Epoched Brownian Motion: Relativistic Dynamics

• Epoched Brownian Motion is a relativistic stochastic process with two independent proper-time parameters—one for random impulses (stopping times) and one for deterministic geodesic motion—ensuring causal behavior.
• It employs Lorentz-covariant probability distributions, yielding 4D Gaussian displacement laws and Maxwell–Jüttner-type velocity distributions, which rigorously conform to relativistic constraints.
• The model’s hyperbolic PDE and quaternion formalism provide a robust framework for simulating diffusion in high-energy astrophysics and curved spacetime scenarios.

Epoched Brownian Motion (EBM) is a relativistic stochastic process defined on a pseudo-Riemannian manifold, encapsulating the interplay between random thermal "kicks" (impulses) and deterministic free motion along geodesics, as developed by O’Hara and Rondoni (O'Hara et al., 2013). EBM is characterized by two independent proper-time variables: a discrete sequence marking random collision events and a continuous parameter governing geodesic, collision-free evolution between events. The statistical properties of EBM are Lorentz covariant, yielding temperature-dependent four-dimensional Gaussian distributions in position space and Maxwell–Jüttner–type distributions for four-velocity, with dynamics governed by a hyperbolic partial differential equation incorporating both time variables.

1. Structure of the EBM Process: Stopping Times and Kicks

EBM introduces a sequence of stopping times defined with respect to a global laboratory inertial frame: $0 = T_0 < T_1 < T_2 < \cdots$, where each $T_i$ specifies the instant of a Brownian kick or impulse. The time increments $\Delta T_i := T_i - T_{i-1}$ are independent and identically distributed (IID) random variables under a heat bath at fixed temperature $\theta$, subject to $E[\Delta T_i] = \mu_T$ and $\operatorname{Var}(\Delta T_i) = \sigma_T^2$. By the strong Markov property, these increments encode all irreducible randomness and, being differences of proper times in the particle rest frame, are Lorentz invariant. The sequence confers a discrete counting index $n = i$ that tracks the cumulative number of kicks, representing one of the two essential time-parameters in EBM.

2. Geodesic Dynamics Between Stopping Times

Between each pair of kicks $(T_{i-1}, T_i)$, the EBM particle undergoes deterministic free motion along a timelike geodesic in the underlying pseudo-Riemannian manifold, which may be locally identified with Minkowski space. The laboratory metric is $ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2$, and geodesics are parametrized by the particle's continuous proper time $s$, related to laboratory time $T$ via $ds = a_i dT$ on the $i$th interval. The geodesic segment is characterized by constant four-velocity $u^a = dx^a/ds$ with vanishing four-acceleration, ensuring strictly causal (sub-luminal) motion in each segment.

3. Dual Proper-Time Formulation

EBM fundamentally rests on two independent time variables: (1) the continuous parameter $s$ (or equivalently, laboratory time $T$) governing deterministic motion along each geodesic segment, and (2) the discrete label $n$ (number of kicks), indexing stochastic events. These time variables are statistically independent: $T$ (or $s$) determines the location along a geodesic segment, while $n$ indexes the cumulative number of random reorientations. This dual structure distinguishes EBM from classical Brownian models, allowing clear separation of deterministic and stochastic evolution.

4. Probability Distributions: Quaternion Gaussians and Maxwell–Jüttner Law

EBM statistics are formulated covariantly using quaternions to represent Minkowski space. The 4-vector displacement in the $i$th segment is $\Delta X_i^a = V^a(T_i) \Delta T_i$, where $V^a$ is the initial four-velocity post-kick at $T_{i-1}$, with $E[\Delta X_i^a]=0$ and $\operatorname{Var}(\Delta X_i^a)=\sigma_a^2<\infty$ by isotropy. After $n$ kicks, the total displacement $S_n = \sum_{i=1}^n \Delta X_i \in \mathbb{Q}$, the quaternion algebra. By the central limit theorem, $S_n$ approaches a 4D Gaussian:

$$f_S(q;n) \approx \frac{1}{(2\pi n)^2 \sqrt{\det\Sigma}} \exp\left[-\frac{1}{2n} q^\mu (\Sigma^{-1})_{\mu\nu} q^\nu\right],$$

where $$\Sigma = \mathrm{diag}(\sigma_T^2, \sigma_X^2, \sigma_Y^2, \sigma_Z^2)$$, and the signature $(+, -, -, -)$ relates to $$(c t)^2/\sigma_T^2 - x^2/\sigma_X^2 - y^2/\sigma_Y^2 - z^2/\sigma_Z^2$$.

Jumps in four-velocity, $$\Delta V_i^a = V^a(T_i) - V^a(T_{i-1}) + \int_{T_{i-1}}^{T_i} A^a(s) dT$$, encompass both the discontinuity at each kick and any acceleration within a segment. Letting $V_n = \sum_{i=1}^n \Delta V_i$, its distribution also converges to a Gaussian:

$$f_V(q;n) \approx \frac{1}{(2\pi n)^2 \sqrt{|\psi|} \sigma_{\dot T}} \exp\left[ -\frac{1}{2n |\psi|} \left( \frac{c^2 \dot T^2}{\sigma_{\dot T}^2} - \frac{\dot X^2}{\sigma_{\dot X}^2} - \frac{\dot Y^2}{\sigma_{\dot Y}^2} - \frac{\dot Z^2}{\sigma_{\dot Z}^2} \right) \right ],$$

formally reproducing the Maxwell–Jüttner equilibrium for relativistic gases, now generalized to admit two proper-time scales and arbitrary four-acceleration on each free-flight segment.

5. Governing Hyperbolic Partial Differential Equation

The joint distribution $F = F(t, x, y, z; n)$, representing the state of the system after $n$ kicks at a continuous proper time $t$ since the last kick, satisfies a hyperbolic ("telegraph-type") partial differential equation intertwining the dual time variables:

$$\left(\partial_t^2 - \partial_x^2 - \partial_y^2 - \partial_z^2 \right)F(t, \mathbf{x}; n) = \frac{2c^2}{\sigma_X^2} \partial_n F(t, \mathbf{x}; n).$$

Here, $\partial_n$ denotes a derivative with respect to the discrete index $n$, analytically continued for large $n$, while $\partial_t$, $\partial_x$, etc. are the conventional derivatives along the continuous geodesic variable. The constraint that each flight is a timelike geodesic ensures that no superluminal (acausal) propagation can arise.

6. Comparison with Classical Brownian Motion

Feature	Classical Brownian Motion	Epoched Brownian Motion (EBM)
Time Variables	Single (continuous, $t$)	Two: continuous ($T$/$s$), discrete ($n$)
Differentiability of Paths	Almost surely nowhere differentiable	Piecewise geodesic, differentiable except at $\{T_i\}$
Causality	Allows arbitrarily fast steps	Each segment timelike; no causality violation
Governing PDE	Parabolic (heat equation)	Hyperbolic (telegraph-type equation)
Equilibrium Distribution	Maxwell–Boltzmann	4D Lorentz-invariant Gaussian (positions), Maxwell–Jüttner (velocities)

Distinctions are pronounced. EBM’s piecewise-geodesic trajectories are smooth between kicks, with singularities confined to the countable set of impulsive events. Classical diffusion permits unbounded speeds over infinitesimal intervals, in contrast to EBM’s strict adherence to relativistic causality. The governing equation for EBM is hyperbolic, in contrast to the parabolic diffusion equation of classical Brownian motion. In thermodynamic equilibrium, EBM yields Lorentz-covariant (temperature-dependent) Gaussian laws in position space and the Maxwell–Jüttner distribution in velocity space, replacing the Maxwell–Boltzmann law of classical theory.

7. Significance and Theoretical Implications

EBM provides a Lorentz-covariant generalization of Brownian motion suitable for applications in relativistic statistical mechanics (O'Hara et al., 2013). The explicit encoding of randomness via a discrete, Lorentz-invariant proper-time index enables a precise separation of stochastic and deterministic evolution. The hyperbolic PDE governing EBM dynamics enforces causality and reflects the physical constraint that energy and information propagate at or below the speed of light. The adoption of quaternionic formalism and the recovery of familiar equilibrium distributions within a fully relativistic framework mark EBM as a mathematically robust, physically motivated extension of classical stochastic processes. A plausible implication is improved modeling of diffusion processes in high-energy astrophysics or in curved spacetimes, where relativistic effects are non-negligible.