# Bethe ansatz solution of zero-range process with nonuniform stationary state

###### Abstract

The eigenfunctions and eigenvalues of the master-equation for zero range process with totally asymmetric dynamics on a ring are found exactly using the Bethe ansatz weighted with the stationary weights of particle configurations. The Bethe ansatz applicability requires the rates of hopping of particles out of a site to be the -numbers . This is a generalization of the rates of hopping of noninteracting particles equal to the occupation number of a site of departure. The noninteracting case can be restored in the limit . The limiting cases of the model for correspond to the totally asymmetric exclusion process, and the drop-push model respectively. We analyze the partition function of the model and apply the Bethe ansatz to evaluate the generating function of the total distance travelled by particles at large time in the scaling limit. In case of non-zero interaction, , the generating function has the universal scaling form specific for the Kardar-Parizi-Zhang universality class.

###### pacs:

05.40.-a, 02.50.-r, 64.60.Ht## I Introduction

The Bethe ansatz bethe is one of the most powerful tools to get exact results for the systems with many interacting degrees of freedom in low dimensions. The exact solutions of one-dimensional quantum spin chains and two-dimensional vertex models are classical examples of its application baxter . In the last decade, the Bethe ansatz was shown to be useful to study one-dimensional stochastic processes gs ; schutz . The first and most explored example is the asymmetric simple exclusion process (ASEP), which serves as a testing ground for many concepts of the nonequilibrium statistical physics derrida . Yet several other Bethe ansatz solvable models of nonequilibrium processes have been proposed such as the asymmetric diffusion models sw ; sw2 , generalizations of the drop-push model srb ; karim1 ; karim2 , and the asymmetric avalanche process (ASAP) piph .

Most of models studied by the coordinate Bethe ansatz have a common property. That is, a system evolves to the stationary state, where all the particle configurations occur with the same probability. This property can be easily understood from the structure of the Bethe eigenfunction. Indeed, the stationary state is given by the groundstate of evolution operator, which is the eigenfunction with zero eigenvalue and momentum. Such Bethe function does not depend on particle configuration at all and results in the equiprobable ensemble. Except for a few successful attempts to apply the Bethe ansatz to systems with nonuniform stationary state, such as the ASEP with blockage schutz1 or defect particle de , there is no much progress in this direction.

On the other hand many interesting physical phenomena such as condensation in nonequilibrium systems evans , boundary induced phase transitions Krug ; sd or intermittent-continuous flow transition pph3 become apparent from non-trivial form of the stationary state, which change dramatically from one point of phase space to another. Typical example is the zero-range process (ZRP), served as a prototype of a one-dimensional nonequilibrium system exhibiting the condensation transition evans . While its stationary measure has been studied in detail gss , the full dynamical description is still absent.

The aim of this paper is to obtain the Bethe ansatz solution of the ZRP. The paper is organized as follows. In section II we use the Bethe ansatz to solve the eigenfunction and eigenvalue problem for the master equation of the ZRP and show that its applicability requires the rates of hopping of particles out of a site to be the -numbers. We show that with this choice of the rates the model is equivalent to the q-boson totally asymmetric diffusion model. In section III we obtain the partition function of the ZRP with the rates obtained and evaluate some stationary correlations. In section IV we apply the equations obtained from the Bethe ansatz solution to derive the expression for the generating function of the total distance travelled by particles in the large time limit. We summarize the results in section V.

## Ii Master equation of zero range process

Let us consider the system of particles on a ring consisting of sites. Every site can hold an integer number of particles. Every moment of time, one particle can leave any occupied site, hopping to the next site clockwise. The rate of hopping depends only on the occupation number of the site of departure . The stationary measure of such a process has been found to be a product measure evans , i.e. the probability of configuration , specified by the occupation numbers , is, up to the normalization factor, given by the weight

(1) |

where one-site weight is

(2) |

if , and .

Let us consider the probability for sites to have occupation numbers at time . It obeys the master equation defined by the dynamics described.

(3) |

Here, cyclic boundary condition, , is imposed.

Another way to specify the configuration of system is to use the set of coordinates of particles , rather than occupation numbers , the two ways being completely equivalent. While the explicit form of the master equation in such notations is more complicated, it turns out more appropriate for an analytic solution. The main idea of the solution is to use the Bethe ansatz for the function related with the solution of the master equation as follows

(4) |

### ii.1 Two-particle sector

To explain the details we first consider the case subsequently generalizing it to the case of arbitrary . Without lost of generality we can define

(5) |

Now, we are going to show that the solution, , of the master equation for noninteracting particles,

(6) | |||||

is related with the solution of the master equation for the ZRP in the domain through the relation (4), provided that the former satisfy certain constraint. Indeed, when , the equation for probability for the ZRP coincides with one for noninteracting particles (6). For the equation for the ZRP

(7) |

can be obtained from (6) if we define

(8) |

which is consistent with (4). In the case the equation (6) contains the term which is beyond the ”allowed” region and, thus, does not carry any physical content. To restore the correct equation for the ZRP

(9) |

we can redefine this term to compensate the difference between the equations (6) and (9). As a result we get the following constraint on

(10) |

Thus, the solution of the free equation (6), which satisfies the constraint (10), gives the solution of the master equation for the ZRP in the domain . Now, we can use the Bethe ansatz for the eigenfunction of the free equation (6)

(11) |

Its substitution to the equation (6) results in the expression for the eigenvalue.

(12) |

The ansatz (11) to be consistent with the constraint (10), the amplitudes and should satisfy the relation

(13) |

which together with the cyclic boundary conditions, , results in the system of two algebraic equations. The first one is the following

(14) |

while the second can be obtained by the change .

### ii.2 Many-particle sector

To generalize these results to the case of arbitrary let us consider the configuration with two neighboring sites and having occupation numbers and respectively. Let us explicitly write down the terms of the master equation corresponding to the transition into and from this configuration due to a particle jump into and from the site respectively,

(15) |

Here denotes successive arguments of the function . In terms of , which is related to according to (4), this equation looks as follows

(16) |

Note, that in this form the coefficient before the term in square brackets is equal to , i.e. does not depend on the number of particles in the site . Thus, the r.h.s of the master equation expressed through is the sum of one-site factors similar to those in (16) for all nonempty sites. Equating such term with one from the equation for noninteracting particles,

(17) |

we obtain the following constraint for ,

(18) |

Such relations for all are to be understood as a redefinition for the terms outside of allowed region . The Bethe ansatz to be applicable, the relation (18) should be reducible to the two particle constraint (10). This can be proved by induction. To this end, we assume that similar relations including are reducible for all . Then starting from the relation (18), which includes rate , we apply (10) to every pair of arguments of the function under the sum and require the result to be similar relation for . We obtain the following recurrent formula for the rates

(19) |

which can be solved in terms of -numbers

(20) |

where

(21) |

Further generalization of two-particle results is straightforward. We use the Bethe ansatz for the eigenfunction of the equation (17)

(22) |

Here are complex numbers, the summation is taken over all permutations of the numbers , and the coordinates of particles are ordered in the increasing order .

Substituting (22) into (17) we obtain the expression for the eigenvalue,

(23) |

The numbers are to be defined from the Bethe equations,

(24) |

which follow from the condition of compatibility of cyclic boundary conditions with the constraint (10).

For the sake of convenience in the following discussion we use the parameter defined in (21) rather than . We should note that the appearance of q-numbers as the condition of the Bethe ansatz integrability is not unexpected. The notion of q-deformation naturally appears in context of Bethe ansatz solvable models characterized by the trigonometric R-matrix. In algebraic language this is the consequence of the of the fundamental Yang-Baxter equation which leads to q-commutation relations between the local operators constituting the transfer matrices faddeev . One of such models, q-boson totally asymmetric diffusion model sw2 , turns out to be directly related to the model we consider. In order to see the correspondence, let us formally write the distribution as a vector of state

(25) |

where is a configuration of particles on the lattice, , and the summation is over all configurations. Consider the algebra generated by the operators , which act on the occupation number of each site of the lattice as follows:

(26) | |||||

(27) | |||||

(28) |

The state plays the role of vacuum state

(29) |

Then, the master equation (3) with the rates given by (20) can be written in the from of the imaginary time Schrdinger equation

(30) |

where the hamiltonian H is given in terms of the operators (LABEL:B,26)

(31) |

One can directly check that the operators satisfy the following commutation relations

(32) | |||||

(33) | |||||

(34) |

These commutation relations and the hamiltonian (31) give us, up to the change of notations , the definition of the q-boson totally asymmetric diffusion model. Obviously, the dynamical rules of the ZRP with the rates given by (20) is nothing but the explicit realization of this hamiltonian. Its integrability has been shown in sw2 via algebraic Bethe ansatz and two particle diffusion on the infinite lattice has been considered. Note that while in sw2 the q-boson totally asymmetric diffusion model was initially defined in terms of q-boson operators, we started from the ZRP with arbitrary rates and come to q-numbers as the integrability condition.

Let us first take a qualitative look at the behaviour of the ZRP resulted by the choice (20) of the rates for different values of . In the limit the -numbers degenerate into simple numbers. Therefore, the rates are given by , which corresponds to the diffusion of noninteracting particles. The Bethe equations in this case decouple to the form as is expected in noninteracting case. In the domain the rates grow exponentially with , which corresponds to the interaction between particles effectively accelerating free diffusive motion, i.e. the higher density of particles the faster is their mean velocity. In the limit the model is equivalent to drop push model srb , which is confirmed by the same form of Bethe equations karim1 . In the domain the rates grow monotonously from for to for , resulting in the interaction between particles slowing down the particle flow comparing to the one of noninteracting diffusing particles. When , all the rates do not depend on the number of particles, i.e. . This case, (also referred to as phase model BIK ), can be mapped on the totally asymmetric ASEP by insertion of one extra bond before every particle. At last, in the domain the rates also saturate to the constant with growth of , though oscillating around this value. As it has been mentioned above, the ZRP served as an example of the nonequilibrium system with the condensation transition. In our case however the condensation is absent as the rates defined above do not satisfy Evans criteria, according to which the condensation in the ZRP occurs if the rates saturate to a constant slower than .

It is interesting, that the recursion relation (19) rewritten in terms of the quantity coincide with one for the toppling probabilities , imposed by the requirement of the Bethe ansatz integrability in the ASAP. In fact, the ASAP can be represented as a special case of the discrete time ZRP viewed from the reference frame moving together with an avalanche. This situation, however, is quite different from one considered here. In the moving reference frame all particles hop definitely to the next site except of one from an active site, which is the only site with multiple occupation. In that case quantity plays the role of probability of hopping of a particle out of this site. This dynamics leads to the picture similar to the ZRP on an inhomogeneous lattice with one attractive site. Such ZRP was shown to exhibit the condensation transition. In terms of the avalanche processes it is the transition from the intermittent to continuous flow. Despite the nonuniform stationary state of the ASAP in discrete time pph3 , its Bethe ansatz solution was based on the continuous time picture considered on the ensemble of equiprobable configurations with at most one particle occupation. The avalanches were accounted in the rates of transitions between these configurations, generating infinite series in the master equation.

To use the eigenfunctions obtained for the construction of particular solutions one should first question if they form complete orthogonal basis. In general this question is not easy to answer, as the set of solutions of the Bethe equations is not known. Some arguments have been given, gs ; kim , for the the totally asymmetric exclusion process due to its connection with the asymmetric six-vertex modelbaxter1 . The long time characteristics of the process can, however, be analysed without discussing this question. To this end, we can use the properties of the largest eigenvalue for slightly modified equation, which describes the generating function of total distance travelled by particles, dl . The advantages of this approach are first, that the uniqueness of the largest eigenvalue is guarantied by Perron-Frobenius theorem. Second, corresponding solution of the Bethe equations can be easily identified as it corresponds to the stationary state of the process. To give an example of application of the above results we perform this analysis in the section IV.

## Iii Structure of the stationary state

Before going to the results of the analysis of the Bethe equations, let us first look at the structure of the stationary measure of the model. It is characterized by the partition function

(35) |

where one site weight , defined in (2), is expressed through the -factorial . In the limit , -factorial turns into simple factorial, as it should be in the noninteracting case. The sum in (35) can be presented in form of the contour integral

(36) |

where the series can be summed to the infinite product due to the -binomial theorem aar : for

(37) |

and for

Here, we used the notation for shifted -factorial. The above -series are known as -analogs of the exponential function , which can be restored in the limit . The presence of -analogs of the functions, which appear in the case of noninteracting particles, is a direct consequence of the replacement of simple numbers by -numbers in the expression of the rates of hopping. In the thermodynamic limit, , the integral (36) can be calculated in the saddle point approximation. The equation for the saddle point ,

(39) |

contains the logarithmic derivative of , which can be evaluated using the product form of (37,III). As a result we obtain the following relations between and .

(40) |

(41) |

for and respectively. Below the same equations will appear in a different context from the analysis of Bethe ansatz equations. Then, the partition function,

(42) |

can be used to calculate stationary state correlations such as the speed, , i.e. the average hopping rate out of a site

(43) |

or the probability distribution of the number of particles in a site

(44) |

## Iv The long time behaviour from the Bethe equations

To obtain any results beyond the stationary correlations one needs to analyze the above Bethe ansatz solution. Since similar analysis has been done several times before bs ; kim ; lk ; pph2 , we do not give the detailed calculations here. Instead we outline the main points of the solution to emphasize the connection with the formulas obtained from the analysis of stationary measure.

Consider the generating function , where is the joint probability for the system to be in the configuration at time the total distance travelled by particles being . The only difference of the equation for from the equation (3) is the coefficient before the first term under the sum in the r.h.s., which corresponds to the increasing of travelled distance by one due to the hopping of one particle. At large time, , the behaviour of the generating function of the distance travelled by particles up to time , , is determined by the largest eigenvalue of the equation for .

(45) |

Using the ansatz (4,22) for the eigenfunction of the equation for we can repeat all the above arguments. Then, if we make the variable change , the eigenvalue and the Bethe equations will simplify to the following form.

(46) |

(47) |

In these variables the r.h.s. of the Bethe equations coincides with one for the partially asymmetric ASEP and the ASAP. This allows us to modify the techniques developed for the analysis of these processes.

In the thermodynamic limit, , we assume that the roots of the Bethe equations (47) are distributed in the complex plain along some continuous contour with the analytical density , so that the sum of values of a function at the roots is given by

(48) |

After taking the logarithm and replacing the sum by the integral, the system of equation (47) can be reformulated in terms of single integral equation for the density. The particular solution corresponding to the largest eigenvalue is specified by the appropriate choice of branch of the logarithm. Then the integral equation should be solved for a particular form of the contour, which is not known a priori, and being first assumed should be self-consistently checked after the solution has been obtained. In practice, however, analytical solution is possible in the very limited number of cases. Particularly, one, corresponding to the contour closed around zero, yields the density

(49) |

for and

(50) |

for . This case corresponds to and hence . Since is analytic in the ring , the integration of it along any contour closed in this ring does not depend on its form. Therefore, to fix the form of the contour additional constraints are necessary. Such constraint appears if we require that the density preserves its analyticity when deviates from zero and the contour becomes discontinuous. This constraint implies that the density vanishes at the break point , which is a crossing point of the contour and the real axis.

(51) |

This equation was first obtained by Bukman and Shore as a conical point condition for the asymmetric six-vertex model bs . It is remarkable, that after the resummation of series in (49,50), the equation (51) coincides with eqs.(41,40) up to the replacements

(52) |

for and respectively. Remember that was shown to coincide with the speed . It will be clear below that the same relation between and follows directly from the expression for obtained from the Bethe equations, without appealing to the partition function.

The further analysis is related with the calculation of finite size corrections to the above expression of , which makes possible to probe into the nonzero values of . This can be done with the help of method developed in kim ; dl . Its essential part, is the construction of the inverse expansion to the function of the number of roots near the break point . Since the derivative of in the thermodynamic limit vanishes (see eq.(51)), its inverse expansion reveals the square root singularity, which in its turn becomes the origin of terms in the finite size expansion of . As a result we obtain the following parametric dependence of on , both being represented as functions of the same parameter .

(53) | |||||

(54) | |||||

Here and are the Loraunt coefficients of and respectively defined as follows

(55) |

and are the coefficients of in and respectively, where are the coefficients of the inverse expansion near the point . The location of is to be self-consistently defined from the equation . For the first three orders of expansion the coefficients can be obtained from the inverse expansion of zero order function , while has been obtained above. To evaluate the sum over the roots, one needs to integrate along the contour , which can be obtained from the initial closed contour by cutting out small segment connecting two roots closest to the point . For a function which is defined by the expansion

(56) |

this yields

(57) |

minus and plus in power of being for and respectively. Finally the point enters to all the results trough the coefficients and . It is related with the physical quantities through eqs.(40,41,43,52). We use this relation, to write the final results as a function of speed , density , and the parameter . Below we give the expression for the largest eigenvalue in the scaling limit

(58) |

Here the function has the following parametric form

(59) | |||||

(60) |

with is the function of polylogarithm and the constants are

(61) | |||||

(62) |

for , and

(63) | |||||

(64) |

for , were

(65) |

The scaling form of the function was suggested to be universal for Kardar-Parisi-Zhang (KPZ) universality class kpz ; da . Using the generating function obtained one can evaluate all the cumulants of the travelled distance

(66) |

The large deviation function, can be also obtained as a Legendre transformation of .

## V Summary and discussion

To summarize, we apply the Bethe ansatz to solve zero range process with the totally asymmetric dynamics on a ring. The eigenfunctions of the master equation have the form of the Bethe function weighted with the stationary weights of corresponding particle configurations. The requirement of Bethe ansatz integrability leads to the special choice of the rates of hopping of particle out of a site. The rates should be -numbers, , generalizing the case of noninteracting diffusing particles, where the rate is equal to , the number of particles at a site of departure. The noninteracting case can be restored in the limit . Two other limiting cases, and reproduce well known totally asymmetric exclusion process and drop-push model respectively. The case of general is shown to be equivalent to the q-boson totally asymmetric diffusion model. Continuing analogy with noninteracting case, we show that many quantities characterizing the stationary state correlations of the model turns out -analogs of corresponding functions appearing in the noninteracting case. To provide an example of application of the Bethe ansatz solution obtained, we derive the expression for the large time limit of the generating function of cumulants of the total distance travelled by particles. It has a universal form specific for KPZ universality class. The question whether the q-boson totally asymmetric diffusion model belongs to KPZ class was addressed in concluding remarks in sw2 . The result (58-60) is an argument in favour of this assumption.

In connection with above results the following questions appear. First, is it possible to generalize the proposed combination of the Bethe ansatz with stationary weights to any other processes with nonuniform stationary state, say asymmetric exclusion process with parallel update? The consideration of the associated vertex models is also of interest. The different weights of vertex configurations depending on the order of vertices would result in the appearance of non-local interaction. Second, can one apply the matrix product method to study the exclusion process with long range interaction associated with zero range process considered here to probe into spatially inhomogeneous situation, e.g. at the open chain. Appearance of -numbers seems to be an indication of this possibility. We expect that the matrix product ansatz should be again appropriately weighted with stationary weights of some homogeneous system. The consideration of such a system is attractive, as the extra parameter could result in a reacher phase diagram comparing to the usual totally asymmetric exclusion process. Third, it is interesting to establish correspondence of the large scale behaviour of the proposed process with KPZ equation.

###### Acknowledgements.

The author is grateful to V.B. Priezzhev and V.P. Spiridonov for stimulating discussion. The work was supported partially by FCT grant SFR/BPD/11636/2002 and by the grant of Russian Foundation for Basic Research No.03-01-00780.## References

- (1) H. Bethe, Zeitschrift fr Physik 71 205 b (1931)
- (2) R.J. Baxter 1982 Exactly Solved Models in Statistical Mechanics Academic Press
- (3) L.H. Gwa and H. Spohn Phys. Rev. A46 844 (1992)
- (4) G.M. Schtz Phase Transitions and Critical Phenomena 19 ed Domb C and Lebowitz J (London: Academic) (2001)
- (5) B.Derrida Phys. Rep.301, 65 (1998)
- (6) T. Sasamoto and M. Wadati Phys. Rev. E 58 (1998) 4181
- (7) T. Sasamoto and M. Wadati J. Phys. A: Math. Gen. 31 (1998) 6057
- (8) G.M. Schtz, Ramaswamy R and Barma M J. Phys. A: Math. Gen. 29 837 (1996)
- (9) M. Alimohammadi, V. Karimipour and M. Khorrami Phys. Rev. E 57 6370 (1998)
- (10) M. Alimohammadi, V. Karimipour and M. Khorrami J. Stat. Phys. 97 373 (1999)
- (11) V.B. Priezzhev, E.V. Ivashkevich, A.M. Povolotsky and C.-K. Hu, Phys. Rev. Lett. 87 084301 (2001)
- (12) G. Schtz J. Stat. Phys. 71 471 (1993)
- (13) B. Derrida and M.R. Evans J. Phys. A: Math. Gen. 32 4833 (1999 )
- (14) M.R. Evans Brazilian Journal of Physics 30 42 (2000)
- (15) J. Krug Phys. Rev. Lett. 67 1882 (1991)
- (16) G. Schtz and E. Domany J. Stat. Phys. 72, 277 (1993)
- (17) A.M. Povolotsky, V.B. Priezzhev and C.-K. Hu Phys. Rev. Lett. 91, 255701 (2003)
- (18) S. Grosskinsky, G.M. Schtz, H. Spohn J. Stat. Phys. 113 389 (2003)
- (19) L.D. Faddeev, Les-Houches lectures, preprint hep-th/9605187 (1996)
- (20) N.M. Bogoliubov, A.G. Izergin and N.A. Kitanin, Nucl.Phys B 516 501 (1998)
- (21) D. Kim Phys. Rev. E 52 3512 (1995)
- (22) R. J. Baxter J. Stat. Phys. 108 1 (2002)
- (23) B. Derrida, J. L. Lebowitz, Phys. Rev. Lett. 80, 209-213 (1998)
- (24) G.E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and Its Applicatons (Cambridge University Press, Cambridge, 1999)
- (25) D.J.Bukman and J.D.Shore, J.Stat.Phys. 78 1277 (1995)
- (26) D.-S. Lee and D. Kim, Phys. Rev. E 59, 6476 (1999)
- (27) A. M. Povolotsky, V. B. Priezzhev and Chin-Kun Hu, J. Stat. Phys. 111 1149 (2003)
- (28) M. Kardar, G. Parisi and Y.-C. Zhang Phys. Rev. Lett. 56 889 (1986)
- (29) B. Derrida and C. Appert, J. Stat. Phys. 94 1 (1999)