# Presentation: 380706

## Slide 1

## 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**
** ****Can we use to estimate viscosity, relaxation time, etc**
**   ****in relativistic causal hydrodynamics?**

## Relativistic Brownian Motion

- Invariant phase volume vs. Boltzmann Dist.
- Langevin Equation approach
- Multiplicative noise and discretization scheme
- Jüttner distribution vs. Invariant Phase volume distribution
- Perspectives

**Outline**

## Covariant cascade in momentum space

** ****Consider *****N***** particles with four momenta **
**            ****{*****p******i*****,***** ******i=1,..,N*****}**
**2.  Choose randomly a pair *****(i,j) ***
**3.  Perform elastic collision *****p******i******+p******j******= p’******i******+p’******j***
**4.  Repeat the procedure (back to 2).**

**After a large number of collisions, the system **
**eventually thermalizes.**

## dN/dp3

***dN/dp******3***

## E x dN/dp3

***E ***x*** dN/dp******3***

## Invariant Phase Volume

## For

## On the other hand, thermal Distribution (Boltzmann, or good-old Cooper-Frye) says:

**Note no 1/E factor.**
**Phase Volume Spectrum **
**                ********   ****MicroCanonical Ensemble ??**

**dx dp  is still a covariant measure**

## 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**

## dN/dp3

***dN/dp******3***

**Covariant Cascade Single Particle Spectrum**

**Note! Without 1/E**

## What is the difference?

- Integral Measure

**vs.**

## Langevin Approach

**Many works for low and medium energy heavy ion physics**
**J. Randrup, Csernai+Kapusta,... **

## Langevin Approach

**Observe the trajectory of a particle in the rest frame of the bath**

## Langevin Approach

**dt**

## 1-D case for simplicity

**1-D case for simplicity**

**Average over events for the same initial condition**

## 1-D case for simplicity

**1-D case for simplicity**

**Average momentum transfer**

**Fluctuation around the average**

## 1-D case for simplicity

**1-D case for simplicity**

## 1-D case for simplicity

**1-D case for simplicity**

**Average momentum transfer per unit time**

**Gaussian random noise**

## Slide 22

## Dependence on Noise Interpretation

- Ito scheme
- Stratonovich-Fisk scheme
- Hänggi-Klimontovich (Phys. Usp. 37, 737,1994), Helv. Phys. Acta 51, 183 (1978)

## Ito scheme:

**Ito scheme:**

**dt**

## Stratonovich-Fisk  scheme:

**Stratonovich-Fisk  scheme:**

**dt**

## Hänggi  scheme:

**Hänggi  scheme:**

**dt**

## Fokker-Planck Equation

**Sum over initial distribution**

**Average over ensemble of events**

## Fokker-Planck Equation

- where

**Ito**

**Stratonovich-Fisk**

**Hänggi-Klimontovich**

## Equilibrium distribution

- Homogeneous and static

## Equilibrium distribution 

- Homogeneous and static

## Equilibrium distribution 

## Ito scheme: =0

- Stratonovich-Fisk scheme: =1/2

- Stratonovich-Fisk scheme: =1

## Slide 33

## Lorentz boost of the bath

- Non covariance of noise and viscous term
- On-mass-shell requirement

## Notation:

**With *********  : in the rest frame of the bath**

**Without ********* : in the observational frame where the bath is boosted by ******

## 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***

## Requirement for noise transformation

**In general,**

**but**

**where**

## Covariance of equilibrium distribution

**If we require:**

**For Hänggi scheme ******** = 1**

## Covariance of equilibrium distribution

**If we require:**

**For Ito scheme ******** = 0**

## 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).

## PERSPECTIVES ....

**PERSPECTIVES ....**