Probability Theory Compatible with the New Conception of Modern Thermodynamics. Economics and Crisis of Debts

We show that G\"odel's negative results concerning arithmetic, which date back to the 1930s, and the ancient "sand pile" paradox (known also as "sorites paradox") pose the questions of the use of fuzzy sets and of the effect of a measuring device on the experiment. The consideration of these facts led, in thermodynamics, to a new one-parameter family of ideal gases. In turn, this leads to a new approach to probability theory (including the new notion of independent events). As applied to economics, this gives the correction, based on Friedman's rule, to Irving Fisher's "Main Law of Economics" and enables us to consider the theory of debt crisis.Comment: 48p., 14 figs., 82 refs.; more precise mathematical explanations are added. arXiv admin note: significant text overlap with arXiv:1111.610

On the superfluidity of classical liquid in nanotubes

In 2001, the author proposed the ultra second quantization method. The ultra second quantization of the Schr\"odinger equation, as well as its ordinary second quantization, is a representation of the N-particle Schr\"odinger equation, and this means that basically the ultra second quantization of the equation is the same as the original N-particle equation: they coincide in 3N-dimensional space. We consider a short action pairwise potential V(x_i -x_j). This means that as the number of particles tends to infinity, $N\to\infty$, interaction is possible for only a finite number of particles. Therefore, the potential depends on N in the following way: $V_N=V((x_i-x_j)N^{1/3})$. If V(y) is finite with support $\Omega_V$, then as $N\to\infty$ the support engulfs a finite number of particles, and this number does not depend on N. As a result, it turns out that the superfluidity occurs for velocities less than $\min(\lambda_{\text{crit}}, \frac{h}{2mR})$, where $\lambda_{\text{crit}}$ is the critical Landau velocity and R is the radius of the nanotube.Comment: Latex, 20p. The text is presented for the International Workshop "Idempotent and tropical mathematics and problems of mathematical physics", Independent University of Moscow, Moscow, August 25--30, 2007 and to be published in the Russian Journal of Mathematical Physics, 2007, vol. 15, #

Gapped Phases of Quantum Wires

We investigate possible nontrivial phases of a two-subband quantum wire. It is found that inter- and intra-subband interactions may drive the electron system of the wire into a gapped state. If the nominal electron densities in the two subbands are sufficiently close to each other, then the leading instability is the inter-subband charge-density wave (CDW). For large density imbalance, the interaction in the inter-subband Cooper channel may lead to a superconducting instability. The total charge-density mode, responsible for the conductance of an ideal wire, always remains gapless, which enforces the two-terminal conductance to be at the universal value of 2e^2/h per occupied subband. On the contrary, the tunneling density of states (DOS) in the bulk of the wire acquires a hard gap, above which the DOS has a non-universal singularity. This singularity is weaker than the square-root divergency characteristic for non-interacting quasiparticles near a gap edge due to the "dressing" of massive modes by a gapless total charge density mode. The DOS for tunneling into the end of a wire in a CDW-gapped state preserves the power-law behavior due to the frustration the edge introduces into the CDW order. This work is related to the vast literature on coupled 1D systems, and most of all, on two-leg Hubbard ladders. Whenever possible, we give derivations of the important results by other authors, adopted for the context of our study.Comment: 30 pages, 6 figures, to appear in "Interactions and Transport Properties of Lower Dimensional Systems", Lecture Notes in Physics, Springe

Conductance of a Mott Quantum Wire

We consider transport through a one-dimensional conductor subject to an external periodic potential and connected to non-interacting leads (a "Mott quantum wire"). For the case of a strong periodic potential, the conductance is shown to jump from zero, for the chemical potential lying within the Mott-Hubbard gap, to the non-interacting value of 2e^2/h, as soon as the chemical potential crosses the gap edge. This behavior is strikingly different from that of an optical conductivity, which varies continuously with the carrier concentration. For the case of a weak potential, the perturbative correction to the conductance due to Umklapp scattering is absent away from half-filling.Comment: 4 pages, RevTex, 1 ps figure included; published versio

On the Thermal Stability of Graphone

Molecular dynamics simulation is used to study thermally activated migration of hydrogen atoms in graphone, a magnetic semiconductor formed of a graphene monolayer with one side covered with hydrogen so that hydrogen atoms are adsorbed on each other carbon atom only. The temperature dependence of the characteristic time of disordering of graphone via hopping of hydrogen atoms to neighboring carbon atoms is established directly. The activation energy of this process is found to be Ea=(0.05+-0.01) eV. The small value of Ea points to extremely low thermal stability of graphone, this being a serious handicap for practical use of the material in nanoelectronics.Comment: 3 figure

q-Legendre Transformation: Partition Functions and Quantization of the Boltzmann Constant

In this paper we construct a q-analogue of the Legendre transformation, where q is a matrix of formal variables defining the phase space braidings between the coordinates and momenta (the extensive and intensive thermodynamic observables). Our approach is based on an analogy between the semiclassical wave functions in quantum mechanics and the quasithermodynamic partition functions in statistical physics. The basic idea is to go from the q-Hamilton-Jacobi equation in mechanics to the q-Legendre transformation in thermodynamics. It is shown, that this requires a non-commutative analogue of the Planck-Boltzmann constants (hbar and k_B) to be introduced back into the classical formulae. Being applied to statistical physics, this naturally leads to an idea to go further and to replace the Boltzmann constant with an infinite collection of generators of the so-called epoch\'e (bracketing) algebra. The latter is an infinite dimensional noncommutative algebra recently introduced in our previous work, which can be perceived as an infinite sequence of "deformations of deformations" of the Weyl algebra. The generators mentioned are naturally indexed by planar binary leaf-labelled trees in such a way, that the trees with a single leaf correspond to the observables of the limiting thermodynamic system

Peculiarities of dynamics of Dirac fermions associated with zero-mass lines

Zero-mass lines result in appearance of linear dispersion modes for Dirac fermions. These modes play an important role in various physical systems. However, a Dirac fermion may not precisely follow a single zero-mass line, due to either tunneling between different lines or centrifugal forces. Being shifted from a zero-mass line the Dirac fermion acquires mass which can substantially influence its expected "massless" behavior. In the paper we calculate the energy gap caused by the tunneling between two zero-mass lines and show that its opening leads to the delocalization of linear dispersion modes. The adiabatic bending of a zero-mass line gives rise to geometric phases. These are the Berry phase, locally associated with a curvature, and a new phase resulting from the mass square asymmetry in the vicinity of a zero-mass line.Comment: 6 pages, 4 figures. In the second version some references were added and minor changes were made in the introductio