Statistical Equilibrium Research Articles

AXIOMATIC PRINCIPLES AND DETAILED BALANCE

This article presents an analysis of scientific papers and research on axiomatic principles and detailed balance, we are convinced that even today, in accordance with modern requirements, there is still a need to address axiomatic principles and detailed balance, methods and methods for their study. As mentioned earlier, the axiomatics of statistical physics is reduced to the principle of equal probability of the existence of an equilibrium closed physical system in all microstates accessible to it. This postulate, in turn, suggests that the time-averaged probability of the direct and reverse transition of the system between two selected groups of microstates in the accessible region of the phase space must be the same in both directions (this follows from the definition of an equilibrium state, which contains the requirement of stationarity). In fairness, it should be noted that the principle of detailed balance is not a consequence of the axiomatic principles of statistical physics, and therefore, strictly speaking, is not required to be applied within the framework of this axiomatics. There are situations and outcomes we have given in this article, when the principle of equal probability of available microstates in a closed system is satisfied, but the principle of detailed statistical equilibrium is not.

Fermi gas and modified gravity

We derive the equations of state of Fermi gas by maximizing the Fermi-Dirac entropy in modified gravity in relativistic and non-relativistic case. It will be demonstrated that the microphysics must depend on a given theory of gravity in order to consistently describe a physical system at a statistical equilibrium state.

Kinetic theory of two-dimensional point vortices and fluctuation–dissipation theorem

We complete the kinetic theory of two-dimensional (2D) point vortices initiated in previous works. We use a simpler and more physical formalism. We consider a system of 2D point vortices submitted to a small external stochastic perturbation and determine the response of the system to the perturbation. We derive the diffusion coefficient and the drift by polarization of a test vortex. We introduce a general Fokker–Planck equation involving a diffusion term and a drift term. When the drift by polarization can be neglected, we obtain a secular dressed diffusion equation sourced by the external noise. When the external perturbation is created by a discrete collection of N point vortices, we obtain a Lenard–Balescu-like kinetic equation reducing to a Landau-like kinetic equation when collective effects are neglected. We consider a multi-species system of point vortices. We discuss the process of kinetic blocking in the single and multi-species cases. When the field vortices are at statistical equilibrium (thermal bath), we establish the proper expression of the fluctuation–dissipation theorem for 2D point vortices relating the power spectrum of the fluctuations to the response function of the system. In that case, the drift coefficient and the diffusion coefficient satisfy an Einstein-like relation and the Fokker–Planck equation reduces to a Smoluchowski-like equation. We mention the analogy between 2D point vortices and stellar systems. In particular, the drift of a point vortex in 2D hydrodynamics (Chavanis in Phys Rev E 58:R1199, 1998) is the counterpart of the Chandrasekhar dynamical friction in astrophysics. We also consider a gas of 2D Brownian point vortices described by N coupled stochastic Langevin equations and determine its mean and mesoscopic evolution. In the present paper, we treat the case of unidirectional flows, but our results can be straightforwardly generalized to axisymmetric flows.

Non-local thermodynamic equilibrium analysis of the methylidyne radical molecular lines in metal-poor stellar atmospheres

Aims. An analysis of the methylidyne (CH) radical in non-local thermodynamic equilibrium (NLTE) is performed for the physical conditions of cool stellar atmospheres typical of red giants (log ɡ = 2.0, Teff = 4500 K) and the Sun. The aim of the present work is to explore whether the G band of the CH molecule, which is commonly used in abundance diagnostics of carbon-enhanced metal-poor stars, is sensitive to NLTE effects. Methods. LTE and NLTE theoretical spectra were computed with the MULTI code. We used one-dimensional (1D) LTE hydrostatic MARCS model atmospheres with parameters representing eleven red giant stars with metallicities ranging from [Fe/H] = −4.0 to [Fe/H] = 0.0 and carbon-to-iron ratios of [C/Fe] = 0.0, +0.7, +1.5, and +3.0. The CH molecule model was represented by 1981 energy levels, 18 377 radiative bound-bound transitions, and 932 photo-dissociation reactions. The rates due to transitions caused by collisions with free electrons and hydrogen atoms were computed using classical recipes. Results. Our calculations suggest that NLTE effects in the statistical equilibrium of the CH molecule are significant and cannot be neglected for precision spectroscopic analysis of C abundances. The NLTE effects are mostly driven by radiative over-dissociation, owing to the very low dissociation threshold of the molecule and significant resonances in the photo-dissociation cross-sections. The NLTE effects in the G band increase with decreasing metallicity. When comparing the C abundances determined from the CH G band in LTE and in NLTE, we show that the C abundances are always under-estimated if LTE is assumed. The NLTE corrections to C abundance inferred from the CH feature range from +0.04 dex for the Sun to +0.21 dex for a red giant with metallicity [Fe/H] = −4.0. Conclusions. Departures from the LTE assumption in the CH molecule are non-negligible, and NLTE effects have to be taken into account in the diagnostic spectroscopy based on the CH lines. We show here that the NLTE effects in the optical CH lines are non-negligible for the Sun and red giant stars, but further calculations are warranted to investigate the effects in other regimes of stellar parameters.

Progress towards a 3D Monte Carlo radiative transfer code for outflow wind modelling

Context. Radiative transfer modelling of expanding stellar envelopes is an important task in their analysis. To account for inhomogeneities and deviations from spherical symmetry, it is necessary to develop a 3 D approach to radiative transfer modelling. Aims. We present a 3 D Monte Carlo code for radiative transfer modelling, which is aimed to calculate the plasma ionisation and excitation state with the statistical equilibrium equations, moreover, to implement photon-matter coupling. As a first step, we present our Monte Carlo radiation transfer routines developed and tested from scratch. Methods. The background model atmosphere (the temperature, density, and velocity structure) can use an arbitrary grid referred to as the modGrid. The radiative transfer was solved using the Monte Carlo method in a Cartesian grid, referred to as the propGrid. This Cartesian grid was created based on the structure of the modGrid; correspondence between these two grids was set at the beginning of the calculations and then kept fixed. The propGrid can be either regular or adaptive; two modes of adaptive grids were tested. The accuracy and calculation speed for different propGrids was analysed. Photon interaction with matter was handled using the Lucy’s macroatom approach. Test calculations using our code were compared with the results obtained by a different Monte Carlo radiative transfer code. Results. Our method and the related code for the 3 D radiative transfer using the Monte Carlo and macroatom methods offer an accurate and reliable solution for the radiative transfer problem, and are especially promising for the inclusion and treatment of 3 D inhomogeneities.

Slow Time Phenomena in Heterogeneous Materials: from Microscopic Fluctuations to Macroscopic Relaxation

Slow time relaxation of elastic moduli with typically logarithmic time dependence is observed in many media interesting for materials science. This phenomenon is related to internal structure and is, hence, important for the development of present-day materials. Here, we provide a general explanation showing a close link between slow time phenomena and fluctuations on the microscopic and mesoscopic scales. We look for the origin of slow time phenomena in random walk or diffusion processes on microscopic scales. Some bonds occurring in the metastable state make a transition through the energy barriers due to small fluctuations slightly perturbing the statistical equilibrium. If the number of the excited bonds is small compared to the total number of bonds in a heterogeneous material, the process of the transition as a whole can be considered as mesoscopic fluctuations. Averaging over all transient bonds or states is revealed in the observed macroscopic relaxation of elastic moduli, velocities, and others. The functional dependence on time in the relaxation process has been shown to be controlled by the profile of energy barriers. The results obtained point to their possible applications in materials science.

Investigating cohesive sediment dynamics in open waters via grain-resolved simulations

Cohesive particulate flows play an important role in environmental fluid dynamics, as well as in a wide variety of civil and process engineering applications. However, the scaling laws, constitutive equations and continuum field descriptions governing such flows are currently less well understood than for their non-cohesive counterparts. Grain-resolved simulations represent an essential tool for addressing this shortcoming, along with theoretical investigations, laboratory experiments and field studies. Here we provide a tutorial introduction to simulations of fine-grained sediments in viscous fluids, along with an overview of some representative insights that this approach has yielded to date. After a brief review of the key physical concepts governing van der Waals forces as the main cohesive effect for subaqueous sediment suspensions, we discuss their incorporation into particle-resolved simulations based on the immersed boundary method. We subsequently describe simulations of cohesive particles in several model turbulent flows, which demonstrate the emergence of a statistical equilibrium between the growth and break-up of aggregates. As a next step, we review the influence of cohesive forces on the settling behaviour of dense suspensions, before moving on to submerged granular collapses. Throughout the article, we highlight open research questions in the field of cohesive particulate flows whose investigation may benefit from grain-resolved simulations.

A New Approach for Improving GNSS Geodetic Position by Reducing Residual Tropospheric Error (RTE) Based on Surface Meteorological Data

Positioning error components related to tropospheric and ionospheric delays are caused by the atmosphere in positioning determined by global navigation satellite systems (GNSS). Depending on the user’s requirements, the position error caused by tropospheric influences, which is commonly referred to as zenith tropospheric delay (ZTD), must be estimated during position determination or determined later by external tropospheric corrections. In this study, a new approach was adopted based on the reduction of residual tropospheric error (RTE), i.e., the unmodeled part of the tropospheric error that remains included in the total geodetic position error, along with other unmodeled systematic and random errors. The study was performed based on Global Navigation Satellite System (GLONASS) positioning solutions and accompanying meteorological parameters in a defined and harmonized temporal-spatial frame of three locations in the Republic of Croatia. A multidisciplinary approach-based analysis from a navigational science aspect was applied. The residual amount of satellite positioning signal tropospheric delay was quantitatively reduced by employing statistical analysis methods. The result of statistical regression is a model which correlates surface meteorological parameters with RTE. Considering the input data, the model has a regional character, and it is based on the Saastamoinen model of zenith tropospheric delay. The verification results show that the model reduces the RTE and thus increases the geodetic accuracy of the observed GNSS stations (with horizontal components of position accuracy of up to 3.8% and vertical components of position of up to 4.37%, respectively). To obtain these results, the Root Mean Square Error (RMSE) was used as the fundamental parameter for position accuracy evaluation. Although developed based on GLONASS data, the proposed model also shows a considerable degree of success in the verification of geodetic positions based on Global Positioning System (GPS). The purpose of the research, and one of its scientific contributions, is that the proposed method can be used to quantitatively monitor the dynamics of changes in deviations of X, Y, and Z coordinate values along coordinate axes. The results show that there is a distinct interdependence of the dynamics of Y and Z coordinate changes (with almost mirror symmetry), which has not been investigated and published so far. The resultant position of the coordinates is created by deviations of the coordinates along the Y and Z axes—in the vertical plane of space, the deviations of the coordinate X (horizontal plane) are mostly uniform and independent of deviations along the Y and Z axes. The proposed model shows the realized state of the statistical position equilibrium of the selected GNSS stations which were observed using RTE values. Although of regional character, the model is suitable for application in larger areas with similar climatological profiles and for users who do not require a maximum level of geodetic accuracy achieved by using Satellite-Based Augmentation Systems (SBAS) or other more advanced, time-consuming, and equipment-consuming positioning techniques.

Irreversible energy extraction from negative-temperature two-dimensional turbulence.

The formation and transition of patterns of two-dimensional turbulent flows observed in various geophysical systems are commonly explained in terms of statistical mechanics. Different from ordinary systems, for a two-dimensional flow, the absolute temperature defined for a statistical equilibrium can take negative values. In a state of negative temperature, the second law of thermodynamics predicts that energy in microscopic fluctuations is irreversibly converted to a macroscopic form. This study explores the possibility of this one-way energy conversion in a two-dimensional flow using a basic conceptual model. We consider an inviscid incompressible fluid contained in a bounded domain, the shape of which is distorted by an externally imposed force. Unlike the usual fixed boundary cases, the flow energy within the domain is exchanged with the external system via pressure work through the moving lateral boundary. Concurrently, the flow field remains constrained by vorticity conservation. Beginning from a state of Kraichnan's grand-canonical ensemble, when the domain shape is distorted from one shape to another in a finite time, the Jarzynski equality is established. This equality states that, on average, the direction of a net energy flow through the boundary during a cycle of domain distortion changes with the sign of the initial temperature of the system. Numerical experiments are carried out to verify this theoretical argument and to investigate the parameter dependence of the energy exchange rate.

Systematic Evaluation for the Impact of the Geological Conditions on the Adsorption Affinities of Calcite as an Adsorbent of Zn2+ Ions from Aqueous Solutions: Experimental and Theoretical Studies

Three samples of calcite (calcite crystal (CA), calcite of limestone (L.CA), and metamorphosed calcite (marble) (M.CA)) were assessed as adsorbents of Zn (II) to consider the impact of the different geological conditions. The three samples exhibit remarkable changes in their Zn (II) retention capacities (Qsat = 384.6 mg/g (CA), 274.5 mg/g (L.CA), and 512.6 mg/g (M.CA)). The retention systems of the three calcite samples were described on the basis of the suggested statistical physics-based equilibrium studies as well as the traditional kinetic and isotherm models. However, the M.CA samples exhibited the best retention capacity, the steric properties reflecting a higher active site density of CA (Nm (Zn) = 113.46 mg/g) than both M.CA (Nm (Zn) = 82.8 mg/g) and L.CA (Nm (Zn) = 52.4 mg/g) at 323 K. This was assigned to the controlling effect of the sequestered numbers of Zn (II) per site on the surfaces of the calcite phase (n(Zn) = 3.39 (CA), 5.24 (L.CA), and 6.19 (M.CA)) in addition to the higher surface area and ion exchange of the metamorphosed and deformed M.CA. The previous n(Zn) values suggested the retention of Zn (II) by a multi-ionic mechanism in a vertical orientation. The Gaussian energies (8 to 16 KJ/mol) and retention energies (˂40 KJ/mol) of Zn (II) by CA and L.CA suggested complex physical and weak chemical mechanisms involving ion exchange, hydrogen bonding, dipole bonding forces, electrostatic attractions, and van der Waals forces. The thermodynamic properties were illustrated on the basis of the internal energy, free enthalpy, and entropy functions, which validate the endothermic and spontaneous nature of the Zn (II) retention system by the three calcite samples.

Branching random walk in a random time-independent environment

ABSTRACT In a lattice population model, particles move randomly from one site to another as independent random walks, split into two offspring, or die. If duplication and mortality rates are equal and take the same value over all lattice sites, the resulting model is a critical branching random walk (characterized by a mean total number of offspring equal to ). There exists an asymptotical statistical equilibrium, also called steady state. In contrast, when duplication and mortality rates take independent random values drawn from a common nondegenerate distribution (so that the difference between duplication and mortality rates has nonzero variance), then the steady state no longer exists. Simultaneously, at all lattice sites, if the difference between duplication and mortality rates takes strictly positive values with strictly positive probability, the total number of particles grows exponentially. The lattice includes large connected sets where the duplication rate exceeds the mortality rate by a positive constant amount, and these connected sets provide the growth of the total population. This is the supercritical regime of branching processes. On the other hand, if the difference between duplication and mortality rates is almost surely negative or null except when it is almost surely zero, then the total number of particles vanishes asymptotically. The steady state can be reached only if the difference between duplication and mortality rates is almost surely zero.

Theoretical and Numerical Study of Self-Organizing Processes in a Closed System Classical Oscillator and Random Environment

A self-organizing joint system classical oscillator–random environment is considered within the framework of a complex probabilistic process that satisfies a Langevin-type stochastic differential equation. Various types of randomness generated by the environment are considered. In the limit of statistical equilibrium (SEq), second-order partial differential equations (PDE) are derived that describe the distribution of classical environmental fields. The mathematical expectation of the oscillator trajectory is constructed in the form of a functional-integral representation, which, in the SEq limit, is compactified into a two-dimensional integral representation with an integrand: the solution of the second-order complex PDE. It is proved that the complex PDE in the general case is reduced to two independent PDEs of the second order with spatially deviating arguments. The geometric and topological features of the two-dimensional subspace on which these equations arise are studied in detail. An algorithm for parallel modeling of the problem has been developed.

Statistical Equilibrium Principles in 2D Fluid Flow: From Geophysical Fluids to the Solar Tachocline

An overview is presented of several diverse branches of work in the area of effectively 2D fluid equilibria which have in common that they are constrained by an infinite number of conservation laws. Broad concepts, and the enormous variety of physical phenomena that can be explored, are highlighted. These span, roughly in order of increasing complexity, Euler flow, nonlinear Rossby waves, 3D axisymmetric flow, shallow water dynamics, and 2D magnetohydrodynamics. The classical field theories describing these systems bear some resemblance to perhaps more familiar fluctuating membrane and continuous spin models, but the fluid physics drives these models into unconventional regimes exhibiting large scale jet and eddy structures. From a dynamical point of view these structures are the end result of various conserved variable forward and inverse cascades. The resulting balance between large scale structure and small scale fluctuations is controlled by the competition between energy and entropy in the system free energy, in turn highly tunable through setting the values of the conserved integrals. Although the statistical mechanical description of such systems is fully self-consistent, with remarkable mathematical structure and diversity of solutions, great care must be taken because the underlying assumptions, especially ergodicity, can be violated or at minimum lead to exceedingly long equilibration times. Generalization of the theory to include weak driving and dissipation (e.g., non-equilibrium statistical mechanics and associated linear response formalism) could provide additional insights, but has yet to be properly explored.

Importance of radiative pumping for the excitation of the H2O submillimeter lines in galaxies

H2O submillimeter emission is a powerful diagnostic of the molecular interstellar medium in a variety of sources, including low- and high-mass star-forming regions of the Milky Way, and from local to high-redshift galaxies. However, the excitation mechanism of these lines in galaxies has been debated, preventing a basic consensus on the physical information that H2O provides. Radiative pumping due to H2O absorption of far-infrared photons emitted by dust and collisional excitation in dense shocked gas have both been proposed to explain the H2O emission. Here we propose two basic diagnostics to distinguish between the two mechanisms: First, the ortho-H2O 321 − 212 75 μm and the para-H2O 220 − 111 101 μm rotational lines in shock-excited regions are expected to be in emission, while when radiative pumping dominates, the two far-infrared lines are expected to be in absorption. Second, the radiative pumping scenario predicts, based on the statistical equilibrium of H2O level populations, that the apparent isotropic net rate of far-infrared absorption in the 321 ← 212 (75 μm) and 220 ← 111 (101 μm) lines should be higher than or equal to the apparent isotropic net rate of submillimeter emission in the 321 → 312 (1163 GHz) and 220 → 211 (1229 GHz) lines, respectively. Applying both criteria to all 16 galaxies and several Galactic high-mass star-forming regions in which the H2O 75 μm and submillimeter lines have been observed with Herschel/PACS and SPIRE, we show that in most (extra)galactic sources, the H2O submillimeter line excitation is dominated by far-infrared pumping, combined in some cases with collisional excitation of the lowest energy levels. Based on this finding, we revisit the interpretation of the correlation between the luminosity of the H2O 988 GHz line and the source luminosity in the combined Galactic and extragalactic sample.

Thermodynamical Aspects of the LGGR Approach for Hadron Energy Spectra

The local growth global reset (LGGR) dynamical model is reviewed and its performance in describing the hadron energy spectra in relativistic collisions is demonstrated. It is shown that even for dynamical processes a temperature-like quantity can be defined and distributions resembling statistical equilibrium can be reached. With appropriate growth and reset rates the LGGR model is capable of describing the right energy spectra. These findings draw a certain picture of quark–gluon plasma development with random hadronization and re-swallowing steps and signals the fact that observing an exponential spectrum does not necessarily prove thermal equilibrium in the experiment.

Potential spectral lines of ethanimine (CH[formula omitted]CHNH) - an important possible precursor of amino acids

Accurate computational characterization of ethanimine (CH3NH), considered as an important possible precursor of amino acids, is now available. Ethanimine has two E and Z isomers and each isomer has two A and E substates. Some lines of Green Bank Telescope PRIMOS radio astronomy survey spectra in Sagittarius B2 North (Sgr B2(N)) have been assigned to the ethanimine. For known values of accurate energies of rotational levels in the ground vibrational state, and the spectroscopic data (rotational and centrifugal distortion constants, and electric dipole moment), we have calculated Einstein A and B coefficients (radiative transition probabilities) for rotational transitions for each substate. These radiative transition probabilities along with collisional rate coefficients are calculated using a scaling law have been used for solving a set of statistical equilibrium equations coupled with the equations of radiative transfer for each substate. We have found a number of weak MASER lines and anomalous absorption lines for each substate. These lines may play important role for identification of ethanimine in a cosmic object. Corresponding lines in two substates in a isomer have been found to show similar behaviour, except different values of their frequencies.

Predictive model of fermionic dark matter halos with a quantum core and an isothermal atmosphere

We develop a thermodynamical model of fermionic dark matter halos at finite temperature. Statistical equilibrium states may be justified by a process of violent collisionless relaxation in the sense of Lynden-Bell or from a collisional relaxation of nongravitational origin if the fermions are self-interacting. The most probable state (maximum entropy state) generically has a "core-halo" structure with a quantum core (fermion ball) surrounded by an isothermal atmosphere. The quantum core is equivalent to a polytrope of index $n=3/2$. The Pauli exclusion principle creates a quantum pressure that prevents gravitational collapse and solves the core-cusp problem of the cold dark matter model. The isothermal atmosphere (which is similar to the NFW profile of cold dark matter) accounts for the flat rotation curves of the galaxies at large distances. We numerically solve the equation of hydrostatic equilibrium with the Fermi-Dirac equation of state and determine the density profiles and rotation curves of fermionic dark matter halos.

Disilicon carbide (Si2C) in the interstellar medium

The Si2C and SiC2 both are considered to play key role in the formation of the SiC dust grains in the atmosphere of the carbon-rich stars. The molecule of our interest Si2C has been detected in the envelope of the red supergiant star IRC+10216 first time. It is an asymmetric top molecule having electric dipole moment of 1 Debye along the b-axis of inertia. Because of zero nuclear spin of both the Carbon and Silicon atoms, it has only paratransitions. Using the given spectroscopic data (rotational and centrifugal distortion constants and electric dipole moment), for the para-Si2C, we have calculated energies of 200 lower rotational levels (having energy up to 217.8 cm-1) and the Einstein A and B coefficients for 867 radiative transitions between the levels. We have solved a set of 200 statistical equilibrium equations coupled with 867 equations of radiative transfer (Sobolev analysis), where the collisional rate coefficients are taken from a scaling law. Out of 867 radiative transitions, 13 transitions have been found showing weak MASER action, and 19 transitions showing anomalous absorption. One transition 808-717 is found to show both the MASER action as well as the anomalous absorption. These transitions in addition to the observed transitions may play important role in the identification of Si2C in the cosmic objects.

An Improved Method for Coupling Hydrodynamics with Astrophysical Reaction Networks

Reacting astrophysical flows can be challenging to model, because of the difficulty in accurately coupling hydrodynamics and reactions. This can be particularly acute during explosive burning or at high temperatures where nuclear statistical equilibrium is established. We develop a new approach, based on the ideas of spectral deferred corrections (SDC) coupling of explicit hydrodynamics and stiff reaction sources as an alternative to operator splitting, that is simpler than the more comprehensive SDC approach we demonstrated previously. We apply the new method to a double-detonation problem with a moderately sized astrophysical nuclear reaction network and explore the time step size and reaction network tolerances, to show that the simplified-SDC approach provides improved coupling with decreased computational expense compared to traditional Strang operator splitting. This is all done in the framework of the Castro hydrodynamics code, and all algorithm implementations are freely available.

Statistical Equilibrium of Large Scales in Three-Dimensional Hydrodynamic Turbulence.

We investigate experimentally three-dimensional (3D) hydrodynamic turbulence at scales larger than the forcing scale. We manage to perform a scale separation between the forcing scale and the container size by injecting energy into the fluid using centimetric magnetic particles. We measure the statistics of the fluid velocity field at scales larger than the forcing scale (energy spectra, velocity distributions, and energy flux spectrum). In particular, we show that the large-scale dynamics are in statistical equilibrium and can be described with an effective temperature, although not isolated from the turbulent Kolmogorov cascade. In the large-scale domain, the energy flux is zero on average but exhibits intense temporal fluctuations. Our Letter paves the way to use equilibrium statistical mechanics to describe the large-scale properties of 3D turbulent flows.