- A Brownian random walk in which a single time variable determines both the stopping times and the time between jumps. For example, one could adapt Keller’s analysis  of the model of a particle moving along the x axis such that during a time interval of duration Δτ = 1 from τ = i − 1 to τ = i, i = 1, 2,…, the particle moves with velocity ν = +1 or ν = −1, each with probability 1/2.
- A Brownian random walk in which two independent time variables are used, the stopping times and the the proper time difference between two consecutive stopping times, both of which are measured with respect to the universal time τ.
3. Brownian Motion from the Perspective of the Laboratory Frame
4. Concluding Remarks
A. Properties of Curves
B. Central Limit Theorem in Complex Space
Conflicts of Interest
- Einstein, A. On the Movement of Small Particles Suspended in a Stationary Liquid Demanded by the Molecular-kinetic Theory of Heat. Ann. Phys. 1905, 17, 549–560. [Google Scholar]
- Einstein, A. Zur Theorie der brownschen Bewegung. Ann. Phys. 1906, 19, 371–381. [Google Scholar]
- Duplantier, B. Brownian Motion, “Diverse and Undulating”. In In Eistein, 1905–2005: Poincaré Seminar 2005; Progress in Mathematical Physics; Birkhäuser: Basel, Switzerland, 2006; Volume 47, pp. 201–293. [Google Scholar]
- Marini Bettolo Marconi, U.; Puglisi, A.; Rondoni, L.; Vulpiani, A. Fluctuation-dissipation: Response Theory in Statistical Physics. Phys. Rep. 2008, 461, 111–195. [Google Scholar]
- Dunkel, J.; Hänggi, P. Relativistic Brownian Motion. Phys. Rep. 2009, 471, 1–73. [Google Scholar]
- Cecconi, F.; del-Castillo-Negrete, D.; Falcioni, M.; Vulpiani, A. The Origin of Diffusion: The Case of Non-chaotic Systems. Physica D. 2003, 180, 129–139. [Google Scholar]
- Equivalent situations are realized with particles tracing deterministic trajectories in regular environments, if correlations decay in time and space, making inapplicable a deterministic description. This happens, for instance, in the so-called periodic Lorentz gas, consisting of point particles moving in a periodic array of convex (typically circular) scatterers, in which position and velocity correlations decay at an exponential rate, [31,32]. Another example is given by polygonal billiards, in which correlations do not decay exponentially fast . In that case, one observes a different class of phenomena, which imply anomalous rather than standard diffusion.
- Stueckelberg, E.C.G. La Signification du Temps Propre en Mecanique Ondulatoire. Helv. Phys. Acta. 1941, 14, 322–322. [Google Scholar]
- Stueckelberg, E.C.G. Remarque à propos de la creation de paires de particules en théorie de relativite. Helv. Phys. Acta. 1941, 14, 588–594. [Google Scholar]
- Oron, O.; Horwitz, L.P. Relativistic Brownian Motion and Gravity as an Eikonal Approximation to a Quantum Evolution Equation. Found. Phys. 2005, 35, 1181–1203. [Google Scholar]
- Proper time is given by s/c where ds2 = c2dt2−dx2−dy2−dz2 in the coordinate system (t, x, y, z) and is invariant under Lorentz transformations. Note that t corresponds to local time and should not be confused with the proper time.
- In the event that the universal time is a non-affine parameter of the proper time the theory could also be extended to include accelerations. For an affine parameter there is no acceleration by definition.
- This is often done. For instance, in order to formulate relativistically the quantum mechanical measurements, a piece of matter may be viewed as a “galaxy” of events, i.e., of space-time points (called “flashes”) at which the wave function collapses . Flashes constitute the random part of the dynamics, while the unitary evolution of the wave function between flashes constitutes the systematic part. In the classical mechanics of particles, where there is an obvious choice for the universal time, one speaks of “event-driven” dynamics: practically random collisions (events) separate the (systematic) free flight evolutions.
- Keller, J.B. Diffusion at Finite Speed and Random Walks. Proc. Natl. Acad. Sci. USA 2004, 101, 1120–1122. [Google Scholar]
- The Strong Markov property by definition is an extension of the standard Markov property (see e.g.,  and  for a very recent study) to processes indexed by stopping times, also called optional times, see e.g., Sections 1.3 and 2.3 in .
- Oron and Horwitz state “Brownian motion, thought of as a series of “jumps” of a particle along its path, necessarily involves an ordered sequence. In the nonrelativistic theory, this ordering is naturally provided by the Newtonian time parameter. In a relativistic framework, the Einstein time t does not provide a suitable parameter. If we contemplate jumps in spacetime, to accomodate a covariant formulation, a possible spacelike interval between two jumps may appear in two orderings in different Lorentz frames. We therefore adopt the invariant parameter introduced by Stueckelberg in his construction of a relativistically covariant classical and quantum dynamics.”
- This may not be reasonable in General Relativity, since the density of the kicks may be affected by the gravitational field. For example, in the case of the Schwartschild metric, one might expect that for a closed system in equilibrium the Brownian kicks would have the same intensity on the hypersurface given by r = h(t, θ, ϕ) where h is a given function and r is constant. However, for different values of r it will not be so. From the perspective of the heat bath it would mean that it is difficult to maintain a constant temperature except on the hypersurface. A more detailed discussion of Brownian Motion in the context of General Relativity can be found in .
- Feller, W. An Introduction to Probability Theory and Its Applications, 3rd ed; Wiley: New York, NY, USA, 1968. [Google Scholar]
- Almaguer, J.; Larralde, H. A Relativistically Covariant Random Walk. J. Stat. Mech. 2007, 8, P08019. [Google Scholar]
- Cercignani, C.; Kremer, G.M. The Relativistic Boltzmann Equation: Theory and Application; Birkhäuser: Basel, Switzerland, 2000. [Google Scholar]
- Dunkel, J.; Talkner, P.; Hänggi, P. Relativistic Diffusion Processes and Random Walk Models. Phys. Rev. D. 2007, 75, 043001. [Google Scholar]
- Jospeh, D.D.; Preziosi, L. Heat Waves. Rev. Mod. Phys. 1989, 61. [Google Scholar] [CrossRef]
- For instance, in the low density limit, in which the interaction (potential) energy is negligible compared to the kinetic energy,  describes a relativistic gas as a collection of particles which move according to special relativity from collision to collision, and treats as classical the “randomly” occurring collisions among particles. In this way  provides numerically a dynamical justification of the hypothesis of molecular chaos underlying the validity of the relativistic Boltzmann equation and of its equilibrium solution known as the Maxwell-Jüttner distribution [20,47].
- Gallavotti, G. Statistical Mechanics: A Short Treatise; Springer: Berlin, Germany, 1999. [Google Scholar]
- Aliano, A.; Rondoni, L.; Morriss, G.P. Maxwell-Jüttner Distributions in Relativistic Molecular Dynamics. Eur. Phys. J. B. 2006, 50, 361–365. [Google Scholar]
- Ghodrat, M.; Montakhab, A. Molecular Dynamics Simulation of a Relativistic Gas: Thermostatistical Properties. Comp. Phys. Comm. 2011, 182, 1909–1913. [Google Scholar]
- This has been investigated in great detail by Hakim [41,48] who defines relativistic stochastic processes in where is the Minkowski space-time and U4 is the space of velocity 4-vectors but shows it is not suitable for defining a Markov process. Indeed, with the exception of a non-trivial time-discre relativistic Markov model found in , certain relativistic generalizations and their Gaussian solutions must necessarily be non-Markovian or reduce to singular functions [21,41] (cf. the excellent Review , and references therein).
- Weinberg, S.; Dicke, R.H. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Am. J. Phys. 1973, 41, 598–599. [Google Scholar]
- O’Hara, P. Constants of the Motion, Universal Time, and the Hamilton-Jacobi Function in General Relativity. J. Phys. Conf. Ser. 2013, 437, 012007. [Google Scholar]
- Horwitz, L.P.; Piron, C. Relativistic Dynamics. Helv. Phys. Acta. 1973, 46, 316–326. [Google Scholar]
- Morriss, G.P.; Rondoni, L. Periodic Orbit Expansions for the Lorentz Gas. J. Stat. Phys. 1994, 75, 553–584. [Google Scholar]
- Lloyd, J.; Niemeyer, M.; Rondoni, L.; Morriss, G.P. The Nonequilibrium Lorentz Gas. Chaos 1995, 5, 536–551. [Google Scholar]
- Jepps, O.G.; Rondoni, L. Thermodynamics and Complexity of Simple Transport Phenomena. J. Phys. A. 2006, 39. [Google Scholar] [CrossRef]
- The universal time is a standardized time defined within the space. It is not absolute time in the Newtonian sense.
- Ibe, O.C. Markov Processes for Stochastic Modeling; Elsevier: Amsterdam, The Netherlands, 2013. [Google Scholar]
- A Lorentz invariant inner product for two arbitrary vectors x1 and x2 in Minkoski space can be defined by
- Mizrahi, S.; Daboul, J. Squeezed States, Generalized Hermitz Polynomials and Pseudo-diffusion Equation. Physica A. 1992, 189, 635–650. [Google Scholar]
- Castiglione, P.; Falcioni, M.; Lesne, A.; Vulpiani, A. Chaos and Coarse Graining in Statistical Mechanics; Cambridge University Press: Cambridge, UK, 2008. [Google Scholar]
- Goldstein, S. On Diffusion by Discontinuous Movements, and on the Telegraph Equation. Q. J. Mech. Appl. Math. 1951, 4, 129–156. [Google Scholar]
- Dudley, R.M. Lorentz-invariant Markov Processes in Relativistic Phase Space. Ark. Mat. 1965, 6, 241–268. [Google Scholar]
- Hakim, R. Relativistic Stochastic Processes. J. Math. Phys. 1968, 9, 1805–1818. [Google Scholar]
- In practice one only excludes sequences of Brownian kicks that keep the Brownian particle close to the surface of the light cone for a long time. At the same time, it is unlikely that the average speed of a massive particle is close to the speed of light after a large number of kicks. A corresponding theory for spacelike events can be developed. However, requiring spacelike events to be physical would also require timelike events to be “unphysical” and vice-versa. In order words, we cannot use analytical continuity to pass from timelike to spacelike events in any physically meaningful way. A more detailed discussion can be found in [10,49].
- Tumulka, R. A Relativistic Version of the Ghirardi-Rimini-Weber Model. J. Stat. Phys. 2006, 125, 825–840. [Google Scholar]
- Cafaro, C.; Lord, W.M.; Sun, J; Bollt, E.M. Causation Entropy from Symbolic Representations of Dynamical Systems. Chaos 2015, 25, 043106. [Google Scholar]
- Chung, K.L. Lectures from Markov Processes to Brownian Motion; Springer: New York, NY, USA, 1982. [Google Scholar]
- O’Hara, P.; Rondoni, L. Brownian Motion and General Relativity 2013. arXiv: 1304.0405.
- De Groot, S.R.; van Leeuwen, W.A.; van Weert, Ch.G. Relativistic Kinetic Theory: Principles and Applications; North-Holland: Amsterdam, 1980. [Google Scholar]
- Hakim, R. Introduction to Relativistic Statistical Mechanics: Classical and Quantum; World Scientific: Singapore, Singapore, 2011. [Google Scholar]
- Oran, O.; Horwitz, L.P. Relativistic Brownian Motion, 2003; arXiv:0212036.
© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).