Fluids, characterized by broken time-reversal and parity symmetries, exhibit odd transport phenomena where longitudinal drivings can induce transverse fluxes. Recently, a mesoscale model called chiral stochastic rotation dynamics (CSRD) has been developed to simulate odd fluids with high computational efficiency. In this work, we verify the Green-Kubo relations for both normal and odd transport coefficients in this model, confirming that this model correctly captures the underlying statistical relationship between macroscopic transport and microscopic fluctuations in odd fluids. This work solidifies the physical foundation of the CSRD model, paving the way for its application in studying the statistical physics and nonequilibrium behavior of odd fluids.

Pharmaceutical residues, particularly urinary analgesics and their metabolites, are emerging contaminants that pose potential environmental and health risks. This study reports the synthesis of a novel nanocomposite (GOHD) obtained by citric acid crosslinking of H. dulcis with PCT, and its application for adsorbing PhP and PCT from fresh synthetic and real human urine. The material was characterized using various characterization techniques, indicating the formation of a heterogeneous, mesoporous structure with the presence of different functional groups. Adsorption studies revealed that GOHD reached a maximum capacity of 204.82 mg g −1 at the lowest dosage (0.125 g L −1 ). PhP uptake was favored at pH 5.0, and the general-order model well described the kinetic data. Equilibrium analysis revealed that the Freundlich isotherm effectively described the data. At the same time, the M5 statistical physics model provided a deeper interpretation, indicating the presence of two to three adsorption layers with a parallel molecular arrangement ( n < 0.5). Thermodynamic parameters confirmed that the process was spontaneous, exothermic, and governed by weak interactions ( Δ G < 0, Δ H = −12.74 kJ mol −1 ). This is corroborated by the adsorption energy values, which confirm the occurrence of physisorption. In the binary system, GOHD showed high selectivity for PhP, increasing its removal to 80.98% ( q e = 100.73 mg g −1 ), while PCT removal decreased to 11.86% ( q e = 3.75 mg g −1 ), evidencing competitive adsorption. In human urine, PhP removal reached 14.29% ( q e = 29.39 mg g −1 ), whereas PCT adsorption was not observed. The adsorption energy obtained by DFT ( E ads = 49.3 kJ mol −1 ) corroborates the statistical physics energies for the first adsorption layer Δε₁ = 8.27–13.73 kJ mol −1 ), confirming stable, surface-confined adsorption dominated by strong physical interactions. DFT calculations revealed that PhP exhibits higher chemical softness (η = 1.64 eV), electrophilicity (ω = 4.36 eV), and a lower HOMO–LUMO gap (3.27 eV) than PCT, explaining its stronger affinity and preferential adsorption on GOHD. These findings demonstrate that GOHD is a promising, sustainable adsorbent for the adsorption of PhP. • Hovenia dulcis croslinked graphene oxide for drug adsorption in synthetic urine; • A maximum adsorption capacity of 204.82 mg g −1 for phenazopyridine (PhP); • The nanocomposite (GOHD) was selective for phenazopyridine.

From microstates to macroscales: A critical review of maximum entropy modeling and energy landscape analysis in functional MRI.

Abstract Community detection, which aims to reveal the underlying structure of communities within a network, is a fundamental task in network analysis. In this paper, we employ Free Energy Machine (FEM) framework to maximize modularity, thereby identifying optimal community partitions. FEM is grounded in the principles of statistical physics, in the concept of free-energy minimization, and integrates ideas from mean-field theory and simulated annealing, while leveraging modern computational techniques such as automatic differentiation and gradient-based optimization. Experimental results on a wide range of real-world and synthetic networks show that FEM achieves an improvement in modularity compared with other prominent algorithms. These findings demonstrate that FEM is an effective tool for solving the modularity optimization problem.

The Hotelling-Downs model considers parties changing policy to maximise their vote-share. Where policy position lies on a left-right axis, it describes a tendency for political parties to move towards centrist platforms. This is in contrast with widely observed political polarisation. We extend the model to two dimensions, with many parties and with single and multiple-peaked voter distribution. We find that a two party system reduces polarisation, even if voters are polarised with a bimodal distribution. By contrast, multiparty systems induce polarisation, even when most voters favour moderate position. We model the effect of turnout and activists as influences on the parties, showing that this results in more polarisation, even in a two-party system. This suggests that polarisation of parties can be driven by abstention, intra-party politics and turnout on the extremes. In the two-party case, the winning party’s positions are more moderate than the views of their supporters but better representative of the electorate as a whole. With polarisation, individual voters are better able to find a party which represents their views, but the government (winning part or coalition) is less representative of the population, even when the population has a clear consensus on all issues.

A self-consistent theory of bulk electrolytes incorporating electrostatic and hard-core interactions on an equal level is applied to the two-dimensional Coulomb liquid with finite ion size. The ionic pair distributions, the structure factors, and the thermodynamic functions of the formalism are compared with extensive Monte Carlo simulation results from the literature. At moderate salt densities, our computational approach can accurately describe the thermodynamics of two-dimensional solutions across weak to intermediate coupling strengths. The improved accuracy of the present theory with respect to continuum approaches stems mainly from its ability to account for the non-uniform screening of electrostatic interactions associated with the impenetrability of the charged hard disks by their ionic atmosphere. Due to the underestimation of the ionic clusters emerging in the dilute regime, the validity domain of our self-consistent formalism shrinks with the decrease of the salt density. As a result, our approach cannot reach the critical coupling domain where the conductor-insulator transition of two-dimensional charged hard disks occurs. This indicates that approaching the low-temperature dielectric phase via the present formalism will require extending the underlying self-consistent approximation at least to the next cumulant order.

In this perspective article, we discuss the scenario of dynamically emergent correlation (DEC) arising in classical and quantum noninteracting systems when they are subjected to a common fluctuating stochastic environment. The key property of such systems is that the strong correlations between different particles emerge from the dynamics and not from built-in interactions. In many cases, these strong correlations persist even at long times in the stationary state. Computing observables explicitly for such strongly correlated states in general is very hard. Remarkably, the stationary states in several models of DEC exhibit an interesting analytical structure that allows to compute physical observables, despite being strongly correlated. Recent experiments on trapped colloidal particles have established that these DEC in the stationary state can in fact be measured. DEC is a rapidly emerging domain of strongly correlated out-of-equilibrium statistical physics, with both theoretical and experimental, as well as classical and quantum, components.

Activated carbon was produced from olive waste to investigate the adsorption behavior of methylene blue (MB) and methyl orange (MO) dyes in aqueous solutions. The adsorbent was characterized using X-ray diffraction and scanning electron microscopy, and adsorption isotherms were measured at three different temperatures. A statistical physics framework was employed to analyze the adsorption mechanisms of the dyes on the carbon surface. The steric, energetic, and thermodynamic characteristics of the most relevant advanced were analyzed in detail. The number of active sites for MB was determined to range from 1.32 to 1.78, while for MO it ranged from 1.23 to 1.62, indicating that the orientation of dye molecules on the adsorbent is influenced by temperature. Overall, MB shows a slightly higher adsorption capacity than MO. Moreover, increasing the temperature reduces adsorption, indicating that the process is exothermic for both dyes. Moreover, adsorption energy values ranging from 9.85 to 18.90kJ/mol indicate that physical interactions are responsible for the adsorption process. Thermodynamic characteristics demonstrated that deyes adsorption onto the AC adsorbent is spontaneous and feasible.

The discharge of dye-contaminated wastewater from textile and related industries represents a serious environmental concern due to the persistence, toxicity, and resistance of synthetic dyes to natural degradation. In this study, Lantana camara derived biochar (LCB) was investigated as a sustainable and low-cost adsorbent for the removal of Acid Black 172 (AB172) dye from aqueous solutions. The structural and surface properties of LCB were characterized using BET, FTIR, and HRTEM analyses, revealing a mesoporous structure with a surface area of 12.14 m² g⁻¹, pore diameter of 6.8 nm, and total pore volume of 0.031 cm³ g⁻¹. Batch adsorption experiments were conducted to evaluate the effects of pH, contact time, adsorbent dosage, and temperature. The adsorption process showed strong pH dependence, achieving maximum dye removal at pH 3, with equilibrium reached within 90 min. The Langmuir model predicted a maximum adsorption capacity of 303.03 mg g⁻¹, indicating a strong affinity between AB172 molecules and LCB surface sites. Kinetic analysis revealed that the adsorption followed a pseudo-second-order model, while thermodynamic parameters confirmed that the process is spontaneous and endothermic. In addition to this, the use of statistical physics modeling helped understand more about the adsorption process, such as adsorption orientation, adsorption energy, and the mechanism of surface interaction. These aspects set apart the study from other studies on adsorption. In the case of AB172 adsorption, the process was mainly influenced by electrostatic attraction, pore filling, van der Waals force, and hydrogen bonding interactions. Regeneration studies demonstrated high stability of the adsorbent, maintaining over 91% removal efficiency after five adsorption–desorption cycles. Statistical physics modeling and experimental studies on adsorption help in gaining a better insight into the processes involved in the dye-biochar interactions than traditional adsorption experiments do. These findings highlight LCB as a promising, environmentally sustainable, and economically viable adsorbent for the treatment of dye-contaminated wastewater.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-026-57437-x.

Computing molecular thermodynamic properties is instrumental in multiple scientific disciplines, such as statistical physics, N-body simulations, and molecular docking. However, exact thermodynamic calculations are almost always not feasible. In this work, we introduce a versatile algorithm designed to rapidly compute the two-body partition function, its related thermodynamic properties, and the second virial coefficient for anisotropic nanoparticles and proteins under the rigid-body approximation. Our method involves constructing a quasi-regular grid in the 5D angular space between pairs of arbitrary objects and efficiently scanning the radial-angular space between the rigid molecules. Where available, we find excellent agreement with light and X-ray scattering experiments, as well as with Monte Carlo simulations. Our results suggest a correction to current coarse-grained protein force fields, and we further discover a new, counterintuitive effect of temperature on virial coefficients, caused by a population shift in angular space due to the dielectric response of water. Finally, the grid can serve as an interpolation table for N-body simulations, increasing their performance by orders of magnitude.

Analytical response functions for a compressible thin fluid layer with odd viscosity**Article contributed to the topical issue of the New Journal of Physics on ‘Broken symmetries and odd transport in statistical physics’ edited by Erik Kalz, Ralf Metzler, and Abhinav Sharma.

Extreme value statistics and some applications in statistical physics

Experimental and theoretical investigation of radioactive thorium adsorption onto montmorillonite: Equilibrium, statistical physics, and Monte Carlo analysis

Understanding the mechanisms governing radionuclide retention by clay minerals is essential for the long-term safety of geological repositories. This study proposes an operational, multi-criteria assessment framework to interpret the relative contributions of surface adsorption, hydration-state–mediated interlayer uptake (1W→2W transitions), and ion exchange during cesium retention in Wyoming montmorillonite (SWy-2). Rather than relying on any single technique, the framework integrates complementary signatures from batch experiments, X-ray diffraction (used strictly as hydration-state indicators), PHREEQC geochemical modeling, and multilayer statistical-physics fitting. Batch experiments were conducted using both non-radioactive CsCl and 137 Cs across concentrations from 10 -6 to 10 -2 M, pH 4–9, and temperatures of 278–338 K. At trace concentrations (<10 -6 M), cesium retention is consistent with rapid surface adsorption, characterized by short half-lives and strong pH sensitivity. At intermediate concentration (≈10 -4 M), a basal-spacing evolution from 12.5 to 15.2 Å reflects hydration-state transitions accompanying partial Na + /Cs + exchange, supported by slower uptake kinetics and thermodynamic trends. At high concentration (>10 -2 M), ion-exchange signatures predominate, as evidenced by Cs + /Na + selectivity (Kex = 2.8 ± 0.3), Na + release stoichiometry, and persistent 2W domains. Dry-state XRD patterns are interpreted exclusively as hydration-state constraints rather than mechanistic proof, and mechanistic assignments arise only from convergence among kinetic, thermodynamic, structural, and exchange-based evidence. PHREEQC simulations confirm that Cs remains >99.8% as hydrated Cs + across all pH values and that precipitation is thermodynamically unfavorable (SI = –6.22 to –10.21). Statistical-physics modeling reproduces the equilibrium isotherm with R 2 > 0.95 and yields physically consistent layer-site distributions after unit correction. Overall, this integrated approach provides a structured framework for synthesizing multi-signature datasets to interpret cesium retention mechanisms on montmorillonite, while explicitly recognizing the limitations of individual techniques (particularly dry-state XRD) and avoiding over-interpretation of single experimental indicators. • Integrated experimental, XRD, geochemical, and statistical physics approaches to discriminate radionuclide retention mechanisms in montmorillonite • Clear concentration-dependent transition from surface adsorption to intercalation and ion exchange • Quantitative d 001 basal spacing evolution used as a structural fingerprint of intercalation • Kinetic and thermodynamic signatures establish definitive mechanistic criteria • Implications for predictive modeling of radionuclide behavior in geological disposal systems

Learning is a complex dynamical process shaped by a range of interconnected decisions. Careful design of hyperparameter schedules for artificial neural networks or efficient allocation of cognitive resources by biological learners can dramatically affect performance. Yet, theoretical understanding of optimal learning strategies remains sparse, especially due to the intricate interplay between evolving metaparameters and nonlinear learning dynamics. The search for optimal protocols is further hindered by the high dimensionality of the learning space, often resulting in predominantly heuristic, difficult to interpret, and computationally demanding solutions. Here, we combine statistical physics with control theory in a unified theoretical framework to identify optimal learning protocols in prototypical neural network models. In the high-dimensional limit, we derive closed-form ordinary differential equations that track online stochastic gradient descent through low-dimensional order parameters. We formulate the design of learning protocols as an optimal control problem directly on the dynamics of the order parameters with the goal of minimizing the generalization error. This formulation encompasses a variety of learning scenarios, optimization constraints, and control budgets. We apply it to representative cases, including optimal curricula, adaptive dropout regularization and noise schedules in denoising autoencoders. We find nontrivial yet interpretable strategies highlighting how optimal protocols mediate learning trade-offs. Our results establish a principled foundation for understanding and designing optimal protocols and suggest a path toward a theory of meta-learning grounded in statistical physics.

This study details the synthesis and application of a novel one-part geopolymeric hybrid composite (OP-GPHC) derived from glauconite, talc, and olive seed waste–based activated carbon for the efficient sequestration of Congo red (CR) dye from contaminated water. The hybrid binder was synthesized by impregnating activated carbon-based biogenic waste into a thermally treated glauconite/talc matrix, followed by alkali-activation with NaOH. Comprehensive characterization via XRD, FTIR, BET, TG/DTG, FESEM/EDX, and elemental mapping confirmed the material’s exceptional adsorptive properties. A Box–Behnken design (BBD) optimization established the optimal operational conditions: pH 2.0, adsorbent dosage 0.07 g/ 25 ml, contact time 77.5 min, initial CR concentration 150 mg/L, and temperature 328 K, achieving a removal efficiency of 99.2%. Equilibrium data were best described by the Langmuir isotherm, yielding a maximum adsorption capacity of 367 mg/g at 328 K, while kinetic data followed the pseudo-first-order (PFO) model. Advanced statistical physics modeling revealed a multimolecular adsorption mechanism with a vertical orientation of CR molecules at the active sites, independent of temperature, and binding energies in the range of 19.25–21.46 kJ/mol, consistent with physisorption dominated by hydrogen bonding, π–π interactions, and electrostatic forces. Thermodynamic parameters confirmed the endothermic and spontaneous nature of the process. The OP-GPHC adsorbent exhibited excellent reusability (87.8% after five cycles) and a low production cost of $0.032/g. Based on batch-derived data, treatment of 100 L of CR-contaminated water (50 mg/L) is projected to cost approximately $1.68, highlighting its strong potential for industrial-scale tertiary treatment applications.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-026-50411-7.

Characterization of the Docking Mechanism of Fruity Aroma Compounds on Olfactory Receptors Using Molecular Docking Simulation and Statistical Physics Modeling

The approximate calculation of iterated nested expectations is a recurring challenging problem in applications. Nested expectations appear, for example, in the numerical approximation of solutions of backward stochastic differential equations (BSDEs), in the numerical approximation of solutions of semilinear parabolic partial differential equations (PDEs), in statistical physics, in optimal stopping problems such as the approximate pricing of American or Bermudan options, in risk measure estimation in mathematical finance, or in decisionmaking under uncertainty. Nested expectations which arise in the above-mentioned applications often consist of a large number of nestings. However, the computational effort of standard nested Monte Carlo approximations for iterated nested expectations grows exponentially in the number of nestings and it has remained an open question whether it is possible to approximately calculate multiply iterated high-dimensional nested expectations in polynomial time. In this article we tackle this problem by proposing and studying a new class of full-history recursive multilevel Picard (MLP) approximation schemes for iterated nested expectations. Specifically, we prove under suitable assumptions that these MLP approximation schemes can approximately calculate multiply iterated nested expectations with a computational effort growing at most polynomially in the number of nestings K ∈ ℕ = {1,2,3,…}, in the problem dimension d ∈ ℕ, and in the reciprocal 1/ε of the desired approximation accuracy ε ∈ (0, ∞). In particular, the proposed MLP approximation schemes can approximately calculate nested expectations arising in the numerical approximation of solutions of BSDEs and semilinear parabolic PDEs with a computational effort growing at most polynomially in K, d, and 1/ε.

Macro-quantum mechanistic evaluation of magnetron-sputtered W-coated biochar for the adsorption of diclofenac, crystal violet, and Pb(II) using statistical physics and DFT

We develop a model of musical rhythm and meter based on optimizing the trade-off between human psychological preferences for perceiving repeated patterns in time with a desire for variety and complexity. By mapping these competing preferences onto analogous quantities in statistical physics, we define an effective free energy which is minimized in the grand canonical ensemble. Using a mean field approximation, we observe phase transitions in the model from disordered events in time to orderings that closely reproduce those seen in music. We then compare the range of rhythmic characteristics predicted by the model to a data set drawn from compositions by Johann Sebastian Bach, finding generally good quantitative agreement. The results provide a lens through which to study musical rhythm and a method for generatively producing rhythms.