Effective removal of textile dye via synergy of adsorption and photocatalysis over ZnS nanoparticles: Synthesis, modeling, and mechanism.

In this work, we prepared sulfur-zinc nanoparticles (ZnS-TGA) functionalized with thioglycolic acid by a hydrothermal method and tested their photodegradation ability by solar irradiation. ZnS-TGA were characterized by Fourier transform infrared spectroscopy (FTIR), X-ray diffraction (XRD), high-resolution transmission electron microscope (HR-TEM), UV-Vis spectrophotometer and photoluminescence spectroscopy. In the characterization of these nanoparticles, thioglycolic acid proved to be a strong capping ligand, with a specific surface area of 36.82 m2/g and an average size of 7.15 nm. To test the photocatalytic degradability of the product, methylene blue (MB) was used as a model pollutant. Various operational variables were investigated, including pH, amount of nanoparticles, dye concentration, contact time and temperature. The equilibrium adsorption tests, and the statistical physical calculations allowed the analysis of the energetic and steric variables of the adsorption of MB dye molecules on the surface of these nanoparticles. The equilibrium data were well fitted with Langmuir-Freundlich (L-F) and the adsorption kinetics with pseudo-first order. The maximum adsorption capacity of the MB dye removal process was 30.92 mg g-1 at pH 7 and 298 K, and this process was spontaneous and exothermic. The dye molecules and the surface of the nanoparticles exhibited physical interactions with adsorption energies of 23.31-25.92 kJ/mol. The photocatalytic activity of these nanoparticles resulted in a dye degradation efficiency of 91.1 % in 180 min. The photocatalytic efficiency remained almost unchanged after five consecutive degradation cycles, resulting in a methylene blue degradation of 85 %. According to these results, these environmentally friendly nanoparticles have the potential to purify industrial and urban liquids contaminated with harmful organic compounds such as dye molecules.

Non-equilibrium physics of multi-species assembly applied to fibrils inhibition in biomolecular condensates and growth of online distrust.

Self-assembly is a key process in living systems-from the microscopic biological level (e.g. assembly of proteins into fibrils within biomolecular condensates in a human cell) through to the macroscopic societal level (e.g. assembly of humans into common-interest communities across online social media platforms). The components in such systems (e.g. macromolecules, humans) are highly diverse, and so are the self-assembled structures that they form. However, there is no simple theory of how such structures assemble from a multi-species pool of components. Here we provide a very simple model which trades myriad chemical and human details for a transparent analysis, and yields results in good agreement with recent empirical data. It reveals a new inhibitory role for biomolecular condensates in the formation of dangerous amyloid fibrils, as well as a kinetic explanation of why so many diverse distrust movements are now emerging across social media. The nonlinear dependencies that we uncover suggest new real-world control strategies for such multi-species assembly.

EnhancerNet: a predictive model of cell identity dynamics through enhancer selection.

Understanding how cell identity is encoded by the genome and acquired during differentiation is a central challenge in cell biology. I have developed a theoretical framework called EnhancerNet, which models the regulation of cell identity through the lens of transcription factor-enhancer interactions. I demonstrate that autoregulation in these interactions imposes a constraint on the model, resulting in simplified dynamics that can be parameterized from observed cell identities. Despite its simplicity, EnhancerNet recapitulates a broad range of experimental observations on cell identity dynamics, including enhancer selection, cell fate induction, hierarchical differentiation through multipotent progenitor states and direct reprogramming by transcription factor overexpression. The model makes specific quantitative predictions, reproducing known reprogramming recipes and the complex haematopoietic differentiation hierarchy without fitting unobserved parameters. EnhancerNet provides insights into how new cell types could evolve and highlights the functional importance of distal regulatory elements with dynamic chromatin in multicellular evolution.

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.

Eight challenges in developing theory of intelligence.

A good theory of mathematical beauty is more practical than any current observation, as new predictions about physical reality can be self-consistently verified. This belief applies to the current status of understanding deep neural networks including large language models and even the biological intelligence. Toy models provide a metaphor of physical reality, allowing mathematically formulating the reality (i.e., the so-called theory), which can be updated as more conjectures are justified or refuted. One does not need to present all details in a model, but rather, more abstract models are constructed, as complex systems such as the brains or deep networks have many sloppy dimensions but much less stiff dimensions that strongly impact macroscopic observables. This type of bottom-up mechanistic modeling is still promising in the modern era of understanding the natural or artificial intelligence. Here, we shed light on eight challenges in developing theory of intelligence following this theoretical paradigm. Theses challenges are representation learning, generalization, adversarial robustness, continual learning, causal learning, internal model of the brain, next-token prediction, and the mechanics of subjective experience.

Formation of human kinship structures depending on population size and cultural mutation rate.

How does social complexity depend on population size and cultural transmission? Kinship structures in traditional societies provide a fundamental illustration, where cultural rules between clans determine people's marriage possibilities. Here, we propose a simple model of kinship interactions that considers kin and in-law cooperation and sexual rivalry. In this model, multiple societies compete. Societies consist of multiple families with different cultural traits and mating preferences. These values determine interactions and hence the growth rate of families and are transmitted to offspring with mutations. Through a multilevel evolutionary simulation, family traits and preferences are grouped into multiple clans with interclan mating preferences. It illustrates the emergence of kinship structures as the spontaneous formation of interdependent cultural associations. Emergent kinship structures are characterized by the cycle length of marriage exchange and the number of cycles in society. We numerically and analytically clarify their parameter dependence. The relative importance of cooperation versus rivalry determines whether attraction or repulsion exists between families. Different structures evolve as locally stable attractors. The probabilities of formation and collapse of complex structures depend on the number of families and the mutation rate, showing characteristic scaling relationships. It is now possible to explore macroscopic kinship structures based on microscopic interactions, together with their environmental dependence and the historical causality of their evolution. We propose the basic causal mechanism of the formation of typical human social structures by referring to ethnographic observations and concepts from statistical physics and multilevel evolution. Such interdisciplinary collaboration will unveil universal features in human societies.

A statistical-physics approach for codon usage optimisation.

The concept of "codon optimisation" involves adjusting the coding sequence of a target protein to account for the inherent codon preferences of a host species and maximise protein expression in that species. However, there is still a lack of consensus on the most effective approach to achieve optimal results. Existing methods typically depend on heuristic combinations of different variables, leaving the user with the final choice of the sequence hit. In this study, we propose a new statistical-physics model for codon optimisation. This model, called the Nearest-Neighbour interaction (NN) model, links the probability of any given codon sequence to the "interactions" between neighbouring codons. We used the model to design codon sequences for different proteins of interest, and we compared our sequences with the predictions of some commercial tools. In order to assess the importance of the pair interactions, we additionally compared the NN model with a simpler method (Ind) that disregards interactions. It was observed that the NN method yielded similar Codon Adaptation Index (CAI) values to those obtained by other commercial algorithms, despite the fact that CAI was not explicitly considered in the algorithm. By utilising both the NN and Ind methods to optimise the reporter protein luciferase, and then analysing the translation performance in human cell lines and in a mouse model, we found that the NN approach yielded the highest protein expression in vivo. Consequently, we propose that the NN model may prove advantageous in biotechnological applications, such as heterologous protein expression or mRNA-based therapies.

A simplex path integral and a simplex renormalization group for high-order interactions.

Modern theories of phase transitions and scale invariance are rooted in path integral formulation and renormalization groups (RGs). Despite the applicability of these approaches in simple systems with only pairwise interactions, they are less effective in complex systems with undecomposable high-order interactions (i.e. interactions among arbitrary sets of units). To precisely characterize the universality of high-order interacting systems, we propose a simplex path integral and a simplex RG (SRG) as the generalizations of classic approaches to arbitrary high-order and heterogeneous interactions. We first formalize the trajectories of units governed by high-order interactions to define path integrals on corresponding simplices based on a high-order propagator. Then, we develop a method to integrate out short-range high-order interactions in the momentum space, accompanied by a coarse graining procedure functioning on the simplex structure generated by high-order interactions. The proposed SRG, equipped with a divide-and-conquer framework, can deal with the absence of ergodicity arising from the sparse distribution of high-order interactions and can renormalize a system with intertwined high-order interactions at thep-order according to its properties at theq-order (p⩽q). The associated scaling relation and its corollaries provide support to differentiate among scale-invariant, weakly scale-invariant, and scale-dependent systems across different orders. We validate our theory in multi-order scale-invariance verification, topological invariance discovery, organizational structure identification, and information bottleneck analysis. These experiments demonstrate the capability of our theory to identify intrinsic statistical and topological properties of high-order interacting systems during system reduction.

Spectroscopic and statistical physics elucidation for Cu(II) remediation using magnetic bio adsorbent based on Fe3O4-chitosan-graphene oxide.

Efficient harnessing of heavy metal pollution is an urgent environmental task. Herein, magnetic bio adsorbent (MB) based on Fe3O4-chitosan-graphene oxide composite was fabricated via one step co-precipitation for adsorptive remediation of Cu(II). Remediation efficiency was evaluated by batch adsorption, meanwhile adsorption mechanism was elucidated by spectroscopic tests (XPS, UV-Vis absorption and fluorescent emission spectra), statistical physics formalism, isotherm and kinetic fittings. Results show, MB reaches adsorption percent and quantity of 87.61 % and 350.43 mg·g-1 for Cu(II) in 30 min. By virtue of paramagnetism, MB can be readily recovered with a permanent magnet for easy regeneration and cyclic use, thereby retaining adsorption quantity 279.99 mg·g-1 at the fifth cycle. The Freundlich and pseudo second order model satisfactorily describes the adsorption, designating chemical interaction as the rate limiting step. Statistical physics calculation suggests two points. (1) Multi-ionic adsorption mechanism with exothermic, spontaneous and energy lowering feature. (2) Density of adsorption sites increases with temperature, resulting in improved adsorption capacity. Spectroscopic analysis confirms the involvement of CO, CO, -NH2 in Cu(II) uptake via electron donation. These explorations contribute with novel theoretical illumination for understanding both the thermodynamic feature and atomic scale mechanism of common pollutants adsorption by bio adsorbent like Fe3O4-chitosan-graphene oxide.

A Markovian dynamics for Caenorhabditis elegans behavior across scales.

How do we capture the breadth of behavior in animal movement, from rapid body twitches to aging? Using high-resolution videos of the nematode worm Caenorhabditis elegans, we show that a single dynamics connects posture-scale fluctuations with trajectory diffusion and longer-lived behavioral states. We take short posture sequences as an instantaneous behavioral measure, fixing the sequence length for maximal prediction. Within the space of posture sequences, we construct a fine-scale, maximum entropy partition so that transitions among microstates define a high-fidelity Markov model, which we also use as a means of principled coarse-graining. We translate these dynamics into movement using resistive force theory, capturing the statistical properties of foraging trajectories. Predictive across scales, we leverage the longest-lived eigenvectors of the inferred Markov chain to perform a top-down subdivision of the worm's foraging behavior, revealing both "runs-and-pirouettes" as well as previously uncharacterized finer-scale behaviors. We use our model to investigate the relevance of these fine-scale behaviors for foraging success, recovering a trade-off between local and global search strategies.

Discontinuous Shear Thickening in Biological Tissue Rheology.

During embryonic morphogenesis, tissues undergo dramatic deformations in order to form functional organs. Similarly, in adult animals, living cells and tissues are continually subjected to forces and deformations. Therefore, the success of embryonic development and the proper maintenance of physiological functions rely on the ability of cells to withstand mechanical stresses as well as their ability to flow in a collective manner. During these events, mechanical perturbations can originate from active processes at the single-cell level, competing with external stresses exerted by surrounding tissues and organs. However, the study of tissue mechanics has been somewhat limited to either the response to external forces or to intrinsic ones. In this work, we use an active vertex model of a 2D confluent tissue to study the interplay of external deformations that are applied globally to a tissue with internal active stresses that arise locally at the cellular level due to cell motility. We elucidate, in particular, the way in which this interplay between globally external and locally internal active driving determines the emergent mechanical properties of the tissue as a whole. For a tissue in the vicinity of a solid-fluid jamming or unjamming transition, we uncover a host of fascinating rheological phenomena, including yielding, shear thinning, continuous shear thickening, and discontinuous shear thickening. These model predictions provide a framework for understanding the recently observed nonlinear rheological behaviors in vivo.

Explaining neural scaling laws.

The population loss of trained deep neural networks often follows precise power-law scaling relations with either the size of the training dataset or the number of parameters in the network. We propose a theory that explains the origins of and connects these scaling laws. We identify variance-limited and resolution-limited scaling behavior for both dataset and model size, for a total of four scaling regimes. The variance-limited scaling follows simply from the existence of a well-behaved infinite data or infinite width limit, while the resolution-limited regime can be explained by positing that models are effectively resolving a smooth data manifold. In the large width limit, this can be equivalently obtained from the spectrum of certain kernels, and we present evidence that large width and large dataset resolution-limited scaling exponents are related by a duality. We exhibit all four scaling regimes in the controlled setting of large random feature and pretrained models and test the predictions empirically on a range of standard architectures and datasets. We also observe several empirical relationships between datasets and scaling exponents under modifications of task and architecture aspect ratio. Our work provides a taxonomy for classifying different scaling regimes, underscores that there can be different mechanisms driving improvements in loss, and lends insight into the microscopic origin and relationships between scaling exponents.

Neural Activity in Quarks Language: Lattice Field Theory for a Network of Real Neurons.

Brain-computer interfaces have seen extraordinary surges in developments in recent years, and a significant discrepancy now exists between the abundance of available data and the limited headway made in achieving a unified theoretical framework. This discrepancy becomes particularly pronounced when examining the collective neural activity at the micro and meso scale, where a coherent formalization that adequately describes neural interactions is still lacking. Here, we introduce a mathematical framework to analyze systems of natural neurons and interpret the related empirical observations in terms of lattice field theory, an established paradigm from theoretical particle physics and statistical mechanics. Our methods are tailored to interpret data from chronic neural interfaces, especially spike rasters from measurements of single neuron activity, and generalize the maximum entropy model for neural networks so that the time evolution of the system is also taken into account. This is obtained by bridging particle physics and neuroscience, paving the way for particle physics-inspired models of the neocortex.

Adsorptive features of cyclohexane carboxylic naphthenic acid on a novel cross-linked polymer developed from spent coffee grounds.

Naphthenic acids (NA) are organic compounds commonly found in crude oil and produced water, known for their recalcitrance and toxicity. This study introduces a new adsorbent, a polymer derived from spent coffee grounds (SCGs), through a straightforward cross-linking method for removing cyclohexane carboxylic acid as representative NA. The adsorption kinetics followed a pseudo-second-order model for the data (0.007 g min-1 mg-1), while the equilibrium data fitted the Sips model ( q m = 140.55 mg g-1). The process's thermodynamics indicated that the target NA's adsorption was spontaneous and exothermic. The localized sterical and energetic aspects were investigated through statistical physical modeling, which corroborated that the adsorption occurred indeed in monolayer, as suggested by the Sips model, but revealed the contribution of two energies per site ( n 1 ; n 2 ). The number of molecules adsorbed per site ( n ) was highly influenced by the temperature as n 1 decreased with increasing temperature and n 2 increased. These results were experimentally demonstrated within the pH range between 4 and 6, where both C6H11COO-(aq.) and C6H11COOH(aq.) species coexisted and were adsorbed by different energy sites. The polymer produced was naturally porous and amorphous, with a low surface area of 20 to 30 m2 g-1 that presented more energetically accessible sites than other adsorbents with much higher surface areas. Thus, this study shows that the relation between surface area and high adsorption efficiency depends on the compatibility between the energetic states of the receptor sites, the speciation of the adsorbate molecules, and the temperature range studied.

Understanding the adsorption of Reactive red 2 onto metal hydroxide sludge: analysis with physico-statistical steric and energetic perspectives.

This theoretical investigation delves into the analysis of Reactive red 2 (RR-2) adsorption isotherms on metal hydroxide employing a sophisticated double-layer model characterized by dual-energy levels within the realm of physical adsorption phenomena. An examination of five distinct statistical physics frameworks was undertaken to elucidate the modeling and interpretation of equilibrium data. Expression for the physico-chemical parameters involved in the adsorption phenomena was derived based on statistical physics treatment. Fitting experimental adsorption isotherms (308-333 K) to a DAMTBS has revealed the number of anchored molecules per site, occupied receptor site density, and the number of adsorbed layers. The steric parameter n varies between 0.92 and 1.05. More importantly, it is evidenced that the adhesion mechanism of (RR-2) onto metal hydroxide as determined by the estimated adsorption energies (< 40 kJ/mol) supports a spontaneous and exothermic physisorption process. Thermodynamic potential functions such as entropy, Gibbs free energy, and internal energy have been computed based on the most suitable model. This research advances our physical understanding of how metal hydroxide captures dye molecules RR-2 through adsorption reaction for water depollution treatment.

In-depth study of adsorption mechanisms and interactions in the removal of pharmaceutical contaminants via activated carbon: a physicochemical analysis.

This study presents a theoretical analysis of the adsorption process of pharmaceutical pollutants, specifically acetaminophen (ATP) and diclofenac (DFC), onto activated carbon (AC) derived from avocado biomass waste. The adsorption isotherms of ATP and DFC were analyzed using a multilayer model, which revealed the formation of two to four adsorption layers depending on the temperature of the aqueous solution. The saturation adsorption capacities for ATP and DFC were 52.71 and 116.53 mg/g, respectively. A steric analysis suggested that the adsorption mechanisms of ATP and DFC involved a multi-molecular process. The calculated adsorption energies (ΔE1 and ΔE2) varied between 12.86 and 22.58 kJ/mol, with the highest values observed for DFC removal. Therefore, the adsorption of these organic molecules was associated with physisorption interactions: van der Waals forces and hydrogen bonds. These findings enhance the understanding of the depollution processes of pharmaceutical compounds using carbon-based adsorbents and highlight the potential of utilizing waste biomass for environmental remediation.

Morphological, sterical, and localized thermodynamics in the adsorption of CO2 by activated biocarbon from the white rot fungi Trametes gibbosa.

The capture of CO2 by biochar has recently become one of the cornerstones of circular economy models for a sustainable society. In this work, we synthesized an activated biocarbon using Trametes gibbosa (BioACTG) in a one-step synthesis. We investigated CO2 adsorption mechanisms under five different temperatures using a statistical physics approach. The data was better represented by the multilayer model with two distinguished energies, providing more accurate values for the estimated parameters. According to the number of carbon dioxide molecules per site (n) and the densities of the receptor sites (Dzif), the tendency to form a second layer increased as the temperature increased. The adsorption of CO2 on BioACTG was exothermic (the values of Qasat = 15.5 mmol/g at 273 K decrease to 10.5 mmol/g at 353 K), and the temperature influenced CO2 as well as the morphological features of the process. A computational approach was used to investigate the electronic properties of the adsorbate, showing that its lowest unoccupied orbital (LUMO) heavily contributed to the high efficiency of the process which was ruled by pore diffusion mechanisms driven by energetic fluctuations. Other molecules present in CO2-rich mixtures were also investigated, showing that their concentration limited their competitiveness with CO2.

Assessment of chromite ore wastes for methylene blue adsorption: Isotherm, kinetic, thermodynamic studies, ANN, and statistical physics modeling.

This research investigates the adsorption potential of chrysotile and lizardite, two minerals derived from chromite ore wastes, for the uptake of Methylene Blue (MB) dye from waste streams. The characterization of these minerals involves XRD, XRF, FTIR, and SEM. Results confirm the dominance of polymorphic magnesium silicate minerals, specifically chrysotile and lizardite, in the samples. The FTIR spectra reveal characteristic vibration bands confirming the presence of these minerals. The SEM analysis depicts irregular surfaces with broken and bent edges, suggesting favorable morphologies for adsorption. N2 adsorption-desorption isotherms indicate mesoporous structures with Type IV pores in both adsorbents. The Central Composite Design approach is employed to optimize MB adsorption conditions, revealing the significance of contact time, adsorbent mass, and initial MB concentration. The proposed models exhibit high significance, with F-values and low p-values indicating the importance of the studied factors. Experimental validation confirms the accuracy of the models, and the optimum conditions for MB adsorption are determined. The influence of solution acidity on MB uptake is investigated, showing a significant enhancement at higher pH values. Isothermal studies indicate Langmuir and Freundlich models as suitable descriptions for MB adsorption onto chrysotile and lizardite. The maximum adsorption capacities of MB for chrysotile and lizardite were found to be 352.97 and 254.85, respectively. Kinetic studies reveal that the pseudo-first-order model best describes the adsorption process. Thermodynamic analysis suggests an exothermic and spontaneous process. Statistical physics models further elucidate the monolayer nature of adsorption. Additionally, an artificial neural network is developed, exhibiting high predictive capability during training and testing stages. The reusability of chrysotile and lizardite is demonstrated through multiple regeneration cycles, maintaining substantial adsorption potential. Therefore, this research provides comprehensive insights into the adsorption characteristics of chrysotile and lizardite, emphasizing their potential as effiective and reusable sorbents for MB uptake from wastewater.

Machine learning and statistical physics modeling of tetracycline adsorption using activated carbon derived from Cynometra ramiflora fruit biomass.

The current investigation reports the usage of adaptive neuro-fuzzy inference system (ANFIS) and artificial neural network (ANN), the two recognized machine learning techniques in modelling tetracycline (TC) adsorption onto Cynometra ramiflora fruit biomass derived activated carbon (AC). Many characterization methods utilized, confirmed the porous structure of synthesized AC. ANN and ANFIS models utilized pH, dose, initial TC concentration, mixing speed, time duration, and temperature as input parameters, whereas TC removal percentage was designated as the output parameter. The optimized configuration for the ANN model was determined as 6-8-1, while the ANFIS model employed trimf input and linear output membership functions. The obtained results showed a strong correlation, indicated by high R2 values (ANNR2: 0.9939 & ANFISR2: 0.9906) and low RMSE values (ANNRMSE: 0.0393 & ANFISRMSE: 0.0503). Apart from traditional isotherms, the dataset was fitted to statistical physics models wherein, the double-layer with a single energy satisfactorily explained the physisorption mechanism of TC adsorption. The sorption energy was 21.06 kJ/mol, and the number of TC moieties bound per site (n) was found to be 0.42, conclusive of parallel binding of TC molecules to the adsorbent surface. The adsorption capacity at saturation (Qsat) was estimated to be 466.86 mg/g - appreciably more than previously reported values. These findings collectively demonstrate that the AC derived from C. ramiflora fruit holds great potential for efficient removal of TC from a given system, and machine learning approaches can effectively model the adsorption processes.

The Dynamic Spatial Structure of Flocks.

Studies of collective motion have heretofore been dominated by a thermodynamic perspective in which the emergent "flocked" phases are analyzed in terms of their time-averaged orientational and spatial properties. Studies that attempt to scrutinize the dynamical processes that spontaneously drive the formation of these flocks from initially random configurations are far more rare, perhaps owing to the fact that said processes occur far from the eventual long-time steady state of the system and thus lie outside the scope of traditional statistical mechanics. For systems whose dynamics are simulated numerically, the nonstationary distribution of system configurations can be sampled at different time points, and the time evolution of the average structural properties of the system can be quantified. In this paper, we employ this strategy to characterize the spatial dynamics of the standard Vicsek flocking model using two correlation functions common to condensed matter physics. We demonstrate, for modest system sizes with 800 to 2000 agents, that the self-assembly dynamics can be characterized by three distinct and disparate time scales that we associate with the corresponding physical processes of clustering (compaction), relaxing (expansion), and mixing (rearrangement). We further show that the behavior of these correlation functions can be used to reliably distinguish between phenomenologically similar models with different underlying interactions and, in some cases, even provide a direct measurement of key model parameters.