Suppose a system has quartic degrees of freedom, instead of quadratic degrees of freedom: E(x) = cx4 where c is a constant and x is a continuous variable. Find the average energy of this system, in the same way that we derived the equipartition theorem.

Explanation:

The average energy of the system with quartic degrees of freedom. The quartic degrees of freedom is same as biquadratic since it means 4. Systems having quartic degrees of freedom are usually have their energies represented in terms of some variable raised to the power of 4.

The given system with quartic degrees of freedom here has E(x) = cx4 . The standard result from the statistical mechanics will be helpful here in calculating internal energy of the system, which is also its average energy.

U = kT^2\d(lnq)}/dT

Now, to find out q(x) we will use the equation  q(x) = \int^{+\infty}_{-\infty} exp\bigg(\frac{-E(x)}{kT}\bigg)dx = \int^{+\infty}_{-\infty} exp\bigg(\frac{-cx^4}{kT}\bigg)dx

For a quadratic system, you would get a Gaussian integral which has a standard result.

the order for strong to weak forces such as

strong nuclear force → electromegnetic force → weak nuclear force → gravitational force

the fundamental forces are those forces that act between to objects or element to make interactions. there are four types of fundamental forces such as strong and weak nuclear forces (due to protons and electrons), electromagnetic force (due to the motion of electrons) and gravitational force (due to orbital movement of the planet). if these forces are arranged on the basis of their maximum strength then the order is

strong nuclear force → electromegnetic force → weak nuclear force → gravitational force