Nonstationary Generalized TASEP in KPZ and Jamming Regimes

Abstract

We study the model of the totally asymmetric exclusion process with generalized update, which compared to the usual totally asymmetric exclusion process, has an additional parameter enhancing clustering of particles. We derive the exact multiparticle distributions of distances travelled by particles on the infinite lattice for two types of initial conditions: step and alternating ones. Two different scaling limits of the exact formulas are studied. Under the first scaling associated to Kardar–Parisi–Zhang (KPZ) universality class we prove convergence of joint distributions of the scaled particle positions to finite-dimensional distributions of the universal $\hbox {Airy}_2$ and $\hbox {Airy}_1$ processes. Under the second scaling we prove convergence of the same position distributions to finite-dimensional distributions of two new random processes, which describe the transition between the KPZ regime and the deterministic aggregation regime, in which the particles stick together into a single giant cluster moving as one particle. It is shown that the transitional distributions have the Airy processes and fully correlated Gaussian fluctuations as limiting cases. We also give the heuristic arguments explaining how the non-universal scaling constants appearing from the asymptotic analysis in the KPZ regime are related to the properties of translationally invariant stationary states in the infinite system and how the parameters of the model should scale in the transitional regime.

Appendix A: Hydrodynamic, KPZ and Transitional Heuristics from the Stationary State

1.1 A.1  Hydrodynamics

In the nonuniform setting of step IC the particle density c(x, t) and current j(x, t) are related by the continuity equation that expresses the particle conservation law,

$$\begin{aligned} \partial _{t}c(x,t)+\partial _{x}j(x,t)=0. \end{aligned}$$

Then, the local quasi-equilibrium suggests the same stationary state current-density relation $j(x,t)=j_{\infty }(c(x,t))$ holding locally in the varying density landscape, yielding the hyperbolic PDE for the function c(x, t)

$$\begin{aligned} \partial _{t}c+j'_{\infty }(c)\partial _{x}c=0. \end{aligned}$$

(214)

This equation can be solved by the method of characteristics, which in general can be non-trivial due to presence of shocks, see e.g. Ex. 3.6 in [84]. However, for step IC, $c(x,0)={\mathbb {1}}{1}_{x\le 0}$, with $j_{\infty }(c)$ being differentiable, convex function vanishing at the ends of density range

$$\begin{aligned} j_{\infty }(0)=j_{\infty }(1)=0, \end{aligned}$$

(215)

the solution corresponding to the rarefaction fan is straightforward, Ex. 3.7 in [84]. In this case, all nontrivial characteristics of this equation are outgoing from the origin $(x,t)=(0,0)$. Hence the solution $c(x,t)=c(\chi )$ is a function of $\chi =x/t$ given by an inversion of relation

$$\begin{aligned} j'_{\infty }(c)&=\chi , \end{aligned}$$

(216)

in the range $j'(1) \le \chi < j'_{\infty }(0)$ and $c(\chi ) =0$ or 1 when $\chi \ge j'_{\infty }(0)$ or $\chi < j'_{\infty }(1)$ respectively otherwise.

To relate this solution to the position of a particle $x_{n}(t)$ we note that the number $n=\theta t$ is exactly the number of particles to the right of $x_{n}(t)\simeq t\chi (\theta )$, i.e.

$$\begin{aligned} \theta&=\int _{\chi }^{j'_{\infty }(0)}c(y)dy =-\chi c-\int _{\chi }^{j'_{\infty }(0)}yc'(y)dy =-\chi c+j_{\infty }(c), \end{aligned}$$

(217)

where the second equality is an integration by parts and for the third one we used the variable change $y\rightarrow c(y)$ together with (216) and (215). According to (216) and (217) the function $\theta (\chi )$ is nothing but minus the Legendre transform of $j_{\infty }(c)$, which is stated in eq.(11). Being strictly monotonous, when $c\in (0,1)$, it can obviously be inverted.

1.2 A.2  Stationary State and Deterministic Relations

To prepare a translationally invariant steady state we consider first the model on a finite periodic lattice of L sites, ${\mathbb {Z}}/L{\mathbb {Z}}$, implying that L will be sent to infinity in the end. There are two complementary approaches that were shown to be effective in studies of the simplest stationary state of GTASEP. The first one is based on the so called ZRP-ASEP mapping and canonical partition function formalism, while the second works directly with GTASEP though within the framework of grand-canonical ensemble.

Within the first approach we overcome the difficulty connected with the non-locality of dynamical rules of GTASEP by mapping the ASEP-like system, where particles obey the exclusion interaction, to an equivalent zero-range process (ZRP)-like system, where many particles in a site are allowed. To this end, we replace occupied sites of an n-particle cluster plus one empty site ahead with a site occupied by n particles. As a result from the ASEP like system of M particles on the lattice of size L we obtain the ZRP-like system with the same number of particles on the lattice of size $N=L-M$. The dynamical rules of GTASEP prescribe probability $\varphi (m|n)$ form (1) to jumps of m particles out of sites with n particles, all sites being updated simultaneously and independently of the others at every time step.

A crucial observation is that the jumping probabilities (1) can be written in the product form

$$\begin{aligned} \varphi (m|n)=\frac{v(m)w(n-m)}{f(n)}, \end{aligned}$$

where

$$\begin{aligned} v(k)= & {} \mu ^{k}(\delta _{k,0}+(1-\delta _{k,0})(1-\nu /\mu )), \end{aligned}$$

(218)

$$\begin{aligned} w(k)= & {} (\delta _{k,0}+(1-\delta _{k,0})(1-\mu )), \end{aligned}$$

(219)

and

$$\begin{aligned} f(n)=\sum _{k=0}^{n}v(n)w(n-k)=(\delta _{n,0}+(1-\delta _{n,0})(1-\nu )). \end{aligned}$$

(220)

This fact is responsible for the stationary measure of the ZRP obtained from GTASEP on the ring having a factorized form. This is to say that the stationary state probability for $n_{1},\ldots ,n_{N}$ particles to occupy sites $1,\ldots ,N$ respectively is given by product

$$\begin{aligned} P_{st}(n_{1},\ldots ,n_{N})=\frac{1}{Z(M,N)}\prod _{i=1}^{N}f(n_{i}), \end{aligned}$$

where $Z(M,N)=\sum _{||n||=M}\prod _{i=1}^{N}f(n_{i})$ is the partition function, aka sum of the stationary weights over particle configurations ${\varvec{n}}$ constrained by $||{\varvec{n}}||:=n_{1}+\cdots +n_{N}=M$ . The partition function can be represented in the form of a contour integral

$$\begin{aligned} Z\left( M,N\right) =\oint _{\varGamma _{0}}\frac{\left[ F(z)\right] ^{N}}{z^{M+1}} \frac{dz}{2\pi \text {i}}, \end{aligned}$$

(221)

where $F(z)=\sum _{n\ge 0}f(n)z^{n}$ is the generating function of stationary weights, and the contour of integration encircles the origin. An explicit form of F(z) is given by a product $F(z)=V(z)W(z)$ of generating functions of v(k) and w(k),

$$\begin{aligned} V(z)=\frac{1-\nu z}{1-\mu z},\,\,W(z)=\frac{1-\mu z}{1-z},\,\,F(z)=\frac{1-\nu z}{1-z}, \end{aligned}$$

for the stationary weight f(n) being the convolution of the v(k) and w(k). The integral representation suits ideally for the asymptotic analysis we perform in the thermodynamic (large lattice, fixed density) limit

$$\begin{aligned} L\rightarrow \infty ,M\rightarrow \infty ,M/L=c. \end{aligned}$$

In this limit we evaluate integral (221) in the saddle point approximation. The equation for the critical point

$$\begin{aligned} \frac{c}{(1-c)}\frac{1}{z}+\frac{\nu }{1-\nu z}-\frac{1}{1-z}=0 \end{aligned}$$

(222)

has two solutions

$$\begin{aligned} z_{c}^{\pm }=1+\frac{(1-\nu )}{2c\nu }\left( 1\pm \sqrt{1+\frac{4(1-c)c\nu }{1-\nu }}\right) . \end{aligned}$$

from which $z_{c}^{-}$ is the one that brings dominating contribution into the integral. As, the second saddle point does not play any role, we will omit the minus sign for brevity of notations implying $z_{c}\equiv z_{c}^{-}.$

The value of $z_{c}$ being function of the density c increases from zero to one as c decreases from one to zero. Many observables of the stationary state can be represented in the form of similar contour integrals. Then, in the thermodynamic limit they will be the functions of fugacity $z_{c}$.

Now we can change the point of view and consider the stationary state observables as functions of the parameter $z_{c}$, which takes values in the range $z_{c}\in (0,1)$⁵. In particular the total number of particle jumps from a site per one time step (translated into the language of ASEP-like system) having the exact integral representation [59],

$$\begin{aligned} j_{L}=\frac{N/L}{Z\left( M,N\right) }\oint _{\varGamma _{0}} \frac{\left[ F\left( z\right) \right] ^{N}}{z^{M}}\frac{V'(z)}{V(z)} \frac{dz}{2\pi \text {i}}, \end{aligned}$$

(223)

converges to

$$\begin{aligned} j_{\infty }=\big (\frac{1}{1-\mu z_{c}}-\frac{1}{1-\nu z_{c}}\big )(1-c) \end{aligned}$$

(224)

in the thermodynamic limit. In this way we obtain parametric dependence of $j_{\infty }$ on c. Though it can be explicitly resolved to give the current-density relation obtained in [54, 59], the further derivation of $\chi (\theta )$ is possible only in parametric form. Using the relations (216, 217) we obtain the functional dependence (16–19) between $\chi ,\theta$ and c, which are expressed as functions of the parameter $z_{c}$ varying in the range $0<z_{c}<1$. These three functions determine the behavior of GTASEP particles in the deterministic scale. Formally, to express one of them as the function of another, one has to eliminate $z_{c}$ between them by solving the large degree polynomial equation, which is hardly suitable for further calculations. In contrast, the parametric form is the one which is obtained from the asymptotic analysis of the exact distributions and also is enough to proceed with scaling constants defining the fluctuation and correlation scales.

1.3 A.3  KPZ Dimensional Invariants and Model-Dependent Scaling Constants

The next step is to understand the meaning of scaling constants $\kappa _{f}$ and $\kappa _{c}$ defining correlation and fluctuation scales respectively. To this end we refer to the papers [71, 73] and review [85], where predictions for the scaling form of cumulants of the interface height were made on the basis of the analysis of KPZ equation and conjectured to be universal for the large class of models belonging to KPZ class.

To summarize, the large and small-time scaling behavior of height h(x, t) of an interface governed by the KPZ equation

$$\begin{aligned} \frac{\partial h}{\partial t}={\widetilde{\nu }}\varDelta h+{\widetilde{\lambda }}\left( \nabla h\right) ^{2}+\eta , \end{aligned}$$

(225)

with the Gaussian white noise $\eta$ defined by zero mean $\langle \eta (x,t) \rangle =0$ and covariance

$$\begin{aligned} \left\langle \eta (x,t)\eta (x',t')\right\rangle =D\delta (x-x')\delta (t-t'), \end{aligned}$$

depends on two dimensional invariants ${\widetilde{\lambda }}$ and $A=D/2{\widetilde{\nu }}$.⁶ Here and further within this section we use the accepted in physical literature notations $\langle \xi ^n \rangle$ and $\langle \xi ^n \rangle _c$ for n-th moment and cumulant of the random variable $\xi$ respectively.

In the transient (short time) regime, which can be thought of either as the large time evolution of an infinite system or that of the finite system with the large time limit $t\rightarrow \infty$ taken after the large size limit $L\rightarrow \infty$, the fluctuations of the interface height are scaled as $t^{1/3}$ with time units being inverse of $\left| {\widetilde{\lambda }}\right| A^{2}$,

$$\begin{aligned} h-\left\langle h\right\rangle \sim \text {const}\left( \left| {\widetilde{\lambda }}\right| A^{2}t \right) ^{1/3}{\mathcal {X}}. \end{aligned}$$

(226)

Here $\left\langle h\right\rangle$ is the mean value obtained from averaging over the noise realization and ${\mathcal {X}}$ is some universal (parameterless) random variable, which still depends on IC . On the other hand, in the late time regime, where the large time limit is taken in a finite system, which is then supposed to be large, the height deviation from its spacial average ${\bar{h}}=L^{-1}\int hdx$ is purely Gaussian,

$$\begin{aligned} h-{\bar{h}}\sim \text {const}\left( AL\right) ^{1/2}{\mathcal {N}} \end{aligned}$$

(227)

with the variance proportional to the distance measured in the units inverse to A. The notation ${\mathcal {N}}$ is used for the standard normal random variable. These two regimes can be sewed together within the so called Family-Vicek scaling of interface width

$$\begin{aligned} w=\sqrt{\left\langle \left( h-{\bar{h}}\right) ^{2}\right\rangle }=\left( AL\right) ^{1/2}{\mathcal {F}}_{FV}(L/\xi (t)), \end{aligned}$$

(228)

where the asymptotic behavior of the scaling function, ${\mathcal {F}}_{FV}(0)=\text {const}$ and ${\mathcal {F}}_{FV}(x)=O(x^{-1/2})$ as $x\rightarrow \infty$, is dictated by the requirement of attaining both limits (227, 226). The correlation length that would reproduce (226) must have the form

$$\begin{aligned} \xi (t)=\frac{\left( \left| {\widetilde{\lambda }}\right| A^{2}t \right) ^{2/3}}{A}. \end{aligned}$$

(229)

Note that all the dimensional scaling constants are defined up to dimensionless numbers, which being the universal normalization should be chosen consistently with the definitions of limiting random variables and processes, depending on initial and boundary conditions. Qualitatively $\xi (t)$ defines the correlation scale, within which the fluctuations are supposed to be non-trivially correlated, and, in particular, the fluctuations become stationary, when the correlation length is comparable to the system size. In other words we expect that $\xi (t)$ gives the natural spacial correlation scale, i.e. the units of spacial coordinate within the limiting multipoint correlation functions.

The key hypothesis from [73] states that the scaling arguments of the same type are applicable to the wide range systems within the whole KPZ class far beyond the KPZ equation itself, up to the only difference that the a priori unknown parameters ${\tilde{\lambda }}$ and A should be read off from the properties of the stationary state of the models. The recipe of finding them was also proposed in [71, 73]. For growing KPZ interfaces of arbitrary origin the lateral growth parameter ${\widetilde{\lambda }}$, related to the response of the interface velocity to a small tilt $h(x,t)\rightarrow h(x,t)+\kappa x$, can be determined from

$$\begin{aligned} \widetilde{{\lambda }}=\frac{\partial ^{2}v_{\infty }}{\partial \kappa ^{2}}, \end{aligned}$$

(230)

where $v_{\infty }$ is the $L\rightarrow \infty$ limit of stationary interface velocity $v_{L}=\lim _{t\rightarrow \infty }\left\langle \partial h/\partial t\right\rangle ,$ and leading finite size correction to the interface velocity

$$\begin{aligned} b_{v}=\lim _{L\rightarrow \infty }\lim _{t\rightarrow \infty }L\left( \left\langle \partial h/\partial t\right\rangle -v_{\infty }\right) \end{aligned}$$

is expected to be given by

$$\begin{aligned} b_{v}=-\frac{A{\widetilde{\lambda }}}{2}. \end{aligned}$$

Finding these two quantities is enough for defining the two necessary dimensional constants. An independent consistency check can be done with calculation of spacial correlation function

$$\begin{aligned} \lim _{t\rightarrow \infty }\left\langle \left( h(x,t)-h(y,t)\right) ^{2}\right\rangle _{c}=A\left| x-y\right| , \end{aligned}$$

(231)

which amplitude is nothing but A.

All these quantities are accessible for our GTASEP system by identifying it with an interface using mapping

$$\begin{aligned} h_{i+1}-h_{i}=1-2\eta _{i}, \end{aligned}$$

where $\eta _{i}=0,1$ is the occupation number of the ith site, $h_{i}$ is the interface height above the bond connecting sites $i-1$ and i of the lattice, $i=1,\ldots ,L,$ satisfying helicoidal boundary conditions

$$\begin{aligned} h_{i+L}=h_{i}-(L-2M), \end{aligned}$$

which gives a tilt $\kappa =1-2c$ to the interface. Observing that the interface velocity is twice the particle current $v_{L}=2j_{L}(c)$ we obtain

$$\begin{aligned} {\widetilde{\lambda }}&=\frac{1}{2}\frac{d^{2}j_{\infty }}{dc^{2}} =\frac{1}{2}\left( \frac{1}{dc/dz_{c}}\frac{d}{dz_{c}}\right) ^{2}j_{\infty }\nonumber \\&=-\frac{(1-\mu )(\mu -\nu )}{(1-\nu )^{2}}\frac{\left( 1-\nu \left( 2-z_{c} \right) z_{c}\right) {}^{3}\left( 1-\mu \nu z_{c}^{3}\right) }{\left( 1-\mu z_{c}\right) {}^{3}\left( 1-\nu z_{c}^{2}\right) {}^{3}}, \end{aligned}$$

(232)

The first order finite size correction to the current obtained from (223) is

$$\begin{aligned} b_{v}=2\frac{(1-\mu )(\mu -\nu )}{(1-\nu )}\times \frac{\left( 1-z_{c} \right) z_{c}\left( 1-\nu z_{c}\right) \left( 1-\mu \nu z_{c}^{3}\right) }{\left( 1-\mu z_{c}\right) {}^{3}\left( 1-\nu z_{c}^{2}\right) {}^{2}}. \end{aligned}$$

From these formulas ${\widetilde{\lambda }}$ and A can be found in terms of the fugacity $z_{c}.$ Alternatively we could obtain A from knowing correlation function (231), which was derived in [59] within the grand canonical formalism applied directly to the ASEP-like system and was shown to be consistent with the other formulas. We refer the reader to [59] for further details.

Finally to translate the fluctuation and correlation scales, (226) and (229), obtained for the interface to the language of particles, we note that the height increase is twice the number of particles having traversed a particular bond. The ratio of the distance scale and the h-scale is the particle density c, i.e. the ratio of the number of jumps per particle to the number of jumps per bond. Hence we define

$$\begin{aligned} \kappa _{f}&=\frac{1}{2c}\left( \frac{|{\widetilde{\lambda }}|A^{2}}{2}\right) ^{1/3} \end{aligned}$$

(233)

Conversely, to go from x-scale to n-scale we have to multiply $\xi (t)$ by c:

$$\begin{aligned} \kappa _{c}&=\frac{c}{A}\left( \frac{|{\widetilde{\lambda }}|A^{2}}{2}\right) ^{2/3} \end{aligned}$$

(234)

After substitution the explicit expressions for A and ${\tilde{\lambda }}$ in terms of $z_c$ into (233, 234) we arrive at formulas (20, 21). We used the coefficient 1/2 with the dimensional constant $|{\widetilde{\lambda }}|A^{2}$ in these definitions just for aesthetic reasons. As was mentioned above, all the dimensional scales can be defined up to dimensionless numbers, which then can be consistently taken into account in the statements of convergence and definition of limiting processes.