Transient diffusion in a tube with dead ends.

A particle diffusing in a tube with dead ends, from time to time enters a dead end, spends some time in the dead end, and then comes back to the tube. As a result, the particle spends in the tube only a part of the entire observation time that leads to slowdown of its diffusion along the tube. We study the transient diffusion in a tube with periodic identical dead ends formed by cavities of volume V(cav) connected to the tube by cylindrical channels of length L and radius a, which is assumed to be much smaller than the tube radius R and the distance l between neighboring dead ends. Assuming that the particle initial position is uniformly distributed over the tube, we analyze the monotonic decrease of the particle diffusion coefficient D(t) from its initial value D(0)=D, which characterizes diffusion in the tube without dead ends, to its asymptotic long-time value D(infinity)=D(eff)<D. We derive an expression for the Laplace transform of D(t), denoted by D(s), where s is the Laplace parameter. Although the expression is too complicated to be inverted analytically, we use it to find the relaxation time of the process as a function of the geometric parameters of the system mentioned above. To check the accuracy of our results, we ran Brownian dynamics simulations and found the mean squared displacement of the particle as a function of time by averaging over 5x10(4) realizations of the particle trajectory. The time-dependent mean squared displacement found in simulations is compared with that obtained by numerically inverting the Laplace transform of the mean squared displacement predicted by the theory, which is given by 2D(s)/s. Comparison shows excellent agreement between the two time dependences that support the approximations used when developing the theory.