Diffusion in the special theory of relativity.

Area of Science:

• Theoretical Physics
• Statistical Mechanics
• Relativistic Quantum Mechanics

Background:

• Standard Markovian diffusion theory assumes Euclidean velocity spaces.
• Special relativity mandates a hyperbolic velocity space, posing mathematical challenges for diffusion processes.
• Bridging classical diffusion with relativistic constraints is crucial for high-energy physics and cosmology.

Purpose of the Study:

• To generalize Markovian diffusion theory to the framework of special relativity.
• To derive a relativistic diffusion equation applicable in hyperbolic velocity spaces.
• To analyze the properties and solutions of this generalized relativistic diffusion equation.

Main Methods:

• Application of stochastic calculus on Riemannian manifolds, adapted to the hyperbolic velocity space of relativity.
• Definition of a generalized Langevin equation in a fiber space including position, velocity, and orthonormal frames.
• Derivation of the generalized relativistic Kramers equation in phase space with external forces.

Main Results:

• A generalized relativistic Kramers equation was derived, invariant under Lorentz transformations.
• The stationary solution of the derived diffusion equation was identified as the Jüttner distribution.
• A nonstationary analytical solution was obtained for force-free relativistic diffusion.

Conclusions:

• The study successfully extends diffusion theory into the realm of special relativity.
• The derived framework provides a foundation for studying diffusion processes in relativistic regimes.
• The results have implications for understanding particle dynamics in high-energy environments and cosmological models.