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The internal total-energy of the black-body is a physical quantity of paramount importance in the development of modern physics. Accordingly, together with a brief historical development, we report and comment last breaking news (2018-2024) concerning the definition and properties of this quantity. The first comment concerns with the inclusion of the Casimir energy that avoids the vacuum catastrophe implied by he presence of zero-point energy, thus leading to further quantum contributions associated with boundary effects. The second comment concerns with a semi-classical simulation of a one dimensional black-body whose results suggest a possible reconsideration on the role of classical physics on the quantum black-body., Comment: 5 pages, no figures

In this paper we present an extension of Landauer paradigm, conductance is transmission, to the case of macroscopic classical conductors making use of a description of conductance and resistance based on the application of the fluctuation dissipation (FD) theorem. The main result is summarized in the expressions below for conductance $G$ and resistance $R$ at thermodynamic equilibrium, with the usual meaning of symbols. $G$ is given in terms of the variance of total carrier number fluctuations between two ideal transparent contacts in an open system described by a grand canonical ensemble as $$ G =\frac{e^2 \overline{v_x'^2} \tau}{L^2 K_BT} \overline{\delta N^2} %= \frac{e^2 \sqrt{\overline{v_x'^2}} \Gamma }{L K_BT} \overline{\delta %N^2} %= \frac{e^2 \overline{N} \Gamma } {Lm\sqrt{\overline{v_x'^2}}} \ \ \ \ $$ By contrast $R$ is given in terms of the variance of carrier drift-velocity fluctuations due to the instantaneous carrier specular reflection at the internal contact interfaces of a closed system described by a canonical ensemble as $$ R= \frac{(m L)^2}{e^2 K_BT \tau} \overline{\delta v_d^2} %= \frac {Lm\sqrt{\overline{v_x'^2}}} {e^2 \overline{N} \Gamma } $$ The FD approach gives evidence of the duality property of conductance related to transmission and resistance related to reflection. Remarkably, the expressions above are shown to recover the quantum Landauer paradigm in the limit of zero temperature for a one-dimensional conductor.

By analogy with the fundamental quantum units of electrical conductance $G_0^e=\frac{2 e^2}{h}$ and thermal conductance $K_0^t=\frac{2 K_B^2 T}{h}$ we define a fundamental quantum unit of conductance, $G_0^m$, and diffusion of a massive gas of atomic particles, respectively given by $$ G_0^m=\frac{m^2}{h} \ , \ D_0=\frac{h}{m}$$ with $h$ the Planck constant, $K_B$ the Boltzmann constant, $T$ the absolute temperature, $e$ the unit charge and $m$ the mass of the atomic gas particle that move balistically in a one dimensional medium of length $L$. The effect of scattering can be accounted for by introducing an appropriate transmission probability in analogy with the quantum electrical conductance model introduced by Landauer in 1957. For an electron gas $G_0^m=1.25 \times 10^{-27} \ Kg^2/(J s)$ and $D_0 = 7.3 \times 10^{-3} \ m^2/s$, and we found a quantum expression for the generalized Einstein relation that writes $$G_0^e = \frac{2e^2m}{h^2} D_0 $$

Stefan-Boltzmann law was empirically deduced by Stefan in 1874 by fitting existing experiments and theoretically validated by Boltzmann in 1884 on the basis of a classical model involving thermodynamics principles and the Maxwell equations. At first sight the electromagnetic (EM) gas assumed by Boltzmann and identifiable as an ensemble of $N$ classical normal-modes, looks like an extension of the classical model of the massive ideal-gas. Accordingly, for this EM gas the internal total energy, $U$, was taken to be function of volume $V$ and temperature $T$ as $U=U(V,T)$, and the equation of state was given by $U=3PV$, with $P$ the radiation pressure. In addition, Boltzmann implicitly assumed that, for given values of $V$ and $T$, $U$ and $N$ would take finite values. However, from one hand these assumptions are not justified by Maxwell equations since, in vacuum (i.e. far from the EM sources), according to classical statistics, the values of $U$and $N$ diverge. From another hand, Boltzmann derivation of Stefan law is found to be macroscopically compatible with its derivation from quantum statistics announced by Planck in 1901. Accordingly, this letter presents a solution of this puzzling classical/quantum compatibility by noticing that the implicit assumption made by Boltzmann is fully justified by quantum statistics. Furthermore, we shed new light on the interpretation of recent classical simulations of a black-body carried out by Wang, Casati, and Benenti in 2022 who found an analogous puzzling compatibility to induce speculations on classical physics and black-body radiation that are claimed to require a critical reconsideration of the role of classical physics for the understanding of quantum mechanics., Comment: 4 pages

The Stefan-Boltzmann (SB) law relates the emissivity $q$, given in $Wm^{-2}$, of an ideal black-body cavity at thermal equilibrium to the fourth power of the absolute temperature $T$ as $q=\sigma T^4$, with $\sigma = 5.67 \times 10^{-8} \ W m^{-2} K^{-4}$ the SB constant, firstly estimated by Stefan to within $11$ per cent of the actual value. The law is a pillar of modern physics since its microscopic derivation implies the quantization of the energy related to the electromagnetic field. Somewhat astonishing, Boltzmann presented his derivation in 1878 making use only of electrodynamic and thermodynamic classical concepts, apparently without introducing any quantum hypothesis (here called first Boltzmann paradox). By using Planck (1901) quantization of the radiation field in terms of a gas of photons, the SB law received a microscopic interpretation providing also the value of the SB constant on the basis of a set of universal constants including the quantum action constant $h$. However, the successive consideration by Planck (1912) of the zero-point energy contribution was found to be responsible of another divergence of the radiation energy-density for the single photon mode at high frequencies. This divergence is of pure quantum origin and is responsible for a vacuum-catastrophe, to keep the analogy with the well-known ultraviolet catastrophe of the classical black-body radiation spectrum, given by the Rayleigh-Jeans law in 1900. As a consequence, from a rigorous quantum-mechanical derivation we expect the divergence of the SB law (here called second Boltzmann paradox). In this paper we revisit the SB law by accounting for genuine quantum effects associated with Planck energy quantization and Casimir size quantization thus resolving both Boltzmann paradoxes, Comment: 4 pages, no figures. arXiv admin note: text overlap with arXiv:1805.07933

The fluctuation dissipation theorem (FDT) is the basis for a microscopic description of the interaction between electromagnetic radiation and matter.By assuming the electromagnetic radiation in thermal equilibrium and the interaction in the linear response regime, the theorem interrelates the spontaneous fluctuations of microscopic variables with the kinetic coefficients that are responsible for energy dissipation.In the quantum form provided by Callen and Welton in their pioneer paper of 1951 for the case of conductors, electrical noise detected at the terminals of a conductor was given in terms of the spectral density of voltage fluctuations, $S_V({\omega})$, and was related to the real part of its impedance, $Re[Z({\omega})]$, by a simple relation.The drawbacks of this relation concern with: (I) the appearance of a zero point contribution which implies a divergence of the spectrum at increasing frequencies; (ii) the lack of detailing the appropriate equivalent-circuit of the impedance, (iii) the neglect of the Casimir effect associated with the quantum interaction between zero-point energy and boundaries of the considered physical system; (iv) the lack of identification of the microscopic noise sources beyond the temperature model. These drawbacks do not allow to validate the relation with experiments. By revisiting the FDT within a brief historical survey, we shed new light on the existing drawbacks by providing further properties of the theorem, focusing on the electrical noise of a two-terminal sample under equilibrium conditions. Accordingly, we will discuss the duality and reciprocity properties of the theorem, its applications to the ballistic transport regime, to the case of vacuum and to the case of a photon gas., Comment: 16 pages, 8 figures

The role played by zero-point contribution, also called quantum noise or vacuum fluctuations, in the quantum expression of the fluctuation-dissipation theorem (FDT) is a long-standing open problem widely discussed by the physicist community since its announcement by Callen and Welton pioneer paper of 1951 [1]. From one hand, it has the drawbacks of: (i) the expectation value of its energy is infinite, (ii) it produces an ultraviolet catastrophe of the noise power spectral density and, (iii) it lacks of an experimental validation under thermal equilibrium conditions. From another hand, by imposing appropriate boundary conditions and eliminating divergences by regulation techniques, vacuum fluctuations are the source of an attractive force between opposite conducting plates, firstly predicted by Casimir in 1948 [2] and later validated experimentally with increasing accuracy. As a consequence, a quantum formulation of FDT should account for the presence of the Casimir force and of its consequences. In this letter we show that at thermal equilibrium the Casimir force should be balanced by the mechanical reaction of the physical system. As a consequence, no zero-point spectrum can be detected and the power spectrum emitted by the physical system is the same of that calculated by Planck in 1901 [3] for a black-body. Accordingly, the experimental validation of the standard expression of the quantum FDT [1] is prevented in favor of the Nyquist expression that includes the Planck factor [4]., Comment: 4 pages

Aptamers are chemically produced oligonucleotides, able to bind a variety of targets such as drugs, proteins and pathogens with high sensitivity and selectivity. Therefore, aptamers are largely employed for producing label-free biosensors, with significant applications in diagnostics and drug delivery. In particular, the anti-thrombin aptamers are biomolecules of high interest for clinical use, because of their ability to recognize and bind the thrombin enzyme. Among them, the DNA 15-mer thrombin-binding aptamer (TBA), has been widely explored concerning both its structure, which was resolved with different techniques, and its function, especially about the possibility of using it as the active part of biosensors. This paper proposes a microscopic model of the electrical properties of TBA and the aptamer-thrombin complex, combining information from both structure and function. The novelty consists in describing both the aptamer alone and the complex as an impedance network, thus going deeper inside the issues addressed in an emerging electronics branch known as proteotronics. The theoretical results are compared and validated with Electrochemical Impedance Spectroscopy measurements reported in the literature. Finally, the model suggests resistance measurements as a novel tool for testing aptamer-target affinity., Comment: 5 figures, 1 table

By analogy with linear-response we formulate the duality and reciprocity properties of current and voltage fluctuations expressed by Nyquist relations including the intrinsic bandwidths of the respective fluctuations. For this purpose we individuate total-number and drift-velocity fluctuations of carriers inside a conductor as the microscopic sources of noise. The spectral densities at low frequency of the current and voltage fluctuations and the respective conductance and resistance are related in a mutual exclusive way to the corresponding noise-source. The macroscopic variance of current and voltage fluctuations are found to display a dual property via a plasma conductance that admits a reciprocal plasma resistance. Analogously, the microscopic noise-sources are found to obey a dual property and a reciprocity relation. The formulation is carried out in the frame of the grand canonical (for current noise) and canonical (for voltage noise) ensembles and results are derived which are valid for classical as well as for degenerate statistics including fractional exclusion statistics. The unifying theory so developed sheds new light on the microscopic interpretation of dissipation and fluctuation phenomena in conductors. In particular it is proven that, as a consequence of the Pauli principle, for Fermions non-vanishing single-carrier velocity fluctuations at zero temperature are responsible for diffusion but not for current noise, which vanishes in this limit., Comment: 5 pages, 1 figure

The electrical properties of a set of seven-helix transmembrane proteins, whose space arrangement (3D structure) is known, are investigated by using regular arrays of the amino acids. These structures, specifically cubes, have topological features similar to those shown by the chosen proteins. The theoretical results show a good agreement between the predicted current-voltage characteristics obtained from a cubic array and those obtained from a detailed 3D structure. The agreement is confirmed by available experiments on bacteriorhodopsin. Furthermore, all the analyzed proteins are found to share the same critical behaviour of the voltage-dependent conductance and of its variance. In particular, the cubic arrangement evidences a short plateau of the excess conductance and its variance at high voltages. The results of the present investigation show the possibility to predict the I-V characteristics of multiple-protein sample even in the absence of a detailed knowledge of their 3D structure., Comment: 5 pages, 7 figures

We investigate the nature of a class of probability density functions, say G(a), with a the shape parameter, which generalizes the Gumbel distribution. These functions appear in a model of charge transport, when applied to a metal-insulator-metal structure, where the insulator is constituted by a monolayer of bacteriorhodopsin. Current shows a sharp increase above about 3 V, interpreted as the cross-over between direct and injection sequential-tunneling regimes. In particular, we show that, changing the bias value, the probability density function changes its look from bimodal to unimodal. Actually, the bimodal distributions can be resolved in at least a couple of $G(a)$ functions with different values of the shape parameter., Comment: 5 pages, 6 figures

Increasing attention has been recently devoted to protein-based nanobiosensors. The main reason is the huge number of possible technological applications, going from drug detection to cancer early diagnosis. Their operating model is based on the protein activation and the corresponding conformational change, due to the capture of an external molecule, the so-called ligand. Recent measurements, performed with different techniques on human 17-40 olfactory receptor, evidenced a very narrow window of response in respect of the odour concentration. This is a crucial point for understanding whether the use of this olfactory receptor as sensitive part of a nanobiosensor is a good choice. In this paper we investigate the topological and electrical properties of the human olfactory receptor 17-40 with the objective of providing a microscopic interpretation of available experiments. To this purpose, we model the protein by means of a graph able to capture the mean features of the 3D backbone structure. The graph is then associated with an equivalent impedance network, able to evaluate the impedance spectra of the olfactory receptor, in its native and activated state. We assume a topological origin of the different protein electrical responses to different ligand concentrations: In this perspective all the experimental data are collected and interpreted satisfactorily within a unified scheme, also useful for application to other proteins., Comment: 6 pages, 6 figures, DOI:10.1039/c1ra00025

Current voltage (I-V) characteristics in proteins can be sensitive to conformational change induced by an external stimulus (photon, odour, etc.). This sensitivity can be used in medical and industrial applications besides shedding new light in the microscopic structure of biological materials. Here, we show that a sequential tunneling model of carrier transfer between neighbouring amino-acids in a single protein can be the basic mechanism responsible of the electrical properties measured in a wide range of applied potentials. We also show that such a strict correlation between the protein structure and the electrical response can lead to a new generation of nanobiosensors that mimic the sensorial activity of living species. To demonstrate the potential usefulness of protein electrical properties, we provide a microscopic interpretation of recent I-V experiments carried out in bacteriorhodopsin at a nanoscale length., Comment: 4 pages, 4 figures

A trapping-detrapping model is proposed for explaining the current fluctuation behavior in organic semiconductors (polyacenes) operating under current-injection conditions. The fraction of ionized traps obtained from the current-voltage characteristics, is related to the relative current noise spectral density at the trap-filling transition. The agreement between theory and experiments validates the model and provides an estimate of the concentration and energy level of deep traps.

When moving from native to light activated bacteriorhodopsin, modification of charge transport consisting of an increase of conductance is correlated to the protein conformational change. A theoretical model based on a map of the protein tertiary structure into a resistor network is implemented to account for a sequential tunneling mechanism of charge transfer through neighbouring amino acids. The model is validated by comparison with current-voltage experiments. The predictability of the model is further tested on bovine rhodopsin, a G-protein coupled receptor (GPCR) also sensitive to light. In this case, results show an opposite behaviour with a decrease of conductance in the presence of light., Comment: 6 pages, 4 figures

We present a theoretical investigation on possible selection of olfactory receptors (ORs) as sensing components of nanobiosensors. Accordingly, we generate the impedance spectra of the rat OR I7 in the native and activated state and analyze their differences. In this way, we connect the protein morphological transformation, caused by the sensing action, with its change of electrical impedance. The results are compared with those obtained by studying the best known protein of the GPCR family: bovine rhodopsin. Our investigations indicate that a change in morphology goes with a change in impedance spectrum mostly associated with a decrease of the static impedance up to about 60 % of the initial value, in qualitative agreement with existing experiments on rat OR I7. The predictiveness of the model is tested successfully for the case of recent experiments on bacteriorhodopsin. The present results point to a promising development of a new class of nanobiosensors based on the electrical properties of GPCR and other sensing proteins., Comment: 6 pages, 8 figures

We develop a microscopic theory of electron transport in superlattices within the Wannier-Stark approach by including the interaction associated with Zener tuneling among the energy levels pertaining to adjacent quantum wells. By using a Monte Carlo technique we have simulated the hopping motion associated with absorption and emission of polar optical phonons and determined the main transport parameters for the case of a GaAs/GaAlAs structure at room temperature. The interaction among the levels is found to be responsible for a systematic increase of the level energy with respect to the bottom of the quantum well at electric fields above about 20 kV/cm. When compared with the non-interacting case, at the highest fields the average carrier energy evidences a consistent increase which leads to a significant softening of the negative differential value of both the drift velocity and diffusivity.