{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:17:19Z","timestamp":1760217439169,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2015,6,1]],"date-time":"2015-06-01T00:00:00Z","timestamp":1433116800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We construct a model of Brownian motion in Minkowski space. There are two aspects of the problem. The first is to define a sequence of stopping times associated with the Brownian \u201ckicks\u201d or impulses. The second is to define the dynamics of the particle along geodesics in between the Brownian kicks. When these two aspects are taken together, the Central Limit Theorem (CLT) leads to temperature dependent four dimensional distributions defined on Minkowski space, for distances and 4-velocities. In particular, our processes are characterized by two independent time variables defined with respect to the laboratory frame: a discrete one corresponding to the stopping times when the impulses take place and a continuous one corresponding to the geodesic motion in-between impulses. The subsequent distributions are solutions of a (covariant) pseudo-diffusion equation which involves derivatives with respect to both time variables, rather than solutions of the telegraph equation which has a single time variable. This approach simplifies some of the known problems in this context.<\/jats:p>","DOI":"10.3390\/e17063581","type":"journal-article","created":{"date-parts":[[2015,6,1]],"date-time":"2015-06-01T10:55:28Z","timestamp":1433156128000},"page":"3581-3594","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Brownian Motion in Minkowski Space"],"prefix":"10.3390","volume":"17","author":[{"given":"Paul","family":"O'Hara","sequence":"first","affiliation":[{"name":"Department of Mathematics, Northeastern Illinois University, 5500 North St. Louis Avenue, Chicago, IL 60625-4699, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4223-6279","authenticated-orcid":false,"given":"Lamberto","family":"Rondoni","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienze Matematiche and Graphene@PoliTO Lab, Politecnico di Torino, Corso Ducadegli Abruzzi 24, 10129 Torino, Italy"},{"name":"National Institute of Nuclear Physics (INFN), Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,6,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1002\/andp.19053220806","article-title":"On the Movement of Small Particles Suspended in a Stationary Liquid Demanded by the Molecular-kinetic Theory of Heat","volume":"17","author":"Einstein","year":"1905","journal-title":"Ann. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1002\/andp.19063240208","article-title":"Zur Theorie der brownschen Bewegung","volume":"19","author":"Einstein","year":"1906","journal-title":"Ann. Phys."},{"key":"ref_3","first-page":"201","article-title":"Brownian Motion, \u201cDiverse and Undulating\u201d","volume":"47","author":"Duplantier","year":"2006","journal-title":"In Eistein, 1905\u20132005: Poincar\u00e9 Seminar 2005"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.physrep.2008.02.002","article-title":"Fluctuation-dissipation: Response Theory in Statistical Physics","volume":"461","author":"Puglisi","year":"2008","journal-title":"Phys. Rep."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2008.12.001","article-title":"Relativistic Brownian Motion","volume":"471","author":"Dunkel","year":"2009","journal-title":"Phys. Rep."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/S0167-2789(03)00051-4","article-title":"The Origin of Diffusion: The Case of Non-chaotic Systems","volume":"180","author":"Cecconi","year":"2003","journal-title":"Physica D."},{"key":"ref_7","unstructured":"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 [33]. In that case, one observes a different class of phenomena, which imply anomalous rather than standard diffusion."},{"key":"ref_8","first-page":"322","article-title":"La Signification du Temps Propre en Mecanique Ondulatoire","volume":"14","author":"Stueckelberg","year":"1941","journal-title":"Helv. Phys. Acta."},{"key":"ref_9","first-page":"588","article-title":"Remarque \u00e0 propos de la creation de paires de particules en th\u00e9orie de relativite","volume":"14","author":"Stueckelberg","year":"1941","journal-title":"Helv. Phys. Acta."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1007\/s10701-005-6406-z","article-title":"Relativistic Brownian Motion and Gravity as an Eikonal Approximation to a Quantum Evolution Equation","volume":"35","author":"Oron","year":"2005","journal-title":"Found. Phys."},{"key":"ref_11","unstructured":"Proper time is given by s\/c where ds2 = c2dt2\u2212dx2\u2212dy2\u2212dz2 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."},{"key":"ref_12","unstructured":"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."},{"key":"ref_13","unstructured":"This is often done. For instance, in order to formulate relativistically the quantum mechanical measurements, a piece of matter may be viewed as a \u201cgalaxy\u201d of events, i.e., of space-time points (called \u201cflashes\u201d) at which the wave function collapses [43]. 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 \u201cevent-driven\u201d dynamics: practically random collisions (events) separate the (systematic) free flight evolutions."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1120","DOI":"10.1073\/pnas.0307052101","article-title":"Diffusion at Finite Speed and Random Walks","volume":"101","author":"Keller","year":"2004","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_15","unstructured":"The Strong Markov property by definition is an extension of the standard Markov property (see e.g., [35] and [44] 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 [45]."},{"key":"ref_16","unstructured":"Oron and Horwitz state \u201cBrownian motion, thought of as a series of \u201cjumps\u201d 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.\u201d"},{"key":"ref_17","unstructured":"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, \u03b8, \u03d5) 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 [46]."},{"key":"ref_18","unstructured":"Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Wiley. [3rd]."},{"key":"ref_19","first-page":"P08019","article-title":"A Relativistically Covariant Random Walk","volume":"8","author":"Almaguer","year":"2007","journal-title":"J. Stat. Mech."},{"key":"ref_20","unstructured":"Cercignani, C., and Kremer, G.M. (2000). The Relativistic Boltzmann Equation: Theory and Application, Birkh\u00e4user."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"043001","DOI":"10.1103\/PhysRevD.75.043001","article-title":"Relativistic Diffusion Processes and Random Walk Models","volume":"75","author":"Dunkel","year":"2007","journal-title":"Phys. Rev. D."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Jospeh, D.D., and Preziosi, L. (1989). Heat Waves. Rev. Mod. Phys., 61.","DOI":"10.1103\/RevModPhys.61.41"},{"key":"ref_23","unstructured":"For instance, in the low density limit, in which the interaction (potential) energy is negligible compared to the kinetic energy, [25] describes a relativistic gas as a collection of particles which move according to special relativity from collision to collision, and treats as classical the \u201crandomly\u201d occurring collisions among particles. In this way [25] 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-J\u00fcttner distribution [20,47]."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Gallavotti, G. (1999). Statistical Mechanics: A Short Treatise, Springer.","DOI":"10.1007\/978-3-662-03952-6"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1140\/epjb\/e2006-00117-x","article-title":"Maxwell-J\u00fcttner Distributions in Relativistic Molecular Dynamics","volume":"50","author":"Aliano","year":"2006","journal-title":"Eur. Phys. J. B."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1909","DOI":"10.1016\/j.cpc.2011.01.018","article-title":"Molecular Dynamics Simulation of a Relativistic Gas: Thermostatistical Properties","volume":"182","author":"Ghodrat","year":"2011","journal-title":"Comp. Phys. Comm."},{"key":"ref_27","unstructured":"This has been investigated in great detail by Hakim [41,48] who defines relativistic stochastic processes in \u03bc=M4\u00d7U4 where M4 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 [21], certain relativistic generalizations and their Gaussian solutions must necessarily be non-Markovian or reduce to singular functions [21,41] (cf. the excellent Review [5], and references therein)."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"598","DOI":"10.1119\/1.1987308","article-title":"Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity","volume":"41","author":"Weinberg","year":"1973","journal-title":"Am. J. 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(2006). Thermodynamics and Complexity of Simple Transport Phenomena. J. Phys. A., 39.","DOI":"10.1088\/0305-4470\/39\/6\/007"},{"key":"ref_34","unstructured":"The universal time is a standardized time defined within the space. It is not absolute time in the Newtonian sense."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Ibe, O.C. (2013). Markov Processes for Stochastic Modeling, Elsevier.","DOI":"10.1016\/B978-0-12-407795-9.00015-3"},{"key":"ref_36","unstructured":"A Lorentz invariant inner product for two arbitrary vectors x1 and x2 in Minkoski space can be defined by \u2329x1,x2\u232a\u2261=x0y0\u2212x1y1\u2212x2y2\u2212x3y3."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1016\/0378-4371(92)90066-Y","article-title":"Squeezed States, Generalized Hermitz Polynomials and Pseudo-diffusion Equation","volume":"189","author":"Mizrahi","year":"1992","journal-title":"Physica A."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Castiglione, P., Falcioni, M., Lesne, A., and Vulpiani, A. (2008). Chaos and Coarse Graining in Statistical Mechanics, Cambridge University Press.","DOI":"10.1017\/CBO9780511535291"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1093\/qjmam\/4.2.129","article-title":"On Diffusion by Discontinuous Movements, and on the Telegraph Equation","volume":"4","author":"Goldstein","year":"1951","journal-title":"Q. J. Mech. Appl. Math."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/BF02592032","article-title":"Lorentz-invariant Markov Processes in Relativistic Phase Space","volume":"6","author":"Dudley","year":"1965","journal-title":"Ark. Mat."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1805","DOI":"10.1063\/1.1664513","article-title":"Relativistic Stochastic Processes","volume":"9","author":"Hakim","year":"1968","journal-title":"J. Math. 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