The Thermodynamical Arrow of Time: Reinterpreting the Boltzmann–Schuetz Argument

The recent surge of interest in the origin of the temporal asymmetry of thermodynamical systems (including the accessible part of the universe itself) has put forward two possible explanatory approaches to this age-old problem. Hereby we show that there is a third possible alternative, based on the generalization of the classical (Boltzmann–Schuetz) anthropic fluctuation picture of the origin of the perceived entropy gradient. This alternative (which we dub the Acausal-Anthropic approach) is based on accepting Boltzmann's statistical measure at its face value, and accomodating it within the quantum cosmological concept of the multiverse. We argue that conventional objections raised against the Boltzmann–Schuetz view are less forceful and serious than it is usually assumed. A fortiori, they are incapable of rendering the generalized theory untenable. On the contrary, this analysis highlights some of the other advantages of the multiverse approach to the thermodynamical arrow of time.     

Application of Sobolev gradient method to Poisson–Boltzmann system

The idea of a weighted Sobolev gradient, introduced and applied to singular differential equations in [1], is extended to a Poisson–Boltzmann system with discontinuous coefficients. The technique is demonstrated on fully nonlinear and linear forms of the Poisson– Boltzmann equation in one, two, and three dimensions in a finite difference setting. A comparison between the weighted gradient and FAS multigrid is given for large jump size in the coefficient function.     

The international Boltzmann project – the contribution of the PTB

To support new determinations of the Boltzmann constant, which have been requested by the International Committee for Weights and Measures (CIPM) and which are necessary for preparative steps towards new definitions of the kilogram, the ampere, the kelvin and the mole, an iMERAPlus joint research project is coordinating the European activities in this field in Spain (CEM), Denmark (DFM), France (LNE-INM/CNAM, University Paris North), Italy (INRiM, Universities of Naples and Milan), United Kingdom (NPL), Germany (PTB) and in the European Institute for Reference Materials and Measurements (IRMM). In this major European research project, the Boltzmann constant will be determined with various methods. The aims and the progress to date of the PTB contribution are reviewed in this article. To cite this article: B. Fellmuth et al., C. R. Physique 10 (2009).     

Integrable generalization of the associated Camassa–Holm equation

In this paper, we study an integrable generalization of the associated Camassa–Holm equation. The generalized system is shown to be integrable in the sense of Lax pair and the bilinear Bäcklund transformations are presented through the Bell polynomial technique. Meanwhile, its infinite conservation laws are constructed, and conserved densities and fluxes are given in explicit recursion formulas. Furthermore, a Darboux transformation for the system is derived with the help of the gauge transformation between two Lax pairs. As an application, soliton and periodic wave solutions are given through the Darboux transformation.     

Lattice Boltzmann method with the cell-population equilibrium

The central problem of the lattice Boltzmann method （LBM） is to construct a discrete equilibrium. In this paper, a multi-speed 1D cell-model of Boltzmann equation is proposed, in which the cell-population equilibrium, a direct non- negative approximation to the continuous Maxwellian distribution, plays an important part. By applying the explicit one-order Chapman-Enskog distribution, the model reduces the transportation and collision, two basic evolution steps in LBM, to the transportation of the non-equilibrium distribution. Furthermore, 1D dam-break problem is performed and the numerical results agree well with the analytic solutions.     

A new method to obtain the Carnahan–Starling equation and its generalization

We obtain the Carnahan–Starling equation for a system of hard spheres using the Euler method of accelerated series convergence. For this purpose, the virial series is transformed into a new series with coefficients that differ slightly from each other, even when considering the eleven currently known virial coefficients. The method of accelerated convergence was applied to this series; it allowed us to obtain the Carnahan–Starling equation. In this work, this equation is derived for the first time using the method of accelerated convergence. It is generalized to accurately reproduce all of the known virial coefficients and the asymptotic behavior of the free energy at high densities. This also makes it possible to describe a metastable region with a high degree of accuracy and to obtain the equation of state for a homogeneous system of hard spheres with the accuracy of a computer experiment.     

The generalization of the Mermin–Wagner theorem and the possibility of long-range order in the isotropic discrete one-dimensional quantum Heisenberg model

The problem of existence of long-range order in the isotropic quantum Heisenberg model on the D=1 lattice is reconsidered in view of the possibility of sufficiently slow decaying exchange interaction with infinite effective radius. It is shown that the macrosopic arguments given by Landau and Lifshitz and then supported microscopically by Mermin and Wagner fail for this case so that the non-zero spontaneous magnetization may yet exist. This result was anticipated by Thouless on the grounds of phenomenological analysis, and we give its microscopic foundation, which amounts to the generalization of Mermin–Wagner theorem for the case of the infinite second moment of the exchange interaction. Two well known in lattice statistics models – i.e., Kac-I and Kac-II – illustrate our results.     

Non-negativity and stability analyses of lattice Boltzmann method for advection–diffusion equation

Stability is one of the main concerns in the lattice Boltzmann method (LBM). The objectives of this study are to investigate the linear stability of the lattice Boltzmann equation with the Bhatnagar–Gross–Krook collision operator (LBGK) for the advection–diffusion equation (ADE), and to understand the relationship between the stability of the LBGK and non-negativity of the equilibrium distribution functions (EDFs). This study conducted linear stability analysis on the LBGK, whose stability depends on the lattice Peclet number, the Courant number, the single relaxation time, and the flow direction. The von Neumann analysis was applied to delineate the stability domains by systematically varying these parameters. Moreover, the dimensionless EDFs were analyzed to identify the non-negative domains of the dimensionless EDFs. As a result, this study obtained linear stability and non-negativity domains for three different lattices with linear and second-order EDFs. It was found that the second-order EDFs have larger stability and non-negativity domains than the linear EDFs and outperform linear EDFs in terms of stability and numerical dispersion. Furthermore, the non-negativity of the EDFs is a sufficient condition for linear stability and becomes a necessary condition when the relaxation time is very close to 0.5. The stability and non-negativity domains provide useful information to guide the selection of dimensionless parameters to obtain stable LBM solutions. We use mass transport problems to demonstrate the consistency between the theoretical findings and LBM solutions.     

The Vlasov–Poisson–Boltzmann System Near Vacuum

Global classical solutions with small amplitude are constructed for the Cauchy problem to the Vlasov–Poisson–Boltzmann system, which describes the dynamics of charged particles interacting with their self-consistent electrostatic potential as well as with themselves through collisions. Received: 29 September 2000 / Accepted: 27 November 2000     

On-node lattices construction using partial Gauss–Hermite quadrature for the lattice Boltzmann method

A concise theoretical framework,the partial Gauss–Hermite quadrature(pGHQ),is established to construct on-node lattices of the lattice Boltzmann(LB)method under a Cartesian coordinate system.Compared with the existing approaches,the pGHQ scheme has the following advantages:extremely concise algorithm,unifies the constructing procedure for symmetric and asymmetric on-node lattices,and covers a full-range quadrature degree of a given discrete velocity set.We employ the pGHQ scheme to search the local optimal and asymmetric lattices for{n=3,4,5,6,7}moment degree equilibrium distribution discretization on the range[-10,10].The search reveals a surprising abundance of available lattices.Through a brief analysis,the discrete velocity set shows a significant influence on the positivity of equilibrium distributions,which is considered as one of the major impacts of the numerical stability of the LB method.Hence,the results of the p GHQ scheme lay a foundation for further investigations to improve the numerical stability of the LB method by modifying the discrete velocity set.It is also worth noting that pGHQ can be extended into the entropic LB model,even though it was proposed for the Hermite polynomial expansion LB theory.     

Thermodynamics of the small clusters: The Gibbs free-energy derivation for the Honmura–Kaneyoshi method

A cluster density matrix is proposed for calculation of the thermodynamic averages in the effective field theory with correlations. On this basis, derivation of the Gibbs free-energy for the Honmura–Kaneyoshi method is made.     

A generalization of Poisson–Nijenhuis structures

We generalize Poisson–Nijenhuis structures. We prove that on a manifold endowed with a Nijenhuis tensor and a Jacobi structure which are compatible, there is a hierarchy of pairwise compatible Jacobi structures. Furthermore, we study the homogeneous Poisson–Nijenhuis structures and their relations with Jacobi structures.     

Application of the method of static fluctuational approach to the Bogolyubov–Kolesnikov–Shelah model

A method of calculating the equilibrium correlation functions of any arbitrary order for the Baldwin– Kolesnikov–Shelah (BKSh) model is suggested based on the static fluctuational approach. The method based on only one controllable approach allows the so-called equations of long-range coupling to be obtained which contain all information on the sought-after equilibrium correlation functions within the scope of the BKSh model. Calculations of the sought-after equilibrium correlation functions allow one to go beyond the scope of the conventional molecular field approach and to take into account the effect of field fluctuations on the gap behavior and the heat capacity to the left and right of the critical point. For the simplest case disregarding a dependence of the potential on the wave vector, temperature dependences of the energy gap and heat capacity with allowance for the fluctuations are presented. It is demonstrated that in this case, the fluctuations are small for three-dimensional systems, but sharply increase with decreasing dimensionality of the system.     

Determination of the mass spectrum of quarkonia by the Nikiforov–Uvarov method

The Schr?dinger equation for a potential being the sum of a harmonic oscillator potential, a linear potential, and a Coulomb potential has been solved by the Nikiforov–Uvarov method for large and small distances between particles being in the bound state. Asymptotic expansions have been obtained for the energy levels and wave functions, and also the wave function and the energy of the ground state have been found. The mass spectrum of heavy quarkonia and their radius have been calculated.     

Rigorous Proof of the Boltzmann–Gibbs Distribution of Money on Connected Graphs

Models in econophysics, i.e., the emerging field of statistical physics that applies the main concepts of traditional physics to economics, typically consist of large systems of economic agents who are characterized by the amount of money they have. In the simplest model, at each time step, one agent gives one dollar to another agent, with both agents being chosen independently and uniformly at random from the system. Numerical simulations of this model suggest that, at least when the number of agents and the average amount of money per agent are large, the distribution of money converges to an exponential distribution reminiscent of the Boltzmann–Gibbs distribution of energy in physics. The main objective of this paper is to give a rigorous proof of this result and show that the convergence to the exponential distribution holds more generally when the economic agents are located on the vertices of a connected graph and interact locally with their neighbors rather than globally with all the other agents. We also study a closely related model where, at each time step, agents buy with a probability proportional to the amount of money they have, and prove that in this case the limiting distribution of money is Poissonian.     

Determination of the Galaxy age by the method of uranium–thorium–plutonium isotopic ratios

The dependence of the Galaxy age (Т _G), as determined by the method of uranium–thorium isotopic ratios, on the parameters of the nucleosynthesis model is studied within the theory of galactic nucleosynthesis. It is shown that ТG depends strongly both on the scenario of the production of nuclei in the r-process and those features of neutron-rich nuclei that are used in the respective analysis and on galactic-nucleosynthesis parameters. The effect of a sudden nucleosynthesis spike before the formation of a solar system on the Galaxy age is evaluated. The region of admissible values of the parameters of galacticnucleosynthesis theory is discussed. The method of uranium–thorium isotopic ratios is supplemented with the ²⁴⁴Pu/²³⁸U ratio for yet another cosmochronometer pair, and the Galaxy age is estimated on the basis of the model modified in this way.     

Global Existence of Classical Solutions to the Inelastic Vlasov–Poisson–Boltzmann System

We study the global existence and uniqueness of classical solutions to the inelastic Vlasov–Poisson–Boltzmann system for a soft potential in the near vacuum regime. For the global existence of classical solutions, we assume reasonable conditions on the restitution coefficient, which represents the character of inelastic collisions. We use the smallness of initial data and an algebraically decaying weight function in the spatial variable to control the self-consistence force and collision operator.     

Computer data analysis of the oscillating forward–reverse method

Nonequilibrium work theorems have recently gained wide acceptance as useful tools in determining free energy profiles for soft-matter systems. We have recently proposed an extension of the forward–reverse method, called the oscillating forward–reverse method; by introducing an oscillatory drift it enables the user to obtain PMFs from a single nonequilibrium pull. The analysis, although manageable, is non-trivial. We present here the data analysis and the software (OFR-AT) created to construct PMFs and associated uncertainties from the oscillating forward–reverse (OFR) method. The output analyzed by OFR-AT is often from molecular dynamics simulations, but as with the OFR method itself, it can be more generally applied. OFR-AT is a fast and efficient analysis tool that can analyze very large files (larger than 5 GB) in a short time period. We also describe the uncertainty and correlation calculations performed, provide a map of the data flow through the program, and present representative examples of PMF profiles calculated using OFR-AT.