Phase coexistence and spatial correlations in reconstituting $k$-mer models

In reconstituting $k$-mer models, extended objects that occupy several sites on a one-dimensional lattice undergo directed or undirected diffusion, and reconstitute—when in contact—by transferring a single monomer unit from one $k$-mer to the other; the rates depend on the size of participating $k$-mers. This polydispersed system has two conserved quantities, the number of $k$-mers and the packing fraction. We provide a matrix product method to write the steady state of this model and to calculate the spatial correlation functions analytically. We show that for a constant reconstitution rate, the spatial correlation exhibits damped oscillations in some density regions separated, from other regions with exponential decay, by a disorder surface. In a specific limit, this constant-rate reconstitution model is equivalent to a single dimer model and exhibits a phase coexistence similar to the one observed earlier in totally asymmetric simple exclusion process on a ring with a defect.

Figure 1

Polydispersed $k$-mers in one dimension showing drift and reconstitution. Engine of the $k$-mers are marked as open circles. The drift rate $u\left(k\right)$ depends on the length of the corresponding $k$-mers. Reconstitution occurs only among consecutive immobile $k$-mers with a rate $w(k_i,k_{i+1})$, which depends on their lengths. An additional constraint $w(k,k^{\prime })=0$ for $k<2$, conserves the number of $k$-mers in the system.

Figure 2

Disorder surface separating regions where pair correlation function $C\left(r\right)$ decays exponentially (all three eigenvalues are real) from regions having damped oscillations (where the subdominant eigenvalues are complex).

Figure 3

Densities: (a) packing fraction $\eta$ and (b) $k$-mer density $\rho$, as a function of $z$ for $x=0.01,0.1,1,10.$

Figure 4

Correlation functions: (a) For $v=2,$ complex roots appear for $x>z/2$, (b) the corresponding line in $\rho -\eta$ plane, (c) engine-engine correlation function $C\left(r\right)$ for $z=\frac{1}{2},1,\frac{3}{2}$, and (d) $z=\frac{5}{2},3,\frac{7}{2}.$ For (c) and (d), we use $x=1$, and the symbols used here for different $z$ are also marked in (a) and (b).

Figure 5

(a) The density profile $\rho \left(i\right)$: (a) for $L=200,v=2$, and different $z=1,1.5,1.8,1.9,1.99.$ For small $z,\rho \left(i\right)$ saturates to the macroscopic bulk density $v/(v+z).$ For large $z$ near $z_c$, such a saturation can be seen by increasing the system size $L$—this is shown in (b). The profile for $z=1.9$ (marked as dashed line) as a function of $i/L$ shows saturation as $L$ increased.