Equiparition theorem, kinetic energy, temperatur

In summary, according to the equipartition theorem, the temperature for a system of particles should be given by the result of the following derivation: the sum of all p_k vectors is f k_b T.
  • #1
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Hi,

according to the equipartition theorem

[itex]<p_k \frac{\partial H}{\partial p_k}> =k_bT[/itex] (where H depends on f generalized coodinates p and q and f is the number of degrees of freedom)

the temperature for a system of particles should be given by the result of the following derivation

[itex]<\sum_k^f p_k \frac{\partial H}{\partial p_k}> =\sum_k^f k_bT[/itex]
<=>
[itex]<\sum_k^f p_k \frac{\partial H}{\partial p_k}> = f k_b T[/itex]

now using [itex]f=3N-d[/itex] where N is the number of particles and d the number of 'constraints', one gets:
[itex]<\sum_k^{3N-d} p_k \frac{\partial H}{\partial p_k}> = (3N-d) k_b T[/itex]

using [itex]\frac{\partial H}{\partial p_k}=2\frac{p_k}{2m}[/itex] one gets:
[itex]<\sum_k^{3N-d} \frac{p_k^2}{2 m}> = \frac{3N-d}{2} k_b T[/itex]

however, the common result is:
[itex]<\sum_k^{N} \frac{\vec p_k^2}{2 m}> = \frac{3N-d}{2} k_b T[/itex] (note, that p_k is a vector)

That is, what people seem to do is, they don't write the kinetic energy on the left hand side as a sum of the components p_k, but as a sum over the momentum vectors:
[itex]<\sum_k^{N} \frac{\vec p_k^2}{2 m}>[/itex]

I don't see, how they manage the upper bound 3N-d of the sum. If the upper bound was only 3N, then the last equation for the kinetic energy would be obvious to me. That is, I would understand the following
[itex]<\sum_k^{3N} \frac{p_k^2}{2 m}> = <\sum_k^{N} \frac{\vec p_k^2}{2 m}>[/itex]
where the sum on the left hand side is going over the momentum components of all particles and the sum on the right hand side is going over the momentum-vectors of all particles.

but I don' t see, how people manage to get
[itex]<\sum_k^{3N-d} \frac{p_k^2}{2 m}> = <\sum_k^{N} \frac{\vec p_k^2}{2 m}>[/itex]for an example, please see page 11 of http://www.physics.buffalo.edu/phy411-506/topic3/lec-3-1.pdf (where d=3)
 
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  • #2
Because, in the thermodynamic limit, N is of order 1023 and d is only 3.
 
  • #3
i see, only an approximation...

thank you.
 

Related to Equiparition theorem, kinetic energy, temperatur

What is the Equipartition Theorem?

The Equipartition Theorem is a fundamental principle in statistical mechanics that states that for a system in thermal equilibrium, the average energy associated with each degree of freedom is equal to the thermal energy of the system divided by the number of degrees of freedom.

How is kinetic energy related to temperature?

Kinetic energy is directly related to temperature through the Equipartition Theorem. As temperature increases, the average kinetic energy of the particles in a system also increases. This is because the thermal energy of the system is divided equally among all of the available degrees of freedom, including the translational, rotational, and vibrational energies of the particles.

What factors affect the kinetic energy of a system?

The kinetic energy of a system is affected by several factors, including the mass and velocity of the particles, as well as any external forces acting on the system. Additionally, the number of degrees of freedom and the temperature of the system also play a role in determining the kinetic energy.

How does temperature affect the behavior of particles in a system?

Temperature has a significant impact on the behavior of particles in a system. As temperature increases, the average kinetic energy of the particles also increases, causing them to move faster and collide more frequently. This can lead to changes in the physical properties and behavior of the system, such as changes in volume or phase transitions.

Can the Equipartition Theorem be applied to all systems?

The Equipartition Theorem can be applied to most classical systems in thermal equilibrium, as long as the particles in the system have thermal energy and multiple degrees of freedom. However, it does not apply to quantum mechanical systems or systems with only one degree of freedom, such as a monoatomic gas.

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