Quantization of thermal conductance and quantum thermodynamicsThe finnish research group led by Mikko Möttönen working at Aalto University has made several highly interesting contributions to condensed matter physics during last years (see the popular articles about condensed matter magnetic monopoles and about tying quantum knots: both contributions are interesting also from TGD point of view). This morning I read about a new contribution published in Nature ). What has been shown in the recent work is that quantal thermal conductivity is possible for wires of 1 meter when the heat is transferred by photons. This length is by a factor 10^{4} longer than in the earlier experiments. The improvement is amazing and the popular article tells that it could mean a revolution in quantum computations since heat spoling the quantum coherence can be carried out very effectively and in controlled manner from the computer. Quantal thermal conductivity means that the transfer of energy along wire takes place without dissipation. To understand what is involved consider first some basic definitions. Thermal conductivity k is defined by the formula j= k∇ T, where j is the energy current per unit area and T the temperature. In practice it is convenient to use thermal power obtained by integrating the heat current over the transversal area of the wire to get the heat current dQ/dt as analog of electric current I. The thermal conductance g for a wire allowing approximation as 1D structure is given by conductivity divided by the length of the wire: the power transmitted is P= gΔ T, g=k/L. One can deduce a formula for the conductance at the the limit when the wire is ballistic meaning that no dissipation occurs. For instance, superconducting wire is a good candidate for this kind of channel and is used in the measurement. The conductance at the limit of quantum limited heat conduction is an integer multiple of conductance quantum g_{0}= k_{B}^{2}π^{2}T/3h: g=ng_{0}. Here the sum is over parallel channels. What is remarkable is quantization and independence on the length of the wire. Once the heat carriers are in wire, the heat is transferred since dissipation is not present. A completely analogous formula holds true for electric conductance along ballistic wire: now g would be integer multiple of g_{0}=σ/L= 2e^{2}/h. Note that in 2D system quantum Hall conductance (not conductivity) is integer (or more generally some rational) multiple of σ_{0}= e^{2}/h. The formula in the case of conductance can be "derived" heuristically from Uncertainty Principle Δ EΔ t=h plus putting Δ E = eΔ V as difference of Coulomb energy and Δ t= e/I=e L/ΔV=e/g_{0}. The essential prerequisite for quantal conduction is that the length of the wire is much shorter than the wavelength assignable to the carrier of heat or of thermal energy: λ>> L. It is interesting to find how well this condition is satisfied in the recent case. The wavelength of the photons involved with the transfer should be much longer than 1 meter. An order of magnitude for the energy of photons involve and thus for the frequency and wavelength can be deduced from the thermal energy of photons in the system. The electron temperatures considered are in the range of 10100 mK roughly. Kelvin corresponds to 10^{4} eV (this is more or less all that I learned in thermodynamics course in student days) and eV corresponds to 1.24 microns. This temperature range roughly corresponds to thermal energy range of 10^{6}10^{5} eV. The wave wavelength corresponding to maximal intensity of blackbody radiation is in the range of 2.323 centimeters. One can of course ask whether the condition λ >> L=1 m is consistent with this. A specialist would be needed to answer this question. Note that the gap energy .45 meV of superconductor defines energy scale for Josephson radiation generated by superconductor: this energy would correspond to about 2 mm wavelength much below one 1 meter. This energy does not correspond to the energy scale of thermal photons. I am of course unable to say anything interesting about the experiment itself but cannot avoid mentioning the hierarchy of Planck constants. If one has E= h_{eff}f, h_{eff}=n× h instead of E= hf, the condition λ>> L can be easily satisfied. For superconducting wire this would be true for superconducting magnetic flux tubes in TGD Universe and maybe it could be true also for photons, if they are dark and travel along them. One can even consider the possibility that quantal heat conductivity is possible over much longer wire lengths than 1 m. Showing this to be the case, would provide strong support for the hierarchy of Planck constants. There are several interesting questions to be pondered in TGD framework. Could one identify classical spacetime correlates for the quantization of conductance? Could one understand how classical thermodynamics differs from quantum thermodynamics? What quantum thermodynamics could actually mean? There are several rather obvious ideas.
