Models Hydrod. 2nd class Before Question BCRW

Second class particles can perform simplerandom walks (in some cases)

Joint withGyorgy Farkas, Peter Kovacs and Attila Rakos

Marton Balazs

Budapest University of Technology and Economics

BME Stochastics SeminarBudapest, October 22, 2009.

Models Hydrod. 2nd class Before Question BCRW

The modelsAsymmetric simple exclusion processZero range processGeneralized ZRPBricklayers processStationary distributions

Hydrodynamics

The second class particle

Earlier results

The question

Branching coalescing random walk

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Asymmetric simple exclusion

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • •

Particles try to jump

to the right with rate p,to the left with rate q = 1 − p < p.

The jump is suppressed if the destination site is occupied byanother particle.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •••

•

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •••

•

}

ω−3=2

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •••

•

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •••

•

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •••

•

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• ••• •

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• • •

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• • •

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• ••

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• ••

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• ••

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• ••

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• ••

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• ••

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• ••

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• • •

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• • •

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• • •

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• • •

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• • •

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• • •

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric zero range process

-3 -2 -1 0 1 2 3 4 i

•• •• • •

}

ω−3=2

Particles jump to the right from site i with rate r(ωi)

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

A generalized totally asymmetric zero range process:ωi ∈ Z

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi).

(r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

}

ωi+1=−1

(

ωi

ωi+1

)

||(

1−1

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

The totally asymmetric bricklayers process

i i + 1

(

ωi

ωi+1

)

→

(

ωi − 1ωi+1 + 1

)

|| ||(

1−1

)

→

(

00

)

a brick is added with rate r(ωi) + r(−ωi+1).

A mirror-symmetrized version of the extended zero range. Leftand right jumps of the dynamics cooperate, if(r(ω) · r(1 − ω) = 1; r non-decreasing).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Stationary product distributions

For the ASEP: the Bernoulli() distribution is time-stationary forany (0 ≤ ≤ 1).

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Stationary product distributions

For the ASEP: the Bernoulli() distribution is time-stationary forany (0 ≤ ≤ 1).

For zero range, bricklayers: the product of marginals

µθ(ωi) =eθωi

r(ωi)!·

1Z (θ)

is stationary for any θ ∈ R that makes Z (θ) finite.

Models Hydrod. 2nd class Before Question BCRW ASEP TAZRP TAGZRP TABLP Stati distr.

Stationary product distributions

For the ASEP: the Bernoulli() distribution is time-stationary forany (0 ≤ ≤ 1).

For zero range, bricklayers: the product of marginals

µθ(ωi) =eθωi

r(ωi)!·

1Z (θ)

is stationary for any θ ∈ R that makes Z (θ) finite.

Here r(0)! : = 1, and r(z + 1)! = r(z)! · r(z + 1) for all z ∈ Z.

Models Hydrod. 2nd class Before Question BCRW

Hydrodynamics (very briefly)

The density : = E(ω) and the hydrodynamic fluxH : = E[growth rate] both depend on a parameter or θ of thestationary distribution.

Models Hydrod. 2nd class Before Question BCRW

Hydrodynamics (very briefly)

The density : = E(ω) and the hydrodynamic fluxH : = E[growth rate] both depend on a parameter or θ of thestationary distribution.

◮ H() is the hydrodynamic flux function.

Models Hydrod. 2nd class Before Question BCRW

Hydrodynamics (very briefly)

The density : = E(ω) and the hydrodynamic fluxH : = E[growth rate] both depend on a parameter or θ of thestationary distribution.

◮ H() is the hydrodynamic flux function.◮ If the process is locally in equilibrium, but changes over

some large scale (variables X = εi and T = εt), then

∂T (T , X ) + ∂X H((T , X )) = 0 (conservation law).

Models Hydrod. 2nd class Before Question BCRW

Hydrodynamics (very briefly)

For the ASEP, H() = (p − q) · (1 − ), concave.

Models Hydrod. 2nd class Before Question BCRW

Hydrodynamics (very briefly)

For the ASEP, H() = (p − q) · (1 − ), concave.

For the zero range and bricklayers, H() is convex/concave ifthe rate function r is convex/concave.

Models Hydrod. 2nd class Before Question BCRW

Hydrodynamics (very briefly)

For the ASEP, H() = (p − q) · (1 − ), concave.

For the zero range and bricklayers, H() is convex/concave ifthe rate function r is convex/concave.

Special case: r(ω) = const. · eβω; H() is convex.

will be interested in TAGEZRP, TAEBLP.

Models Hydrod. 2nd class Before Question BCRW

Hydrodynamics (very briefly)

For the ASEP, H() = (p − q) · (1 − ), concave.

For the zero range and bricklayers, H() is convex/concave ifthe rate function r is convex/concave.

Special case: r(ω) = const. · eβω; H() is convex.

will be interested in TAGEZRP, TAEBLP.

Either convex or concave, discontinuous shock solutionsexist.

Models Hydrod. 2nd class Before Question BCRW

Hydrodynamics (very briefly)

For the ASEP, H() = (p − q) · (1 − ), concave.

For the zero range and bricklayers, H() is convex/concave ifthe rate function r is convex/concave.

Special case: r(ω) = const. · eβω; H() is convex.

will be interested in TAGEZRP, TAEBLP.

Either convex or concave, discontinuous shock solutionsexist. Let’s look for the corresponding microscopic structure.

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• •• ••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• •• ••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• •• ••

Growth on the right:rate≤rate

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• •• ••

Growth on the right:rate≤ratewith rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•••• •• ••

•••

Growth on the right:rate≤ratewith rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•••• •• ••

•••

Growth on the right:rate≤ratewith rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•••• •• ••

•••

Growth on the right:rate≤ratewith rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•••• • • ••

•••

Growth on the right:rate≤ratewith rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•••• • • ••

•••

Growth on the right:rate≤ratewith rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•••• • • •

•

•••

Growth on the right:rate≤ratewith rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•••• • ••

•

•••

Growth on the right:rate≤ratewith rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•••• • ••

•

•••

Growth on the right:rate≤ratewith rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•••• • ••

•

•••

Growth on the right:rate≤ratewith rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•••• • ••

•

•••

Growth on the right:rate≤ratewith rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• •• ••

Growth on the right:rate≤ratewith rate-rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• •• ••

Growth on the right:rate≤ratewith rate-rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• •• ••

Growth on the right:rate≤ratewith rate-rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• •• ••

Growth on the right:rate≤ratewith rate-rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• • • ••

Growth on the right:rate≤ratewith rate-rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• • • ••

Growth on the right:rate≤ratewith rate-rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• • • •

•

Growth on the right:rate≤ratewith rate-rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• • ••

•

Growth on the right:rate≤ratewith rate-rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• • ••

•

Growth on the right:rate≤ratewith rate-rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• • ••

•

Growth on the right:rate≤ratewith rate-rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

i

•• • •••• • ••

•

Growth on the right:rate≤ratewith rate-rate:

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥rate

i

•• • •••• •• ••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate:

i

•• • •••• •• ••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate:

i

• ••• ••• ••

• •

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate:

i

• ••• ••• ••

• •

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate:

i

• ••• ••• ••

• •

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate:

i

• ••• ••• ••

• •

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate:

i

• ••• ••• ••

• •

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate:

i

• ••• ••• •

•

• •

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate:

i

• ••• ••••

•

••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate:

i

• ••• ••••

•

••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate:

i

• ••• ••••

•

••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate:

i

• ••• ••••

•

••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate-rate:

i

•• • •••• •• ••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate-rate:

i

• •••• •• ••

• •

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate-rate:

i

• •••• •• ••

• •

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate-rate:

i

• •••• •• ••

• •

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate-rate:

i

• •••• •• ••

• •

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate-rate:

i

• •••• •• ••

• •

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate-rate:

i

• •••• •• ••

• •

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate-rate:

i

• •••• •• ••

••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate-rate:

i

• •••• •• ••

••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate-rate:

i

• •••• •• ••

••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate-rate:

i

• •••• •• ••

••

Models Hydrod. 2nd class Before Question BCRW

The second class particle

States ω and ω only differ at one site.

Growth on the left:rate≥ratewith rate-rate:

i

• •••• •• ••

••

A single discrepancy , the second class particle, is conserved.

Models Hydrod. 2nd class Before Question BCRW

Earlier results: as seen by the second class particle

From now on: ASEP, TAGEZRP, TAEBLP only; “E”=exponential.

Theorem (Derrida, Lebowitz, Speer ’97)For the ASEP, the Bernoulli product distribution with densities

-3 -2 -1 0 1 2 3

left 0

right

i

i

is stationary for the process, as seen from the second class particle, if

right · (1 − left)

left · (1 − right)=

pq

.

Models Hydrod. 2nd class Before Question BCRW

Earlier results: as seen by the second class particle

From now on: ASEP, TAGEZRP, TAEBLP only; “E”=exponential.

Theorem (Derrida, Lebowitz, Speer ’97)For the ASEP, the Bernoulli product distribution with densities

-3 -2 -1 0 1 2 3

left 0

1right

i

i

is stationary for the process, as seen from the second class particle, if

right · (1 − left)

left · (1 − right)=

pq

.

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Theorem (Belitsky and Schutz ’02)For the ASEP with the very same parameters, the Bernoulliproduct distribution µ0 with densities

-3 -2 -1 0 1 2 3

left

right

i

i

evolves according to

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Theorem (Belitsky and Schutz ’02)For the ASEP with the very same parameters, the Bernoulliproduct distribution µ0 with densities

-3 -2 -1 0 1 2 3

left

right

i

i

evolves according to

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0].

Multiple shocks and their interactions are also handled.

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate p · leftright

:

-3 -2 -1 0 1 2 3

left

right

i

i

X (t) = 0µ0

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0]

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate p · leftright

:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0]

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate p · leftright

:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0]

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate p · leftright

:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0]

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate p · leftright

:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0]

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate p · leftright

:

-3 -2 -1 0 1 2 3

left

right

i

i

X (t) = −1µ−1

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0]

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate q ·rightleft

:

-3 -2 -1 0 1 2 3

left

right

i

i

X (t) = 0µ0

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0]

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate q ·rightleft

:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0]

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate q ·rightleft

:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0]

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate q ·rightleft

:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0]

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate q ·rightleft

:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0]

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate q ·rightleft

:

-3 -2 -1 0 1 2 3

left

right

i

i

X (t) = 1µ1

ddt

µ0 = p ·left

right· [µ

−1 − µ0] + q ·right

left· [µ1 − µ0]

Models Hydrod. 2nd class Before Question BCRW

Earlier results: as seen by the second class particle

Theorem (B. ’01)For the TAEBLP, the product distribution of marginals µi withdensities

-3 -2 -1 0 1 2 3

left

right

i

i

is stationary for the process, as seen from the second classparticle, if

left − right = 1.

Models Hydrod. 2nd class Before Question BCRW

Earlier results: as seen by the second class particle

Theorem (B. ’01)For the TAEBLP, the product distribution of marginals µi withdensities

-3 -2 -1 0 1 2 3

left

right

i

i

is stationary for the process, as seen from the second classparticle, if

left − right = 1.

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Theorem (B. ’04)For the very same parameters, the product distribution µ0 withdensities

-3 -2 -1 0 1 2 3

left

right

i

i

evolves according to

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Theorem (B. ’04)For the very same parameters, the product distribution µ0 withdensities

-3 -2 -1 0 1 2 3

left

right

i

i

evolves according to

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Multiple shocks and their interactions are also handled.

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate Cleft:

-3 -2 -1 0 1 2 3

left

right

i

i

X (t) = 0µ0

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate Cleft:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate Cleft:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate Cleft:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate Cleft:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate Cleft:

-3 -2 -1 0 1 2 3

left

right

i

i

X (t) = −1µ−1

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate Cright:

-3 -2 -1 0 1 2 3

left

right

i

i

X (t) = 0µ0

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate Cright:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate Cright:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate Cright:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate Cright:

-3 -2 -1 0 1 2 3

left

right

i

i

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate Cright:

-3 -2 -1 0 1 2 3

left

right

i

i

X (t) = 1µ1

ddt

µ0 = Cleft · [µ−1 − µ0] + Cright · [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

The question

Of course, the drift of the walk X (t) is the same as theexpected drift of the second class particle in its stationaryshock distribution,

Models Hydrod. 2nd class Before Question BCRW

The question

Of course, the drift of the walk X (t) is the same as theexpected drift of the second class particle in its stationaryshock distribution,

and also agrees with the Rankine Hugoniot formula for thespeed of shocks.

Models Hydrod. 2nd class Before Question BCRW

The question

Of course, the drift of the walk X (t) is the same as theexpected drift of the second class particle in its stationaryshock distribution,

and also agrees with the Rankine Hugoniot formula for thespeed of shocks.

Is it the second class particle that performs the simple randomwalk in the middle of a shock?

Models Hydrod. 2nd class Before Question BCRW

The question

Of course, the drift of the walk X (t) is the same as theexpected drift of the second class particle in its stationaryshock distribution,

and also agrees with the Rankine Hugoniot formula for thespeed of shocks.

Is it the second class particle that performs the simple randomwalk in the middle of a shock?

In what sense? Annealed w.r.t. the initial shock distribution...But what does this mean?

Models Hydrod. 2nd class Before Question BCRW

Here is the question:

For the ASEP, let ν0 be the Bernoulli product distribution

ν0 =(

⊗

i<0

µleft

)

⊗(

δ)

⊗(

⊗

i>0

µright

)

,

where

µ(ω = ω) =

{

, if ω = 1,

1 − , if ω = 0;δ(0, 1) = 1.

-3 -2 -1 0 1 2 3

left 0

1right

i

i

Models Hydrod. 2nd class Before Question BCRW

Here is the question:

For the ASEP, let ν0 be the Bernoulli product distribution

-3 -2 -1 0 1 2 3

left 0

1right

i

i

Does it satisfy

ddt

ν0 = p ·left

right· [ν

−1 − ν0] + q ·right

left· [ν1 − ν0]

whenright · (1 − left)

left · (1 − right)=

pq

?

Models Hydrod. 2nd class Before Question BCRW

Here is the question:

For the TAEBLP, let ν0 be the product distribution

ν0 =(

⊗

i<0

µleft

)

⊗(

δright

)

⊗(

⊗

i>0

µright

)

,

where

µ(ω = ω) =eθ()·ω

r(ω)!·

1Z (θ())

;

δ(ω, ω + 1) =eθ()·ω

r(ω)!·

1Z (θ())

.

-3 -2 -1 0 1 2 3

left

right

i

i

Models Hydrod. 2nd class Before Question BCRW

Here is the question:For the TAEBLP, let ν0 be the product distribution

ν0 =(

⊗

i<0

µleft

)

⊗(

δright

)

⊗(

⊗

i>0

µright

)

,

-3 -2 -1 0 1 2 3

left

right

i

i

Does it satisfy

ddt

ν0 = Cleft · [ν−1 − ν0] + Cright · [ν1 − ν0].

whenleft − right = 1 ?

Models Hydrod. 2nd class Before Question BCRW

Here is the answer:

Theorem (Gy. Farkas, P. Kovacs, A. Rakos, B. ’09)Yes, and yes. Even more, the thing also works for theTAGEZRP.

Models Hydrod. 2nd class Before Question BCRW

Here is the answer:

Theorem (Gy. Farkas, P. Kovacs, A. Rakos, B. ’09)Yes, and yes. Even more, the thing also works for theTAGEZRP.

The second class particle, annealed w.r.t. the initial shockproduct distribution, does perform a drifted simple random walkin these cases.

Models Hydrod. 2nd class Before Question BCRW

Here is the answer:

Theorem (Gy. Farkas, P. Kovacs, A. Rakos, B. ’09)Yes, and yes. Even more, the thing also works for theTAGEZRP.

The second class particle, annealed w.r.t. the initial shockproduct distribution, does perform a drifted simple random walkin these cases.

This explains both types of the previous results.

Models Hydrod. 2nd class Before Question BCRW

Here is the answer:

Theorem (Gy. Farkas, P. Kovacs, A. Rakos, B. ’09)Yes, and yes. Even more, the thing also works for theTAGEZRP.

The second class particle, annealed w.r.t. the initial shockproduct distribution, does perform a drifted simple random walkin these cases.

This explains both types of the previous results.

The presence of a second class particle in the measuresignificantly simplifies the computations. This is how wediscovered the TAGEZRP.

Models Hydrod. 2nd class Before Question BCRW

Nice, since

This might open up the path for applying methods physicistslike (e.g. Bethe Ansatz).

Models Hydrod. 2nd class Before Question BCRW

Nice, since

This might open up the path for applying methods physicistslike (e.g. Bethe Ansatz).

It also gives a rough tail bound for the second class particle in aflat initial distribution; essential in the t2/3 proofs for theexponential models.

Models Hydrod. 2nd class Before Question BCRW

Interactions:

We also see that shocks+second class particles◮ locally interact by exclusion in ASEP, and don’t locally

interact in TAGEZRP, TAEBLP, but

Models Hydrod. 2nd class Before Question BCRW

Interactions:

We also see that shocks+second class particles◮ locally interact by exclusion in ASEP, and don’t locally

interact in TAGEZRP, TAEBLP, but

◮ their jump rates depend on their position, making themattract each other such that

Models Hydrod. 2nd class Before Question BCRW

Interactions:

We also see that shocks+second class particles◮ locally interact by exclusion in ASEP, and don’t locally

interact in TAGEZRP, TAEBLP, but

◮ their jump rates depend on their position, making themattract each other such that

◮ their center of mass has the Rankine-Hugoniot velocity forthe large shock they jointly represent.

Models Hydrod. 2nd class Before Question BCRW

Interactions:

We also see that shocks+second class particles◮ locally interact by exclusion in ASEP, and don’t locally

interact in TAGEZRP, TAEBLP, but

◮ their jump rates depend on their position, making themattract each other such that

◮ their center of mass has the Rankine-Hugoniot velocity forthe large shock they jointly represent.

Macroscopically it’s one shock after all.

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate p: jump to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate p: jump to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate p: jump to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate p: jump to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate p: jump to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate p: jump to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate p: jump to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate p: jump to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate p: jump to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate p: jump to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate q: jump to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate q: jump to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate q: jump to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate q: jump to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate q: jump to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate q: jump to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate q: jump to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate q: jump to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate q: jump to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate q: jump to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cr : coalescence to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cr : coalescence to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cr : coalescence to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cr : coalescence to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cr : coalescence to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cr : coalescence to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cr : coalescence to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cr : coalescence to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cr : coalescence to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cr : coalescence to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate bl : branching to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate bl : branching to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate bl : branching to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate bl : branching to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate bl : branching to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate bl : branching to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate bl : branching to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate bl : branching to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate bl : branching to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate bl : branching to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cl : coalescence to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cl : coalescence to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cl : coalescence to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cl : coalescence to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cl : coalescence to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cl : coalescence to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cl : coalescence to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cl : coalescence to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cl : coalescence to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate cl : coalescence to the left

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate br : branching to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate br : branching to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate br : branching to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate br : branching to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate br : branching to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate br : branching to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate br : branching to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate br : branching to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate br : branching to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate br : branching to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate br : branching to the right

Models Hydrod. 2nd class Before Question BCRW

A similar result: branching coalescing random walk

-3 -2 -1 0 1 2 3 4 i

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• ••

With rate br : branching to the right

The Bernoulli(∗) distribution is stationary for

∗ =bl + br

bl + br + cl + cr.

Models Hydrod. 2nd class Before Question BCRW

Earlier results: as seen by the rightmost particle

TheoremFor the BCRW, the Bernoulli product distribution with densities

-3 -2 -1 0 1 2 3

∗

= 0

i

i

is stationary for the process, as seen from the rightmostparticle.

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Theorem (Krebs, Jafarpour and Schutz ’03)For the BCRW with the very same parameters, the Bernoulliproduct distribution µ0 with densities

-3 -2 -1 0 1 2 3

∗

= 0

i

i

evolves according to

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate cl ·(bl+br )+q·(cl+cr )bl+br +cl+cr

:

-3 -2 -1 0 1 2 3

∗

= 0

i

i

X (t) = 0µ0

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate cl ·(bl+br )+q·(cl+cr )bl+br +cl+cr

:

-3 -2 -1 0 1 2 3

∗

= 0

i

i

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate cl ·(bl+br )+q·(cl+cr )bl+br +cl+cr

:

-3 -2 -1 0 1 2 3

∗

= 0

i

i

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate cl ·(bl+br )+q·(cl+cr )bl+br +cl+cr

:

-3 -2 -1 0 1 2 3

∗

= 0

i

i

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate cl ·(bl+br )+q·(cl+cr )bl+br +cl+cr

:

-3 -2 -1 0 1 2 3

∗

= 0

i

i

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate cl ·(bl+br )+q·(cl+cr )bl+br +cl+cr

:

-3 -2 -1 0 1 2 3

∗

= 0

i

i

X (t) = −1µ−1

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate p · bl+br +cl+crcl+cr

:

-3 -2 -1 0 1 2 3

∗

= 0

i

i

X (t) = 0µ0

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate p · bl+br +cl+crcl+cr

:

-3 -2 -1 0 1 2 3

∗

= 0

i

i

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate p · bl+br +cl+crcl+cr

:

-3 -2 -1 0 1 2 3

∗

= 0

i

i

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate p · bl+br +cl+crcl+cr

:

-3 -2 -1 0 1 2 3

∗

= 0

i

i

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate p · bl+br +cl+crcl+cr

:

-3 -2 -1 0 1 2 3

∗

= 0

i

i

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

Earlier results: random walking shocks

Interpretation: random walking shock µ(t) = µX(t):

with rate p · bl+br +cl+crcl+cr

:

-3 -2 -1 0 1 2 3

∗

= 0

i

i

X (t) = 1µ1

ddt

µ0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [µ

−1 − µ0]

+ p ·bl + br + cl + cr

cl + cr· [µ1 − µ0].

Models Hydrod. 2nd class Before Question BCRW

The question:

Is it the rightmost particle that performs the random walk?

Models Hydrod. 2nd class Before Question BCRW

Here is the question:For the BCRW, let ν0 be the Bernoulli product distribution

ν0 =(

⊗

i<0

µ∗)

⊗(

δ)

⊗(

⊗

i>0

µ0)

,

where δ(0) = 1.

-3 -2 -1 0 1 2 3

∗

= 0

i

i

1

Does it satisfy

ddt

ν0 =cl · (bl + br ) + q · (cl + cr )

bl + br + cl + cr· [ν

−1 − ν0]

+ p ·bl + br + cl + cr

cl + cr· [ν1 − ν0]?

Models Hydrod. 2nd class Before Question BCRW

The answer

◮ ... is, of course, yes again. [Gy. Farkas, P. Kovacs, A.Rakos, B. ’09]

Models Hydrod. 2nd class Before Question BCRW

The answer

◮ ... is, of course, yes again. [Gy. Farkas, P. Kovacs, A.Rakos, B. ’09]

◮ Fronts of the other direction: 0 − 1 − ∗ can also behandled.

Models Hydrod. 2nd class Before Question BCRW

Thank you.