Braffort–Marshall equation

The Braffort–Marshall equation is the fundamental equation of motion in stochastic electrodynamics (SED), extending the Abraham–Lorentz equation by adding a stochastic driving force from the zero-point field (ZPF). It was independently proposed by P. Braffort (1954) and T. W. Marshall (1963) and takes the form:

m x ¨ = F ext + m τ x... + e E zpf (x, t) {\displaystyle m{\ddot {\mathbf {x} }}=\mathbf {F} _{\text{ext}}+m\tau {\overset {...}{\mathbf {x} }}+e\mathbf {E} _{\text{zpf}}(\mathbf {x},t)} $m{\ddot {\mathbf {x} }}=\mathbf {F} _{\text{ext}}+m\tau {\overset {...}{\mathbf {x} }}+e\mathbf {E} _{\text{zpf}}(\mathbf {x},t)$

m d 2 x d t 2 = F ext + m τ d 3 x d t 3 + e E zpf (x, t) {\displaystyle m{\frac {d^{2}\mathbf {x} }{dt^{2}}}=\mathbf {F} _{\text{ext}}+m\tau {\frac {d^{3}\mathbf {x} }{dt^{3}}}+e\mathbf {E} _{\text{zpf}}(\mathbf {x},t)} $m{\frac {d^{2}\mathbf {x} }{dt^{2}}}=\mathbf {F} _{\text{ext}}+m\tau {\frac {d^{3}\mathbf {x} }{dt^{3}}}+e\mathbf {E} _{\text{zpf}}(\mathbf {x},t)$

where E zpf {\displaystyle \mathbf {E} _{\text{zpf}}} $\mathbf {E} _{\text{zpf}}$ is the electric field of the zero-point radiation field, treated as a classical random field with spectral density ρ (ω) = ℏ ω 3 / (2 π 2 c 3) {\displaystyle \rho (\omega)=\hbar \omega ^{3}/(2\pi ^{2}c^{3})} $\rho (\omega)=\hbar \omega ^{3}/(2\pi ^{2}c^{3})$ .

The Braffort-Marshall equation is the starting point for all SED derivations. When applied to the harmonic oscillator, it reproduces the quantum ground-state energy 1 2 ℏ ω {\displaystyle {\tfrac {1}{2}}\hbar \omega } ${\tfrac {1}{2}}\hbar \omega$ as the equilibrium between radiation damping and ZPF driving — without invoking quantization. This is the core claim of SED: quantum behavior emerges from classical electrodynamics coupled to a real, physical zero-point radiation background.

Braffort–Marshall equation

References