Can Brownian Motion Be Formulated in Both Probability and Classical Terms?

[lokofer]

Brownian motion...

Hello ..since it has several application to physics i would like to hear about Brownian motion..in fact i think you can approach it by means of a functional integral of the form:

$$\int D[x] e^{-a \int_{a}^{b} L(x,\dot x, t)}$$ and that from this you derive the "difussion" process... my question is about the trajectories (it'll sound to you familiar to QM if known)

- What's the PDE equation satisfied by the functional integral above?..

- Can the "Brownian motion" by formulated only in probability terms or are there "classical trajectories" that can be obtained?..i heard (wikipedia) about the equation for Brwonian motion in the form:

$$m \ddot x = -\nabla V + \eta (t)$$ where the last extra term was some kind of "random" force applied to the particles.

 

[mmwave]

Thank you for your interest in Brownian motion and for sharing your thoughts on the topic. I would be happy to provide some insights into the questions you have raised.

Firstly, the functional integral you have mentioned is indeed a common way to approach Brownian motion in physics. It is known as the Feynman path integral and is used to calculate the probability amplitude for a particle to travel from one point to another in a given time interval. The Lagrangian, L, represents the kinetic and potential energies of the particle, and the integral is taken over all possible paths that the particle can take.

Regarding your first question, the functional integral for Brownian motion satisfies the Fokker-Planck equation, which is a partial differential equation (PDE) that describes the time evolution of the probability distribution of a particle undergoing diffusion. It is given by:

∂P/∂t = D∇^2P - ∇(F(x)P)

where P(x,t) is the probability density, D is the diffusion coefficient, and F(x) is the force acting on the particle.

As for your second question, Brownian motion can be described in terms of both classical trajectories and probability. The classical trajectories can be obtained by solving the Langevin equation, which is the Newton's second law of motion with the addition of a random force (as you have mentioned in your post). This equation is given by:

m(d^2x/dt^2) = -∇V(x) + η(t)

where m is the mass of the particle, V(x) is the potential energy, and η(t) is the random force with zero mean and a variance proportional to the temperature.

However, in reality, the trajectories of particles undergoing Brownian motion are not deterministic due to the random nature of the forces acting on them. Therefore, it is more accurate to describe Brownian motion in terms of probability, using the Fokker-Planck equation mentioned above.

I hope this helps to answer your questions. Brownian motion is a fascinating phenomenon with applications in various fields of science, and I am glad to see your interest in it. If you have any further questions, please feel free to ask. Thank you.

 

[tacman]

Thank you for your question about Brownian motion. Brownian motion is a phenomenon observed in physics where small particles, such as dust or pollen, suspended in a fluid exhibit random motion. This motion is due to the constant bombardment of the particles by the molecules in the fluid, causing them to move in a random manner.

To answer your first question, the functional integral you mentioned is actually a path integral, which is a mathematical tool used in quantum field theory to calculate probabilities of different outcomes. In the case of Brownian motion, the path integral is used to calculate the probability of a particle's trajectory over a period of time. The PDE equation satisfied by this path integral is known as the Fokker-Planck equation, which describes the evolution of the probability distribution of a particle undergoing Brownian motion.

As for your second question, Brownian motion can be described in both classical and probabilistic terms. In classical mechanics, the motion of a particle is determined by its initial conditions and the forces acting upon it. In the case of Brownian motion, the random forces exerted by the fluid molecules on the particle result in a random trajectory. However, in probabilistic terms, the behavior of the particle is described by the probability distribution of its position and velocity, rather than a specific trajectory. This is where the Fokker-Planck equation comes into play, as it allows us to calculate the probability of a particle's position and velocity at any given time.

In summary, Brownian motion is a fascinating phenomenon that can be described in both classical and probabilistic terms. The path integral and Fokker-Planck equation are important mathematical tools used to study and understand this phenomenon. I hope this helps answer your questions about Brownian motion.