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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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**Relativistic Brownian Motion**ISMD 07 Berkeley**Relativistic Brownian Motion**G.S. Denicol, T. Koide and T.K. Univ. of Rio de Janeiro ISMD 07 Berkeley**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**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**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**dN/dp3**ISMD 07 Berkeley**E x dN/dp3**ISMD 07 Berkeley**Invariant Phase Volume**ISMD 07 Berkeley**For**ISMD 07 Berkeley**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**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**Covariant Cascade Single Particle Spectrum**dN/dp3 Note! Without 1/E ISMD 07 Berkeley**What is the difference?**• Integral Measure vs. ISMD 07 Berkeley**Langevin Approach**Many works for low and medium energy heavy ion physics J. Randrup, Csernai+Kapusta,… ISMD 07 Berkeley**Langevin Approach**Observe the trajectory of a particle in the rest frame of the bath ISMD 07 Berkeley**Langevin Approach**dt ISMD 07 Berkeley**1-D case for simplicity**Average over events for the same initial condition ISMD 07 Berkeley**1-D case for simplicity**Average momentum transfer Fluctuation around the average ISMD 07 Berkeley**1-D case for simplicity**ISMD 07 Berkeley**1-D case for simplicity**Gaussian random noise Average momentum transfer per unit time ISMD 07 Berkeley**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**Ito scheme:**dt ISMD 07 Berkeley**Stratonovich-Fisk scheme:**dt ISMD 07 Berkeley**Hänggi scheme:**dt ISMD 07 Berkeley**Fokker-Planck Equation**Sum over initial distribution Average over ensemble of events ISMD 07 Berkeley**Fokker-Planck Equation**where Ito Stratonovich-Fisk Hänggi-Klimontovich ISMD 07 Berkeley**Equilibrium distribution**• Homogeneous and static ISMD 07 Berkeley**Equilibrium distribution**Homogeneous and static ISMD 07 Berkeley**Equilibrium distribution**ISMD 07 Berkeley**Ito scheme: a=0**Stratonovich-Fisk scheme: a=1/2 Stratonovich-Fisk scheme: a=1 ISMD 07 Berkeley**Lorentz boost of the bath**Non covariance of noise and viscous term On-mass-shell requirement ISMD 07 Berkeley**Notation:**With * : in the rest frame of the bath Without * : in the observational frame where the bath is boosted by b ISMD 07 Berkeley**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**Requirement for noise transformation**In general, but where ISMD 07 Berkeley**Covariance of equilibrium distribution**If we require: For Hänggi scheme a = 1 ISMD 07 Berkeley**Covariance of equilibrium distribution**If we require: For Ito scheme a = 0 ISMD 07 Berkeley**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**PERSPECTIVES ….**ISMD 07 Berkeley

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