Multibalance conditions in nonequilibrium steady states

We consider a periodic ZRP lattice in two dimensions of size ($L\times L$). Each site $(i,j)$ with, $i=1,2,\cdots L,j=1,2,\cdots L$, can be vacant or it can be occupied by one or more particle $n_{i,j}\leq N$ $\left(N=\sum_{i=1}^{L}\sum_{j=1}^{L}n_{i,j}\right)$. A particle from any randomly chosen site can hop to its nearest neighbours (right, left, up and down) with rates $u_{r}(n_{i,j})$, $u_{l}(n_{i,j})$, $u_{u}(n_{i,j})$ and $u_{d}(n_{i,j})$ respectively as shown in Fig. 1.

ZRP does not have an exact steady state when hop rates in all four directions are different. The model is solvable using (a) DB, when the rates $u_{r}(n)=u_{l}(n)=\alpha u(n)$ and $u_{u}(n)=u_{d}(n)=\beta u(n)$, where both $\alpha$ and $\beta$ are constants or $u_{r}(n)=u_{l}(n)=u_{u}(n)=u_{d}(n)=u(n)$ and (b) PWB, when all rates $u_{r}(n)$, $u_{l}(n)$, $u_{u}(n)$ and $u_{d}(n)$ differ by a multiplicative constant i.e., the ratios of the rates are independent of $n$, assuming that the model evolves to a FSS

where, the canonical partition function

$N$ is the total number of particles and the density of the system $\rho=\frac{N}{L^{2}}$ is conserved by the dynamics. The steady state weight is defined as

Is it possible to obtain an exact steady state using other flux cancellation schemes for this asymmetric hopping when the hop rates in all four directions are different, that increases the regime of solvability. We consider a new balance condition namely multibalance (MB).

We consider a periodic ZRP lattice in three dimensions of size ($L\times L\times L$) (see Fig. 4). Each site, represented by $(i,j,k)$ can be either vacant or it can be occupied by one or more particles denoted by $n_{i,j,k}\leq N$ $\left(N=\sum_{i,j,k=1}^{L}n_{i,j,k}\right)$. From a randomly chosen site $(i,j,k)$, a particle can hop to nearest neighbours (up, down, right, left, frnt and back) with rates $u_{u}(n_{i,j,k})$, $u_{d}(n_{i,j,k})$, $u_{r}(n_{i,j,k})$, $u_{l}(n_{i,j,k})$, $u_{f}(n_{i,j,k})$ and $u_{b}(n_{i,j,k})$.

The steady state probability can be defined as

When all rates are different, no general solution is available. One can obtain the FSS as in Eq. (1) for this model using MB condition when the hop rates satisfy the condition

and the steady state weight is defined as $f(n)=\prod_{\nu=1}^{n}u(\nu)^{-1}$. As an example, let us define a simple choices of hop rates

The grand canonical partition function can be expressed as $Z_{L}(z)=\sum_{N=0}^{\infty}z^{N}Q_{L,N}=[F(z)]^{L}$ with $F(z)=\sum_{n=0}^{\infty}z^{n}f(n)=(z)^{-1}Li_{b}(z)$ and $Li_{b}(z)$ is the Polylogarithm function defined by $Li_{b}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{b}}$. Using our choices of rates in Eqs. (17) - (19) we can calculate the currents $J_{rl}$ (right - left direction) and $J_{fb}$ (front - back direction) as

In Fig. 5(a) and Fig. 5(b) we have plotted the currents $J_{rl}$ and $J_{fb}$ with the parameter $a$ for particle density $\rho=0.5$ and $b=1.1$. As expected currents in both directions increase linearly with the parameter $a$.

We will try to get the steady states of this model for the asymmetric rates. Consider the balance conditions as

1. 1.

   A PWB condition, where the flux generated due to a particle hopping from site $i$ of a configuration $C\equiv\{\cdots,n_{i-1},n_{i},n_{i+1},\cdots\}$ to site $(i+1)$, can be balanced with the flux obtained by a particle hopping from site $(i-1)$ of another configuration $C^{\prime}\equiv\{\cdots,n_{i-1}+1,n_{i}-1,n_{i+1},\cdots\}$ to site $i$.

   	$\displaystyle u_{R}(n_{i-K},\cdots,n_{i},\cdots n_{i+K})P(\{n_{i}\})=$	$\displaystyle u_{R}(n_{i-K-1},\cdots,n_{i-1}+1,n_{i}-1,\cdots n_{i+K-1})$			(24)
   		$\displaystyle\times P(\cdots,n_{i-1}+1,n_{i}-1,n_{i+1},\cdots).$			(25)

   Following Eq. (22), one can check that PWB condition as in Eq. (25) will be satisfied if the hop rate at site $i$ satisfies the condition as like FRP model [19]

   $$u_{R}(n_{i-K},\cdots,n_{i},\cdots,n_{i+K)})=\prod_{k=1}^{K}\frac{g(\widetilde{n}_{i-K+k},\widetilde{n}_{i-K+1+k},\cdots,\widetilde{n}_{i+k})}{g(n_{i-K+k},n_{i-K+1+k},\cdots,n_{i+k})}$$

   (26)

   where $\widetilde{n}_{j}=n_{j}-\delta_{ji}$
   

2. 2.

   A MB condition, where fluxes generated due to a particle hopping from site $i$ of a configuration $C\equiv\{\cdots,n_{i-1},n_{i},n_{i+1},\cdots\}$ to the sites $(i+2)$ and $(i-2)$, can be balanced with the flux obtained by a particle hopping from site $(i+1)$ of another configuration $C^{\prime\prime}\equiv\{\cdots,n_{i-1},n_{i}-1,n_{i+1}+1,\cdots\}$ to its left nearest neighbour site $i$. The flux balance scheme in Eq. (4) gives the following equation

   	$\displaystyle[~{}U_{R}(n_{i-K},\cdots,n_{i},\cdots n_{i+K})+U_{L}(n_{i-K},\cdots,n_{i},\cdots n_{i+K})~{}]~{}P(\{n_{i}\})$		(27)
   	$\displaystyle=u_{L}(n_{i-K+1},\cdots,n_{i-1},n_{i}-1,n_{i+1}+1,\cdots n_{i+K+1})~{}P(\cdots,n_{i-1},n_{i}-1,n_{i+1}+1,\cdots).$		(28)
   			(29)

MB condition as in Eq. (29) for this asymmetric FRP model will be satisfied and one can obtain a cluster-factorized form of steady state (CFSS) as in Eq. (22), when $u_{R}(.)=u_{L}(.)=u(.)$ and the hop rates satisfy the condition

where $\widetilde{n}_{j}=n_{j}-\delta_{ji}$

For $K=1$, we can obtain the steady states in pair-factorized form using MB, if the hop rates satisfy, following Eq. (31),

Let us consider that the weight function $g(n_{i},n_{i+1})$ can be written as the inner product of two 2-dimensional vectors [19]

In grand canonical ensemble where the fugacity $z$ controls the density $\rho$, the partition sum can be written as $Z_{L}(z)=\sum_{N=0}^{\infty}Q_{L,N}z^{N}=\Tr[T(z)^{L}]$ with

Now, for a particular choice of the steady state weight $g(n_{i},n_{i+1})$, one can construct the transfer matrix $T(z)$ to calculate the density of the system $\rho=z\left(\frac{Z_{L}^{\prime}(z)}{Z_{L}(z)}\right)$ and the correlation function by transfer matrix method in pair-factorized steady state [19].

The model is defined on an one dimensional periodic lattice with sites labeled by $i=1,2,\cdots,L$ (see Fig. 7). At each site $i$, there are $n_{i}$ particles of species $A$ (coloured red) and $m_{i}$ particles of species $B$ (cloured blue). Total number of particle $A$ is $N$ and that of particle $B$ is $M$. A particle of any species, from any randomly chosen site $i$ can hop to its right nearest neighbour with a rate $u(.)$ for species $A$ and $v(.)$ for species $B$. These two rates depend on the number of particles at all the sites which are within a range $K$ w.r.t the departure site like the FRP model [19].

Dynamics of this model can be described as for species $A$

and the dynamics of species $B$

For $K=0$, this model is identical to two species zero range process [7] with hop rate $u(.)$ for particles of species $A$, and $v(.)$ for particles of species $B$, an exactly solvable nonequilibrium model that evolves to a FSS

with the steady state weight

We can add a directional asymmetry in two species FRP model (see Fig. 8), by adding conditions, from a randomly chosen site $i$, the particle of species $A$, can hop to its right and left nearest neighbours with rates $u_{R}(.)$ and $u_{L}(.)$, it can hop to right and left next nearest neighbours with rates $U_{R}(.)$ and $U_{L}(.)$. Similarly, particle of species $B$, can hop to its right and left nearest neighbours with rates $v_{R}(.)$ and $v_{L}(.)$, to right and left next nearest neighbours with rates $V_{R}(.)$ and $V_{L}(.)$. All these rates depend on the number of particles at all the sites which are within a range $K$ w.r.t the departure site Fig. 8.