Can you please explain your notation? In particular, what is [tex]x_t[/tex], what is [tex]\dot x[/tex], and what is [tex]\mathcal D[/tex]?tpm said:HI, i would need some help to solve the Brownian motion given by the Weiner Integral(over paths):
[tex] \int \mathcal D [x_{t}]exp(-\int dt (m/2(\dot x)^{2}-V(x)) [/tex]
for the case [tex] V(x)=\delta (x) +\delta (x-1)+\delta (x-2) [/tex]
any help would be appreciated, thanks
Brownian Motion is a phenomenon in which small particles in a fluid or gas move randomly and unpredictably due to collisions with molecules in the medium.
The Weiner Integral is a mathematical tool used to calculate the total displacement of a particle undergoing Brownian Motion. It takes into account the random and continuous nature of the particle's movement.
Delta Functions are mathematical functions that are used to represent the concentration of particles at a specific point in time or space. They are often used in conjunction with the Weiner Integral to understand and model Brownian Motion.
The Weiner Integral and Delta Functions are used in mathematical models to analyze and predict the behavior of particles undergoing Brownian Motion. They take into account the randomness and unpredictability of the motion and allow for more accurate predictions and analysis.
Understanding Brownian Motion is crucial in many scientific fields, including physics, chemistry, and biology. It has practical applications in the study of diffusion, heat transfer, and the behavior of molecules in a variety of systems, such as the movement of pollutants in water or the spread of diseases in a population.