Linear Potential and Thermal Equilibrium

[JonathanT]

"Linear" Potential and Thermal Equilibrium

Homework Statement

Consider a classical particle moving back and forth along the x-axis while restrained by a "Linear" potential V(x) = b|x|. If the particle is in thermal equilibrium with the environment at temperature T, calculate the mean value of the potential energy b|x|.

Homework Equations

Equipartition Theorem: Each term in the energy proportional to the square of a velocity or a coordinate contributes $\frac{1}{2}$kT to the mean energy at thermal equilibrium.

Where k is the Boltzmann Constant
T is the Temperature in Kelvin

The Attempt at a Solution

Basically I'm just confused by the question. Maybe its the wording or I'm oversimplifying it.

If I used the Equipartition Theorem for the mean potential energy in the system the answer would just be $\frac{1}{2}$kT right? Seeing as the system only has 1 "degree of freedom." Obviously I'm missing the point of the equation given in the problem. V(x) = b|x|. Or how to manipulate it in order to get a logical conclusion rather then just stating the answer.

 

[mark726]

Hello,

Thank you for your question. Let's break down the problem and see if we can come up with a solution.

First, we have a classical particle moving back and forth along the x-axis. This means that the particle has a certain amount of kinetic energy due to its motion.

Next, we have a "Linear" potential, V(x) = b|x|. This means that the potential energy of the particle is directly proportional to its position along the x-axis. As the particle moves further away from the origin (x=0), the potential energy increases linearly.

Now, we are told that the particle is in thermal equilibrium with the environment at temperature T. This means that the particle is experiencing collisions with other particles in the environment, which leads to a distribution of kinetic energies among the particles.

Using the Equipartition Theorem, we know that at thermal equilibrium, each term in the energy (kinetic and potential) contributes \frac{1}{2}kT to the mean energy.

So, to calculate the mean value of the potential energy b|x|, we need to consider the distribution of positions along the x-axis for the particle. This distribution will follow a Boltzmann distribution, and we can use integration to find the mean value of the potential energy.

In other words, we need to find the average position of the particle along the x-axis, and then plug that value into the potential energy equation V(x) = b|x| to find the mean potential energy.

I hope this helps you in solving the problem. Don't hesitate to ask for further clarification if needed. Good luck!