than the Drude picture of unrestricted free particles; this laid the groundwork for the nearly free electron model, where weak periodic potentials perturb the free electron states to produce band gaps.^[29] Despite these quantum developments, the Drude formula persists empirically in contemporary applications, such as modeling the frequency-dependent relaxation time \tau(\omega) for optical properties in plasmonics, where it describes the dielectric function \epsilon(\omega) = 1 - \omega_p^2 / (\omega(\omega + i/\tau)) for metals like gold in nanostructures.^[30] In disordered systems, the model remains a core for semiclassical transport calculations, fitting measured mobilities.^[31] For stronger electron correlations beyond the Drude and free electron approximations, the Hubbard model introduces on-site repulsion U to capture Mott insulation and other phenomena, yet retains the Drude-like mean-field core for weakly interacting regimes.

References

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