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Register a new userExact Solution for a 1-dimensional model for Reptation
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We discuss the exact solution for the properties of the recently introduced ``necklace model for reptation. The solution gives the drift velocity, diffusion constant and renewal time for asymptotically long chains. Its properties are also related to a special case of the Rubinstein-Duke model in one dimension.
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Through a series of exact mappings we reinterpret the Bernoulli model of sequence alignment in terms of the discrete-time totally asymmetric exclusion process with backward sequential update and step function initial condition. Using earlier results from the Bethe ansatz we obtain analytically the exact distribution of the length of the longest common subsequence of two sequences of finite lengths $X,Y$. Asymptotic analysis adapted from random matrix theory allows us to derive the thermodynamic limit directly from the finite-size result.
The exact nonequilibrium steady state solution of the nonlinear Boltzmann equation for a driven inelastic Maxwell model was obtained by Ben-Naim and Krapivsky [Phys. Rev. E 61, R5 (2000)] in the form of an infinite product for the Fourier transform of the distribution function $f(c)$. In this paper we have inverted the Fourier transform to express $f(c)$ in the form of an infinite series of exponentially decaying terms. The dominant high energy tail is exponential, $f(c)simeq A_0exp(-a|c|)$, where $aequiv 2/sqrt{1-alpha^2}$ and the amplitude $A_0$ is given in terms of a converging sum. This is explicitly shown in the totally inelastic limit ($alphato 0$) and in the quasi-elastic limit ($alphato 1$). In the latter case, the distribution is dominated by a Maxwellian for a very wide range of velocities, but a crossover from a Maxwellian to an exponential high energy tail exists for velocities $|c-c_0|sim 1/sqrt{q}$ around a crossover velocity $c_0simeq ln q^{-1}/sqrt{q}$, where $qequiv (1-alpha)/2ll 1$. In this crossover region the distribution function is extremely small, $ln f(c_0)simeq q^{-1}ln q$.
A simple one-dimensional model is constructed for polymer motion. It exhibits the crossover from reptation to Rouse dynamics through gradually allowing hernia creation and annihilation. The model is treated by the density matrix technique which permits an accurate finite-size-scaling analysis of the behavior of long polymers.
We investigate the Rubinstein-Duke model for polymer reptation by means of density-matrix renormalization group techniques both in absence and presence of a driving field. In the former case the renewal time tau and the diffusion coefficient D are calculated for chains up to N=150 reptons and their scaling behavior in N is analyzed. Both quantities scale as powers of N: $tau sim N^z$ and $D sim 1/N^x$ with the asymptotic exponents z=3 and x=2, in agreement with the reptation theory. For an intermediate range of lengths, however, the data are well-fitted by some effective exponents whose values are quite sensitive to the dynamics of the end reptons. We find 2.7 <z< 3.3 and 1.8 <x< 2.1 for the range of parameters considered and we suggest how to influence the end reptons dynamics in order to bring out such a behavior. At finite and not too small driving field, we observe the onset of the so-called band inversion phenomenon according to which long polymers migrate faster than shorter ones as opposed to the small field dynamics. For chains in the range of 20 reptons we present detailed shapes of the reptating chain as function of the driving field and the end repton dynamics.
The equilibrium properties of a Janus fluid confined to a one-dimensional channel are exactly derived. The fluid is made of particles with two faces (active and passive), so that the pair interaction is that of hard spheres, except if the two active faces are in front of each other, in which case the interaction has a square-well attractive tail. Our exact solution refers to quenched systems (i.e., each particle has a fixed face orientation), but we argue by means of statistical-mechanical tools that the results also apply to annealed systems (i.e., each particle can flip its orientation) in the thermodynamic limit. Comparison between theoretical results and Monte Carlo simulations for quenched and annealed systems, respectively, shows an excellent agreement.
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