Nearest-neighbor Exclusion Research Articles

Density and Correlation in a Random Sequential Adsorption Model

We consider the random sequential adsorption process on the one- dimensional lattice with nearest-neighbor exclusion. In this model, each site s 2 Z starts empty and a particle will be deposited in it at time ts, where (ts)s2Z is a sequence of independent random variables uniformly distributed on the interval [0; 1]. The site will be occupied if both of its neighbors are vacant. Analytical expressions for the density of occupied sites and the pair correlation function, for all time t, are well-established and have been obtained through methods such as generating functions and differential equations. In this study, we present a method based on probabilistic arguments for the calculation of these expressions.

Gumbel Laws in the Symmetric Exclusion Process

We consider the symmetric exclusion particle system on $\mathbb{Z}$ starting from an infinite particle step configuration in which there are no particles to the right of a maximal one. We show that the scaled position $X_t/(\sigma b_t) - a_t$ of the right-most particle at time $t$ converges to a Gumbel limit law, where $b_t = \sqrt{t/\log t}$, $a_t = \log(t/(\sqrt{2\pi}\log t))$, and $\sigma$ is the standard deviation of the random walk jump probabilities. This work solves a problem left open in Arratia (1983). Moreover, to investigate the influence of the mass of particles behind the leading one, we consider initial profiles consisting of a block of $L$ particles, where $L \to \infty$ as $t \to \infty$. Gumbel limit laws, under appropriate scaling, are obtained for $X_t$ when $L$ diverges in $t$. In particular, there is a transition when $L$ is of order $b_t$, above which the displacement of $X_t$ is similar to that under a infinite particle step profile, and below which it is of order $\sqrt{t\log L}$. Proofs are based on recently developed negative dependence properties of the symmetric exclusion system. Remarks are also made on the behavior of the right-most particle starting from a step profile in asymmetric nearest-neighbor exclusion, which complement known results.

Steady-state thermodynamics: Description equivalence and violation of reservoir independence.

For stochastic lattice models in spatially uniform nonequilibrium steady states, an effective thermodynamic temperature T and chemical potential μ can be defined via coexistence with heat and particle reservoirs. We verify that the probability distribution P_{N} for the number of particles in the driven lattice gas with nearest-neighbor exclusion in contact with a particle reservoir with dimensionless chemical potential µ^{*} possesses a large-deviation form in the thermodynamic limit. This implies that the thermodynamic properties determined in isolation (fixed particle number representation) and in contact with a particle reservoir (fixed dimensionless chemical potential representation) are equal. We refer to this as description equivalence. This finding motivates investigation of whether the effective intensive parameters so obtained depend on the nature of the exchange between system and reservoir. For example, a stochastic particle reservoir is usually taken to insert or remove a single particle in each exchange, but one may also consider a reservoir that inserts or removes a pair of particles in each event. In equilibrium, equivalence of pair and single-particle reservoirs is guaranteed by the canonical form of the probability distribution on configuration space. Remarkably, this equivalence is violated in nonequilibrium steady states, limiting the generality of steady-state thermodynamics based on intensive variables.

How SipA, a toxin from Salmonella, increases stability of F-actin.

Salmonella infection is a disease that is mainly limited to the intestinal tract, but can also be systemic and may be deadly for young children or people with a weakened immune system. Salmonella strains produce protein toxins that target cytoplasmic actin to promote bacterial uptake by a host cell, which is essential for pathogenicity. The SipA protein is a Salmonella effector delivered into the host cell by bacterial type III secretion system. SipA has been reported to stabilize F-actin upon binding, and to promote actin polymerization in vitro. Only very low resolution reconstructions from negative stain existed to show how SipA binds to actin. Here, we present a 3.5Å cryo-EM reconstruction of the SipA425-685/F-actin complex. Our reconstruction shows that SipA binds actin with nearest neighbor exclusion along actin's long-pitch strand. The globular SipA domain binds actin at subunit n through ionic and hydrophobic interactions with both actin strands with nanomolar affinity. The extended C-terminal tail is inserted in between the two actin strands at subunit n+2 preventing the binding of another SipA molecule at this site. Thus, SipA will be bound at n, n+1, n+4, n+5, etc. The C-terminal tail of SipA binds to the same site as two F-actin toxins, phalloidin (Pha) and jasplakinolide (Jasp). All three molecules interact with actin mainly through hydrophobic interactions, however SipA binds F-actin with higher affinity than Pha or Jasp, and outcompetes the latter two in competitive binding assay. The SipA binding involves conformational changes of actin's D-loop which provides additional intra-strand interactions. The observed stabilization of actin filaments upon SipA binding is therefore likely due to the introduction of additional inter- and intra-strand interactions within the actin filament.

Hard core lattice gas with third next-nearest neighbor exclusion on triangular lattice: One or two phase transitions?

We obtain the phase diagram of the hard core lattice gas with third nearest neighbor exclusion on the triangular lattice using Monte Carlo simulations that are based on a rejection-free flat histogram algorithm. In a recent paper [Darjani et al., J. Chem. Phys. 151, 104702 (2019)], it was claimed that the lattice gas with third nearest neighbor exclusion undergoes two phase transitions with increasing density with the phase at intermediate densities exhibiting hexatic order with continuously varying exponents. Although a hexatic phase is expected when the exclusion range is large, it has not been seen earlier in hard core lattice gases with short range exclusion. In this paper, by numerically determining the entropies for all densities, we show that there is only a single phase transition in the system between a low-density fluid phase and a high density ordered sublattice phase and that a hexatic phase is absent. The transition is shown to be first order in nature, and the critical parameters are determined accurately.

Measuring pressure in equilibrium and nonequilibrium lattice-gas models.

We develop an algorithm based on the method proposed by Dickman for directly measuring pressure in lattice-gas models. The algorithm gives the possibility to access the equation of state with a single run by adding multiple ghost sites to the original system. This feature considerably improves calculations and makes the algorithm particularly efficient for systems with inhomogeneous density profiles, both in equilibrium and nonequilibrium steady states. We illustrate its broad applicability by considering some paradigmatic systems of statistical mechanics such as the lattice gas under gravity, nearest-neighbor exclusion models in finite dimension and on regular random graphs, and the boundary-driven simple symmetric exclusion process.

Steady-state entropy: A proposal based on thermodynamic integration

Defining an entropy function out of equilibrium is an outstanding challenge. For stochastic lattice models in spatially uniform nonequilibrium steady states, definitions of temperature T and chemical potential μ have been verified using coexistence with heat and particle reservoirs. For an appropriate choice of exchange rates, T and μ satisfy the zeroth law, marking an important step in the development of steady-state thermodynamics. These results suggest that an associated steady-state entropy S_{th} be constructed via thermodynamic integration, using relations such as (∂S/∂E)_{V,N}=1/T, ensuring that derivatives of S_{th} with respect to energy and particle number yield the expected intensive parameters. We determine via direct calculation the stationary nonequilibrium probability distribution of the driven lattice gas with nearest-neighbor exclusion, the Katz-Lebowitz-Spohn driven lattice gas, and a two-temperature Ising model so that we may evaluate the Shannon entropy S_{S} as well as S_{th} defined above. Although the two entropies are identical in equilibrium (as expected), they differ out of equilibrium; for small values of the drive, D, we find |S_{S}-S_{th}|∝D^{2}, as expected on the basis of symmetry. We verify that S_{th} is not a state function: changes ΔS_{th} depend not only on the initial and final points, but also on the path in parameter space. The inequivalence of S_{S} and S_{th} implies that derivatives of S_{S} are not predictive of coexistence. In other words, a nonequilibrium steady state is not determined by maximizing the Shannon entropy. Our results cast doubt on the possibility of defining a state function that plays the role of a thermodynamic entropy for nonequilibrium steady states.

Independent sets on the Towers of Hanoi graphs

The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. In this article, we mainly study the number of independent sets i(Hn) on the Tower of Hanoi graph Hn at stage n, and derive the recursion relations for the numbers of independent sets. Upper and lower bounds for the asymptotic growth constant μ on the Towers of Hanoi graphs are derived in terms of the numbers at a certain stage, where μ = limv → ∞(lni(G)) / v(G) and v(G) is the number of vertices in a graph G. Furthermore, we also consider the number of independent sets on the Sierpiński graphs which contain the Towers of Hanoi graphs as a special case.

Reentrant disorder-disorder transitions in generalized multicomponent Widom-Rowlinson models.

In the lattice version of the multicomponent Widom-Rowlinson (WR) model, each site can be either empty or singly occupied by one of M different particles, all species having the same fugacity z. The only nonzero interaction potential is a nearest-neighbor hard-core exclusion between unlike particles. For M<M(0) with some minimum M(0) dependent on the lattice structure, as z increases from 0 to ∞ there is a direct transition from the disordered (gas) phase to a demixed (liquid) phase with one majority component at z>z(d) (M). If M≥M(0), there is an intermediate ordered "crystal phase" (composed of two nonequivalent even and odd sublattices) for z lying between z(c)(M) and z(d)(M) which is driven by entropy. We generalize the multicomponent WR model by replacing the hard-core exclusion between unlike particles by more realistic large (but finite) repulsion. The model is solved exactly on the Bethe lattice with an arbitrary coordination number. The numerical calculations, based on the corner transfer matrix renormalization group, are performed for the two-dimensional square lattice. The results for M=4 indicate that the second-order phase transitions from the disordered gas to the demixed phase become of first order, for an arbitrarily large finite repulsion. The results for M≥M(0) show that, as the repulsion weakens, the region of the crystal phase diminishes itself. For weak enough repulsions, the direct transition between the crystal and demixed phases changes into a separate pair of crystal-gas and gas-demixed transitions; this is an example of a disorder-disorder reentrant transition via an ordered crystal phase. If the repulsion between unlike species is too weak, the crystal phase disappears from the phase diagram. It is shown that the generalized WR model belongs to the Ising universality class.

Mixed-Sequence Recognition of Double-Stranded DNA Using Enzymatically Stable Phosphorothioate Invader Probes.

Development of probes that allow for sequence-unrestricted recognition of double-stranded DNA (dsDNA) continues to attract much attention due to the prospect for molecular tools that enable detection, regulation, and manipulation of genes. We have recently introduced so-called Invader probes as alternatives to more established approaches such as triplex-forming oligonucleotides, peptide nucleic acids and polyamides. These short DNA duplexes are activated for dsDNA recognition by installment of +1 interstrand zippers of intercalator-functionalized nucleotides such as 2′-N-(pyren-1-yl)methyl-2′-N-methyl-2′-aminouridine and 2′-O-(pyren-1-yl)methyluridine, which results in violation of the nearest neighbor exclusion principle and duplex destabilization. The individual probes strands have high affinity toward complementary DNA strands, which generates the driving force for recognition of mixed-sequence dsDNA regions. In the present article, we characterize Invader probes that are based on phosphorothioate backbones (PS-DNA Invaders). The change from the regular phosphodiester backbone furnishes Invader probes that are much more stable to nucleolytic degradation, while displaying acceptable dsDNA-recognition efficiency. PS-DNA Invader probes therefore present themselves as interesting probes for dsDNA-targeting applications in cellular environments and living organisms.

Gibbs measures for the fertile three-state hard-core models on a Cayley tree

We study translation-invariant splitting Gibbs measures (TISGMs, tree-indexed Markov chains) for the fertile three-state hard-core models with activity $$\lambda >0$$?>0 on the Cayley tree of order $$k\ge 1$$k?1. There are four such models: wrench, wand, hinge, and pipe. These models arise as simple examples of loss networks with nearest-neighbor exclusion. It is known that (i) for the wrench and pipe cases $$\forall \lambda >0$$??>0 and $$k\ge 1$$k?1, there exists a unique TISGM; (ii) for hinge (resp. wand) case at $$k=2$$k=2 if $$\lambda 9/4$$?>9/4 (resp. $$\lambda >1$$?>1), there exist three TISGMs. In this paper we generalize the result (ii) for any $$k\ge 2$$k?2, i.e., for hinge and wand cases we find the exact critical value $$\lambda _\mathrm{cr}(k)$$?cr(k) with properties mentioned in (ii). Moreover, we find some regions for the $$\lambda $$? parameter ensuring that a given TISGM is extreme or non-extreme in the set of all Gibbs measures. For the Cayley tree of order two, we give explicit formulae and some numerical values.

Failure of steady-state thermodynamics in nonuniform driven lattice gases.

To be useful, steady-state thermodynamics (SST) must be self-consistent and have predictive value. Consistency of SST was recently verified for driven lattice gases under global weak exchange. Here I verify consistency of SST under local (pointwise) exchange, but only in the limit of a vanishing exchange rate; for a finite exchange rate the coexisting systems have different chemical potentials. I consider the lattice gas with nearest-neighbor exclusion on the square lattice, with nearest-neighbor hopping, and with hopping to both nearest and next-nearest neighbors. I show that SST does not predict the coexisting densities under a nonuniform drive or in the presence of a nonuniform density provoked by a hard wall or nonuniform transition rates. The steady-state chemical potential profile is, moreover, nonuniform at coexistence, contrary to the basic principles of thermodynamics. Finally, I discuss examples of a pair of systems possessing identical steady states but which do not coexist when placed in contact. The results of these studies confirm the validity of SST for coexistence between spatially uniform systems but cast serious doubt on its consistency and predictive value in systems with a finite rate of particle exchange between coexisting regions exhibiting a nonuniform particle density.

Inconsistencies in steady-state thermodynamics

We address the issue of extending thermodynamics to nonequilibrium steady states. Using driven stochastic lattice gases, we ask whether consistent definitions of an effective chemical potential μ, and an effective temperature Te, are possible. μ and Te are determined via coexistence, i.e., zero flux of particles and energy between the driven system and a reservoir. In the lattice gas with nearest-neighbor exclusion, temperature is not relevant, and we find that the effective chemical potential, a function of density and drive strength, satisfies the zeroth law, and correctly predicts the densities of coexisting systems. In the Katz-Lebowitz-Spohn driven lattice gas both μ and Te need to be defined. We show analytically that in this case the zeroth law is violated for Metropolis exchange rates, and determine the size of the violations numerically. The zeroth law appears to be violated for generic exchange rates. Remarkably, the system-reservoir coupling proposed by Sasa and Tasaki [J. Stat. Phys. 125, 125 (2006)] is free of inconsistencies, and the zeroth law holds. This is because the rate depends only on the state of the donor system, and is independent of that of the acceptor.

Ising-like transitions in theO(n)loop model on the square lattice

We explore the phase diagram of the O(n) loop model on the square lattice in the (x,n) plane, where x is the weight of a lattice edge covered by a loop. These results are based on transfer-matrix calculations and finite-size scaling. We express the correlation length associated with the staggered loop density in the transfer-matrix eigenvalues. The finite-size data for this correlation length, combined with the scaling formula, reveal the location of critical lines in the diagram. For n>>2 we find Ising-like phase transitions associated with the onset of a checkerboardlike ordering of the elementary loops, i.e., the smallest possible loops, with the size of an elementary face, which cover precisely one-half of the faces of the square lattice at the maximum loop density. In this respect, the ordered state resembles that of the hard-square lattice gas with nearest-neighbor exclusion, and the finiteness of n represents a softening of its particle-particle potentials. We also determine critical points in the range -2≤n≤2. It is found that the topology of the phase diagram depends on the set of allowed vertices of the loop model. Depending on the choice of this set, the n>2 transition may continue into the dense phase of the n≤2 loop model, or continue as a line of n≤2 O(n) multicritical points.

Large deviations for the current and tagged particle in 1D nearest-neighbor symmetric simple exclusion

Laws of large numbers, starting from certain nonequilibrium measures, have been shown for the integrated current across a bond, and a tagged particle in one-dimensional symmetric nearest-neighbor simple exclusion [Ann. Inst. Henri Poincare Probab. Stat. 42 (2006) 567-577]. In this article, we prove corresponding large deviation principles and evaluate the rate functions, showing different growth behaviors near and far from their zeroes which connect with results in [J. Stat. Phys. 136 (2009) 1-15].

Asymptotic enumeration of independent sets on the Sierpinski gasket

The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. We study the number of independent sets md,b(n) on the generalized Sierpinski gasket SGd,b(n) at stage n with dimension d equal to two, three and four for b = 2, and layer b equal to three for d = 2. Upper and lower bounds for the asymptotic growth constant, defined as ZSGdb = limu(∞ ln md,b (n)/v where v is the number of vertices, on these Sierpinski gaskets are derived in terms of the numbers at a certain stage. The numerical values of these ZSGdb are evaluated with more than a hundred significant figures accurate. We also conjecture upper and lower bounds for the asymptotic growth constant ZSGd2 with general d, and an approximation of ZSGd2 when d is large.

Entanglement spectra of the quantum hard-square model: Holographic minimal models

We study the entanglement properties of a quantum lattice-gas model for which we can find the exact ground state (of the Rokhsar-Kivelson type). The ground state can be expressed as a superposition of states, each of which is characterized by a particle configuration with nearest-neighbor exclusion. We show that the reduced density matrix of the model on a ladder is intimately related to the transfer matrix of the classical hard-square model. The entanglement spectra of the model on square and triangular ladders are critical when parameters are chosen so that the corresponding classical hard-square models are critical. A detailed analysis reveals that the critical theories for the entanglement Hamiltonians are $c&lt;1$ minimal conformal field theories. We further show that the entanglement Hamiltonian for the triangular ladder is integrable despite the fact that the original quantum lattice-gas model is nonintegrable.

Discontinuous phase transition in a dimer lattice gas

I study a dimer model on the square lattice with nearest neighbor exclusion as the only interaction. Detailed simulations using tomographic entropic sampling show that as the chemical potential is varied, there is a strongly discontinuous phase transition, at which the particle density jumps by about 18% of its maximum value, 1/4. The transition is accompanied by the onset of orientational order, to an arrangement corresponding to the {1/2, 0, 1/2} structure identified by Phares et al. [Physica B 409, 1096 (2011)] in a dimer model with finite repulsion at fixed density. Using finite-size scaling and Binder's cumulant, the expected scaling behavior at a discontinuous transition is verified in detail. The discontinuous transition can be understood qualitatively given that the model possesses eight equivalent maximum-density configurations, so that its coarse-grained description corresponds to that of the q = 8 Potts model.

Complete high-precision entropic sampling

Monte Carlo simulations using entropic sampling to estimate the number of configurations of a given energy are a valuable alternative to traditional methods. We introduce tomographic entropic sampling, a scheme which uses multiple studies, starting from different regions of configuration space, to yield precise estimates of the number of configurations over the full range of energies, without dividing the latter into subsets or windows. Applied to the Ising model on the square lattice, the method yields the critical temperature to an accuracy of about 0.01%, and critical exponents to 1% or better. Predictions for system sizes L=10-160, for the temperature of the specific heat maximum, and of the specific heat at the critical temperature, are in very close agreement with exact results. For the Ising model on the simple cubic lattice the critical temperature is given to within 0.003% of the best available estimate; the exponent ratios β/ν and γ/ν are given to within about 0.04% and 1%, respectively, of the literature values. In both two and three dimensions, results for the antiferromagnetic critical point are fully consistent with those of the ferromagnetic transition. Application to the lattice gas with nearest-neighbor exclusion on the square lattice again yields the critical chemical potential and exponent ratios β/ν and γ/ν to good precision.

Hydrodynamic Limit for a Type of Exclusion Process with Slow Bonds in Dimension d ≥ 2

Let Λ be a connected closed region with smooth boundary contained in the d-dimensional continuous torus T d . In the discrete torus N -1 T d N , we consider a nearest-neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on Λ in the following way: if both sites are in Λ or Λc, the exchange rate is 1; if one site is in Λ and the other site is in Λc, and the direction of the bond connecting the sites is e j , then the exchange rate is defined as N -1 times the absolute value of the inner product between e j and the normal exterior vector to ∂Λ. We show that this exclusion-type process has a nontrivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by ∂Λ. Thus, the model represents a system of particles under hard-core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementary regions.