Relativistic Stochastic Dynamics: A Review

1. Relativistic Stochastic Dynamics:A Review F. Debbasch LERMA-ERGA Université Paris 6

2. Why is the construction of relativistic stochastic dynamics a problem? Why is the construction of relativistic stochastic dynamics interesting? The Relativistic Ornstein-Uhlenbeck Process (ROUP) Special relativistic applications General relativistic applications Other relativistic stochastic processes Towards stochastic geometry

3. Why is the Construction of Relativistic Stochastic Dynamics a Problem? • The basic tool of Galilean stochastic models is Brownian motion • If dxt = l dBt/t then ∂t n = cD n with c = l2 2 t

4. If n (t = 0, x) = d (x), then, for all t > 0, n (t, x) = G(t, x) ~ exp ( - x2/4 c t)  Faster than light particle (mass, energy) transfer

5. Why is the Construction ofRelativistic Stochastic DynamicsInteresting? • General theoretical interest • Practical problems involving special or general relativistic diffusions: Plasma Physics, Astrophysics, Cosmology, … • Toy model of relativistic irreversible behavior • Stochastic geometry

6. Standard Relativistic Fluid Models Relativistic Boltzman Equation Causal Chapman-Enskog expansion Grad expansion (No small parameter) First order hydrodynamics (relativistic Navier-Stokes model, …) Extended Thermodynamics Causal, but contradicted by experiments Non-causal

7. The Relativistic Ornstein-Uhlenbeck Process (Debbasch, Mallick, Rivet, 1997) • Models the diffusion of a particle of mass m in a fluid characterized by a temperature and a velocity field • Simplest case: Flat space-time. Uniform and constant fluid temperature and velocity • The rest-frame of the fluid is a preferred reference frame for the process i.e. the equations are a priori simpler in this frame • But the whole treatment is covariant

8. 1 g(p) (p) p2 where g(p) = (1 + )1/2 m2c2 The Relativistic Ornstein-Uhlenbeck Process Idea: Brownian motion in momentum space In flat space-time, for uniform and constant fluid temperature and velocity fields: p dxt = dt m in the rest frame of the fluid dpt = - a p dt +D dBt

9. The relativistic Ornstein-Uhlenbeck Process: Alternative definition via the transport equation • In flat space-time, for uniform and constant fluid temperature and velocity fields • In the rest frame of the fluid, dn = P(t, x, p) d3x d3p and P(t, x, p) verifies the forward Kolmogorov equation: D2 p ∂tP + ∂x( P) + ∂p(-a(p)pP) = DpP mg(p) 2

10. The Relativistic Ornstein-Uhlenbeck Process: Fluctuation-Dissipation Theorem The coefficients a(p) and D are constrained by imposing that the Jüttner distribution PJ(p) at temperature q, PJ (p) ~ exp(- bg(p) mc2), b = 1/(kBq), is a solution of the transport equation a(p) = a0/g(p) D/a0 = m kBq and

11. The Relativistic Ornstein-Uhlenbeck process: General Transport Equation (Barbachoux, Debbasch, Rivet, 2001, 2004) • Manifestly covariant formalism  Extended phase space with (xm, pm) as coordinates  P(t, x, p) • Fluid characterized by U(x), a0(x) and D(x) • Basic objects: 1. Derivative with respect to p at constant x: ∂pm • 2. Derivative with respect to x at p covariantly constant: Dm = m + Gb pb ∂pa • 3. Projector Dmn (x) = gmn(x) - Um(x)Un(x) f(x, p) ma

12. The Relativistic Ornstein-Uhlenbeck process: General Transport Equation Kolmogorov equation reads L(f) = 0 with • L(f) = Dm (gmn(x)pnf) + ∂pm(mc Fm(x, p)f) + N(f) • Fm(x, p) =  lmn(x, p) = 1 lab(x, p) papb pm- gab(x) papblmn(x)pn m2c2 2 mc a0(x) Dmn (x) p.U(x)

13. The Relativistic Ornstein-Uhlenbeck Process:General Transport Equation • N(f) = Kmrbn (x) ∂pr • Kmrbn(x) = UmUbDrn - UmUnDrb + 2 D (x) pmpb ∂pn f 2 p.U(x) UrUnDmb - UrUbDmn

14. Special Relativistic Applications:Near-equilibrium, large-scale Diffusion • The fluid has constant and uniform temperature  and velocity field • Study in the proper frame of the fluid • Chapman-Enskog expansion of a near equilibrium situation • The diffusion is completely determined by the density n(t, x) (Debbasch, Rivet, 1998)

15. Near-equilibrium, large scale Special Relativistic Diffusion • Microscopic time-scale t = 1/a0 • Microscopic length-scale l = vth(q) t • Density n varies on characteristic scales T and L, t/T = O(), l/L = O() • Then h = e2 • ∂t n = cD n with  = l2 : APPARENT PARADOX! 2t

16. Near-equilibrium, Large-Scale Special Relativistic Diffusion: Paradox Resolved • Green function G(t, x) of the diffusion equation: G(0, x) = d(x) • The conditions t/T = O(), l/L = O() applied to G(t, x) lead to: t >> t x /t << c

17. General conclusion on relativistic irreversible phenomena • In the local rest-frame of a continuous medium, all non-Galilean irreversible phenomena are microscopic • In the local rest-frame of a continuous medium, all macroscopic irreversible phenomena are Galilean • There can be no coherent relativistic hydrodynamics of viscous fluids • Purely relativistic irreversible phenomena can only be described through statistical physics, e.g. Boltzmann equation

18. General Relativistic Applications (Rigotti, Debbasch, 2004, 2005) • Diffusion in an expanding universe • Diffusion around a black-hole, in an accretion disk,…. • H-theorem: One can construct out of any two distributions f and h a conditional entropy current Sf/h .Sf/h ≥ 0 in any Lorentzian space-time, even those with naked singularity and/or closed time-like curves Are these space-times physical after all?

19. Other Relativistic Stochastic Processes: ‘Intrinsic’ Brownian Motion • The diffusing particle is NOT surrounded by a fluid • Possible physical cause of diffusion: microscopic degrees of freedom of the space-time itself • In its proper frame, the equation of motion of the particle is at any proper time s: dp = D dBs (Dudley, 1965/67; Dowker, Henson, Sorkin, 2004; Franchi, Le Jan, 2004)

20. Other Relativistic Stochastic Processes: ‘Intrinsic’ Brownian Motion • The diffusion is at any (proper) time isotropic in the proper rest-frame of the particle • Main application, as of today: Diffusion in the vacuum Schwarzschild space-time • Main conclusion: The particle can enter the future Schwarzschild horizon and then escape the holeby crossing the past Schwarzschild horizon

21. Other Relativistic Stochastic Processes: The ‘Relativistic Brownian Motion’ of Haenggi and Dunkel (2004/5) • The particle is diffusing in an isotropic fluid • At any proper time, in the proper frame of the diffusing particle: dp = - a*(p)p ds + D dBs • The coefficient a*(p) is adjusted for the process to have the same equilibrium Jüttner distribution as the ROUP

22. Other Relativistic Stochastic Processes: The ‘Relativistic Brownian Motion’ of Haenggi and Dunkel (2004/5) • Main problem: The diffusion in an isotropic fluid is characterized by two tensors (a* and D), which are not isotropic in the proper rest frame of this fluid, but in the instantaneous and therefore time-dependent proper frame of the diffusing particle • No construction in curved space-time (yet?) • No application (yet?)

23. Other Relativistic Stochastic Processes: The ‘Relativistic Brownian Motion’ of Oron and Horwitz (2003) • Special Relativistic model with both time-like and space-like trajectories for the diffusing particle • Diffusion equation replaced by d’Alembert wave equation • No general relativistic extension, no application

24. Towards stochastic (classical) geometry: Mean field theory for General Relativity • Geometry is encoded in the metric g and the connection G • G = Levi-Civitta connection of g • g is linked to the stress-energy tensor T by Einstein’s equation • The whole theory is non-linear (Debbasch, Chevalier, Ollivier, Bustamante, 2003/4/5)

25. Towards stochastic (classical) geometry: Mean field theory for General Relativity • Statistical ensembles of general relativistic space-times: g(w), G(w), T(w) • Averaged motion of test matter = motion in the mean field  mean gravitational field described by: • Metric g (x) = < g (x, w) > • The Levi-Civitta connection of g, G < G >

26. Towards stochastic (classical) geometry: Mean field theory for General Relativity • The mean metric and connection define through Einstein’s equation the mean or apparent large-scale stress-energy tensor T  < T > • The separation between matter and gravitational field is scale-dependent • Similar effect on other gauge fields, which mix with charges upon averaging

27. Towards stochastic (classical) geometry: Mean field theory for General Relativity • In particular, a fluctuating vacuum space-time appears as filled with matter when observed on scales much larger than the fluctuation scales • Is this the origin of (part of the) dark energy? Original idea by Debbasch (2003) recently developed pertubatively by Kolb et al. (2005) • Non perturbative astrophysical application presented by C. Chevalier at the Einstein Symposium (Paris, 2005)

28. Stochastic (classical) geometry dg = ? dG = ? dT = ?

29. The Future • Relativistic classical diffusion: further applications • Relativistic quantum processes: under construction • Classical stochastic geometry: slow progress is being made • Quantum stochastic ‘geometry’: ?

30. Other Relativistic Stochastic Processes: The Lorentz invariant diffusion process of Dowker, Henson, Sorkin (2004) • Variant of the Haenggi-Dunkel process, interpreted as an ‘intrinsic’ Brownian motion (no fluid) • dp = - a0p ds + D dBs • No way of physically justifying the model (notably the dissipative term) since nothing is known of the ‘microphysics’ • No general relativistic construction