Totally asymmetric simple exclusion process

The totally asymmetric simple exclusion process, or TASEP, describes particles that jump in one direction and cannot share a site. A finite periodic ring has a simple stationary distribution despite a nonzero circulating current. Its generator also admits Bethe eigenvector constructions, providing a concrete connection between stochastic dynamics and integrability. This record defines the homogeneous continuous-time ring model, gives its exact finite stationary observables, and separates this regime from open boundaries and scaling limits.

Required background. Use probabilities on a finite state space and interpret a rate; Probability and Markov generators develops these ideas. Helpful background. The stochastic conventions distinguish probability generators, observable generators, currents, and normalization.

Right-moving exclusion on a finite ring

Choose an integer number of sites $N\geq2$, an integer number of particles $0\lt M\lt N$, and a homogeneous jump rate $r\gt0$. A configuration is a binary occupation vector

$$\begin{aligned} \eta&=(\eta_1,\ldots,\eta_N),\qquad \eta_i\in\{0,1\},\\ \Omega_{N,M}&=\left\{\eta:\sum_{i=1}^N\eta_i=M\right\}, \qquad |\Omega_{N,M}|=\binom NM. \end{aligned}$$

Indices are periodic: $\eta_{N+1}=\eta_1$. The directed bond $(i,i+1)$ permits the update

$$10\longrightarrow01 \quad\text{at rate }r.$$

All other local patterns produce no jump. In particular, the directed closing bond is $(N,1)$, and an occupied destination blocks the move. The process conserves $M$ and treats particles as indistinguishable. Lattice positions are counted in sites; $r$ has units of inverse time. A physical lattice spacing $a$ would multiply a mean hopping speed in sites per time by $a$.

This is the periodic exclusion process of Golinelli and Mallick 2006, § II.A, pp. 2–3, arXiv v1 PDF, specialized to right jumps. Their $L$ sites, $N$ particles, and rates $p=1,q=0$ become our $N$ sites, $M$ particles, and right rate $r$. Their normalization $p+q=1$ of the microscopic right and left rates fixes a time unit; it does not set the total escape rate of every configuration to one.

Generator and initial data

Let $\eta^{i,i+1}$ be the configuration obtained by swapping occupations at sites $i$ and $i+1$, and define the number of eligible bonds

$$b(\eta)=\sum_{i=1}^N\eta_i(1-\eta_{i+1}).$$

For two different configurations, the probability generator is

$$Q_{\eta',\eta} =r\sum_{i=1}^N\eta_i(1-\eta_{i+1}) \mathbf1_{\{\eta'=\eta^{i,i+1}\}}, \qquad Q_{\eta,\eta}=-rb(\eta).$$

Thus columns sum to zero and off-diagonal entries are nonnegative. For a probability column $p(t)$ and any initial distribution $p_0$,

$$\dot p(t)=Qp(t),\qquad p(0)=p_0, \qquad p(t)=e^{tQ}p_0,$$

where $p_0(\eta)\geq0$ and $\sum_\eta p_0(\eta)=1$. No special initial condition is built into the model. A deterministic configuration uses $p_0=\delta_{\eta_0}$; a stationary calculation uses the distribution below. The eigenvalues of $Q$ have units of inverse time. Writing $Q=rQ_1$ shows that the dimensionless evolution time is $rt$.

The total escape rate from a configuration is $rb(\eta)$, so its residence time is exponentially distributed with that parameter. Conditional on a jump, each of the $b(\eta)$ eligible bonds is chosen with probability $1/b(\eta)$. This is continuous-time dynamics; moving every particle once per discrete time step defines a different update rule.

For an observable $f:\Omega_{N,M}\to\mathbb R$, the corresponding operator is

$$\begin{aligned} (Q^{\mathsf T}f)(\eta) &=r\sum_{i=1}^N\eta_i(1-\eta_{i+1})\\ &\quad\times\bigl[f(\eta^{i,i+1})-f(\eta)\bigr]. \end{aligned}$$

It gives $d\mathbb E[f(\eta(t))]/dt=\mathbb E[(Q^{\mathsf T}f)(\eta(t))]$. The transpose changes what the operator acts on; it is not optional notation.

A local matrix definition

Use the ordered two-site occupation basis $(11,10,01,00)$. The probability generator of a single directed bond is

$$q=r\begin{pmatrix} 0&0&0&0\\ 0&-1&0&0\\ 0&1&0&0\\ 0&0&0&0 \end{pmatrix}.$$

On $(\mathbb C^2)^{\otimes N}$, embed this matrix at sites $(i,i+1)$, use the directed order $(N,1)$ for the seam, and sum:

$$Q=\left.\sum_{i=1}^N q_{i,i+1}\right|_{\Omega_{N,M}}.$$

The tensor notation represents classical configuration vectors; it does not introduce quantum probabilities. This is Golinelli and Mallick 2006, § III.A, p. 5, equations (20)–(21), arXiv v1 PDF with the rates converted as above. Keeping the basis order explicit prevents a silent reversal of $10\to01$.

Stationary measure and finite-size current

The stationary distribution in a fixed-$M$ sector is uniform:

$$\pi(\eta)=\frac1{\binom NM},\qquad \eta\in\Omega_{N,M}.$$

For each ring configuration, the number of $10$ interfaces equals the number of $01$ interfaces. Indeed,

$$\sum_i\bigl[\eta_i(1-\eta_{i+1}) -(1-\eta_i)\eta_{i+1}\bigr] =\sum_i(\eta_i-\eta_{i+1})=0.$$

Incoming transitions have one possible predecessor per $01$ interface; outgoing transitions have one possible destination per $10$ interface. Uniform weights therefore give equal total incoming and outgoing probability flux at every configuration, proving $Q\pi=0$. The finite ring with $0\lt M\lt N$ is irreducible, so this stationary probability distribution is unique; see Golinelli and Mallick 2004, § 2.1, p. 3, arXiv v1 PDF.

This is a balance of total fluxes. It is generally not detailed balance, which would require $Q_{\eta',\eta}\pi(\eta)=Q_{\eta,\eta'}\pi(\eta')$ for each pair separately. On the four-site, two-particle ring, $12\to13$ is allowed and $13\to12$ is not. A uniform distribution can sustain a directed current. The small case $N=2,M=1$ is exceptional: the two directed ring bonds connect the two states in opposite directions with equal rates.

Occupations, correlations, and the current

Let $\rho=M/N$. Counting configurations under $\pi$ gives, for distinct sites $i,j$,

$$\begin{aligned} \mathbb E_\pi[\eta_i]&=\rho,\\ \mathbb E_\pi[\eta_i\eta_j]&=\frac{M(M-1)}{N(N-1)},\\ \operatorname{Cov}_\pi(\eta_i,\eta_j) &=-\frac{\rho(1-\rho)}{N-1}. \end{aligned}$$

Fixing the total number of particles produces anticorrelation, even though all allowed configurations have equal weight. The instantaneous expected rate through bond $(i,i+1)$ is

$$j_i(t)=r\,\mathbb E[\eta_i(t)(1-\eta_{i+1}(t))].$$

In stationarity every bond has the same rate, and exactly $\binom{N-2}{M-1}$ configurations have occupation pattern $10$ there. Hence

$$\boxed{ j=r\frac{\binom{N-2}{M-1}}{\binom NM} =r\frac{M(N-M)}{N(N-1)} =r\rho(1-\rho)\frac N{N-1}. }$$

This is a mean number of right crossings per unit time through one bond. The mean total number of jumps per unit time is $Nj$, and the mean number of jumps per particle per unit time is

$$\frac{Nj}{M}=r\frac{N-M}{N-1}.$$

Replacing the joint occupation probability by $\rho(1-\rho)$ would miss the finite-size factor. At fixed limiting density, that factor approaches one as $N\to\infty$; this simple limit says nothing by itself about time-dependent fluctuations or their scaling.

The full counting derivation and practice problems are in TASEP on a finite ring. The laboratory tests the rates with both finite-state evolution and trajectories sampled at fixed observation times.

Exact checks and Bethe structure

Several limits check the meaning of the current. With one particle, $j=r/N$ and $Nj=r$: the particle is never blocked. With one hole, $j=r/N$ again and precisely one bond is eligible at any instant. At $N=4,M=2$, the stationary distribution is uniform on six states and

$$j=\frac r3,\qquad Nj=\frac{4r}{3},\qquad \frac{Nj}{M}=\frac{2r}{3}.$$

The Bethe structure concerns eigenvectors of the generator, beyond the stationary counting argument. In the two-particle sector, a regular coordinate ansatz takes the form

$$\psi(x,y)=A_{12}z_1^xz_2^y+A_{21}z_2^xz_1^y.$$

For distinct nonzero $z_1,z_2$ different from $1$, its contact relation and periodicity require

$$\begin{aligned} S_{12}=\frac{A_{21}}{A_{12}} &=-\frac{1-z_2}{1-z_1},\\ z_1^N=S_{12}^{-1},\qquad z_2^N&=S_{12},\\ E&=r(z_1^{-1}+z_2^{-1}-2). \end{aligned}$$

Here $A_{12}\ne0$, and the reconstructed vector must be nonzero. These are the two-particle specialization of Golinelli and Mallick 2004, § 2.2, p. 4, equations (4)–(6), arXiv v1 PDF. Their lowercase spectral variables agree with ours and their eigenvalue is $E/r$.

The Library derivation proves the contact and closing-bond equations and constructs the exact four-site mode

$$v=(1,-2,1,1,-2,1)^{\mathsf T},\qquad Qv=-3rv,$$

in the configuration order $(12,13,14,23,24,34)$. This signed vector is a relaxation mode, not a probability distribution. The uniform stationary state must be treated separately from the regular root formula. The finite construction neither establishes a complete Bethe basis for every size nor supplies every observable. In particular, this mode is not the slowest four-site decay: the smallest nonzero value of $-\operatorname{Re}E$ is $r$.

Boundaries and update rules are part of the model

The formulas here assume a finite periodic ring, fixed $M$, homogeneous rate $r$, and continuous time. Opening the ring and adding particle injection or removal changes the state space and stationary problem. Adding left jumps gives ASEP with two rates. Bond-dependent rates change the balance argument. Parallel or sequential discrete-time updates change both the transition rule and its time normalization. None of those changes is covered by substituting a new symbol into the current formula above.

The excluded sectors $M=0$ and $M=N$ each contain one absorbing configuration with zero current. The case $r=0$ freezes every configuration and loses uniqueness of the stationary distribution. Infinite lattices and thermodynamic limits require their own initial data, observables, and limiting procedures; the finite ring record makes no scaling or universality claim.

References

• Golinelli, Olivier, and Kirone Mallick. “Bethe Ansatz calculation of the spectral gap of the asymmetric exclusion process.” Journal of Physics A: Mathematical and General 37 (2004), 3321–3331. DOI. Author version arXiv:cond-mat/0312371v1, submitted 2003, PDF. Locators above refer to the printed pages of that version.
• Golinelli, Olivier, and Kirone Mallick. “The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics.” Journal of Physics A: Mathematical and General 39 (2006), 12679–12705. DOI. Author version arXiv:cond-mat/0611701v1, PDF.