Equipartition Thm: mv_x^2/2 = k_B T/2

Thread starter michael892
Start date Mar 25, 2013
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Theorem

In summary, the Equipartition Theorem is a principle in statistical mechanics that states that at thermal equilibrium, the total energy of a system is equally distributed among all of its available degrees of freedom. The equation mv_x^2/2 = k_B T/2 represents the average kinetic energy of a single particle in a system at thermal equilibrium and is used to calculate the average energy and predict the behavior of gases and other systems. The assumptions of the Equipartition Theorem include thermal equilibrium, no external forces, and the classical regime. However, it has limitations and is not applicable to all systems, as it does not consider quantum effects.

Mar 25, 2013

michael892

show that
[tex]< \frac{mv_{x}^{2}}{2}> \equiv s\int_{-\infty }^{\infty} \frac{mv_{x}^{2}}{2} \rho (v_{x}) dv_{x}=\frac{k_{B}T}{2}[/tex]

im stuck

Last edited: Mar 26, 2013

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Mar 29, 2013

Chandra Prayaga

Science Advisor

Plug in the expression for ρ(v_x) inside the integral and then actually calculate the integral, or look it up in a table of integrals.

What is the Equipartition Theorem?

The Equipartition Theorem is a principle in statistical mechanics that states that, at thermal equilibrium, the total energy of a system is equally distributed among all of its available degrees of freedom.

What is the significance of the mv_x^2/2 = k_B T/2 equation in the Equipartition Theorem?

This equation represents the average kinetic energy of a single particle in a system at thermal equilibrium. It shows that the kinetic energy is directly proportional to the temperature and the mass of the particle.

How is the Equipartition Theorem used in statistical mechanics?

The Equipartition Theorem is used to calculate the average energy of a system at thermal equilibrium by considering the number of degrees of freedom and their associated energies. It is also used to predict the behavior of gases and other systems at thermal equilibrium.

What are the assumptions of the Equipartition Theorem?

The Equipartition Theorem assumes that the system is in thermal equilibrium, meaning that the temperature is constant throughout and there are no external forces acting on the particles. It also assumes that the system is in the classical regime, where quantum effects can be neglected.

Are there any limitations to the Equipartition Theorem?

Yes, the Equipartition Theorem does not hold for all systems. It is only applicable to systems at thermal equilibrium and in the classical regime. It also does not take into account quantum effects, which can be significant for small particles or at low temperatures.