Tham khảo tài liệu 'mathematical method in science and engineering episode 18', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 20 GREEN S FUNCTIONS and PATH INTEGRALS In 1827 Brown investigates the random motions of pollen suspended in water under a microscope. The irregular movements of the pollen particles are due to their random collisions with the water molecules. Later it becomes clear that many small objects interacting randomly with their environment behave the same way. Today this motion is known as Brownian motion and forms the prototype of many different phenomena in diffusion colloid chemistry polymer physics quantum mechanics and finance. During the years 1920 1930 Wiener approaches Brownian motion in terms of path integrals. This opens up a whole new avenue in the study of many classical systems. In 1948 Feynman gives a new formulation of quantum mechanics in terms of path integrals. In addition to the existing Schrodinger and Heisenberg formulations this new approach not only makes the connection between quantum and classical physics clearer but also leads to many interesting applications in field theory. In this Chapter we introduce the basic features of this technique which has many interesting existing applications and tremendous potential for future uses. BROWNIAN MOTION AND THE DIFFUSION PROBLEM Starting with the principle of conservation of matter we can write the diffusion equation as D 2p -r t 633 634 GREEN S FUNCTIONS AND PATH INTEGRALS where is the density of the diffusing material and D is the diffusion constant which depends on the characteristics of the medium. Because the diffusion process is also many particles undergoing Brownian motion at the same time division of by the total number of particles gives the probability w r t of finding a particle at T and t as w rV Naturally t also satisfies the diffusion equation D 2w T t . For a particle starting its motion from r 0 we have to solve Equation with the initial condition lim In one dimension we write Equation as 9w x t d2w x t dT - D dx2 20-5 and by using the Fourier transform
