Specific heat theories of Debye’s phonons and Einstein’s atomic vibrations including modification thereof by Raman are modified by quantum mechanics to include zero specific heat at the nanoscale.

**Background**

Specific heat is thought to be an intensive thermophysical property independent of the amount of the substance. Given the amount of the substance in a body is proportional to its volume, specific heat should therefore be independent of whether the body dimensions are macroscopic or nanoscopic. In contrast, specific heat that depends on the amount of the substance is an extensive property dependent on the dimensions of the body. See http://en.wikipedia.org/wiki/Specific_heat_capacity

**Classical Specific Heat at the Nanoscale**

Currently, specific heat at the nanoscale is considered an intensive property having the same value as for macroscopic bodies. The Debye and Einstein macroscopic theories of specific heat including modifications thereof by Raman are generally assumed in simulating heat transfer in nanostructures. See Thumbnail of “Macroscopic Specific Heat at the Nanoscale?”in http://www.prlog.org/10609553-materials-at-the-nanoscale-have-zero-specific-heat.html. What this means is the classical oscillators of statistical mechanics all having the same kT energy are used to model specific heat at the nanoscale.

**Specific Heat by Quantum Mechanics**

Contrarily, quantum mechanics (QM) embodied in the Einstein-Hopf relation for the harmonic oscillator shows the QM states do not have the same kT energy. At ambient temperature, the average Planck energy of QM states is kT only at thermal wavelengths greater than about 50 microns while at shorter wavelengths is less than kT and vanishes for nanostructures at submicron wavelengths. See *Paper* and *Presentation* at http://www.nanoqed.org at “Zero Specific Heat”, 2010.

Since the Planck energy at a given wavelength is the amount of thermal energy that can be stored in the QM oscillator, and since the only thermal wavelengths that can fit into nanostructures are submicron, QM requires zero specific heat capacity at the nanoscale, the consequence of which is absorbed heat cannot be conserved in nanostructures by an increase in temperature. Conservation may only proceed by the QED induced frequency up-conversion of absorbed heat to non-thermal EM radiation at the fundamental EM confinement frequency of the nanostructure, typically in the UV and beyond. The EM confinement is quasi-bound allowing leakage of QED induced radiation from the nanostructure to be absorbed in the macroscopic surroundings. See *Ibid*.

But QED emission in the UV and beyond from nanostructures is not readily observed – even by standard photomultipliers because of the UV cut-off, and therefore heat balances of nanostructures do not include QED emissions as heat losses. Hence, thermal conductivity is inferred to be reduced from that of the bulk to be consistent with the measured temperature difference across the body, e.g., as in thin films. If QED emissions are included in heat losses, the bulk conductivity need not be reduced for consistency with temperature differences thereby precluding any modification of Fourier’s theory of heat conduction by the Boltzmann transport equation (BTE). See *Ibid*.

**Molecular Dynamics and Periodic Boundaries
**

Molecular Dynamics (MD) describes the classical solution of atomic motion based on Newton’s equations. To determine bulk transport properties, there are no QM restrictions on kT energy of atoms, i.e., atoms are assumed to have kT energy because the MD solution for the bulk is obtained by imposing periodic boundary conditions on the computational box. Historically, Monte Carlo (MC) preceded MD simulations, however. MC simulations of spherical particles in a submicron computational square with periodic boundaries were used to determine the 2D virial coefficients for the PVT equation of state. See Metropolis et al. *Ibid*. For a discrete nanostructure, periodic boundaries do not apply, and therefore the atoms in the nanostructure are subject to QM restrictions of zero kT energy.

Heat transfer of discrete nanostructures which are unambiguously not periodic is generally simulated by MD on the invalid assumption the atoms have kT energy. For nanocars, see e.g., http://pubs.acs.org/doi/full/10.1021/ct7002594 Extending specific heat from macroscopic samples to the nanoscale is just as invalid as extending the Dulong-Petit law for specific heat at ambient temperature to low temperatures about 200 years ago. Nevertheless, MD simulations of nanostructures today are proudly displayed in the belief they provide precise atomistic explanations of conduction heat transfer when in fact they are not valid because the simulations are performed on the assumption the atoms have finite kT energy. See http://www.scienceblog.com/cms/blog/8209-quantum-mechanics-questions-molecular-dynamics-submicron-structures-25639.html

**Conclusions**

1. QM requires zero specific heat capacity at the nanoscale be specified as a new thermophysical property of all materials.

2. The classification of specific heat as an intensive thermophysical property of a body should be changed to an extensive property depending on the dimensions of the body.

3. Nanoscale heat transfer based on the assumption of macroscopic specific heat is likely to produce unphysical results, e.g., reduced thermal conductivity in thin films.

4. There is no need for the BTE to determine the thermal conductivity in thin films as bulk conductivity may be assumed without any loss in accuracy.

5. Macroscopic Debye and Einstein theories should be revised to include zero specific heat at the nanoscale.

6. Lacking specific heat at the nanoscale, absorbed EM energy is not conserved by an increase in temperature, but rather by the emission of non-thermal QED induced EM radiation.

7. MD and MC simulations of bulk thermal conductivity based on full kT energy of atoms in submicron computational boxes under periodic boundary conditions are consistent with QM.

8. Zero specific heat is required for atoms in MD and MC simulations of discrete nanostructures without periodic constraints.

9. Absorbed EM energy in discrete nanostructures may be *a priori* assumed to be emitted as high frequency EM radiation that and absorbed in the macroscopic surroundings, thereby obviating any need to perform MD and MC simulations of the nanostructure itself.