Equipartition formula

This result is analogous to the equipartition formula for the average thermal energy of a system with m degrees of freedom, whose Hamilton function is quadratic in the attendant momenta and coordinates, cf. (5.85). Equation (7.17) shows that the average entropy fluctuation in a system containing m fluctuating quantities is contributed in increments of —ksl l by each of these quantities. 

At high temperatures, E(ui,T) takes its equipartition value of E(u,T) kT, (kT >> Hu) to yield the so-called Nyquist formula that describes voltage fluctuations (Nyquist noise) generated in a linear electrical system, i.e. 

Temperature and pressure are calculated from standard formulas from statistical mechanics (equipartition kinetic energy and virial, respectively), and are regulated at preset values by computational thermostats and barostats. 

In the limit of long wave-lengths and hi temperatures the principle of equipartition of energy is valid. This leads to the classical formula of... 

From the formula for the pressure of a perfect gas, p = and the relation pV — NkT, it follows that the kinetic energy in the three translational degrees of freedom is pT. The allocation for each is thus kT. The equipartition law provides that where the energy is shared between s square terms, the average amount in a molecule is skT. The molecular heat, C , is therefore... 

The quantity , = hcv/k is called the vibrational temperature, wha-e h is Planck s constant, c is the speed of light, k is Boltzmann s constant, and v is the vibrational frequency in wave numbers (cm ). The vibrational frequencies for F2 and I2 are 919 and 214 cm, respectively. Use the formula above, together with the equipartition values for the translational and rotational contributions, to calculate the constant-pressure heat capacity for these two diatomic molecules. Compare your results with the expraimental values given in Table 7.4. 

The de Gennes formulae [28, 29] establish a relation between the elastic coefficients and the scattering intensities. Small thermal director fluctuations can be expressed in terms of two eigenmodes, the splay-bend mode Srii and the twist-bend mode Sn2. The equipartition theorem gives the intensities... 

The variation in potential energy, AE, is estimated by the difference in average energies over NVT simulations at different temperatures. 3R accounts for the kinetic equipartition contribution (see Section 7.2), and Cv(vibr) corrects for the vibrational constraints imposed in the simulation (for example by the SHAKE algorithm), and can be estimated by the usual quantum statistical mechanics formulas (equation 7.30) if the vibrational frequencies are known. 

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