1,673 research outputs found
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
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
Quasithermodynamics and a Correction to the Stefan--Boltzmann Law
We provide a correction to the Stefan--Boltzmann law and discuss the problem
of a phase transition from the superfluid state into the normal state.Comment: Latex, 9page
Mathematical Conception of "Phenomenological" Equilibrium Thermodynamics
In the paper, the principal aspects of the mathematical theory of equilibrium
thermodynamics are distinguished. It is proved that the points of degeneration
of a Bose gas of fractal dimension in the momentum space coincide with critical
points or real gases, whereas the jumps of critical indices and the Maxwell
rule are related to the tunnel generalization of thermodynamics. Semiclassical
methods are considered for the tunnel generalization of thermodynamics and also
for the second and ultrasecond quantization (operators of creation and
annihilation of pairs). To every pure gas there corresponds a new critical
point of the limit negative pressure below which the liquid passes to a
dispersed state (a foam). Relations for critical points of a homogeneous
mixture of pure gases are given in dependence on the concentration of gases.Comment: 37 pages, 9 figure, more precise explanations, more references. arXiv
admin note: substantial text overlap with arXiv:1202.525
Analytic description of monodromy oscillons
We develop precise analytic description of oscillons - long-lived
quasiperiodic field lumps - in scalar field theories with nearly quadratic
potentials, e.g. the monodromy potential. Such oscillons are essentially
nonperturbative due to large amplitudes, and they achieve extreme longevities.
Our method is based on a consistent expansion in the anharmonicity of the
potential at strong fields, which is made accurate by introducing a
field-dependent "running mass." At every order, we compute effective action for
the oscillon profile and other parameters. Comparison with explicit numerical
simulations in (3+1)-dimensional monodromy model shows that our method is
significantly more precise than other analytic approaches.Comment: 9 pages, 9 figures; v2: Introduction, Sec. 4 and Discussion extended;
journal versio
Semiclassical Estimates of Electromagnetic Casimir Self-Energies of Spherical and Cylindrical Metallic Shells
The leading semiclassical estimates of the electromagnetic Casimir stresses
on a spherical and a cylindrical metallic shell are within 1% of the field
theoretical values. The electromagnetic Casimir energy for both geometries is
given by two decoupled massless scalars that satisfy conformally covariant
boundary conditions. Surface contributions vanish for smooth metallic
boundaries and the finite electromagnetic Casimir energy in leading
semiclassical approximation is due to quadratic fluctuations about periodic
rays in the interior of the cavity only. Semiclassically the non-vanishing
Casimir energy of a metallic cylindrical shell is almost entirely due to
Fresnel diffraction.Comment: 12 pages, 2 figure
Initial Conditions for Semiclassical Field Theory
Semiclassical approximation based on extracting a c-number classical
component from quantum field is widely used in the quantum field theory.
Semiclassical states are considered then as Gaussian wave packets in the
functional Schrodinger representation and as Gaussian vectors in the Fock
representation. We consider the problem of divergences and renormalization in
the semiclassical field theory in the Hamiltonian formulation. Although
divergences in quantum field theory are usually associated with loop Feynman
graphs, divergences in the Hamiltonian approach may arise even at the tree
level. For example, formally calculated probability of pair creation in the
leading order of the semiclassical expansion may be divergent. This observation
was interpretted as an argumentation for considering non-unitary evolution
transformations, as well as non-equivalent representations of canonical
commutation relations at different time moments. However, we show that this
difficulty can be overcomed without the assumption about non-unitary evolution.
We consider first the Schrodinger equation for the regularized field theory
with ultraviolet and infrared cutoffs. We study the problem of making a limit
to the local theory. To consider such a limit, one should impose not only the
requirement on the counterterms entering to the quantum Hamiltonian but also
the requirement on the initial state in the theory with cutoffs. We find such a
requirement in the leading order of the semiclassical expansion and show that
it is invariant under time evolution. This requirement is also presented as a
condition on the quadratic form entering to the Gaussian state.Comment: 20 pages, Plain TeX, one postscript figur
Semiclassical Description of Wavepacket Revival
We test the ability of semiclassical theory to describe quantitatively the
revival of quantum wavepackets --a long time phenomena-- in the one dimensional
quartic oscillator (a Kerr type Hamiltonian). Two semiclassical theories are
considered: time-dependent WKB and Van Vleck propagation. We show that both
approaches describe with impressive accuracy the autocorrelation function and
wavefunction up to times longer than the revival time. Moreover, in the Van
Vleck approach, we can show analytically that the range of agreement extends to
arbitrary long times.Comment: 10 pages, 6 figure
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