Abstract: We suggest a mathematical model of a chaotic dynamics of a polymer chain in water. The model consists of a parabolic equation that is derived according to the selfconsistent field approach. This model is employed for the numerical simulation of a biological sensor that detects the presence of a specific protein in the fluid. The equation is interesting from the mathematical point of view. It is nonlocal in time and contains a term, called the interaction potential, that depends on the time-integral of the solution over the entire interval of solving the problem. In fact, one needs to know the “future” to determine the coefficient in this term, that is, the causality principle is violated. The point is that the time in this parabolic equation is, in fact, the arc length parameter along the polymer chain. The unknown function is the density of probability that the segment of the chain is at a point. Since each segment of the chain interacts with all other segments through the surrounding fluid, the equation contains an interaction term which includes an integral over the entire chain. The existence of a weak solution of the initial boundary value problem is proven. The interaction potential satisfies fairly general conditions and can have arbitrary growth at infinity. The uniqueness of this solution is established with restrictions on the length of the considered time interval, which is unusual for parabolic problems. The mathematical part of this work related to the proof of the existence and uniqueness of the solution was supported by Russian Science Foundation (project No.19-11-00069).

Cite: Starovoitov V.N. , Starovoitova B.N.
Modeling chaotic dynamics of a polymer chain in water solution and a nonlocal parabolic equation
Workshop on Mathematical Modeling and Scientific Computing Technische Uneversitaet Muenchen, November 19–20, 2020 19-20 Nov 2020