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Relativistic Brownian Motion

Relativistic Brownian Motion. Relativistic Brownian Motion. G.S. Denicol, T. Koide and T.K. Univ. of Rio de Janeiro. Relativistic Brownian Motion. G.S. Denicol, T. Koide and T.K. Univ. of Rio de Janeiro. Non equilibrium effects in heavy ion physics

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Relativistic Brownian Motion

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  1. ISMD 07 Berkeley

  2. Relativistic Brownian Motion ISMD 07 Berkeley

  3. Relativistic Brownian Motion G.S. Denicol, T. Koide and T.K. Univ. of Rio de Janeiro ISMD 07 Berkeley

  4. Relativistic Brownian Motion G.S. Denicol, T. Koide and T.K. Univ. of Rio de Janeiro • Non equilibrium effects in heavy ion physics • Can we use to estimate viscosity, relaxation time, etc • in relativistic causal hydrodynamics? ISMD 07 Berkeley

  5. Relativistic Brownian Motion Outline • Invariant phase volume vs. Boltzmann Dist. • Langevin Equation approach • Multiplicative noise and discretization scheme • Jüttner distribution vs. Invariant Phase volume distribution • Perspectives ISMD 07 Berkeley

  6. Covariant cascade in momentum space • Consider N particles with four momenta • {pi,i=1,..,N} • 2. Choose randomly a pair (i,j) • 3. Perform elastic collision pi+pj= p’i+p’j • 4. Repeat the procedure (back to 2). After a large number of collisions, the system eventually thermalizes. ISMD 07 Berkeley

  7. dN/dp3 ISMD 07 Berkeley

  8. E x dN/dp3 ISMD 07 Berkeley

  9. Invariant Phase Volume ISMD 07 Berkeley

  10. For ISMD 07 Berkeley

  11. On the other hand, thermal Distribution (Boltzmann, or good-old Cooper-Frye) says: Note no 1/E factor. Phase Volume Spectrum ¹ MicroCanonical Ensemble ?? dx dp is still a covariant measure ISMD 07 Berkeley

  12. How to recover the statistical spectrum? Space-time (covariant) cascade calculation • Collision Criteria • Covariant Impact parameter - T.K. et al, Phys. Rev. C29 (1984) • Time ordering of collision events ISMD 07 Berkeley

  13. Covariant Cascade Single Particle Spectrum dN/dp3 Note! Without 1/E ISMD 07 Berkeley

  14. What is the difference? • Integral Measure vs. ISMD 07 Berkeley

  15. Langevin Approach Many works for low and medium energy heavy ion physics J. Randrup, Csernai+Kapusta,… ISMD 07 Berkeley

  16. Langevin Approach Observe the trajectory of a particle in the rest frame of the bath ISMD 07 Berkeley

  17. Langevin Approach dt ISMD 07 Berkeley

  18. 1-D case for simplicity Average over events for the same initial condition ISMD 07 Berkeley

  19. 1-D case for simplicity Average momentum transfer Fluctuation around the average ISMD 07 Berkeley

  20. 1-D case for simplicity ISMD 07 Berkeley

  21. 1-D case for simplicity Gaussian random noise Average momentum transfer per unit time ISMD 07 Berkeley

  22. ISMD 07 Berkeley

  23. Dependence on Noise Interpretation • Ito scheme • Stratonovich-Fisk scheme • Hänggi-Klimontovich (Phys. Usp. 37, 737,1994), Helv. Phys. Acta 51, 183 (1978) ISMD 07 Berkeley

  24. Ito scheme: dt ISMD 07 Berkeley

  25. Stratonovich-Fisk scheme: dt ISMD 07 Berkeley

  26. Hänggi scheme: dt ISMD 07 Berkeley

  27. Fokker-Planck Equation Sum over initial distribution Average over ensemble of events ISMD 07 Berkeley

  28. Fokker-Planck Equation where Ito Stratonovich-Fisk Hänggi-Klimontovich ISMD 07 Berkeley

  29. Equilibrium distribution • Homogeneous and static ISMD 07 Berkeley

  30. Equilibrium distribution Homogeneous and static ISMD 07 Berkeley

  31. Equilibrium distribution ISMD 07 Berkeley

  32. Ito scheme: a=0 Stratonovich-Fisk scheme: a=1/2 Stratonovich-Fisk scheme: a=1 ISMD 07 Berkeley

  33. ISMD 07 Berkeley

  34. Lorentz boost of the bath Non covariance of noise and viscous term On-mass-shell requirement ISMD 07 Berkeley

  35. Notation: With * : in the rest frame of the bath Without * : in the observational frame where the bath is boosted by b ISMD 07 Berkeley

  36. Noise term Ensemble average in the rest frame of the bath at t Ensemble average in the boosted frame at t const Due to an finite time span dt ISMD 07 Berkeley

  37. Requirement for noise transformation In general, but where ISMD 07 Berkeley

  38. Covariance of equilibrium distribution If we require: For Hänggi scheme a = 1 ISMD 07 Berkeley

  39. Covariance of equilibrium distribution If we require: For Ito scheme a = 0 ISMD 07 Berkeley

  40. Conclusion • For relativistic Brownian motion, we have intrinsically multiplicative noise • Due to the finite time span dt (coarse graining), the Lorentz transformation properties of Langevin equation is not trivial. • Depending on the interaction scheme, we have to be careful for the descretization scheme. • Cascade type process -> existence of a natural integration scheme. • We expect that such an approach will clarify non-equilibrium effects in hydrodynamic approach (which also requires a finite size coarse graining in space-time). ISMD 07 Berkeley

  41. PERSPECTIVES …. ISMD 07 Berkeley

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