Derivation of Two-Fluid Model Based on Onsager Principle

Using the Onsager variational principle, we study the dynamic coupling between the stress and the composition in a polymer solution. In the original derivation of the two-fluid model of Doi and Onuki the polymer stress was introduced a priori; therefore, a constitutive equation is required to close the equations. Based on our previous study of viscoelastic fluids with homogeneous composition, we start with a dumbbell model for the polymer, and derive all dynamic equations using the Onsager variational principle.     

Onsager principle as a tool for approximation

Onsager principle is the variational principle proposed by Onsager in his celebrated paper on the reciprocal relation.The principle has been shown to be useful in deriving many evolution equations in soft matter physics.Here the principle is shown to be useful in solving such equations approximately.Two examples are discussed:the diffusion dynamics and gel dynamics.Both examples show that the present method is novel and gives new results which capture the essential dynamics in the system.     

The large deviation principle and some models of an interacting boson gas

This is a study of the equilibrium thermodynamics of the Huang-Yang-Luttinger model of a boson gas with a hard-sphere repulsion using large deviation methods; we contrast its properties with those of the mean field model. We prove the existence of the grand canonical pressure in the thermodynamic limit and derive two alternative expressions for the pressure as a function of the chemical potential. We prove the existence of condensate for values of the chemical potential above a critical value and verify a prediction of Thouless that there is a jump in the density of condensate at the critical value. We show also that, at fixed mean density, the density of condensate is an increasing function of the strength of the repulsive interaction. In an appendix, we give proofs of the large deviation results used in the body of the paper.     

Derivation of Treder-type gravitational equations from a variational principle

A single variation principle for the gravitational field and nongravitating matter leading to Treder-type gravitational equations is formulated. It is shown that when Treder's theory is modified in this manner the resulting special-relativistic theory of gravitation with 16-component gravitational potentials h _µ ⁽⁾ predicts 7/12 of the perihelion advance and 1/2 of the light deflection and light delay predicted by general relativity.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 36–39, January, 1976.I thank Professor D. D. Ivanenko for a helpful discussion.     

Derivation of quantum statistics from Gauss's principle and the second law

Quantum statistical laws are derived from bona fide stationary probability distributions of physical stochastic processes. These distributions are shown to be the laws of error for which the average occupation numbers are the most probable values. They determine uniquely the statistical entropy functions and the second law gives the quantum statistical distributions.     

Thin film dynamics in coating problems using Onsager principle

A new variational method is proposed to investigate the dynamics of the thin film in a coating flow where a liquid is delivered through a fixed slot gap onto a moving substrate. A simplified ODE system has also been derived for the evolution of the thin film whose thickness h_f is asymptotically constant behind the coating front. We calculate the phase diagram as well as the film profiles and approximate the film thickness theoretically, and agreement with the well-known scaling law as Ca~(2/3) is found.     

A perturbed mean field model of an interacting boson gas and the large deviation principle

This is a study of the equilibrium thermodynamics of a mean-field model of an interacting boson gas perturbed by a term quadratic in the occupation numbers of the free-gas energy-levels. We prove the existence of the pressure in the thermodynamic limit. We obtain also a variational formula for the pressure; this enables us to compare the effect of a smooth quadratic perturbation with that of a singular one (the Huang-Yang-Luttinger model). The proof uses a large deviation result for the occupation measure of the free boson gas which is of independent interest.     

Equivalences of the large deviation principle for Gibbs measures and critical balance in the Ising model

In this paper we obtain the equivalence of the large deviation principle for Gibbs measures with and without an external field. For the Ising model, the equivalence allows us to study the result of competing influences of a positive external fieldh and a negative boundary condition in the cube ((B/h) ash0 for variousB. We find a critical balance at a valueB ₀ ofB.     

Derivation of the density functional theory from the cluster expansion

The density functional theory is derived from a cluster expansion by truncating the higher-order correlations in one and only one term in the kinetic energy. The formulation allows self-consistent calculation of the exchange correlation effect without imposing additional assumptions to generalize the local density approximation. The pair correlation is described as a two-body collision of bound-state electrons, and modifies the electron- electron interaction energy as well as the kinetic energy. The theory admits excited states, and has no self-interaction energy.     

Derivation of the principle of equivalence for antimatter

In view of the announcement of some experiments testing the principle of equivalence for antimatter, we give here stringent arguments, based on elementary and well-established physical principles, that these experiments will turn out negative. The question is important because disproving the principle of equivalence (equality of inertial and gravitating mass) would entail a breakdown of general relativity. (There is only one type of geodesics and there are no antigeodesics for antimatter).     

Derivation of an adiabatic time-dependent Hartree-Fock formalism from a variational principle

A derivation of the adiabatic time-dependent Hartree-Fock formalism is given, which is based on a variational principle analogous to Hamilton's principle in classical mechanics. The method leads to a Hamiltonian for collective motion which separates into a potential and a kinetic energy and gives mass and potential parameters in terms of the nucleon-nucleon interaction. The adiabatic approximation assumes slow motion but not small amplitudes and can therefore describe anharmonic effects. The RPA is a limiting case where both amplitudes and velocities are small. The variational approach provides a consistent way of extracting coordinates and momenta from the density matrix and of obtaining equations of motion when particular trial forms for this density matrix are chosen. One such choice leads to Thouless-Valatin formula. An other choice leads to irrotational hydrodynamics.     

Large deviation principle for spatial density of local observables from Gibbs distributions

We establish the large deviation principle characterising, in the thermodynamic limit, the exponential decay rates for the probabilities of macroscopic fluctuations of spatial densities generated by local observables from Gibbs lattice systems with absolutely summable interactions.     

Positive deviation from debye‘s theory in small—angle x—ray scattering

The small-angle x-ray scattering (SAXS) resulting from a non-ideal two-phase system having sharply defined phase boundaries but with micro-fluctuations of electron density within any phase shows a positive deviation from Debye's theory. We have found a simple method suitable for fitting the slit-smeared SAXS data and correcting the positive deviation. The validity of this procedure is tested on porous ZrO_2 xerogel prepared using the sol-gel process.     

Derivation of relativistic charge-monopole and monopole-monopole potentials from field theory

Relativistic potentials (generalizing the Breit-potential of quantum electrodynamics) between spin $\text{1}\text{2}\text{-}\text{electric}$ and magnetic charges are presented, each monopole g_j having its own singularity-string along some direction n_j. The monopole potentials involve integrations along the singularities. By using suitable gauge transformations and limiting procedures a simple form of the potential independent of n_j is derived, if the string connects two monopoles.     

Derivation of fluid dynamics from kinetic theory with the 14-moment approximation

We review the traditional derivation of the fluid-dynamical equations from kinetic theory according to Israel and Stewart. We show that their procedure to close the fluid-dynamical equations of motion is not unique. Their approach contains two approximations, the first being the so-called 14-moment approximation to truncate the single-particle distribution function. The second consists in the choice of equations of motion for the dissipative currents. Israel and Stewart used the second moment of the Boltzmann equation, but this is not the only possible choice. In fact, there are infinitely many moments of the Boltzmann equation which can serve as equations of motion for the dissipative currents. All resulting equations of motion have the same form, but the transport coefficients are different in each case.     

Optimal hedging via large deviation

The criterion of minimizing the cumulative hedged returns’ probability of underperforming a benchmark provides a framework for evaluating short-term hedges that are rolled over to produce longer-term hedges. Large deviations theory can be used to either parametrically or nonparametrically estimate underperformance probabilities for cumulative hedged returns produced by roll-overs, providing a straightforward way to find optimal hedge ratios. Optimal hedges using soybean futures are constructed to illustrate the procedures, and their relationship to the popular hedging criteria that are motivated by normality.     

Comment on “Towards a large deviation theory for strongly correlated systems”

I comment on a recent paper by Ruiz and Tsallis [Phys. Lett. A 376 (2012) 2451] claiming to have found a “q-exponential” generalization of the large deviation principle for strongly correlated random variables. I show that the basic scaling results that they find numerically can be reproduced with a simple example involving independent random variables, and are not specifically related to the q-exponential function. In fact, identical scaling results can be obtained with any other power-law deformations of the exponential. Thus their results do not conclusively support their claim of a q-exponential generalization of the large deviation principle.