A Some examples of \hat{\phi}(s) in the rational form

# Exact gap statistics for the random average process on a ring with a tracer

## Abstract

We study statistics of the gaps in Random Average Process (RAP) on a ring with particles hopping symmetrically, except one tracer particle which could be driven. These particles hop either to the left or to the right by a random fraction of the space available till next particle in the respective directions. The random fraction is chosen from a distribution . For non-driven tracer, when satisfies a necessary and sufficient condition, the stationary joint distribution of the gaps between successive particles takes an universal form that is factorized except for a global constraint. Some interesting explicit forms of are found which satisfy this condition. In case of driven tracer, the system reaches a current-carrying steady state where such factorization does not hold. Analytical progress has been made in the thermodynamic limit, where we computed the single site mass distribution inside the bulk. We have also computed the two point gap-gap correlation exactly in that limit. Numerical simulations support our analytical results.

## I Introduction

Interacting particle problems have been widely studied in statistical mechanics as examples of many-body systems that usually exhibit rich types of behavior (1). For equilibrium systems all the stationary properties are in principle known through the Gibbs-Boltzmann distribution, and the dynamics close to equilibrium can be described in very general terms (2). Out of equilibrium, studying such interacting many-particle systems analytically in dimensions other than one is usually difficult. In one dimension, one of the most studied example is the Simple Exclusion Process (SEP), where each lattice site is occupied by one hardcore particle or it is empty. In every small time interval , each particle moves to the neighboring site on the right (left) with probability () iff the target site is empty. A large number of results are known for this system (see (3); (4) and references therein) and these are obtained using sophisticated analytical tools such as Bethe Ansatz, Matrix Product Ansatz or mappings to growth models (4); (5). Another widely studied model of interacting particle systems is the Random Average Process (RAP). In the RAP particles move on a one dimensional continuous line in contrast to SEP where hardcore particles move on a lattice. Each particle moves to the right (left) by a random fraction of the space available till the next nearest particle on the right (left) with some rate. Thus the jumps in one direction is a random fraction of the gap to the nearest particle in that direction where the random number is chosen from some distribution .

Many different results have been reported for this model in the literature. For example, dynamical properties like diffusion coefficient, variance of the tracer position and the pair correlation between the positions of two particles have been studied on an infinite line  (6); (7); (8). In particular, it has been shown that the variance of the tagged particle position in the steady state grows at late times as for the asymmetric case (jumping rates to the right or to the left are different), whereas for the symmetric case it grows as . For the fully asymmetric case, this result was first derived by Krug and Garcia (6) using a heuristic hydrodynamic description. Later, the variance as well as other correlation functions, both for the symmetric and asymmetric cases, were computed rigorously by Rajesh and Majumdar (8). The RAP model has also been shown to be linked to the porous medium equation (9) and appears in a variety of problems like the force propagation in granular media (10); (11), in models of mass transport (11); (6), models of voting systems (12); (13), models of wealth distribution (14) and in the generalized Hammersley process (15). The RAP can be shown to be equivalent, up to an overall translation, to Mass Transfer (MT) models (6) when one identifies the gaps or the inter-particle spacings in the RAP picture with the masses. In terms of MT picture, statistics of masses/gaps has been well studied. For uniform distributions , invariant measures of the masses have been computed for a general class of RAP (or equivalently MT) models (11) and for a totally asymmetric version of the model (6) defined on infinite line with different dynamics. For parallel updating scheme a mean-field calculation has been shown to be exact for certain parameters (16) and a Matrix Product Ansatz has been developed (17). Finally, it has been shown that a condensation transition occurs in the related mass transfer model if one imposes a cutoff on the transferred mass ( i.e on the amount of jump made by the particles in RAP picture) (18).

Recently there has been a considerable interest in studying the motion of a special driven tagged particle in presence of other non-driven interacting particles. Such special particle is called the tracer particle. In experimental studies, driven tracers in quiescent media have been used to probe rheological properties of complex media such as DNA (19), polymers (20), granular media (21); (22) or colloidal crystals (23). Some practical examples of biased tracer are : charged impurity being driven by applied electric field or a colloidal particle being pulled by optical tweezer in presence of other colloid particles performing random motion. On the theoretical side, problems with driven tracer have been studied in the context of SEP where both particle number conservation and non-conservation (absorption/desorption) have been considered  (24); (25); (26); (27). In presence of a driven tracer on an infinite line, the particle density profile is inhomogeneous around the tracer and in absence of absorption or desorption, the current flowing across the system vanishes in the large time limit, as does the velocity of the tracer (24); (25); (26). Other quantities like the mean and the variance of the position of the tracer have also been studied (28). In this contribution we derive new results for RAP concerning the statistics of the gaps between particles on a ring in presence of a tracer particle which may be driven.

More precisely we consider particles moving according to RAP on a ring of size . In a small time duration , any particle, except the tracer particle, jumps to the right or left by a random fraction of the space available up to the neighboring particle on the right or left, respectively, with equal probability . On the other hand the tracer particle moves to the right or left by a random fraction of the space available in the respective direction with probability and , see fig. 1. In this paper we study the statistics of the gaps between neighboring particles in the stationary state (SS) for the following two cases : (i) (ii) . Note that the dynamics does not satisfy detailed balance even in the case, hence keeps the system always out of equilibrium. Using a mapping from RAP model of particles to an equivalent (except for a global translation) MT model, we find various interesting exact results related to gap statistics. The summary of these results are given below :

• When i.e. the tracer particle is not driven, all the particles are moving symmetrically. In this case, for a large class of jump distributions that satisfy some necessary and sufficient condition, we find that the stationary joint distribution of the gaps takes the following universal form :

 PN,L(g1,g2,...,gN)=1ZN,L(β)N∏i=1gβ−1i  δ(N∑i=1gi−L), (1)

where is the normalization constant and the delta function represents the global constraint due to total mass conservation in the MT picture. The parameter is given by , where is the moment of the jump distribution . Similar factorized joint distributions with power law weight functions have also been obtained in the context of the q-model of force fluctuations in granular media (10) as well as in a totally asymmetric version of the RAP on an infinite line with parallel dynamics (16). For arbitrary jump distribution we find that the average mass profile is given by and the two point gap-gap (mass-mass) correlation is expressed in terms of and as

 di,j=(μ2ω20μ1+(N−1)(μ1−μ2))[Nδi,j−1]. (2)
• In the second case when i.e. the tracer particle is driven, the factorization form of the joint distribution of the gaps does not hold. In this case the stationary state has a global current associated with the non-zero mean velocity of the tracer, which supports an inhomogeneous mean mass profile in the MT picture. We compute this average mass profile and it is given by

 mi=ω0(p+q)(N−1)+1 [2(p−q)i+(2qN−2p+1)]. (3)

We also compute the pair correlation in the steady state. In the thermodynamic limit i.e. for both, and limit, while keeping the mean gap density fixed, we find that the average mass , variance and the correlation scale as

 mi = ω0 M(iN)+o(1/N), (4) di,j = ω20 1N D(iN, jN)+o(1/N),i≠j (5) di,i = ω20[C0(iN)+1N C1(iN)+1N D(iN, iN)+o(1/N)], (6)

where represents order smaller that . We find explicit expressions of the scaling functions , and in (40), (43) and (47), respectively. The explicit expression of is given in (52) for whereas the expression for general can be obtained following the analysis given in Appendix C.

The paper is organized as follows. In section II the model and basic equations are written. We then study the RAP without driven tracer in section III, where we focus on factorized stationary distributions with a global constraint and on the jump distributions for which they occur. In the next section IV we study the influence of a driven tracer. For this case, we obtain single site gap distribution, mean gap and two-point correlations of the gaps in the thermodynamic limit. Finally, section V concludes the paper. Some details are given in the appendices A, B and C.

## Ii Model and basic equations

We consider particles moving on a ring of size ( see Fig.1) which are labeled as . Among these particles, let us consider the first particle as the tracer particle which may be driven. Their positions at time are denoted by for . The dynamics of the particles are given as follows. In an infinitesimal time interval to , any particle (say ) other than the tracer, jumps from , either to the right or to the left with probability and with probability it stays at . The tracer particle jumps from to the right with probability , to the left with probability and does not jump with probability . The jump length, either to the right or to the left, made by any particle is a random fraction of the space available between the particle and its neighboring particle to the right or to the left. For example, the particle jumps by an amount to the right and by an amount to the left. The random variables are independently chosen from the interval and each is distributed according to the same distribution . For most of our calculations in this paper we consider arbitrary , unless it is specified. Following the dynamics of the particles for the homogeneous case from (8), one can write down the stochastic evolution equations for the locations of the particles as:

 xi(t+dt) = ⎧⎪ ⎪⎨⎪ ⎪⎩xi(t)+ηri(xi+1(t)−xi(t)),  with~{}Prob.~{} R(ηri)dηri dt2,xi(t)+ηli(xi−1(t)−xi(t)),  with~{}Prob.~{} R(ηli)dηli dt2,     i≠1,xi(t),                              with~{}Prob.~{}   1−dt, (7) and x1(t+dt) = ⎧⎪⎨⎪⎩x1(t)+ηr1(x2(t)−x1(t)),   with~{}Prob.~{} R(ηr1)dηr1 pdt,x1(t)+ηl1(x0(t)−x1(t)),   with~{}Prob.~{} R(ηl1)dηl1 qdt,x1(t),                             with~{}Prob.~{} 1−(p+q)dt, (8)

where to impose the periodicity in the problem, we introduced two auxiliary particles with positions and for all . Note that the dynamics is invariant under simultaneous dilation of and of all the positions. In this paper, the results are however stated for general .

Since we are interested in the statistics of the gaps , it is convenient to work with an equivalent and appropriate mass transfer (MT) model with sites. The mass transfer model is defined as follows. Corresponding to the particles in the RAP, we consider a periodic one dimensional lattice of sites with a mass at each site. Particles from the RAP picture are mapped to the links between lattice sites in the MT picture. For example, the particle in RAP corresponds to the link between sites and in MT model whereas the mass at site is equal to the gap between the and particle in RAP. As a result for each configuration of the positions of the particles in the RAP we have a unique mass configuration in the MT model where . The opposite is however not true, as the instantaneous configuration of the masses leaves freedom for a global translation of the system in the RAP. Such mapping from particle model to mass model have been considered in other contexts like from the exclusion process to the zero range process (29).

Concerning the dynamics, a hop of the particle by towards the particle in the RAP corresponds to a transfer of mass from site to the site in the MT model whereas a hop of the particle by towards the particle in the RAP corresponds to a transfer of mass from site to the site. More precisely, the updating rules for the configurations in the MT model are given by :

 gi(t+dt)=gi(t)+σi+1rηi+1gi+1(t)+σi−1lηi−1gi−1(t)−(σil+σir) ηigi(t), (9)

where the variables are independent and identically distributed according to and are with probability and otherwise except for and . The random variable is with probability and with probability . Similarly, is with probability and with probability . The periodicity is imposed by and , for all .

Steady State Master Equation : The dynamics described in (9) will take the system eventually to a stationary state. If represents the steady state joint probability distribution of the configuration , then it satisfies the following master equation

 (N+p+q−1)PN,L(G)=N∑i=1∫∞0dg′i∫∞0dg′i+1∫10dη R(η) PN,L(G′i,i+1) × [( 1/2+(q−1/2)δi,N) δ(gi−g′i+ηg′i) δ(gi+1−g′i+1−ηg′i) (10) + ( 1/2+(p−1/2)δi,N) δ(gi−g′i−ηg′i+1) δ(gi+1−g′i+1+ηg′i+1)],

where we used the shorthand notation . The conservation of the number of particles follows naturally from the fact that the lattice size is fixed. Moreover, the dynamics clearly conserves the total mass i.e. we have , at all times. As stated in the introduction we are interested in the statistics of the gaps/masses in SS for the following two cases : (i) and (ii) . In the next section we analyze the first case.

## Iii Unbiased Tracer Case : p=q

In this case the tracer particle is not driven and all the particles are moving symmetrically. In the large time limit, the inter-particle spacings or gaps will reach a stationary state. To understand completely the statistics of the gaps in SS, one would require to find the joint probability distribution function (JPDF) of all the gaps . Clearly, this JPDF could be different for different choices of jump distributions . Finding this JPDF for arbitrary is generally a hard task. We ask a relatively simpler question : for what choices of the jump distribution does the stationary JPDF have a factorized form

 PN,L(G)=1ZN,LN∏i=1wi(gi) δ(N∑i=1gi−L)=WN,L(G)ZN,L, (11)

except for a global mass conservation condition represented by the delta function. Here ensures normalization. Analogous questions have also been asked in different other contexts, for example in mass transport models (6); (11); (16), in zero range processes (30); (31) and in finite range processes (32). In the following subsection we will see that there exists a large class of for which the above form for JPDF is true. Moreover we will see that the weights corresponding to individual sites are independent and the weight functions have universal form.

### iii.1 Determination of the weight functions wi(g)

To obtain the weight functions , we first insert from (11) in the SS master equation (II) and then integrate over all s except . We get

 2[1+(p−1/2)(δi,1+δi,N)] W(1)N,L(gi) (12) =∫∞0dgi−1∫∞0dg′i∫10dη R(η) [1/2+(p−1/2)δi,1][δ(gi−g′i+ηg′i)+δ(gi−g′i−ηgi−1)] W(2)N,L(gi−1,g′i) +∫∞0dgi+1∫∞0dg′i∫10dη R(η) [1/2+(p−1/2)δi,N][δ(gi−g′i+ηg′i)+δ(gi−g′i−ηgi+1)] W(2)N,L(g′i,gi+1),

where

 W(1)N,L(gi)=⎛⎝N∏j≠i∫∞0dgj ⎞⎠WN,L(G),  and  W(2)N,L(gi,gk)=⎛⎝N∏j≠i≠k∫∞0dgj ⎞⎠WN,L(G). (13)

We define the Laplace transform of any function as where is the Laplace conjugate of . Taking Laplace transform over as well as over on both sides of (12) we get

 2^wi(s+s′)^wi(s) = ∫1η=0dη R(η)^wi(s+(1−η)s′)^wi(s)+^wi(s+s′)^wi(s)∫1η=0dη R(η)[1/2+(p−1/2)δi,11+(p−1/2)(δi,1+δi,N)^wi−1(s+ηs′)^wi−1(s) (14) +1/2+(p−1/2)δi,N1+(p−1/2)(δi,1+δi,N)^wi+1(s+ηs′)^wi+1(s)],

where is Laplace conjugate to and is Laplace conjugate to . While deriving the above equation we have assumed that the Laplace transform of , defined by , exists. Equation (14) provides the condition satisfied by , in order to get a factorized JPDF as in (11). To find the solution for , let us expand both sides of (14) in powers of and equate coefficients of each power on both sides. One can easily see that at order , (14) is automatically satisfied because of the normalization . At order , we get

 2^w′i(s)^wi(s)=(^w′i−1(s)^wi−1(s)+^w′i+1(s)^wi+1(s))+(2p−1)(δi,1−δi,N)(^w′N(s)^wN(s)−^w′1(s)^w1(s)). (15)

The general solution of the above equation is where and are dependent constants. From the boundary conditions at and , we determine . Hence we find that is independent of site index . To proceed further, we now look at the expansion of (14) at order and get

 (μ2−μ1)^w′′(s)^w(s)+μ1(^w′(s)^w(s))2=0. (16)

The above equation can be easily solved to get

 ^w(s)=A0(s+B0)−β,   where,   β=μ1−μ2μ2, (17)

and , are constants. Taking inverse Laplace transform we get where is the Gamma function. Inserting this form of in (11) and absorbing all the constants in the normalization constant we arrive at the result

 PN,L(g1,g2,...,gN)=1ZN,L(β)N∏i=1gβ−1i  δ(N∑i=1gi−L), (18)

as stated in (1). The normalization constant can be computed to give

 ZN,L=ZN,L(β)=LβN−1Γ[β]NΓ[βN]. (19)

### iii.2 Jumping distributions that yield (18)

In the previous section we have seen that if there exists a jump distribution for which JPDF is in the form (11) then, should always be equal to for all where . Now we assume (18) to be true and find what conditions should satisfy. To get that condition we start with (12) which implies (14). Using now in (14) we find,

 2(1+u)β=∫10dη R(η)(1+(1−η)u)β+1(1+u)β∫10dη R(η)(1+ηu)β, (20)

for all . This is a necessary condition that should satisfy to get a factorized JPDF in SS. For distributions satisfying (20) such that the system is ergodic, we expect the stationary state to be unique. Consequently the condition (20) is sufficient to ensure that the steady state joint gap distribution is given by (18). Analogous conditions satisfied by hopping rates, have been obtained in other mass transport models (30); (33) and in finite range process (32).

We next find some solutions of (20) for particular values of , as well as a family of solutions that covers the whole range.

• Let us first consider the case. Formally, corresponds to , i.e. . For this choice of system breaks ergodicity and the stationary state is a trivial one with all the particles at a single point. As a result, .

• A very special case is , for which the stationary distribution  (18) is uniform over the allowed configurations. For this case some simple examples of the solutions are , which can be easily verified by directly inserting them in (20) with . Interestingly, one can find all the possible solutions of (20) in this particular case. Defining and , elementary manipulations show that for , eq. (20) is equivalent to

 ∫10dη (1−η)η2(2η−1)1+vη(1−η)[f(η)−f(1−η)]=0,∀ v>0. (21)

The factor multiplying on the RHS is always positive, which implies that is symmetric with respect to . Hence all possible solutions are of the form with for all . The converse is shown to be true iff the integral of is equal to . Therefore, the set of solutions for are

 R(η)=(1−η)f(η),  with,  f(η)=f(1−η),  % and  ∫10f(η) dη=2. (22)
• General : One can find a simple family of solutions that spans the whole range of values,

 R(η)=2β(1−η)2β−1, β>0. (23)

In particular, taking gives the uniform distribution.

• The situation is quite interesting for . One can check that the following scaling form solves  (20) for , where is a real, positive function for and its Laplace transform satisfies

 ∫∞0dy ϕ(y) (e−y(1+u)+e−y1+u)=^ϕ(1+u)+^ϕ(11+u)=2,   for   u≥0. (24)

Note that . There exists an infinite number of solutions of the above equation. For example, one class of solution can be chosen as where, both and , are polynomials of same order, say . There are constants corresponding to coefficients of different powers of associated to the two polynomials among which only are independent, as they are linked by (24). Now these coefficients have to be chosen such that and is real and positive for . The simplest of them is which gives the solution

 R(η)=2βe−2βη   for   β→∞. (25)

Some other examples of solutions in form are given in appendix (A).

The above analysis suggests that there are possibly infinitely many solutions of the (20) for any . Although we found all the solutions for and , and a somewhat simpler characterization of them for large , a proper characterization of all the possible solutions for arbitrary seems difficult. We are however able to make a few general remarks about the properties of the solutions of  (20).

• If and both are solutions of (20) corresponding to the same , then any normalized linear combination of them i.e. for is also a solution.

• Secondly, the system is invariant by dilation of time. Replacing by is equivalant to replacing by for . Indeed, with this latter choice a chosen particle hops with probability only, and nothing happens with probability . Clearly, the values of and depend on the value of however should not appear in the stationary distribution. The stationary distribution therefore cannot depend on and independently, but only through a particular combination in which cancels, i.e . All the solutions given above are understood up to a dilation of time.

Equation (20) necessary for getting the stationary state (18) is quite robust, as the same equation can be obtained in the case where all the particles are identical but asymmetrically moving. In a much more general case where each particle has its own hopping rates and to the right and to the left, respectively, it can be shown that one still obtains (20) as the condition for getting the factorized SS (18). But when all the particles have arbitrary jumping rates, possibly there is no jump distribution available for which such factorized states exists and this most general case has not been studied in the literature.

### iii.3 Single mass distribution

In this subsection we compute the marginal mass distribution and compare it to numerically obtained distributions for different jump distributions . The marginal distribution can be readily computed form by integrating out all s except . For factorized stationary states we get from (18) that

 PN,L(g)=1LΓ[βN]Γ[β]Γ[β(N−1)](gL)β−1(1−gL)βN−β−1,   0≤g≤L,  1≤i≤N. (26)

The corresponding moments of the above distribution for arbitrary and are

 ⟨gk⟩=LkΓ[β+k]Γ[βN]Γ[β]Γ[βN+k],   ∀k≥0. (27)

In the limit and with fixed density the marginal mass distribution in (26) becomes

 limN→∞Pω0N,N(g)=ββω0Γ[β](gω0)β−1e−gω0β, (28)

which matches with the previously known result (11) for the uniform case (). Similar gamma distribution for the inter-particle spacings has been obtained by Zielen and Schadschneider in the context of totally asymmetric RAP with parallel update on an infinite line (16). As in our case their stationary distribution of the gaps, also depends on jump distribution only through a particular combination of and given by , which is analogous to our parameter . However, in their case particles are moving on an infinite line. As a result their stationary state is completely factorized in contrast to our almost factorized form.

In Fig. 2, we compare the single site marginal gap/mass distribution in (26) with the same obtained numerically for different jump distributions . The details are given in the caption. The agreement between theory and numerics is perfect for the power law, sine and exponential distributions as expected, since they satisfy condition (20). Finally, for certain classes of jump distribution, such as that doesn’t satisfy the condition (20), the fact that the gap distribution still is numerically very close to (26) (see fig. 2: case ’AR’) is indeed very interesting. Similar (in spirit) results were also found in (34).

### iii.4 Variance and correlations of the gaps for arbitrary jump distribution R(η)

For arbitrary jump distributions that do not satisfy (20), the almost factorization of the stationary JPDF for the gaps does not hold in general and finding exact expression of JPDF is somewhat difficult. However, one can compute the mean, variance and correlations of the gaps in the stationary state for arbitrary . Multiplying both sides of the SS master equation (10) by and then integrating over all s, one can easily find that the average mass profile satisfies

 mi+1−2mi+mi−1=(2p−1)(δi,1−δi,N)(m1−mN),  for  i=1,2,...,N. (29)

This equation can be easily solved to get for all . Similarly multiplying both sides of (10) by and then integrating over all s one obtains the equation satisfied by the correlation function . The equation is given by

 μ1(4di,j−di,j+1−di,j−1−di+1,j−di−1,j)=μ2(δj,i−δj,i+1) (di+1,i+1+di,i)+μ2(δj,i−δj,i−1)(di−1,i−1+di,i) +μ1(δi,N−δi,1)(2p−1)(dj,1−dj,N)+μ1(δj,N−δj,1)(2p−1)(d1,i−dN,i) (30)