Lorentz-boosted diffusion: initial value formulation and exact solutions

Abstract

It is well known that the diffusion equation, when treated as a stand-alone partial differential equation, exhibits exponential instabilities in boosted frames, which render the corresponding initial-value problem ill-posed. Recently, however, it was shown that Fick-type diffusion arises as the exact hydrodynamic sector of relativistic Fokker-Planck kinetic theory. In this work, we exploit this kinetic embedding to formulate a modified initial-value problem for one-dimensional Lorentz-boosted diffusion. We show that the resulting dynamics are well posed both forward and backward in time, provided the boosted density profiles admit a kinetic-theory realization. Such profiles form a space of band-limited functions, within which the evolution can be expressed as a discrete superposition of spatially sampled initial data, weighted by a Shannon-Whittaker-type Green function defined on the full Minkowski plane. The Green function is obtained in closed analytic form.

Figures (5)

• Figure 1: Imaginary part (left) and real part (right) of the boosted dispersion relations ${\Tilde{\omega}}({\Tilde{k}}):\mathbb{R}\to\mathbb{C}$, obtained by applying a Lorentz boost to Fick’s law of diffusion $\omega=-ik^{2}$. The boost velocity is chosen as $v=1/2$, but the qualitative features are the same for all $v>0$. The blue branch is stable, while the red branch is unstable. The shaded region in the left panel indicates the kinetically inadmissible domain, which must be excluded on physical grounds (the dashed lines correspond to $\mathfrak{Im}\, \omega=\pm 1/(\gamma v)$). As expected, the unstable branch lies entirely within this region. In addition, the kinetic bound excludes the high-wavenumber portion of the stable branch, leading to an upper cutoff in ${\Tilde{k}}$ (for $v=1/2$, the cutoff wavenumber is $\Lambda=4$).
• Figure 2: Wavenumber cutoff $\Lambda$ as a function of the boost velocity $v$, as determined by \ref{['cutoffone']}. For $|{\Tilde{k}}|>\Lambda$, solutions of the boosted diffusion equation \ref{['diffusionBoosted']} no longer admit a realization within the underlying Fokker-Planck kinetic theory, because the kinetic integral in \ref{['densitone']} diverges.
• Figure 3: Snapshot at $t=0$, in the rest frame of the medium, of the fundamental solution $\mathcal{K}$ associated with the boosted initial-value problem for an observer moving with velocity $-1/2$. The profile is modulated by an exponential factor $e^{-x}$, which arises due to relativity of simultaneity, and reflects the saturation of the kinetic admissibility bound \ref{['thebound']}. The corresponding analytic formula is given in equation \ref{['iniziamofund']}.
• Figure 4: Snapshots of the forward (left panel) and backward (right panel) time evolution of the Green function $\mathcal{K}(\tilde{t},\tilde{x})$, obtained by boosting \ref{['TheAnswer']}. To compensate for advection by the moving medium, each snapshot covers a region shifted by $v\tilde{t}$. For $\tilde{t}>0$, the evolution exhibits asymmetric diffusion and rapid suppression of oscillations associated with the cutoff scale. Under backward time evolution, those same oscillatory features are progressively amplified. Again, we chose $v=1/2$.
• Figure 5: Exact solutions \ref{['dnAfinal']} of the boosted diffusion equation \ref{['diffusionBoosted']} for the initial data \ref{['initialdataexampleapproximae']} at boost velocity $v=1/2$. Left panels: forward evolution. We see a progressive loss of sensitivity to the detailed structure of the initial profile, and convergence toward a universal (asymmetric) diffusive shape. Right panels: backward evolution. Here, antidiffusion amplifies modes near the cutoff, leading to the emergence or enhancement of spatial oscillations on the scale $1/\Lambda$.