Obtaining the Drude Equation for Electrons in Metals Using a Fractional Variational Principle

TL;DR

This paper introduces a fractional variational principle with derivatives of order 1/2, deriving a modified Euler-Lagrange equation that reproduces the Drude model for electrons in metals, linking fractional calculus to condensed matter physics.

Contribution

It develops a fractional variational framework to derive the Drude equation, connecting fractional derivatives with electron dynamics in metals for the first time.

Findings

Derived a fractional Euler-Lagrange equation for systems with fractional derivatives.

Reproduced the Drude model for electrons in metals using fractional kinetic energy.

Validated the approach with classical dissipative systems like damped oscillators and RLC circuits.

Abstract

A fractional variational principle was derived in order to be used with lagrangians containing fractional derivatives of order 1/2. By forcing the action associated to this type of lagrangian to be stationary, a modified fractional Euler-Lagrange equation was obtained. This was shown to reproduce the equations of motion of two basic 1-dimensional energy-dissipative systems: a spring-mass system damped by friction, and a RLC circuit connected in series. Finally, by using the fractional Euler-Lagrange equation, the Drude relationship for electrons in metals was recovered when a fractional kinetic energy was taken into consideration in the electron's associated energies.