F. Debbasch

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Relativistic Stochastic DynamicsA Review

• F. Debbasch
• LERMA-ERGA
• Université Paris 6

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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

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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 c D n with c

l2
2 t
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• If n (t 0, x) d (x), then, for all t gt 0,
• n (t, x) G(t, x) exp ( - x2/4 c t)
• ?
• Faster than light particle (mass, energy)
• transfer

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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

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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
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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

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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
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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)p P)
DpP
m g(p)
2
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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(- b g(p) mc2), b
1/(kB q), is a solution of the transport
equation
a(p) a0/g(p)
D/a0 m kBq
and
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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
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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) papb lmn(x)pn
m2c2
2
mc
a0(x)
Dmn (x)
p.U(x)
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The Relativistic Ornstein-Uhlenbeck
ProcessGeneral Transport Equation
2
D (x)
pmpb

• N(f) Kmrbn (x) ?pr
• Kmrbn(x) UmUbDrn - UmUnDrb

?pn f
2
p.U(x)
UrUnDmb - UrUbDmn
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Special Relativistic ApplicationsNear-equilibriu
m, large-scale Diffusion
(Debbasch, Rivet, 1998)

• 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)

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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 c D n with ?

l2
APPARENT PARADOX!
2t
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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

1. t gtgt t
2. x /t ltlt c

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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

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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?

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Other Relativistic Stochastic Processes
Intrinsic Brownian Motion
(Dudley, 1965/67 Dowker, Henson, Sorkin,
2004 Franchi, Le Jan, 2004)

• 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

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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

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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

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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?)

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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 dAlembert wave
  equation
• No general relativistic extension, no application

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Towards stochastic (classical) geometry Mean
field theory for General Relativity
(Debbasch, Chevalier, Ollivier, Bustamante,
2003/4/5)

• 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
  Einsteins equation
• The whole theory is non-linear

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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) lt g (x, w) gt
• The Levi-Civitta connection of g, G? lt G gt

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Towards stochastic (classical) geometry Mean
field theory for General Relativity

• The mean metric and connection define through
  Einsteins equation the mean or apparent
  large-scale stress-energy tensor
• T ? lt T gt
• The separation between matter and gravitational
  field is scale-dependent
• Similar effect on other gauge fields, which mix
  with charges upon averaging

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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)

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Stochastic (classical) geometry
dg ? dG ? dT ?
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The Future

• Relativistic classical diffusion further
  applications
• Relativistic quantum processes under
  construction
• Classical stochastic geometry slow progress is
  being made
• Quantum stochastic geometry ?

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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 - a0 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