Abstract
We present a model of an AB-diblock random copolymer sequential self-packaging with local quenched interactions on a one-dimensional infinite sticky substrate. It is assumed that the A-A and B-B contacts are favorable, while A-B are not. The position of a newly added monomer is selected in view of the local contact energy minimization. The model demonstrates a self-organization behavior with the nontrivial dependence of the total energy, $E$ (the number of unfavorable contacts), on the number of chain monomers, $N$: $E\sim N^{3/4}$ for quenched random equally probable distribution of A- and B-monomers along the chain. The model is treated by mapping it onto the "lamplighter" random walk and the diffusion-controlled chemical reaction of $X+X\to 0$ type with the subdiffusive motion of reagents.
Highlights
In this letter we propose a simple one-dimensional model of stochastic dynamic system possessing a local optimization in quenched environment
The system considered below should not be regarded as a model of any specific physical system, but it rather highlights the principles of local optimization of a specific one-dimensional stochastic system in quenched random environment, which could lead to a nontrivial self-organization
We have already mentioned that our model is reminiscent to the so-called “lamplighter random walk”, widely considered in the mathematical literature
Summary
In this letter we propose a simple one-dimensional model of stochastic dynamic system possessing a local optimization in quenched environment. Let us imagine that someone knows nothing about the future and adapts his or her own behavior in each current time moment only on the basis of the knowledge about the best solution at a given narrow time slice Such a behavior leads to an optimal local strategy when only a partial (current) knowledge is accessible, though it might be far from generic optimal one if the knowledge about the future is available. The system considered below should not be regarded as a model of any specific physical system (though it has some features of a protein folding), but it rather highlights the principles of local optimization of a specific one-dimensional stochastic system in quenched random environment, which could lead to a nontrivial self-organization.
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