Periodic potential. Brownian particle

In understanding experimental studies where a particle in an optical trap could be considered as a Brownian particle, FRs based on the stochastic Langevin equations were developed. This allowed analytic expressions for the entropy production and its probability to be obtained, and numerical predictions to be made. A similar approach has been used to study a Brownian particle diffusing in a periodic potential under steady state conditions and useful information characterising the fluctuations have been obtained analytically and from numerical calculations. ... 

A relatively unexplored extension of the Kramers theory is the escape of a Brownian particle out of a potential well in the presence of an external periodic force. Processes such as multiphoton dissociation and isomerization of molecules in high-pressure gas or in condensed phases/ laser-assisted desorption/ and transitions in current-driven Josephson junctions under the influence of microwaves " may be described with such a model, where the pieriodic force results from the radiation field. 

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. 

Applications are then presented in Section IV. These examples should served as a guide as to what kinds of problems can be studied with these techniques and the limitations and possibilities for these methods. We present three examples (1) a dynamical test of the centroid quantum transition-state theory for electron transfer (ET) reactions in the crossover regime between adiabatic and nonadiabatic electron transfer, (2) the primary electron transfer reaction in bacterial photosynthesis, and (3) the diffusion kinetics of a Brownian particle in a periodic potential. Finally, Section V offers an outlook and a perspective of the current status of the field from our vantage point. 

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... 

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. 

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