Has Relativistic Brownian Motion Been Studied?

[kleinwolf]

Consider the density probability function following the diffusion equation with diffusion parameter D, with the initial condition $$f(x,t=0)=\delta(x)$$ :

$$f(x,t)=\frac{e^{-\frac{x^2}{4Dt}}}{\sqrt{4\pi Dt}}$$

From this : if t=0, then the particle is at x=0.

Consider a very small t>0...then we get $$P(A<x<A+dx)>0 \forall A>0, dx>0$$.

Which means that : the particle has a non-vanishing prob. of having traveled a distance as big as you want in a time as small as you want.

Does anybody know if relativistic brownian motion was studied ?

There the function f should be compact...or something like that ?

 

[eljose79]

Thank you for bringing up this interesting topic. As a scientist studying diffusion and statistical mechanics, I can provide some insights on this issue.

First, let's clarify the initial condition f(x, t=0)=δ(x). This means that at t=0, the particle is located at a specific point x=0 with certainty. This is represented by the Dirac delta function, which is a mathematical way of describing a point particle. However, in reality, particles have a finite size and cannot be located at a single point. Therefore, this initial condition is somewhat of an idealization and should be interpreted carefully.

Now, moving on to the density probability function f(x,t). As you correctly pointed out, at t=0, the particle is at x=0. However, as time progresses, the particle starts to diffuse and its position becomes less certain. The function f(x,t) describes the probability of finding the particle at a specific position x at a given time t. As t increases, the function spreads out and becomes more peaked around the mean position of the particle.

Regarding your statement about the particle having a non-vanishing probability of traveling any distance in a very small time, this is indeed true. This is a consequence of the diffusion process, where the particle has a random motion and can travel a certain distance in a given time. However, as the time interval becomes smaller and smaller, the probability of traveling a larger distance decreases. This is due to the fact that the diffusion process is a random walk, and the distance traveled in a given time follows a Gaussian distribution, which has a finite variance.

As for the concept of relativistic Brownian motion, it has been studied in the context of relativistic statistical mechanics. In this case, the diffusion equation is modified to take into account the effects of special relativity. However, the concept of a compact density function does not apply here, as the relativistic diffusion equation has a different form and does not have a delta function as an initial condition.

I hope this clarifies some of your questions. If you are interested in learning more about diffusion and statistical mechanics, I recommend reading some introductory textbooks on the subject. Thank you for your contribution to the forum.

 

[charng]

Thank you for sharing this interesting concept. Brownian motion, also known as random motion, is a phenomenon observed in many different systems, including the movement of particles in a fluid. The diffusion equation and its density probability function provide a mathematical framework for understanding and predicting the behavior of these particles over time.

In the context of Brownian motion, the diffusion parameter D represents the rate at which particles spread out over time. As you mentioned, the initial condition f(x,t=0)=\delta(x) means that at t=0, the particle is located at x=0. As time passes, the particle's location becomes increasingly uncertain and it is possible for it to travel large distances in a short amount of time.

It is interesting to consider the behavior of Brownian motion in the context of relativity. While I am not aware of any specific studies on relativistic Brownian motion, it is certainly a topic worth exploring. In the relativistic case, the density probability function f(x,t) may have different properties, such as being compact as you mentioned. This could have implications for the behavior of particles in a relativistic system.

Overall, Brownian motion is a fascinating concept that has been extensively studied in various fields of science. Its application to different systems and its potential connection to relativity make it an intriguing area of research. Thank you for bringing this topic to our attention.