Equipartition theory

Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

MCAT won t test tJis equipartition theory This is a variation of an equation from Lecture 2. The equipartition theorem is... 

This expression is consistent with the equipartition theory of energy, which states that the contribution to the average energy of each quadratic term is (l/2)kB7 . In fact, the translational energy that in classical mechanics is in the form ... 

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

There is, however, a fatal objection to the theory of Boltzmann. At very low temperatures the oscillations will be small, and should conform to the theory. But the atomic heats, instead of approaching the limit 5 955 at low temperatures, diminish very rapidly, and in the case of diamond the specific heat is already inappreciable at the temperature of liquid air. A new point of view is therefore called for, and it is a priori very probable that this will consist of a replacement of the hypothesis of Equipartition of Energy adopted by Boltzmann. This supposition has been verified, and the new law of partition of energy derived... 

Quantum theory therefore correctly predicts the equipartition principle in the classical limit. 

In emphasizing the need for satisfying the equipartition theorem, the linear response theory provides a connection for stationary processes through the fluctuation-dissipation theorem. 

Hendrik Antoon Lorentz, from Leyden (Holland), presided the conference, whose general theme was the Theory of Radiation and the Quanta. The conference5 was opened with speeches by Lorentz and Jeans, one on Applications of the Energy Equipartition Theorem to Radiation, the other on the Kinetic Theory of Specific Heat according to Maxwell and Boltzmann. In their talks, the authors explored the possibility of reconciling radiation theory with the principles of statistical mechanics within the classical frame. Lord Rayleigh, in a letter read to the... 

Analogous difficulties are encountered in the gas theory itself.218 While the law of equipartition of energy can very well be applied to the translational and in some... 

Each of the vibrational degrees of freedom given by Eq. (6.1) would have a mean kinetic and potential energy of kT according to equipartition, and would contribute an amount R to the specific heat. As with diatomic molecules, however, the quantum theory tells us, and wc find... 

The most general vibrational motion of our solid is one in which each overtone vibrates simultaneously, with an arbitrary amplitude and phase. But in thermal equilibrium at temperature T, the various vibrations will be excited to quite definite extents. It proves to be mathematically the case that each of the overtones behaves just like an independent oscillator, whose frequency is the acoustical frequency of the overtone. Thus we can make immediate connections with the theory of the specific heats of oscillators, as we have done in Chap. XIII, Sec. 4. If the atoms vibrated according to the classical theory, then we should have equipartition, and at temperature T each oscillation would have the mean energy kT. This means that each of the N overtones would have equal... 

For polyatomic molecules, theory based on the equipartition theorem allows one to calculate only limiting values for Cy by either completely ignoring all vibrational contributions or assuming that the vibrational contributions achieve their full classical value. For monatomic gases and all ordinary diatomic molecules (where the vibrational contribution is not important at room temperature and can be ignored), definite Cy values can be calculated. For a brief discussion of a more accurate calculation of C (vib), see Exp. 37. 

Because of equipartition all frequencies should be equally weighted (i.e. fV((o) = 1 for all ). This clearly contradicts our understanding of dynamics from quantum mechanics (QM) theory. Since QM is essential to the treatment of certain phenomena, quantum corrections must be introduced, regardless of whether structural or dynamic simulations are required. This discrepancy is illustrated if we calculate the total kinetic energy of the system quantum-mechanically ... 

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