F. DebbaschLERMA-ERGA

Université Paris 6

Relativistic Stochastic Dynamics:A Review

1. Why is the construction of relativistic stochastic dynamics a problem?

2. Why is the construction of relativistic stochastic dynamics interesting?

3. The Relativistic Ornstein-Uhlenbeck Process (ROUP)

4. Special relativistic applications

5. General relativistic applications

6. Other relativistic stochastic processes

7. Towards stochastic geometry

• The basic tool of Galilean stochastic models is Brownian motion

• If dxt = dBt/

then

∂t n = n with =

Why is the Construction ofRelativistic Stochastic Dynamics

a Problem?

€

2

• If n (t = 0, x) = (x), then, for all t > 0,

n (t, x) = G(t, x) ~ exp ( - x2/4 t)

Faster than light particle (mass, energy)

transfer

Why is the Construction ofRelativistic Stochastic Dynamics

Interesting? • General theoretical interest

• Practical problems involving special or general relativistic diffusions: Plasma Physics, Astrophysics, Cosmology, …

• Toy model of relativistic irreversible behavior

• Stochastic geometry

Standard Relativistic Fluid Models

Relativistic Boltzman Equation

First order hydrodynamics(relativistic Navier-Stokes

model, …)

Extended Thermodynamics

Chapman-Enskog expansion Grad expansion(No small parameter)

Causal

Non-causal

Causal, but contradicted by experiments

The Relativistic Ornstein-Uhlenbeck Process

• 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

(Debbasch, Mallick, Rivet, 1997)

The Relativistic Ornstein-Uhlenbeck Process

dxt =p

mdt

dpt = - p dt +D dBt

1

(p)

(p)

p2

where (p) = (1 +m2c2

)1/2

In flat space-time, for uniform and constant fluid temperature and velocity fields:

Idea: Brownian motion in momentum space

in the rest frame of the fluid

The relativistic Ornstein-Uhlenbeck Process: Alternative definition via

the transport equationIn flat space-time, for uniform and constant fluid temperature and velocity fields

In the rest frame of the fluid, dn = (t, x, p) d3x d3p and (t, x, p) verifies the forward Kolmogorov equation:

∂t + ∂x( ) + ∂p(-(p)p ) = pp

m (p)

D2

2

The Relativistic Ornstein-Uhlenbeck Process:

Fluctuation-Dissipation Theorem

The coefficients (p) and D are constrained by imposing that the Jüttner distribution J(p) at temperature ,

J (p) ~ exp(- (p) mc2), = 1/(kB ),

is a solution of the transport equation

and(p) = 0/(p) D/0 = m kB

The Relativistic Ornstein-Uhlenbeck process: General Transport Equation

• Manifestly covariant formalism Extended phase space with (x, p) as coordinates

(t, x, p) • Fluid characterized by U(x), 0(x) and D(x)• Basic objects:

1. Derivative with respect to p at constant x: ∂p

2. Derivative with respect to x at p covariantly constant: D = + p ∂p

3. Projector (x) = g(x) - UxUx

f(x, p)

(Barbachoux, Debbasch, Rivet, 2001, 2004)

The Relativistic Ornstein-Uhlenbeck process: General Transport Equation

• L(f) = D (g(x)pf) + ∂p(mc F(x, p)f) +

N(f)

• F(x, p) =

(x, p) =

Kolmogorov equation reads L(f) = 0 with

1

m2c2 (x, p) pp p- g(x) pp (x)p

0(x)mc

p.U(x)

2

(x)

The Relativistic Ornstein-Uhlenbeck Process:General Transport Equation

• N(f) = K (x) ∂p

• K(x) = UU - UU +

pp

p.U(x)∂p

f

UU - UU

D (x)2

2

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)

Near-equilibrium, large scale Special Relativistic Diffusion

• Microscopic time-scale = 1/0

• Microscopic length-scale = vth() • Density n varies on characteristic scales T and L, /T = O(), /L = O()

• Then = 2

• ∂t n = n with =2

: APPARENT PARADOX!

Near-equilibrium, Large-Scale Special Relativistic Diffusion:

Paradox Resolved• Green function G(t, x) of the diffusion equation:

G(0, x) = (x)• The conditions /T = O(), /L = O() applied

to G(t, x) lead to:

i. t >> ii. x /t << c

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

General Relativistic Applications

• Diffusion in an expanding universe• Diffusion around a black-hole, in an accretion

disk,….• H-theorem:

(Rigotti, Debbasch, 2004, 2005)

i. One can construct out of any two distributions f and h a conditional entropy current Sf/h

ii. .Sf/h ≥ 0 in any Lorentzian space-time, even those with naked singularity and/or closed time-like curves

iii. Are these space-times physical after all?

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)

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 hole by crossing the past Schwarzschild horizon

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 = - (p) p ds + D dBs

• The coefficient (p) is adjusted for the process to have the same equilibrium Jüttner distribution as the ROUP

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 ( * 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?)

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

Towards stochastic (classical) geometry: Mean field theory for

General Relativity

• Geometry is encoded in the metric g and the connection

• = 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)

Towards stochastic (classical) geometry: Mean field theory for

General Relativity

• Statistical ensembles of general relativistic space-times: g(), (), T()

• Averaged motion of test matter = motion in the mean field mean gravitational field described by:

• Metric g (x) = < g (x, ) >

• The Levi-Civitta connection of g, < >

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

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)

Stochastic (classical) geometry

dg = ?

d = ?

dT = ?

The Future

• Relativistic classical diffusion: further applications

• Relativistic quantum processes: under construction

• Classical stochastic geometry: slow progress is being made

• Quantum stochastic ‘geometry’: ?

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 = - 0 p 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