Brownian motion periodic potentials

Zatsepin V. M. Time correlation functions of one-dimensional rotational Brownian motion in n-fold periodical potential. Theor. and Math. Phys. 

Besides the remarkable directionality of the motion, the images also demonstrate a periodic variation of the cluster from an elongated to a circular shape (Fig. 39). The diagrams in Fig. 39 depict the time dependence of the displacement and the cluster size. Until the cluster was finally trapped, the speed remained fairly constant as can be seen from the constant slope in Fig. 39 a. The oscillatory variation of the cluster shape is shown in Fig. 39b. Although a coarse model for the motion has been presented in Fig. 39, the actual cause of the motion remains unknown. The ratchet model proposed by J. Frost requires a non-equiUb-rium variation in the energetic potential to bias the Brownian motion of a molecule or particle under anisotropic boundary conditions [177]. Such local perturbations of the molecular structure are believed to be caused by the mechanical contact with the scaiming tip. A detailed and systematic study of this question is still in progress. 

Theory for the self- and tracer-diffusion of a diblock copolymer in a weakly ordered lamellar phase was developed by Fredrickson and Milner (1990). They modelled the interactions between the matrix chains and a labelled tracer molecule as a static, sinusoidal, chemical potential field and considered the Brownian dynamics of the tracer for small-amplitude fields. For a macroscopically-oriented lamellar phase, they were able to account for the anisotropy of the tracer diffusion observed experimentally. The diffusion parallel and perpendicular to the lamellae was found to be sensitive to the mechanism assumed for the Brownian dynamics of the tracer. If the tracer has sufficiently low molecular weight to be unentangled with the matrix, then its motion can be described by a Rouse model, with an added term representing the periodic potential (Fredrickson and Bates 1996) (see Fig. 2.50). In this case, motion parallel to the lamellae does not change the potential on the chains, and Dy is unaffected by... 

Temperatiue can be incorporated into the PT model by including random forces in the equation of motion, Eq. (22), similar to those introduced in the rigid-wall model, Eqs. (12) and (13). The additional complication in the PT model is the elastic coupling of the central particle to a (moving) equilibrium site, which makes it difficult to apply the rigorous concepts developed for the Brownian motion of a particle in a periodic potential to the PT model. 

As the third application, we consider the motion of a Brownian particle in a periodic potential. Schmid [69] was the first to study this problem, and many others have since then applied a variety of techniques to this model [70-72]. The importance of the model stems from its widespread... 

Sakaguchi, H. GeneraKzed Einstein relation for Brownian motion in tilted periodic potential. 

Movement of the motor protein can be modeled by the Smoluchowski equation by assuming that the center of mass of the motor protein as a Brownian motion with the presence of a periodic energy potential ... 

Another typical example of the stochastic resonance system is the nonlinear bistable doublewell dynamic system, which describes the overdamped motion of a Brownian jjartide in a symmetric double-well potential in the presence of noise and periodic forcing as shown in... 

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