Statistical Physics Theory Research Articles

Constructing binuclear sites to modulate the charge distribution of MIL-101 for enhanced toluene adsorption performance: experimental and theoretical studies

Toluene is a class of volatile organic compounds (VOCs) in the air that harm environmental quality and human health. To enhance the adsorption performance of adsorbents for toluene, based on the principle that unsaturated metal sites in MOFs can be replaced, nanoscale M@MIL-101(M represents Mg, Ca, Sr) with binuclear sites was prepared by a one-step solvothermal method. The second metal was successfully introduced into the topology of MIL-101 by partially substituting Cr sites. The results of the dynamic adsorption experiments showed that 20%Sr@MIL-101 exhibited the highest toluene adsorption capacity, which was about 60% higher than that of the undoped MIL-101. Analysis using a monolayer and multilayer model based on statistical physics theory indicated that the Sr/Cr binuclear sites could enhance the interaction strength of the adsorption system, promoting the adsorption of more layers of toluene molecules on the surface of MIL-101. Characterization and simulation together confirmed that constructing Sr/Cr binuclear sites in MIL-101 can cause significant redistribution of charge within the pores, enhancing the van der Waals interactions, π-π stacking interactions, hydrogen bonding, and Lewis acid-base interactions between the pore interior and the adsorbate molecules. Additionally, density functional theory (DFT) revealed that the adsorption sites of toluene in MIL-101 were distributed in a trigonal prism arrangement and were symmetrically aligned with a 180° rotation of the MIL-101 triangular cage.

Microscopic and macroscopic analysis of hexavalent chromium adsorption on polypyrrole-polyaniline@rice husk ash adsorbent using statistical physics modeling

This paper contributed with new findings to understand and characterize a heavy metal adsorption on a composite adsorbent. The synthesized polypyrrole–polyaniline@rice husk ash (PPY–PANI@RHA) was prepared and used as an adsorbent for the removal of hexavalent chromium Cr(VI). The adsorption isotherms of Cr(VI) ions on PPY–PANI@RHA were experimentally determined at pH 2, and at different adsorption temperatures (293, 303, and 313 K). Multi-layer model developed using statistical physics formalism was applied to theoretically analyze and characterize the different interactions and ion exchanges during the adsorption process for the elimination of this toxic metal from aqueous solutions, and to attribute new physicochemical interpretation of the process of adsorption. The physicochemical structures and properties of the synthesized PPY–PANI@RHA were characterized via Fourier transform infrared spectroscopy (FTIR). Fitting findings showed that the mechanism of adsorption of Cr(VI) on PPY–PANI@RHA was a multi-ionic mechanism, where one binding site may be occupied by one and two ions. It may also be noticed that the temperature augmentation generated the activation of more functional groups of the composite adsorbent, facilitating the interactions of metal ions with the binding sites and the access to smaller pore. The energetic characterization suggested that the mechanism of adsorption of the investigated systems was exothermic and Cr(VI) ions were physisorbed on PPY-PANI@RHA surface via electrostatic interaction, reduction of Cr(VI) to Cr(III), hydrogen bonding, and ion exchange. Overall, the utilization of the theory of statistical physics provided fruitful and profounder analysis of the adsorption mechanism. The estimation of the pore size distribution (PSD) of the polypyrrole-polyaniline@rice husk ash using the statistical physics approach was considered stereographic characterization of the adsorbent (here PPY–PANI@RHA was globally a meso-porous adsorbent). Lastly, the mechanism of Cr(VI) removal from wastewater using PPY-PANI@RHA as adsorbent was macroscopically investigated via the estimation of three thermodynamic functions.

Steric and energetic investigations of adsorption isotherms of methylene blue on new polymeric multi-layered material-based k-carrageenan and alginate

In this research study, we examined the adsorption behavior of methylene blue (MB) on a novel biosorbent made of polyelectrolyte multilayers (PEM) biopolymeric material at three different temperatures. Various models derived from statistical physics formalism were employed to analyze the adsorption isotherms. An alternative approach based on the application of multilayer models from statistical physics theory was used to better investigate the adsorption mechanisms of MB on the PEM grafting material. We adjusted various parameters within this model across temperatures of 295K, 313 K, and333K, and these parameters played a crucial role in comprehending the adsorption mechanism. In addition, this approach revealed that the MB dye is linked through a multilayer process on the PEM grafting material with ‘non-parallel’ configurations and multimolecular process. The PEM grafting material showed maximum adsorption capacities ranging from 632.23 mg⁄g to 463.76 mg⁄g for the removal of MB over a pH of 6. The calculated adsorption energies ranged from 11.6 to 18.65kJ/mol, indicating that the MB adsorption process was primarily of a physisorption nature influenced by both hydrogen bonding and van der Waals interactions. Thus, the theoretical analysis indicated that this low-cost adsorbent material (PEM grafting) is a promising material for the removal of MB dye from the solution. Additionally, we examined the adsorption behavior by calculating the entropy, the free energy and the internal energy which govern the adsorption process of MB onto the PEM grafting material.

Statistical physics investigation of the docking process of fruity odorants on Machilis hrabei MhOR5: New microscopic interpretations

In this paper, the putative phenomenon of adsorption involved in the insect olfactory perception of four fruity odorants on Machilishrabei olfactory receptor MhOR5 was studied at a molecular scale. Thus, the experimental concentration–response curves of propyl acetate, butyl acetate, isobutyl acetate, and prenyl acetate on MhOR5 were adjusted using the monolayer model of ideal gas established using the powerful statistical physics approach, which was selected as the most appropriate model among the others. The fitted model parameters were applied to stereographic and energetic analyze the four olfactory systems. Indeed, the modeling results indicated that the tested fruity key food odorants (KFOs) were docked as 2- or 3-aggregated molecules on one MhOR5 binding site via a nonparallel position (i.e., multi-molecular adsorption mechanism). Energetic parameters were used to characterize the adsorption process involved in the studied systems via the determination of the molar adsorption energies (ranging from 12.91 to 17.52 kJ/mol). Hence, the values of the molar adsorption energies are positive and inferior to 40 kJ/mol; this means that the putative adsorption mechanism presented an exothermic character and a physical type. The docking simulation results proved that the adsorption of propyl acetate, butyl acetate, isobutyl acetate, and prenyl acetate on MhOR5 may occur via physical interactions (weak interactions) like alkyl, pi-alkyl, van der Waals, carbon hydrogen bond, and conventional hydrogen bond. The statistical physics theory may also be utilized to quantitatively characterize Machilishrabei MhOR5 activated by propyl acetate, butyl acetate, isobutyl acetate, and prenyl acetate odorants via the estimation of the different site size distributions (SSDs) and site energy distributions (SEDs).

Theoretical and experimental study of strain localization phenomenon based on phase transition theory

In the study of large-scale localized strain features, localized strain can lead to energy release and seismic effects. From the macroscopic process of localized strain, it is found that localized deformation is accompanied by structural weakening, which is due to phase changes in rocks. The phase change corresponds to the secondary phase transition process in physics. The present study illustrates this phenomenon from the perspective of secondary phase transition theory in statistical physics, combining the localized strain features on a large scale. Theoretical analysis and experimental studies were carried out using three brittle rocks, including marble, granite and red sandstone. First, two perturbation methods, i.e., Krylov–Bogoliubov method and Poincare method, are used to calculate the higher-order control equations in analytical model. The influences of control equation coefficients on the strain localization process are analyzed. Then the uniaxial compression tests are carried out on the three rocks to record the strain process. Finally, the theoretical and experimental results are compared to analyze the strain localization phenomenon. The comparison results show that both the Krylov–Bogoliubov solution and the Poincare solution in the analytical model can well describe the evolution characteristics of localized strain. It indicates that the theoretical model is valid and has high accuracy. The Poincare method better simulates the nonlinear phenomenon of strain localization. The present study provides a new theoretical method to better understand the strain localization phenomenon.

A Theory of Best Choice Selection through Objective Arguments Grounded in Linear Response Theory Concepts

In this study, we propose how to use objective arguments grounded in statistical mechanics concepts in order to obtain a single number, obtained after aggregation, which would allow for the ranking of “agents”, “opinions”, etc., all defined in a very broad sense. We aim toward any process which should a priori demand or lead to some consensus in order to attain the presumably best choice among many possibilities. In order to specify the framework, we discuss previous attempts, recalling trivial means of scores—weighted or not—Condorcet paradox, TOPSIS (Technique for Order Preference by Similarity to Ideal Solution), etc. We demonstrate, through geometrical arguments on a toy example and with four criteria, that the pre-selected order of criteria in previous attempts makes a difference in the final result. However, it might be unjustified. Thus, we base our “best choice theory” on the linear response theory in statistical physics: we indicate that one should be calculating correlations functions between all possible choice evaluations, thereby avoiding an arbitrarily ordered set of criteria. We justify the point through an example with six possible criteria. Applications in many fields are suggested. Furthermore, two toy models, serving as practical examples and illustrative arguments are discussed.

Physicochemical assessment of ammonium adsorption using a palm shell-based adsorbent activated with acetic acid: experimental and theoretical studies.

The adsorption of ammonium from water was studied on an activated carbon obtained using raw oil palm shell and activated with acetic acid. The performance of this adsorbent was tested at different operating conditions including the solution pH, adsorbent dosage, and initial ammonium concentration. Kinetic and equilibrium studies were carried out, and their results were analyzed with different models. For the adsorption kinetics, the pseudo-first order equation was the best model to correlate this system. Calculated adsorption rate constants ranged from 0.071 to 0.074 g/mg min. The ammonium removal was 70-80% at pH 6-8, and it was significantly affected by electrostatic interaction forces. Ammonium removal (%) increased with the adsorbent dosage, and neutral pH condition favored the adsorption of this pollutant. The best ammonium adsorption conditions were identified with a response surface methodology model where the maximum removal was 91.49% with 2.27 g/L of adsorbent at pH 8.11 for an initial ammonium concentration of 36.90 mg/L. The application of a physical monolayer model developed by statistical physics theory indicated that the removal mechanism of ammonium was multi-ionic and involved physical interactions with adsorption energy of 29 kJ/mol. This activated carbon treated with acetic acid is promising to depollute aqueous solutions containing ammonium.

Advanced analysis of the adsorption mechanism for sweet odorant on mouse and human olfactory receptors via statistical physics theory

In this work, the one layer statistical physics model (M1) was applied to more deeply comprehend the phenomenon of adsorption, which is putatively involved in the olfactory sensation of acetophenone (AP) on mouse and human olfactory receptors. On one hand, the proposed model was utilized to microscopically characterize the studied olfactory systems in order to obtain advanced stereographic and energetic information concerning the studied olfactory systems in terms of a molecular view point. Based on the values of the number of acetophenone molecules docked on mouse and human binding sites n, it was found that the studied sweet odorant was linked with a mixed orientation. Energetically speaking, the values of the molar adsorption energy ΔE1, which ranged from 11.81 to 17.67 kJ/mol for mouse olfactory receptors (Olfr62, Olfr1093, Olfr1094, Olfr895, and Olfr876) and from 14.66 to 20.58 kJ/mol for human olfactory receptors (OR2C1, OR1A1, and OR2J2), indicated that acetophenone molecules were exothermically physisorbed. The adsorption energy distributions relative to AP may also be estimated to understand the olfactory sense in the investigated olfactory systems via the calculation of olfactory bands (i. e., AED bands), which may be defined between 4.80 and 27 kJ/mol and between 5.50 and 27 kJ/mol for mouse and human olfactory receptors, respectively. On the other hand, M2 was used to determine three thermodynamic potentials (i.e., the internal energy, the Gibbs free energy, and the adsorption entropy), which macroscopically characterized the investigated olfactory systems via a detailed thermodynamic analysis of the different olfactory systems.

Non-Hermitian non-equipartition theory for trapped particles

The equipartition theorem is an elegant cornerstone theory of thermal and statistical physics. However, it fails to address some contemporary problems, such as those associated with optical and acoustic trapping, due to the non-Hermitian nature of the external wave-induced force. We use stochastic calculus to solve the Langevin equation and thereby analytically generalize the equipartition theorem to a theory that we denote the non-Hermitian non-equipartition theory. We use the non-Hermitian non-equipartition theory to calculate the relevant statistics, which reveal that the averaged kinetic and potential energies are no longer equal to kBT/2 and are not equipartitioned. As examples, we apply non-Hermitian non-equipartition theory to derive the connection between the non-Hermitian trapping force and particle statistics, whereby measurement of the latter can determine the former. Furthermore, we apply a non-Hermitian force to convert a saddle potential into a stable potential, leading to a different type of stable state.

Selective Extraction and Determination of Hydrocortisone and Dexamethasone in Skincare Cosmetics: Analytical Interpretation Using Statistical Physics Formalism

Molecularly imprinted polymers (MIPs), as magnetic extraction adsorbents, are used for the selective, rapid determination and extraction of dexamethasone and hydrocortisone in skincare products. Therefore, in this paper, magnetic molecularly imprinted polymers (MMIPs) and magnetic non-molecularly imprinted polymers (MNIPs) were utilized as adsorbents to describe the adsorption phenomena of dexamethasone and hydrocortisone. This interpretation, based on a statistical physics theory, applies the multilayer model with saturation to comprehend the adsorption of the drugs. Results obtained via numerical simulation revealed that dexamethasone and hydrocortisone adsorption happens via a non-parallel orientation on the surfaces of MMIPs and MNIPs, and they also showed that the adsorption amount of the MMIPs for the template molecule was notably greater than that of the MNIPs at the same initial concentration. The adsorption energy values retrieved from the data analysis ranged between 7.65 and 15.77 kJ/mol, indicating that the extraction and determination of dexamethasone and hydrocortisone is a physisorption process. Moreover, the distribution of a site’s energy was calculated to confirm the physical nature of the interactions between adsorbate/adsorbent and the heterogeneity of the surfaces of the MMIPs and MNIPs. Finally, the thermodynamic interpretation confirmed the exothermicity and spontaneous nature of the adsorption of these drugs on the tested adsorbents.

Adsorption of food dyes from aqueous solution on a sweet potato residue-derived carbonaceous adsorbent: Analytical interpretation of adsorption mechanisms via adsorbent characterization and statistical physics modeling

The adsorption of three food dyes, namely allura red AC, carmine and tartrazine, on a sweet potato residue-derived activated carbon (SPAC) was analyzed via experimental and theoretical tools. In a first step, SPAC was characterized by BET, FTIR and XRD to unravel its textural and surface properties. Adsorption isotherms of the three dyes were obtained at different temperatures ranging from 30 to 50 °C. Classical models were first applied to model the adsorption isotherms, but they were unable to describe the adsorption mechanisms. An alternative approach based on the application of mono- and multi-layer models from statistical physics theory was then used to better investigate the adsorption mechanisms. This alternative approach indicated that carmine and allura red AC adsorption followed a monolayer behavior, whereas the adsorption of tartrazine was a multi-layer process. These dyes were adsorbed via different configurations (horizontal, non-horizontal, horizontal and non-horizontal at the same time), depending on the system and the temperature. Tartrazine dye was removed via an aggregation process at different temperatures, while carmine was aggregated only at high temperatures. This aggregation process was absent for the allura red AC adsorption. Endothermic physisorption processes was observed for dye adsorption under these experimental conditions. The theoretical analysis indicated that SPAC is a promising material for the removal of these dyes, particularly in the case of allura red AC dye molecule. Overall, this study reports a combination of experimental and theoretical results to provide a new perspective at the molecular level of the removal of dye molecules using a low-cost activated carbon.

Adsorption of antibiotics by bentonite-chitosan composite: Phenomenological modeling and physical investigation of the adsorption process

The increased use of antibiotics worldwide turned into a serious preoccupation due to their environmental and health impacts. Since the majority of antibiotic residuals are hardly eliminated from wastewater, based on usual methods, other treatments receive considerable attention. Adsorption is known as the most effective method of the treatment of antibiotics. In this paper, the adsorption isotherms of doripenem, ampicillin, and amoxicillin on bentonite-chitosan composite are determined at three temperatures, T = 303.15, 313.15 and 323.15 K, which are used to achieve a theoretical investigation of the removal phenomenon, based on a statistical physics theory. Three analytical models are utilized to describe the AMO, AMP, and DOR adsorption phenomena at the molecular level. From the fitting results, all antibiotic adsorption on a BC adsorbent is associated with the monolayer formation with one type of site. Concerning the number of adsorbed molecules per site (n), it is concluded that multi-docking (n < 1) and multi-molecular (n > 1) phenomena are feasible for AMO, AMP, and DOR adsorption on BC. The adsorption amounts at saturation of the BC adsorbent, deduced by the monolayer model, are found to be 70.4–88.0 mg/g for doripenem, 57.8–79.2 mg/g for ampicillin and 38.6–67.5 mg/g for amoxicillin indicating that the antibiotics adsorption performance of BC was greatly depended on temperature where the adsorption capacities increased with the increment of this operating variable. All adsorption systems are demonstrated by a calculation of the energy of adsorption, considering that the extrication of these pollutants implies physical interactions. The thermodynamic interpretation confirms the spontaneous and feasible nature of the adsorption of the three antibiotics on BC adsorbent. In brief, BC sample is regarded as a promising adsorbent to extract antibiotics from water and presents important potentials to be effected in wastewater handling at industrial level.

Adsorption of rare earth elements onto diatomite M45: Experimental investigations and modeling with statistical physics theory

The separation of rare earth elements using diatomite M45 from aqueous solutions was studied. The experimental isotherms for the adsorption of trivalent lanthanum, cerium, and neodymium cations on this adsorbent were quantified under strongly acidic conditions (pH 2) at 298–328 K. The adsorption equilibria of these earth elements were analyzed using two statistical physics models (homogeneous and heterogeneous monolayer models). The results show that the adsorption of these ions implies a multi-ionic mechanism, which is exothermic. Si-containing functional groups are responsible for the adsorption of these rare-earth elements on the diatomite surface. A heterogeneous statistical physics model confirms that two Si-based functional groups participate in the separation of these cations. The calculated adsorption capacities at saturation follow the order: neodymium > cerium > lanthanum. Calculated interaction energies range from 28600 to 40100 J/mol, indicating physical adsorption on diatomite M45. This study demonstrates that diatomite M45 is a promising separation medium that can be used for the recovery of REEs dissolved in aqueous solutions via adsorption.

Parking Search in the Physical World: Calculating the Search Time by Leveraging Physical and Graph Theoretical Methods

Parking plays a central role in transport policies and has wide-ranging consequences: While the average time spent searching for parking exceeds dozens of hours per driver every year in many Western cities, the associated cruising traffic generates major externalities, by emitting pollutants and contributing to congestion. However, the laws governing the parking search time remain opaque in many regards, which hinders any general understanding of the problem and its determinants. Here, we frame the problem of parking search in a very generic, but mathematically compact formulation that puts the focus on the role of the street network and the unequal attractiveness of parking spaces. This problem is solved in two independent ways, valid in any street network and for a wide range of drivers’ behaviours. Numerically, this is done by means of a computationally efficient and versatile agent-based model. Analytically, we leverage the machinery of Statistical Physics and Graph Theory to derive a generic mean-field relation giving the parking search time as a function of the occupancy of parking spaces; an expression for the latter is obtained in the stationary regime. We show that these theoretical results are applicable in toy networks as well as in complex, realistic cases such as the large-scale street network of the city of Lyon, France. Taken as a whole, these findings clarify the parameters that directly control the search time and provide transport engineers with a quantitative grasp of the parking problem. Besides, they establish formal connections between the parking issue in realistic settings and physical problems. Funding: This work was supported by IDEXLYON (IDEXLYON 2020–2021); Institut Rhonalpin des Systèmes Complexes (IXXI) (Vulnerabilite). Supplemental Material: The e-companion is available at https://doi.org/10.1287/trsc.2023.1206 .

Regeneration Analysis of Bone Char Used in Water Defluoridation: Chemical Desorption Route, Surface Chemistry Analysis and Modeling

High concentrations of fluoride (F−) in drinking water represent a public health threat, and consequently, effective and sustainable methods are required to improve the water quality, mainly in developing and low-income countries. This study focused on the thermodynamics of fluoride adsorption on bone char regenerated with NaOH for water defluoridation. A detailed analysis of the number of fluoride adsorption/desorption cycles, their impact on the performance and surface chemistry of bone char using different NaOH concentrations, and modeling of the adsorption mechanism using statistical physics theory was carried out. The results showed that 0.075 mol/L NaOH was effective in recuperating the defluoridation properties of bone char with a regeneration efficiency higher than 90% during five adsorption/desorption cycles. Bone char regeneration efficiency decreased up to 64% after ten adsorption/desorption cycles with a maximum fluoride adsorption capacity of 0.18 mmol/g. NaOH restored the bone char surface properties for ligand exchange of the fluoride anions via the hydroxyapatite functionalities contained in this adsorbent. It was calculated that around 0.25–0.46 mmol/g hydroxyapatite ligand exchange sites of regenerated bone char samples could be involved in the fluoride adsorption, which was also expected to be a mono-ligand mechanism. The reduction in defluoridation properties of bone char during the regeneration cycles was attributed to the decrease in the ligand exchange capacity as well as the deactivation and blocking of some functional groups of hydroxyapatite, which limited their participation in consecutive adsorption processes. This study contributes to the optimization of the recycling and reuse of bone char for fluoride removal from water to reduce the operating defluoridation costs, thus enhancing the application of this technology in low-income areas where fluorinated water represents a threat to public health.

Utilization of red clay brick waste in the green preparation of an efficient porous nanocomposite for phenol adsorption: Characterization, experiments and statistical physics treatment

Red clay brick waste was utilized, for the first time, in the green preparation of nanoscale zero-valent iron (nZVI) to impregnate with activated charcoal (ACh). The as-synthesized nZVI@ACh composite was characterized via X-ray diffraction (XRD), Fourier transform infrared (FTIR) spectrometry, thermogravimetric analysis (TG) and its derivative thermogravimetry (DTG), field-emission scanning electron microscopy (FESEM), and transmission electron microscopy (TEM). This novel material was employed for phenol remediation at 25, 35, and 45 °C and pH 7. The results indicated that the prepared ZVI nanoparticles could be either trapped or distributed on the ACh structure. Experimental studies and their modeling with statistical physics theory were applied to analyze the phenol adsorption performance and mechanism using the nZVI@ACh. The phenol adsorption data fitted well to the Langmuir model, with maximum adsorption capacities of 122.85, 137.46, and 152.22 mg/g at 25°, 35°, and 45 °C, respectively. Theoretical calculations via statistical physics indicated that phenol molecules were removed on the nZVI@ACh following the hypothesis of the advanced monolayer model. The number of molecules adsorbed per site (n) ranged from 1.13 to 1.26 suggesting the involvement of vertical geometry and multi-molecular mechanism at all temperatures. The increment of the receptor sites density (NM) from 83.47 to 121.40 mg/g and the adsorption capacity at saturation (Qsat) from 104.75 to 137.19 mg/g with increasing temperature suggested the endothermic behavior of this removal process. Energetically, the phenol adsorption was directed by physical forces with adsorption energies (ΔE) ranging from 13.85 to 16.45 kJ/mol. Macroscopically, phenol adsorption onto nZVI@ACh was spontaneous according to the calculated thermodynamic functions. The developed nZVI@ACh adsorbent can be easily regenerated and reused preserving its competitive adsorption performance. Overall, these results offer a new approach that takes the Fe-rich solid wastes to the next standard in the design of outstanding and low-cost adsorbents for wastewater remediation. Also, the experimental study supported by the theoretical analysis generated new insights of the adsorption system at the micro- and macroscopic scales.

A domain free of the zeros of the partial theta function

The partial theta function is the sum of the series \medskip\centerline{$\displaystyle\theta (q,x):=\sum\nolimits _{j=0}^{\infty}q^{j(j+1)/2}x^j$,}\medskip\noi where $q$is a real or complex parameter ($|q|&lt;1$). Its name is due to similaritieswith the formula for the Jacobi theta function$\Theta (q,x):=\sum _{j=-\infty}^{\infty}q^{j^2}x^j$. The function $\theta$ has been considered in Ramanujan's lost notebook. Itfinds applicationsin several domains, such as Ramanujan type$q$-series, the theory of (mock) modular forms, asymptotic analysis, statistical physics, combinatorics and most recently in the study of section-hyperbolic polynomials,i.~e. real polynomials with all coefficients positive,with all roots real negative and all whose sections (i.~e. truncations)are also real-rooted.For each $q$ fixed,$\theta$ is an entire function of order $0$ in the variable~$x$. When$q$ is real and $q\in (0,0.3092\ldots )$, $\theta (q,.)$ is a function of theLaguerre-P\'olyaclass $\mathcal{L-P}I$. More generally, for $q \in (0,1)$, the function $\theta (q,.)$ is the product of a realpolynomialwithout real zeros and a function of the class $\mathcal{L-P}I$. Thus it isan entire function withinfinitely-many negative, with no positive and with finitely-many complexconjugate zeros. The latter are known to belongto an explicitly defined compact domain containing $0$ andindependent of $q$ while the negative zeros tend to infinity as ageometric progression with ratio $1/q$. A similar result is true for$q\in (-1,0)$ when there are also infinitely-many positive zeros.We consider thequestion how close to the origin the zeros of the function $\theta$ can be.In the generalcase when $q$ is complex it is truethat their moduli are always larger than $1/2|q|$. We consider the case when $q$ is real and prove that for any $q\in (0,1)$,the function $\theta (q,.)$ has no zeros on the set $$\displaystyle \{x\in\mathbb{C}\colon |x|\leq 3\} \cap \{x\in\mathbb{C}\colon {\rm Re} x\leq 0\}\cap \{x\in\mathbb{C}\colon |{\rm Im} x|\leq 3/\sqrt{2}\}$$which containsthe closure left unit half-disk and is more than $7$ times larger than it.It is unlikely this result to hold true for the whole of the lefthalf-disk of radius~$3$. Similar domains do not exist for $q\in (0,1)$, Re$x\geq 0$, for$q\in (-1,0)$, Re$x\geq 0$ and for $q\in (-1,0)$, Re$x\leq 0$. We show alsothat for $q\in (0,1)$, the function $\theta (q,.)$ has no real zeros $\geq -5$ (but one can find zeros larger than $-7.51$).

Modeling Method for Semicrystalline Polymers Controlling Aspects of the Morphology at the Molecular Scale for the Study of Mechanical and Physicochemical Properties.

A novel method is presented to build semicrystalline polymer models used in molecular dynamics simulations. The method allows controlling certain aspects of the molecular morphology of the material. It relies on the generation of the polymer sections in the amorphous phase of the semicrystalline structure according to the statistical polymer physics theory proposed by Adhikari and Muthukumar ( J. Chem. Phys.2019, 151, 114905). The amorphous phase is first built based on the method initially developed by Theodorou and Suter ( Macromolecules1985,18 (7), 1467-1478). Then, the amorphous phase is stacked between crystallites, and a connection algorithm proposed by Rigby et al. ( Advanced Composites for Aerospace, Marine, and Land Applications; Springer: Cham, Switzerland, 2014), initially developed to build polymer thermosets, is employed to link the two phases. For a given set of degree of crystallinity, semicrystalline long period, densities of the crystalline and amorphous phases, and polymer molecular weight, the characteristic ratio is used to control the relative fractions of different types of polymer sections in the amorphous phase as well as the distribution of their lengths. There are three types of amorphous polymer sections: the ones that are reentering in the same crystallite called loops, those that are bonding two different crystallites called tie chains, and the chain tails ending in the amorphous region. The higher the imposed characteristic ratio is, the higher the fraction of the tie chains is. The full implementation of the theory is described and then applied to high-density polyethylene (HDPE). Several samples are generated. The obtained structures are characterized. Their elastic coefficients are computed, and high uniaxial deformations are performed. It is shown that the higher the degree of crystallinity, the higher the elastic coefficients. An entanglement analysis shows that the quantity of tie chains is more decisive than the entanglements in acting as stress transmitters to rigidify the structure.

Iterative mean-field approach to the spherical collapse of dark matter haloes

ABSTRACT Gravitational collapse of dark matter overdensities leads to the formation of dark matter haloes which embed galaxies and galaxy clusters. An intriguing feature of dark matter haloes is that their density profiles closely follow a universal form irrespective of the initial condition or the corresponding growth history. This represents a class of dynamical systems with emergent universalities. We propose an ‘iterative mean-field approach’ to compute the solutions of the gravitational collapse dynamics. This approach iteratively searches for the evolution of the interaction field ϕ(t) – in this case the enclosed mass profile M(r, t) – that is consistent with the dynamics, thus that ϕ(t) is the fix-point of the iterative mapping, $\mathcal {H}(\phi) = \phi$. The formalism replaces the N-body interactions with one-body interactions with the coarse-grained interaction field, and thus shares the spirit of the mean-field theory in statistical physics. This ‘iterative mean-field approach’ combines the versatility of numerical simulations and the comprehensiveness of analytical solutions, and is particularly powerful in searching for and understanding intermediate asymptotic states in a wide range of dynamical systems where the solutions can not be obtained through the traditional self-similar analysis.

Insight into the adsorption properties of methoxy bentonite (MXBE) as an environmental adsorbent for malachite green dye: statistical physics modelling

ABSTRACT Natural bentonite was functionalised by methanol to obtain a novel methoxy bentonite structure (MXBE) of enhanced adsorption capacity for malachite green dye (MG.D). The adsorption properties of MXBE adsorbent were illustrated based on the classic equilibrium studies and the advanced equilibrium modelling according to the statistical physics theory. The uptake of MG.D by MXBE follows the kinetic properties of the Pseudo-First order model (R2 > 0.9) and exhibits the equilibrium behaviour of a classic Langmuir isotherm (R2 > 0.9). This demonstrates the physical adsorption of MG.D in homogeneous and monolayer form considering the Gaussian energy value (4.24–6.53 kJ/mol). The description of the adsorption system according to the statistical physics theory and the assumption of the Monolayer model with one energy site was accomplished considering the steric and energetic parameters. The saturation adsorption capacity of MXBE during the uptake of MG.D is 435.12 mg/g (313.13 K) and this value depends on the saturation of the structure with numerous active adsorption sites (Nm = 121 to 240.6 mg/g) and the efficiency of each active site to adsorb up to three MG.D molecules (n = 1.92 to 2.68). Considering the adsorption energy (ΔE = 16.02 to 17.94 kJ/mol), the uptake of MG.D occurred by physical mechanisms involving mainly hydrogen bonding and dipole bonding forces. The recognised thermodynamic functions (entropy, internal energy, free enthalpy) validate the spontaneous and endothermic behaviours of the MG.D adsorption reactions by MXBE.