Relativistic Lévy processes

TL;DR

The paper develops a relativistically consistent framework for one-dimensional velocity distributions by applying the generalized central limit theorem to rapidities, yielding Lorentz-invariant α-stable distributions for velocity. The central result is that the relativistic velocity distribution is ${\mathcal{F}}(\alpha,\lambda;\beta) = \gamma^2(\beta) f(\alpha,\lambda;\sigma(\beta))$, preserving functional form under Lorentz transformations. Two practical quantifiers, concavity at the origin $R(\alpha,\lambda)$ and the probability of measuring relativistic velocities $p_r(\alpha,\lambda)$, are introduced to classify regimes from weak-relativistic to fully relativistic and to guide data interpretation. Supporting evidence from heavy-ion diffusion and antiproton cooling experiments shows good agreement with the relativistic-stable distributions, suggesting these distributions capture essential non-Gaussian relativistic fluctuations in real systems. The work provides a concrete protocol for assessing stochastic relativistic effects across particle physics, plasma physics, and astrophysical contexts, while noting that extensions to higher dimensions require further study.

Abstract

We study sums of independent and identically distributed random velocities in special relativity. We show that the resulting one-dimensional velocity distributions are not only stable under relativistic velocity addition but define a genuinely new class of stochastic processes--relativistic Lévy processes. Given a system, this allows identifying distinct relativistic regimes in terms of the distribution's concavity at the origin and the probability of measuring relativistic velocities. These features provide a protocol to assess the relevance of stochastic relativistic effects in actual experiments. As supporting evidence, we find agreement with previous results about heavy-ion diffusion and show that our findings are consistent with the distribution of momentum deviations observed in measurements of antiproton cooling.

Figures (7)

• Figure 1: Relativistic stable distributions of velocities (solid) and their associated non-relativistic stable distributions (dashed). The change in modal behavior, characterizing distinct statistical velocity regimes (see main text), is observed as $\alpha$ and $\lambda$ vary. (a) For $\alpha=2$, at $\lambda=0.1$, the distribution displays a unimodal trend (non-relativistic regime). As $\lambda$ grows, the distribution transitions to a bimodal shape (relativistic regime), with the threshold being $\lambda > 0.5$. (b) For $\alpha=1$, even at $\lambda=0.1$, the distribution is trimodal (weak relativistic regime). (c) For $\lambda=1$, as $\alpha$ increases, the distribution approaches relativistic regime. The transition to a bimodal shape occurs when $\alpha \geq 1$.
• Figure 2: Relativistic stable distribution of (a) energy and (b) momentum for $\alpha=1$ (solid) and the corresponding non-relativistic cases (dashed). The log-linear plots in the insets highlight that these distributions exhibit a behavior close to heavy-tail, thus contrasting with the distribution of velocities in Fig. \ref{['fig:relmbcauchy']}. Due to the rest energy, the energy distribution is shifted by $mc^2$ (dotted line).
• Figure 3: Variance of relativistic velocity, $\langle \beta^2 \rangle$, as a function of the scale parameter $\lambda$ for several values of $\alpha$. Because the velocity is bounded by $c$, the variance converges as $\lambda \to \infty$, irrespective of $\alpha$.
• Figure 4: (a)-(c) Density plots of the quantifiers $R$ and $p_{r}$ (the latter is reliable for any value of $\alpha$). (d) Distributions for the parameters ($\lambda=0.6$, $\alpha=1.9$) and ($\lambda = \lambda_c^{(1.9)} = 0.514$, $\alpha=1.9$) — indicated by $\times$ in (a) and (c) — displaying a special quadrimodal shape for the former, a trend that cannot be predicted solely by the concavity at $\beta=0$, given by $R$. The (white) dotted-dashed curves represent the set of points $(\lambda_c^{(\alpha)},\alpha)$. In (b), for $\alpha < 2$, the border ${\mathfrak B}$ between the yellow and the other colored regions marks the transition between the relativistic ($p_r=1$) and weak relativistic ($p_r < 1$) regimes. The vertical and horizontal lines indicate special parameter values and are guides to the eye. A blow-up in the region of $\alpha > 1.4$ displayed in (c) evidences a mismatch between $R$ and $p_{r}$ in characterizing the transition from weak relativistic to relativistic regime when $\alpha>1.5$. While $R$ predicts a continuous transition from weak relativistic to relativistic regime for any $\alpha$, $p_{r}$ predicts a continuous (discrete) transition for $\alpha<1.5$ ($\alpha \geq 1.5$). Also, for $\alpha=2$, $\lambda_c^{(2)} = 0.5$, but the transition takes place for $\lambda \approx 0.9$, as indicated by an arrow in (c).
• Figure 5: Data fitting of theoretical and experimental data. (a), (b) Fits of the distribution data from heavy-ion collisions presented in Figs. 3 and 5 of Ref. Wolschin2004Feb, respectively, using Eq. \ref{['eq:stablerapidity']} (black curves). The parameter values are (for $Q$, see main text): $\alpha=1.959$, $\lambda=1.071$, $Q=156.792$ in (a) and $\alpha=1.86$, $\lambda=1.107$, $Q=23.37$ in (b). The figures illustrate the probability distribution of rapidity $y$, which is obtained from $dN(y,t)/dy$ in the limit $t \to \infty$. The insets highlight the strong agreement at the tails. (c) Fitting of the experimental momentum deviation distribution data of antiproton cooling, as presented in Fig. 3 of Ref. Nagaitsev2006Jan, employing Eq. \ref{['eq:rellsmomentum']} (black curve), with $\alpha=1.736$, $\lambda=0.119$, $Q=0.206$. The Gaussian distribution was considered in the original work Nagaitsev2006Jan. Remarkably, the tails observed experimentally are accurately described by the large-deviation regime of the relativistic stable distribution of momentum ${\mathcal{F}}_p(\alpha, \lambda; p)$.