Ideal Gas Molecules Research Articles

An improved chaotic ideal gas molecular movement algorithm for engineering optimization problems

AbstractRecently, the ideal gas molecular movement (IGMM) optimization algorithm was introduced by the authors. It is inspired by the movement and collision behaviour of the ideal gas molecules in an isolated medium. Although the IGMM could perform a high potential in determining the global optimum, improving its convergence behaviour came to the attention of the present investigation to deal with any type of especially highly nonlinear optimization problems. For that purpose, two actions took place hybridly. One was the simulation of the vibrational motion of gas molecules, especially at the early stages of the optimization process. The second simultaneous move concerned the non‐repetitive nature of chaotic maps which could diversify the molecules, reduce the threat of premature convergence and improve the convergence speed of the IGMM algorithm. Therefore, this article investigates 10 different chaotic map functions and a random number generator along with four different Vibrational‐based Chaotic IGMM (VCIGMM) strategies to still improve the speed of convergence. The results of applying the proposed algorithm to various numerical and engineering benchmark problems, intensely show that the chaotic maps, merged with vibrational motion of gas molecules, significantly improved the performance of the IGMM. It could considerably outperform some of the well‐known meta‐heuristic optimization algorithms in the literature.

Using the molecular rotational motion concept to predict the volume expansion coefficients of liquids

ABSTRACT The rotational motion of molecules in the liquid phase has been shown to be one of the main factor that determines the liquid’s thermal expansion and can be used for its calculation based on an expression for the partition function of the rotational motion of ideal gas molecules, taking into account the contribution of intermolecular interactions. This study establishes that the thermal expansion temperature coefficients are grouped in sequences depending on the functional class of compounds. The general hydrocarbon sequence breaks down into separate sequences according to their functional groups, such as saturated or aromatic hydrocarbons. Thus, it is shown a correlation between the thermal expansion of liquids with an average size of molecules, at temperatures well below the boiling point, and the entropy of the rotational motion which is a function of the moments of inertia of molecules and temperature.

Entropy formula of N-body system

We prove a proposition that the entropy of the system composed of finite N molecules of ideal gas is the q-entropy or Havrda–Charvát–Tsallis entropy, which is also known as Tsallis entropy, with the entropic index q=frac{D(N-1)-4}{D(N-1)-2} in D-dimensional space. The indispensable infinity assumption used by Boltzmann and others in their derivation of entropy formulae is not involved in our derivation, therefore our derived formula is exact. The analogy of the N-body system brings us to obtain the entropic index of a combined system q_C formed from subsystems having different entropic indexes q_A and q_B as frac{1}{1-q_C}=frac{1}{1-q_A}+frac{1}{1-q_B}+frac{D+2}{2}. It is possible to use the number N for the physical measure of deviation from Boltzmann entropy.

The Statistical Style of Reasoning and the Invention of Bose-Einstein Statistics.

This paper is a preliminary exploration of the connections between the statistical style of reasoning and the research practices of statistical mechanics in the early period of the long quantum revolution. It suggests that before 1925 the instantiations of the statistical style in physics went through two phases. The first phase consisted of the formulation of the Maxwell-Boltzmann statistics on the basis of the population-gas analogy. The second phase was characterized by the generalization of the Maxwell-Boltzmann statistics through analogies between ideal gas molecules and other microphysical entities, analogies that shaped and were shaped by the rise of quantum theory. Einstein's invention of the Bose-Einstein statistics started a third phase and created the conditions of possibility for a new classification of microphysical entities according to their different statistics.

A formula for the contribution of a resonance to the canonical partition function

An explicit expression for the Laplace transform of a Lorentzian function is applied to evaluate the contribution to the canonical partition function of ideal-gas molecules stemming from resonances. This formula can be used instead of numerical integration. A discussion of potential errors in replacing this contribution by the conventional formula for the Boltzmann weight is given. When generalised to the complex plane, the known expression is shown to be discontinuous and a new formulation is proposed that removes the discontinuity. As a result, an explicit expression is obtained for the Fourier transform of a Lorentzian function that is truncated at a lower energy bound.

Spatially Resolved Thermodynamic Integration: An Efficient Method To Compute Chemical Potentials of Dense Fluids.

Many popular methods for the calculation of chemical potentials rely on the insertion of test particles into the target system. In the case of liquids and liquid mixtures, this procedure increases in difficulty upon increasing density or concentration, and the use of sophisticated enhanced sampling techniques becomes inevitable. In this work, we propose an alternative strategy, spatially resolved thermodynamic integration, or SPARTIAN for short. Here, molecules are described with atomistic resolution in a simulation subregion and as ideal gas particles in a larger reservoir. All molecules are free to diffuse between subdomains adapting their resolution on the fly. To enforce a uniform density profile across the simulation box, a single-molecule external potential is computed, applied, and identified with the difference in chemical potential between the two resolutions. Since the reservoir is represented as an ideal gas bath, this difference exactly amounts to the excess chemical potential of the target system. The present approach surpasses the high density/concentration limitation of particle insertion methods because the ideal gas molecules entering the target system region spontaneously adapt to the local environment. The ideal gas representation contributes negligibly to the computational cost of the simulation, thus allowing one to make use of large reservoirs at minimal expenses. The method has been validated by computing excess chemical potentials for pure Lennard-Jones liquids and mixtures, SPC and SPC/E liquid water, and aqueous solutions of sodium chloride. The reported results well reproduce literature data for these systems.

The solute rejection of a nanoslit in osmosis

Two-dimensional nanoporous membranes have recently attracted great attention in osmosis-related applications due to their high water permeability and salt rejection. The size distribution of the pores on the membrane is closely related to these properties. Circular pores have been the main subject of research. Recently, it was found that the water permeability of a slit pore can be much higher than that of a circular pore, with the same salt rejection. Hence, it is necessary to discuss the osmotic flow through these pores at the nanoscale. In widely used models, two solute parameters are always used to quantify the ability to produce osmotic pressure and the probability of being rejected by the membrane. These are always assumed to have the same value and thus are both called reflection coefficients. However, this assumption is found to be inconsistent with the experimental results when describing osmosis on this scale. In this work, we discuss osmosis through a slit by making an analogy between the solute and the ideal gas molecules in the dilute solution. Expressions for the two reflection coefficients are derived based on the geometrical relationship between the solute and the slit. We find that the two coefficients increase with the size of the solute. Their values are not equal, as found in previous experiments. Extensive molecular dynamics (MD) simulations are performed and a good correlation between the simulation results and our predictions is found. This work may help to understand the osmotic flow through nanoscale pores and enable the design of highly efficient nanopores on membranes in osmosis-related applications.

Semi-empirical model for intense evaporation

A semi-empirical model based on the linear kinetic theory was developed for intense evaporation. The extrapolated drops for pressure and temperature at the condensed phase surface were calculated through summing of linear and squared terms. The analytical dependencies were obtained for gas parameters in gas-dynamic zone as functions of Mach number, condensation coefficient, and the number of degrees of freedom for molecules of ideal gas. The calculations from semi-empirical model are in agreement with results from known analytical and numerical studies.

Enhanced IGMM optimization algorithm based on vibration for numerical and engineering problems

Recently, a new optimization procedure called ideal gas molecular movement (IGMM) algorithm was introduced by the authors. The algorithm is based on the movement of ideal gas molecules in an isolated medium. A close-up track of their movements reveals the fact that the gas molecules have locally motions such as vibration and rotation in addition to their translational motions. In this paper, a modified version of IGMM is introduced by simulating the vibration of each molecule in the form of an operand referred to as molecular operand on vibrational effect (MOVE) which causes a significant escalation in the convergence speed specially at the early stages of the optimization process. A parametric study has also been carried out with regard to alteration of four most effective parameters of the VIGMM algorithm. Consequently, the best ranges of alteration for each parameter are proposed. The results of applying the proposed vibration-based IGMM (VIGMM) to different numerical and engineering benchmark functions and the Wilcoxon’s rank-sum nonparametric statistical test intensely show that the MOVE operand significantly boosts the performance of the IGMM and one could certify the significance of VIGMM, proposed in the present study, over some other meta-heuristic optimization algorithms.

Engineering optimization based on ideal gas molecular movement algorithm

The present work introduces a new metaheuristic optimization method based on the ideal gas molecular movement (IGMM) to solve mathematical and engineering optimization problems. Ideal gas molecules scatter throughout the confined environment quickly. This is embedded in the high speed of molecules, collisions between them and with the surrounding barriers. In IGMM algorithm, the initial population of gas molecules is randomly generated and the governing equations related to the velocity of gas molecules and collisions between those are utilized to accomplish the optimal solutions. To verify the performance of the IGMM algorithm, some mathematical and engineering benchmark optimization problems, commonly used in the literature, are inspected. Comparison of results obtained by IGMM with other optimization algorithms show that the proposed method has a challenging capacity in finding the optimal solutions and exhibits significance both in terms of the accuracy and reduction on the number of function evaluations vital in reaching the global optimum.

Correspondence between Experiment and Theory of Bulk Electrocrystallisation at Solid Electrodes in Aqueous Electrolyte

A model of electrodeposition and electrodissolution at electrode surfaces in aqueous solution is presented. The description is based on the assumption that redox reaction of water is the more important process controlling the electrode kinetics. Chronoamperometric measurements and experiments of cyclic voltammetry indicate that the current fundamentally is proportional to inverse time. It was proposed that redox-active species different from water never touch the surface but they predominantly interact with surface-active hydrogen or oxygen formed at the surface by redox processes of water. An excellent correspondence was found between the number of ideal-gas molecules in a monolayer at the electrode surface and the charge required, as to dissolve one monolayer of electrodeposited metal. The linear relation between standard enthalphy of metal-oxide formation and standard-reduction potential shows that metal oxide formation at the electrode corresponds to oxidation of a metal in an atmosphere of oxygen.

An approximate likelihood of right censoring from Maxwell-Boltzmanndistribution

The Maxwell-Boltzmann distribution is a premier to identify the distribution of speeds of molecules in thermal equilibrium. The shape of this distribution gets wider for higher temperatures and also the peak decreases. Besides, the mean speed is always higher than the most probable speed due to the skewness of the distribution. The most fabulous property is non-analytic cumulative density function. Consequently, censoring based inferences are function of cumulative density function. The aim in this study is to derive the estimating equation of parameters under type-I and type-II censoring of speed of ideal gas molecules. Using the error function approximation procedure the survival function has been approximated and the procedures of estimating parameters and their standard errors under type-I and type-II censoring have been derived. Besides, the simulation studies have been conducted and the validity of the approximation has been provided.

A Model of Slurry Flow in a Fracture

Abstract An accurate calculation of slurry flow in a fracture is an important issue for fracture design. A model taking into account particle dynamics on a micro-level has been developed. The model shows that the slurry dynamics, to a significant extent, is governed by particle fluctuations generated in a high shear-rate flow. Particles migrate from high shear-rate zones at the fracture walls towards the fracture centre. Thus, a slurry flow is characterized by non-uniform concentration across a fracture. Low solids concentration near the walls leads to reduction of slurry-wall friction compared to that predicted by a model not taking particle migration into account. Reduction in the friction leads to a significant decrease in the pressure gradient and the net pressure, respectively. Introduction Hydraulic fracturing is a complicated but well-studied technology for well stimulation(1). There are a number of computational codes for calculating technological parameters of a fracturing procedure. Such codes solve equations describing slurry hydrodynamics coupled with equations modelling formation fracturing. Since the percentage of fracturing job failures is high (above 30%), it is important to enhance the accuracy of modelling of different components of hydraulic fracturing to make this procedure a more reliable operation. In the current research, we suggest a model of slurry flow in a fracture taking into account an important effect of proppant migration to the fracture centre. Model Slurry flow moving in a fracture is usually characterized by a high mean shear rate (up to γm = um/w= 200 1/s), where um is the mean (superficial) slurry velocity and w is the fracture width. In our analysis, we will consider a steady-state slurry flow in a channel of a constant width. We also neglect impact of fluid leakoff into the formation on slurry dynamics. Many fracturing fluids used in practice are characterized by the power-law rheology. Engineers usually employ a known solution of the Navier-Stokes equation for power-law slurry(1) obtained by assuming that particles have the same velocity as a fluid and that the concentration of solids is uniformly distributed across a fracture. Lattice-Bolzmann computations have shown that in a high shear rate slurry, flow particles fluctuate(2). For the present study, the kinetic theory of granular flows(3) was employed for flow modelling. The kinetic theory assumes that particles collide with each other moving like ideal gas molecules and the velocity distribution of particle fluctuations is Maxwellian. The key parameter of this approach is the granular temperature determining intensity of particle fluctuations: Equation 1 Available in Full Paper. where < υ2s> is the mean-square fluctuation velocity of a particle. An equation of a granular energy balance for a flat channel is written as(4): Equation 2 Available in Full Paper.

Quantum molecular dynamics study of the pressure dependence of the ammonia inversion transition

The mechanism of the shift, broadening, and quenching of the ammonia inversion frequency with gas pressure has been a problem of lively interest for over 70 years. A simple quantum model of the ammonia molecule perturbed by collisions with ideal gas molecules displays the essential features of the experimental data for NH{sub 3} and for ND{sub 3}. The model does not display the behavior expected from theories of environmental decoherence. On the other hand, models of perturbed classical oscillators do display behavior similar to our model and to experimental data. The quenching of the ammonia inversion transition cannot therefore be interpreted as spatial localization of the wave function.

Collisions of ideal gas molecules with a rough/fractal surface. A computational study

The frequency of collisions of ideal gas molecules (argon) with a rough surface has been studied. The rough/fractal surface was created using random deposition technique. By applying various depositions, the roughness of the surface was controlled and, as a measure of the irregularity, the fractal dimensions of the surfaces were determined. The surfaces were next immersed in argon (under pressures 2 x 10(3) to 2 x 10(5) Pa) and the numbers of collisions with these surfaces were counted. The calculations were carried out using a simplified molecular dynamics simulation technique (only hard core repulsions were assumed). As a result, it was stated that the frequency of collisions is a linear function of pressure for all fractal dimensions studied (D = 2, ..., 2.5). The frequency per unit pressure is quite complex function of the fractal dimension; however, the changes of that frequency with the fractal dimension are not strong. It was found that the frequency of collisions is controlled by the number of weakly folded sites on the surfaces and there is some mapping between the shape of adsorption energy distribution functions and this number of weakly folded sites. The results for the rough/fractal surfaces were compared with the prediction given by the Langmuir-Hertz equation (valid for smooth surface), generally the departure from the Langmuir-Hertz equation is not higher than 48% for the studied systems (i.e. for the surfaces created using the random deposition technique).

Molecular dynamics study of the equilibrium flux of gas molecules to a fractal/rough surface: Effects of gas atom diameter

The frequency of collisions of ideal gas molecules (argon) with a rough surface has been studied. The rough/fractal surface was created using the random deposition technique. By applying various depositions the surface roughness was controlled and, as a measure of irregularity, the fractal dimensions of the surfaces were determined. The surfaces were next immersed in ideal gas and the numbers of collisions with these surfaces were counted. The calculations were carried out using the simplified molecular dynamics simulation technique (only hard core repulsions were assumed). The calculations were performed for various ratios of gas phase atoms diameter to the surface substrate atoms diameter. The results obtained showed that the size of a gas phase atom has crucial influence on the relation between the frequency of collision and the surface fractal dimension

On the higher order corrections to the Fokker–Planck equation

The Rayleigh model of nonlinear Brownian motion is revisited in which the heavy particle of mass M interacts with ideal gas molecules of mass m ⪡ M via instantaneous collisions. Using the van Kampen method of expansion of the master equation, nonlinear corrections to the Fokker–Planck equation are obtained up to sixth order in the small parameter λ = m / M , improving earlier results. The role and origin of non-Gaussian statistics of the random force in the corresponding Langevin equation are also discussed.

Langevin equation for the Rayleigh model with finite-range interactions.

Both linear and nonlinear Langevin equations are derived directly from the Liouville equation for an exactly solvable model consisting of a Brownian particle of mass M interacting with ideal gas molecules of mass m via a quadratic repulsive potential. Explicit microscopic expressions for all kinetic coefficients appearing in these equations are presented. It is shown that the range of applicability of the Langevin equation, as well as statistical properties of random force, may depend not only on the mass ratio m/M but also on the parameter Nm/M, involving the average number N of molecules in the interaction zone around the particle. For the case of a short-ranged potential, when N<<1, analysis of the Langevin equations yields previously obtained results for a hard-wall potential in which only binary collisions are considered. For the finite-ranged potential, when multiple collisions are important (N>>1), the model describes nontrivial dynamics on time scales that are on the order of the collision time, a regime that is usually beyond the scope of more phenomenological models.

Mapping volume scale for overlapping clusters

The use of the observed frequency of appearance of a cluster in a small simulation cell, immersed in a macrosystem, to specify the number of such clusters in that macrosystem is discussed, for the special case in which the cluster consists of all the molecules in the cell and the volume of the cell is one of the defining parameters of the cluster. In particular, clusters consisting of ideal gas molecules are considered. Furthermore, the clusters are allowed to “overlap” in the sense that the same molecules may be shared by more than one cluster of the same or different type. Recently, the mapping of simulation cell frequencies onto the macrosystem, for this purpose, has been discussed for the case of rare nonideal, nonoverlapping clusters of the type of interest to nucleation theory and a difference of opinion has arisen with respect to the specification of a volume scale to be used in the mapping process. Two scales have been suggested. One scale is simply the volume of the simulation cell. The other is specified by Eq. (1) of the present paper. In this paper we prove that the scale for the overlapping ideal clusters is that of Eq. (1). Since with overlapping clusters it is easy to perform computer experiments that provide a direct count of the clusters in the macrosystem, the system, although having limited physical value, is still of considerable analytical value in deciding between the two scales and, as a corollary, between the two theories underlying these scales. The indicated experiments have been performed and are reported in a following paper. They confirm the applicability of Eq. (1).

The Bakerian Lecture, 1962 The structure of liquids

A satisfactory picture of the structure of liquids has lagged far behind that of other states of matter. Ever since the time of Euler in the eighteenth century or, in a more precise form, since that of Maxwell in the nineteenth, we have had a convincing qualitative and quantitative picture of the chaos that is represented by the movements of the ideal gas molecules. The notion of a crystal or a solid in general as an arrangement of molecules ‘ in rank and file’, as Newton put it, is, in fact, older than Newton yet its quantitative statement was made possible only through the work of Born and others in our own century. But it is admitted even by those who work most in the field that the study of the structure of liquids or any exposition of their properties in atomic terms is still largely to be sought. This is not for want of trying. A vast number of researches have been devoted to attempts to analyze the structure of liquids, either directly by the diffraction methods which have proved so successful in crystalline solids, or, indirectly, through the construction of models and their thermodynamic testing. But we still lack either an adequate picture of the arrangement of molecules in a liquid or the necessary quantitative theory to explain their thermal and other properties.