Equipartition's energy theorem

In summary, the equipartition's energy theorem is only valid for a potential free system. It may work for harmonic potentials at high temperatures, but there is a more general theorem that can be used in other situations. This theorem states that each quadratic degree of freedom in the Hamiltonian contributes 1/2KT, and there is a more general version for other types of Hamiltonians as well.
  • #1
Talker1500
22
0
Hi,

I'm reading about the equipartition's energy theorem (classical statistics), and I was wondering about its validity when applied to different hamiltonians.

The usual case, H=p ^2/2m, it yields 3/2KT in 3D, but what about more complicated H? Like H=|p|c, or a H with a complex V? would the theorem still be available for use?

Thanks
 
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  • #2
Equipartition of energy is only valid for a potential free system.

Beyond that you may want to look up the Virial Theorem
 
  • #3
It seems to work for harmonic potentials, if the temperature is high enough.
See specific heats of solids (in harmonic approximation, high temperature limit).
Or this:
http://en.wikipedia.org/wiki/Equipartition_theorem
 
  • #4
In 1-D is isn't too hard to derive the result for the cases H = |p|, or H = p^4. The derivation closely follows that of the quadratic case. In 3-D I suspect it is much more complicated, but I am often wrong!

jason
 
  • #5

1. What is Equipartition's energy theorem?

Equipartition's energy theorem is a principle in statistical mechanics that states that in thermal equilibrium, the total energy of a system is equally distributed among all of its degrees of freedom. This means that each degree of freedom, such as the kinetic energy of particles or the vibrational energy of molecules, will have an average energy of kT/2, where k is the Boltzmann constant and T is the temperature.

2. How does Equipartition's energy theorem apply to real-world systems?

Equipartition's energy theorem is a useful tool for understanding the behavior of macroscopic systems, such as gases, liquids, and solids. It allows us to predict the average energy levels of different degrees of freedom within these systems, which is important for understanding their thermal and mechanical properties.

3. What are the limitations of Equipartition's energy theorem?

Equipartition's energy theorem is based on certain assumptions, such as the system being in thermal equilibrium and having a large number of particles. These assumptions do not always hold true in real-world systems, especially at very low temperatures or in systems with only a few particles. Additionally, it does not take into account quantum effects, which are important at the atomic and subatomic level.

4. How is Equipartition's energy theorem related to the ideal gas law?

Equipartition's energy theorem is one of the underlying principles of the ideal gas law, which describes the behavior of gases at low pressures and high temperatures. The ideal gas law states that the product of pressure and volume is proportional to the product of the number of particles and the temperature. Equipartition's energy theorem explains why this relationship holds true, as it predicts that each degree of freedom in an ideal gas will contribute an average energy of kT/2.

5. What are some applications of Equipartition's energy theorem?

Equipartition's energy theorem has many practical applications, including in the design of heat engines, refrigeration systems, and other thermal devices. It is also used in the study of thermodynamics, chemical reactions, and the behavior of materials at different temperatures. Additionally, it has been applied in fields such as astrophysics and cosmology to model the behavior of stars and galaxies.

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