Statistical mechanics, structure of matter Nonequilibrium statistical mechanics Example

Totally asymmetric simple exclusion process

Builds on Asymmetric simple exclusion process Interacting particle system

Example

The totally asymmetric simple exclusion process (TASEP) on $\Z$ is the asymmetric simple exclusion process in which particles jump only to the right. A configuration is $\eta \in \set{0, 1}^{\Z}$ , where $\eta(x) = 1$ means that site $x$ holds a particle. Each particle carries an independent rate-one exponential clock. When it rings, the particle at $x$ moves to $x + 1$ if that site is empty, and otherwise stays. Its generator, on functions $f$ that depend on finitely many coordinates, is $Lf(\eta) = \sum_{x \in \Z} \eta(x)\big(1 - \eta(x + 1)\big)\big(f(\eta^{x, x+1}) - f(\eta)\big),$ where $\eta^{x, x+1}$ exchanges the values at $x$ and $x + 1$ . This is an interacting particle system: the ASEP with $p = 1$ and $q = 0$ .

Remarks

For each density $\rho \in [0, 1]$ , the Bernoulli product measure of density $\rho$ is invariant. In that state the mean current across a bond is $\rho(1 - \rho)$ , the rate of a jump times the probability that the site is occupied and its right neighbour empty. On the hydrodynamic scale the density solves the inviscid Burgers equation $\partial_t \rho + \partial_x\big(\rho(1 - \rho)\big) = 0$ . Start from the step initial condition, with all sites $x \le 0$ occupied and all sites $x > 0$ empty. Rost (1981) proved that the macroscopic density at time $t$ is the rarefaction fan $\rho = \tfrac12(1 - x/t)$ for $\abs{x} \le t$ . For the same initial condition, Johansson (2000) proved that the number of particles that have crossed the origin has fluctuations of order $t^{1/3}$ with the Tracy–Widom GUE law. This makes TASEP a basic exactly solvable model of the Kardar–Parisi–Zhang universality class.

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Asymmetric simple exclusion process example
Interacting particle system concept
Totally asymmetric simple exclusion process example

Tag	`1YZQ7H0THE`
URI	`m2://1YZQ7H0THE`
Kind	example
Writing	W0 stub
MSC	82C22, 60K35, 82C23
Instance of	Interacting particle system
Uses	Asymmetric simple exclusion process, Interacting particle system
Changed	3 November 2026

Totally asymmetric simple exclusion process

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