The appeal of Dzugutov’s scheme lies in its microscopic foundation. Rather than reducing transport coefficients by macroscopic parameters such as number density and temperature, as Yosenfeld’s earlier excess-entropy scaling did in 1977, Dzugutov used microscopic reduction parameters: an effective hard-sphere diameter for length and the Enskog collision frequency for inverse time. The hard-sphere picture rests on two propositions. First, at high densities, the transfer of momentum and energy between atoms is dominated by short-range repulsive interactions, which behave essentially like binary collisions of billiard-ball-like hard spheres. Second, the frequency of local structural relaxation—the rate at which atoms escape the transient cages formed by their neighbors—can be calculated from Enskog kinetic theory for hard spheres. The result is a remarkably compact formula in which the normalized diffusivity equals 0.049 times the exponential of the excess entropy, expressed in units of the Boltzmann constant.

The excess entropy itself is a subtle quantity. Defined as the total thermodynamic entropy minus the ideal gas entropy, it can be expanded as a series of contributions from pairs of particles, triplets, and higher-order correlations. The leading two-body term, which depends directly on the pair distribution function, is known to contribute more than 85 percent of the excess entropy across most thermodynamic states, and as much as 95 percent near the triple point of simple liquids. Crucially, the pair distribution function can be obtained from x-ray or neutron diffraction experiments or from computer simulations, making Dzugutov’s scaling unusually practical: it connects atomic dynamics to structure and thermodynamics using quantities that are actually measurable. Later work by Jakse and Pasturel showed that an excess entropy derived from the Carnahan-Starling equation of state—the best analytical description of hard-sphere thermodynamics—can also justify the scaling, provided the temperature dependence of the effective hard-sphere diameter is properly accounted for.

What has been missing, the authors argue, is a systematic test for alloys rather than pure metals. Although Yokoyama and colleagues applied the scaling law to binary alloys in 2007, their analysis was restricted to equiatomic compositions. The new study therefore examines cobalt-iron (CoxFe1−x) and nickel-iron (NixFe1−x) alloys across the full concentration range, from iron-rich to cobalt-rich or nickel-rich, at a temperature of 1833 Kelvin. Iron-based transition metal alloys are not an arbitrary choice: they underpin structural steels, magnetic materials, and even models of planetary cores, and their liquid-state transport properties remain poorly constrained by experiment because of the extreme conditions involved.

On the theoretical side, the team combined the Bretonnet-Silbert pseudopotential—a model specifically designed for liquid transition metals that accounts for both s-p and d band contributions, including the sp-d hybridization that simpler pseudopotentials miss—with linearized Weeks-Chandler-Andersen perturbation theory. The latter provides the effective hard-sphere diameters by solving a transcendental equation that equates the Helmholtz free energy of the real interacting system with that of a hard-sphere reference system. Two different local field correction functions, due to Ichimaru-Utsumi and Vashishta-Singwi, were used to screen the electron-ion interaction, allowing the authors to assess which treatment of electron correlation better describes these dense metallic liquids. This combination of Bretonnet-Silbert potential and LWCA theory had never before been applied to calculate diffusion and viscosity in transition metal alloys.

To anchor the theory, the researchers performed classical molecular dynamics simulations with the LAMMPS code, using modified embedded atom method (MEAM) interatomic potentials that capture both many-body effects and the angular orientation of bonds—features essential for transition metals. Each simulation placed 30,000 atoms in a cubic box 70 angstroms on a side with periodic boundary conditions. The alloys were first equilibrated at 2500 Kelvin, above their melting points, in a constant-pressure ensemble, then cooled to 1833 Kelvin, and finally equilibrated for a full nanosecond at the target temperature. One thousand configurations were extracted from each run, and the pair correlation functions were computed with the OVITO visualization tool and averaged with a Python code. The team also verified that their results were independent of system size by repeating selected simulations with 20,000 and 40,000 atoms; the reduced diffusion coefficients and viscosities changed only in the second decimal place, consistent with the finite-size correction formula proposed by Khrapak.

The calculated structural quantities proved encouraging. Effective hard-sphere diameters obtained from theory and simulation were comparable and consistent with literature values for pure liquid iron, cobalt, and nickel, and the partial pair correlation functions reproduced the expected concentration trends: the peak height of like-atom pairs rises as that component becomes richer, while cross-pair correlations peak at mid-concentrations. When the reduced diffusion coefficients were plotted against excess entropy, however, a nuanced picture emerged. Simulated data followed Dzugutov’s universal scaling line fairly well across all concentrations of both alloy families. The theoretically calculated data, by contrast, obeyed the scaling only for mid-concentrated alloys, deviating systematically when a single component dominated the mixture. The culprit, the authors show, is the overestimated principal peak height of the calculated partial pair correlation functions in single-component-rich alloys, which inflates the two-body excess entropy.

Notably, the Carnahan-Starling excess entropy proved far more reliable than the two-body approximation, remaining stable across concentrations and closer to established values for pure liquid metals, even in single-component-rich compositions where the two-body entropy varied wildly. This suggests that the Bretonnet-Silbert potential combined with LWCA theory is a sound framework for mid-concentrated liquid transition metal alloys, even if the perturbative treatment of structure needs refinement at the compositional extremes. The Ichimaru-Utsumi local field correction consistently outperformed the Vashishta-Singwi alternative, yielding more reliable inter-diffusion coefficients and shear viscosities, which remained nearly constant across all concentrations and agreed reasonably with molecular dynamics results.

The transport coefficients themselves tell an industrially relevant story. Partial self-diffusion coefficients of cobalt or nickel decrease while those of iron increase as the minority component’s concentration grows, and the inter-diffusion coefficients—computed as concentration-weighted sums of the self-diffusivities—fall in the range of roughly 7 to 9 times 10⁻⁹ square meters per second, comparable to simulation data and to scattered theoretical and experimental values for the pure liquid metals. Shear viscosities derived from the Stokes-Einstein relation with slip boundary conditions, and from Yokoyama’s modified upper-bound formula, came out around 1.6 to 2.8 millipascal-seconds—smaller than experimental measurements, which run nearly twice as high, but closer to experiment than earlier theoretical estimates and consistent in trend with prior work on Fe-Co and Fe-Ni liquid alloys.

Perhaps the most striking result appears in the paper’s appendix, where the authors propose a new scaling law of their own. Because both alloy families deviated from Dzugutov’s line at single-component-rich compositions, they refitted the data with independent prefactor and exponential factors and found that both Co-Fe and Ni-Fe melts obey the same alternative relationship, with a reduced diffusion coefficient proportional to the exponential of minus 0.3 times the excess entropy and a much smaller prefactor of 0.005. Until more sophisticated theories or new experimental data for iron-based liquid alloys become available, the authors suggest, this proposed scheme offers young researchers a practical route to predicting atomic diffusion in these industrially vital melts—while the half-century-old interplay between entropy and atomic mobility continues to reveal fresh surprises in the liquid state.

Subject of Research: Testing Dzugutov's excess-entropy scaling law for atomic diffusion in liquid iron-based cobalt-iron and nickel-iron transition metal alloys using pseudopotential theory and molecular dynamics simulation.

Article Title: Test of scaling law for atomic diffusion of Fe based liquid transition metal alloys

Article References: Test of scaling law for atomic diffusion of Fe based liquid transition metal alloys. (n.d.). https://doi.org/10.1016/j.rinp.2026.108743

Image Credits: AI Generated

DOI: 10.1016/j.rinp.2026.108743

Keywords: atomic diffusion, liquid metals, transition metal alloys, excess entropy scaling, Dzugutov scaling law, molecular dynamics simulation, pseudopotential theory, iron-cobalt alloys, iron-nickel alloys, shear viscosity, hard-sphere model, LAMMPS

The research, carried out by Khushbu Zala and Sreemoyee Sarkar of SVKM’s NMIMS University in Mumbai, focuses on a phenomenon known as the r-mode instability. R-modes are non-radial oscillation modes in a rotating star that couple to gravity: as the stellar fluid sloshes back and forth, it emits gravitational radiation that carries away angular momentum. In the absence of any counteracting effect, these modes would grow without bound, spinning the star down dramatically in a runaway process. Yet astronomers observe that many millisecond pulsars—stars rotating hundreds of times per second—remain remarkably stable. Something inside them must be damping the oscillations, and the leading candidates are the two forms of viscosity that govern how dense matter dissipates energy: shear viscosity, which resists the sliding of adjacent fluid layers, and bulk viscosity, which dissipates energy when the fluid is periodically compressed and expanded.

What makes the new work distinctive is its treatment of a star whose core contains a mixed phase, a region where hadronic and quark matter coexist in thermodynamic equilibrium. Below a transition density of about 2.23 times nuclear saturation density, the outer layer is modeled as ordinary baryonic matter described by the relativistic mean-field equation of state known as DDME2. Above that threshold, the researchers assume a hybrid interior in which quarks and hadrons mingle. The problem, as the authors emphasize, is inherently ill-posed: the equation of state at supranuclear densities is poorly constrained, quantum chromodynamics cannot yet be solved reliably in the non-perturbative regime relevant to stellar cores, and the transport coefficients—the viscosities that control damping—are even harder to pin down from first principles.

To cut through these uncertainties, the team turned to Bayesian inference, a statistical framework that updates prior physical knowledge with observational data to produce posterior distributions for unknown parameters. The prior distributions for the key dimensionless coefficients—labeled S-tilde, V-tilde, W-tilde and J-tilde, which encode the shear and bulk viscous response of the mixed phase along with the star’s equilibrium structure—were drawn from existing calculations for neutron stars, strange stars and hybrid stars. The likelihood function combined two independent observational constraints: mass-radius measurements from NASA’s Neutron Star Interior Composition Explorer, or NICER, mission, and the spin-frequency and temperature observations of neutron stars in low-mass X-ray binaries. Using the UltraNest nested sampling algorithm, the researchers explored the parameter space efficiently, focusing on regions of high likelihood and estimating the Bayesian evidence for their model.

The formalism at the heart of the analysis describes how the amplitude of an r-mode evolves in time as an exponential whose decay constant is set by the competition between three timescales: gravitational radiation, which drives the instability, and shear and bulk viscosity, which suppress it. At low temperatures, shear viscosity dominates and stabilizes the star; at high temperatures, bulk viscosity takes over. Between these regimes lies a window where damping is least effective and the star is most vulnerable. The minimum of the instability curve—the lowest spin frequency at which the mode can grow—is therefore an exquisitely sensitive probe of the microphysics inside the star, including the equation of state and the weak-interaction processes, such as the direct Urca reaction, that generate bulk viscosity.

Applying this framework to two hybrid star configurations of 1.5 and 1.75 solar masses, the team obtained concrete estimates for the dissipation timescales. The shear viscous damping time came out as approximately 4.99 times ten to the eight, multiplied by the temperature to the five-thirds power, in seconds, while the bulk viscous timescale follows a more complex dependence on both temperature and spin, scaling inversely with the square of the angular velocity. From the inferred coefficients, the researchers calculated the minima of the instability curves: the critical angular velocity reaches its lowest value of about 451.87 hertz at a temperature of 0.259 megaelectronvolts for the 1.5 solar mass star, and 517.47 hertz at 0.234 megaelectronvolts for the 1.75 solar mass star. Normalized to the Kepler frequency, the maximum spin rate a star can sustain before mass shedding, these minima correspond to ratios of roughly 0.069 and 0.071.

Crucially, the resulting instability window does more than produce elegant numbers—it matches what astronomers actually see. When the team compared their inferred instability curves with the observed spin frequencies and temperatures of real millisecond pulsars, they found that the enhanced viscous dissipation from the mixed hadron-quark phase provides sufficient damping to explain the stability of several well-known objects. Among them are the accreting low-mass X-ray binary sources XTE J0929-314 and XTE J1807-294, and the radio millisecond pulsars J0437-4715 and J2124-3358. These stars all spin faster than 100 hertz, placing them squarely in the frequency range where r-mode physics matters, and all of them sit safely outside the region where the instability would grow—precisely as the two-layer hybrid model predicts.

The posterior distributions themselves carried informative structure. The corner plots of the inferred parameters showed moderate correlations among the shear and bulk viscous coefficients, reflecting the coupled role of the two viscosities in setting the instability boundary, while correlations involving the equilibrium parameter remained comparatively weak. Notably, the posterior contours for the more massive 1.74 solar mass configuration were narrower and more tightly localized than those for the lighter star, indicating that frequency-temperature observations constrain the dense-core dissipation properties more stringently in heavier hybrid stars. The team also found that the position of the instability minimum is remarkably robust: it barely shifts when the equilibrium and viscous parameters vary across their full credible intervals, suggesting that the result is not an artifact of statistical noise.

The broader significance of the work lies in its demonstration that r-mode phenomenology, combined with modern statistical inference, can serve as a practical observational tool for probing phase transitions in ultra-dense matter. Because pulsar rotational frequencies and their time derivatives are among the most precisely measured quantities in all of astrophysics, and because NICER continues to deliver mass-radius constraints, the approach links macroscopic observables directly to microscopic transport physics. If the inferred viscous properties of the mixed phase continue to align with observations, it would strengthen the case that some neutron stars genuinely harbor deconfined quark matter in their cores—a question that has remained open since the earliest theoretical speculations about quark stars.

The authors outline several directions for extending the framework. A more realistic three-layer stellar model could better capture the stratification of a hybrid star’s interior, and generalizing the formalism to derive quantitative constraints on the shear and bulk viscosities of each individual layer would sharpen the physical picture. Perhaps most ambitiously, they aim to characterize the nature of the hadron-quark phase transition itself—determining whether it is first order, second order, or a smooth crossover—by performing statistical inference on transport coefficients constrained by the gravitational-wave signatures that r-mode oscillations generate. As gravitational-wave detectors grow more sensitive, the faint hum of a wobbling hybrid star may one day confirm what this Bayesian analysis already hints at: that the universe’s most extreme matter hides its secrets in the way it dissipates motion.

Subject of Research: Bayesian inference of viscous dissipation timescales governing r-mode instability in hybrid stars with hadron-quark mixed phases

Article Title: Modelling dissipative dynamics of r-mode instability in hybrid stars

Article References: Zala, K., & Sarkar, S. (2026). Modelling dissipative dynamics of r-mode instability in hybrid stars. The European Physical Journal C, 86(9), Article 1064. https://doi.org/10.1140/epjc/s10052-026-16199-6

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16199-6

Keywords: hybrid stars, r-mode instability, neutron stars, bulk viscosity, shear viscosity, Bayesian inference, quark matter, millisecond pulsars, NICER, low-mass X-ray binaries, gravitational waves, dense matter