# Brownian motion: Gaussian distribution

• SoggyBottoms
In summary, the displacement of a pollen grain in a solvent shows Brownian motion and the displacement during 100 seconds can be described by a Gaussian distribution with an average of 0 and a standard deviation of σ.
SoggyBottoms

## Homework Statement

A grain of pollen shows Brownian motion in a solvent, such that the position x(t) on the x-axis varies with time. The displacement during one second, x(t + 1) - x(t), is measured many times and found to have a Gaussian distribution with an average of 0 and standard devation σ. What is the average and standard deviation of the displacement x(t + 100) − x(t) during 100 seconds?

## The Attempt at a Solution

I would say the average would still be 0 and the SD still σ. Is that correct?

Yes, that is correct. The displacement during 100 seconds would still follow a Gaussian distribution with an average of 0 and a standard deviation of σ. This is because the Brownian motion of the pollen grain is a random process, and the average and standard deviation of the displacement remain constant regardless of the time frame being considered.

## 1. What is Brownian motion?

Brownian motion is the random movement of particles suspended in a fluid, caused by collisions with the fluid molecules. It was first observed in 1827 by botanist Robert Brown and is now widely used to study the behavior of particles in various fields of science.

## 2. What is a Gaussian distribution?

A Gaussian distribution, also known as a normal distribution, is a probability distribution that is often used to describe the distribution of a continuous variable. It is characterized by a bell-shaped curve and is a fundamental concept in statistics and probability theory.

## 3. How is Brownian motion related to a Gaussian distribution?

The displacement of particles in Brownian motion follows a Gaussian distribution. This means that the majority of the particles will have small displacements, while a few particles may have larger displacements. The overall distribution of these displacements follows a bell-shaped curve, similar to a Gaussian distribution.

## 4. Why is a Gaussian distribution important in studying Brownian motion?

A Gaussian distribution is important because it allows us to describe and analyze the behavior of particles in Brownian motion. It provides a mathematical model to predict the probability of particles having a certain displacement, which is crucial in understanding and interpreting experimental data.

## 5. Can Brownian motion always be described by a Gaussian distribution?

In most cases, Brownian motion can be described by a Gaussian distribution. However, in some situations, such as when there are strong external forces acting on the particles, the distribution may deviate from a Gaussian shape. In these cases, other probability distributions may be more appropriate to describe the behavior of the particles.

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