Polymers with nearest- and next nearest-neighbor interactions on the Husimi lattice

The exact grand-canonical solution of a generalized interacting self-avoid walk (ISAW) model, placed on a Husimi lattice built with squares, is presented. In this model, beyond the traditional interaction $\omega_1=e^{\epsilon_1/k_B T}$ between (nonconsecutive) monomers on nearest-neighbor (NN) sites, an additional energy $\epsilon_2$ is associated to next-NN (NNN) monomers. Three definitions of NNN sites/interactions are considered, where each monomer can have, effectively, at most 2, 4 or 6 NNN monomers on the Husimi lattice. The phase diagrams found in all cases have (qualitatively) the same thermodynamic properties: a non-polymerized (NP) and a polymerized (P) phase separated by a critical and a coexistence surface that meet at a tricritical ($\theta$-) line. This $\theta$-line is found even when one of the interactions is repulsive, existing for $\omega_1$ in the range $[0,\infty)$, i. e., for $\epsilon_1/k_B T$ in the range $[-\infty,\infty)$. Counterintuitively, a $\theta$-point exists even for an infinite repulsion between NN monomers ($\omega_1=0$), being associated to a coil-"soft globule" transition. In the limit of an infinite repulsive force between NNN monomers, however, the coil-globule transition disappears and only a NP-P continuous transition is observed. This particular case, with $\omega_2=0$, is also solved exactly on the square lattice, using a transfer matrix calculation, where a discontinuous NP-P transition is found. For attractive and repulsive forces between NN and NNN monomers, respectively, the model becomes quite similar to the semiflexible-ISAW one, whose crystalline phase is not observed here, as a consequence of the frustration due to competing NN and NNN forces. The mapping of the phase diagrams in canonical ones is discussed and compared with recent results from Monte Carlo simulations.Comment: 21 pages, 6 figure

Collapse transition in polymer models with multiple monomers per site and multiple bonds per edge

We present results from extensive Monte Carlo simulations of polymer models where each lattice site can be visited by up to $K$ monomers and no restriction is imposed on the number of bonds on each lattice edge. These \textit{multiple monomer per site} (MMS) models are investigated on the square and cubic lattices, for $K=2$ and $K=3$, by associating Boltzmann weights $\omega_0=1$, $\omega_1=e^{\beta_1}$ and $\omega_2=e^{\beta_2}$ to sites visited by 1, 2 and 3 monomers, respectively. Two versions of the MMS models are considered for which immediate reversals of the walks are allowed (RA) or forbidden (RF). In contrast to previous simulations of these models, we find the same thermodynamic behavior for both RA and RF versions. In three-dimensions, the phase diagrams - in space $\beta_2 \times \beta_1$ - are featured by coil and globule phases separated by a line of $\Theta$ points, as thoroughly demonstrated by the metric $\nu_t$, crossover $\phi_t$ and entropic $\gamma_t$ exponents. The existence of the $\Theta$-lines is also confirmed by the second virial coefficient. This shows that no discontinuous collapse transition exists in these models, in contrast to previous claims based on a weak bimodality observed in some distributions, which indeed exists in a narrow region very close to the $\Theta$-line when $\beta_1 < 0$. Interestingly, in two-dimensions, only a crossover is found between the coil and globule phases

Nature of the collapse transition in interacting self-avoiding trails

We study the interacting self-avoiding trail (ISAT) model on a Bethe lattice of general coordination $q$ and on a Husimi lattice built with squares and coordination $q=4$. The exact grand-canonical solutions of the model are obtained, considering that up to $K$ monomers can be placed on a site and associating a weight $\omega_i$ for a $i$-fold visited site. Very rich phase diagrams are found with non-polymerized (NP), regular polymerized (P) and dense polymerized (DP) phases separated by lines (or surfaces) of continuous and discontinuous transitions. For Bethe lattice with $q=4$ and $K=2$, the collapse transition is identified with a bicritical point and the collapsed phase is associated to the dense polymerized phase (solid-like) instead of the regular polymerized phase (liquid-like). A similar result is found for the Husimi lattice, which may explain the difference between the collapse transition for ISAT's and for interacting self-avoiding walks on the square lattice. For $q=6$ and $K=3$ (studied on the Bethe lattice only), a more complex phase diagram is found, with two critical planes and two coexistence surfaces, separated by two tricritical and two critical end-point lines meeting at a multicritical point. The mapping of the phase diagrams in the canonical ensemble is discussed and compared with simulational results for regular lattices.Comment: 12 pages, 13 figure

Kardar-Parisi-Zhang growth on one-dimensional decreasing substrates

Recent experimental works on one-dimensional (1D) circular Kardar-Parisi-Zhang (KPZ) systems whose radii decrease in time have reported controversial conclusions about the statistics of their interfaces. Motivated by this, we investigate here several 1D KPZ models on substrates whose size changes in time as $L(t)=L_0 + \omega t$, focusing on the case $\omega<0$. From extensive numerical simulations, we show that for $L_0 \gg 1$ there exists a transient regime in which the statistics is consistent with that of flat KPZ systems (the $\omega=0$ case), for both $\omega0$. Actually, for a given model, $L_0$ and $|\omega|$, we observe that a difference between ingrowing ($\omega0$) systems arises only at long times ($t \gtrsim t_c=L_0/|\omega|$), when the expanding surfaces cross over to the statistics of curved KPZ systems, whereas the shrinking ones become completely correlated. A generalization of the Family-Vicsek scaling for the roughness of ingrowing interfaces is presented. Our results demonstrate that a transient flat statistics is a general feature of systems starting with large initial sizes, regardless their curvature. This is consistent with their recent observation in ingrowing turbulent liquid crystal interfaces, but it is in contrast with the apparent observation of curved statistics in colloidal deposition at the edge of evaporating drops. A possible explanation for this last result, as a consequence of the very small number of monolayers analyzed in this experiment, is given. This is illustrated in a competitive growth model presenting a few-monolayer transient and an asymptotic behavior consistent, respectively, with the curved and flat statistics.Comment: 5 pages, 3 figure

Solution on the Bethe lattice of a hard core athermal gas with two kinds of particles

Athermal lattice gases of particles with first neighbor exclusion have been studied for a long time as simple models exhibiting a fluid-solid transition. At low concentration the particles occupy randomly both sublattices, but as the concentration is increased one of the sublattices is occupied preferentially. Here we study a mixed lattice gas with excluded volume interactions only in the grand-canonical formalism with two kinds of particles: small ones, which occupy a single lattice site and large ones, which occupy one site and its first neighbors. We solve the model on a Bethe lattice of arbitrary coordination number $q$. In the parameter space defined by the activities of both particles. At low values of the activity of small particles ($z_1$) we find a continuous transition from the fluid to the solid phase as the activity of large particles ($z_2$) is increased. At higher values of $z_1$ the transition becomes discontinuous, both regimes are separated by a tricritical point. The critical line has a negative slope at $z_1=0$ and displays a minimum before reaching the tricritical point, so that a reentrant behavior is observed for constant values of $z_2$ in the region of low density of small particles. The isobaric curves of the total density of particles as a function of $z_1$ (or $z_2$) show a minimum in the fluid phase.Comment: 18 pages, 5 figures, 1 tabl

On the origins of scaling corrections in ballistic growth models

We study the ballistic deposition and the grain deposition models on two-dimensional substrates. Using the Kardar-Parisi-Zhang (KPZ) ansatz for height fluctuations, we show that the main contribution to the intrinsic width, which causes strong corrections to the scaling, comes from the fluctuations in the height increments along deposition events. Accounting for this correction in the scaling analysis, we obtained scaling exponents in excellent agreement with the KPZ class. We also propose a method to suppress these corrections, which consists in divide the surface in bins of size $\varepsilon$ and use only the maximal height inside each bin to do the statistics. Again, scaling exponents in remarkable agreement with the KPZ class were found. The binning method allowed the accurate determination of the height distributions of the ballistic models in both growth and steady state regimes, providing the universal underlying fluctuations foreseen for KPZ class in 2+1 dimensions. Our results provide complete and conclusive evidences that the ballistic model belongs to the KPZ universality class in $2+1$ dimensions. Potential applications of the methods developed here, in both numerics and experiments, are discussed.Comment: 8 pages, 7 figure