In order for close proximity satellites to safely perform their missions, the relative states of all satellites and pieces of debris must be well understood. This presents a problem for ground based tracking and orbit determination since it may not be practical to achieve the required accuracy. Using space-based sensors allows for more accurate relative state estimates, especially if multiple satellites are allowed to communicate. Of interest to this work is the case where several communicating satellites each need to maintain a local catalog of communicating and non-communicating objects using angles-only limited field of view (FOV) measurements. However, this introduces the problem of efficiently scheduling and coordinating observations among the agents. This paper presents a decentralized task allocation algorithm to address this problem and quantifies its performance in terms of fuel usage and overall catalog uncertainty via numerical simulation. It was found that the new method significantly outperforms the uncertainty-fuel Pareto frontier formed by current approaches.

We present high-angular resolution ($\simeq 0\rlap.{''}06$) VLA and ALMA observations of Orion South separated by 15.52 years. The purpose of this study was to search for orbital motions in three close ($\simeq 0\rlap.{''}1$) binary systems in the region. We do not detect changes in the position angle of the binaries but in two of the cases we detect significant changes in their separation in the plane of the sky. We use these changes to estimate that the total mass of the binaries is in the $\simeq$1-2 $M_\odot$ range. We also estimate the disk masses from the mm emission. The dust-to-stellar mass ratio is in the range of 0.04 to 0.18, values consistent with those expected for very early stellar evolution (Class 0) protostars.

This study investigates how target geometry and material influence pion and muon production from an 8 GeV proton beam, in support of target-system design for a muon collider demonstrator. A 2 m long, 0.7 m radius solenoid with a 5 T peak magnetic field is used to capture secondary particles, with the target positioned at its center. We examine how variations in target radius, length, and material affect secondary-beam yield and emittance at the solenoid exit. In parallel, we evaluate temperature rise within the target to assess material limitations and guide future work on thermal and structural survivability. The results provide initial intuition for optimizing both particle yield and target durability in muon collider front-end systems.

We introduce a compact operator-based technique that solves the paraxial wave equation for a broad class of structured light fields. Using the spatial evolution operator to propagate two families of physically apodized inputs, Gaussian apodized Whittaker integrals and Gaussian apodized Helmholtz fields, we derive closed form expressions that retain the Gaussian width and therefore describe finite energy beams. The method unifies and extends the Helmholtz Gauss families and readily generalizes to nonseparable nondiffracting architectures. Experiments on superposed Bessel Gauss beams confirm the predicted transverse rotations, demonstrating that this operator approach is a fast, transparent, and practical alternative to standard diffraction ntegral treatments

p-spin glasses, characterized by frustrated many-body interactions beyond the conventional pairwise case (p>2), are prototypical disordered systems whose ground-state search is NP-hard and computationally prohibitive for large instances. Solving this problem is not only fundamental for understanding high-order disorder, structural glasses, and topological phases, but also central to a wide spectrum of hard combinatorial optimization tasks. Despite decades of progress, there still lacks an efficient and scalable solver for generic large-scale p-spin models. Here we introduce PLANCK, a physics-inspired deep reinforcement learning framework built on hypergraph neural networks. PLANCK directly optimizes arbitrary high-order interactions, and systematically exploits gauge symmetry throughout both training and inference. Trained exclusively on small synthetic instances, PLANCK exhibits strong zero-shot generalization to systems orders of magnitude larger, and consistently outperforms state-of-the-art thermal annealing methods across all tested structural topologies and coupling distributions. Moreover, without any modification, PLANCK achieves near-optimal solutions for a broad class of NP-hard combinatorial problems, including random k-XORSAT, hypergraph max-cut, and conventional max-cut. The presented framework provides a physics-inspired algorithmic paradigm that bridges statistical mechanics and reinforcement learning. The symmetry-aware design not only advances the tractable frontiers of high-order disordered systems, but also opens a promising avenue for machine-learning-based solvers to tackle previously intractable combinatorial optimization challenges.

As autonomous agents powered by LLM are increasingly deployed in society, understanding their collective behaviour in social dilemmas becomes critical. We introduce an evaluation framework where LLMs generate strategies encoded as algorithms, enabling inspection prior to deployment and scaling to populations of hundreds of agents -- substantially larger than in previous work. We find that more recent models tend to produce worse societal outcomes compared to older models when agents prioritise individual gain over collective benefits. Using cultural evolution to model user selection of agents, our simulations reveal a significant risk of convergence to poor societal equilibria, particularly when the relative benefit of cooperation diminishes and population sizes increase. We release our code as an evaluation suite for developers to assess the emergent collective behaviour of their models.

We use large-scale density-matrix renormalization group simulations with bond dimensions up to $20\ 000$ to determine the magnetization curves of spin-1 and spin-$\tfrac{3}{2}$ pyrochlore Heisenberg antiferromagnets. Both models exhibit a robust half-magnetization plateau, and we find that the same 16-site state (quadrupled unit cell) is selected in both cases on the largest 64-site cubic cluster we consider for the plateau state. This contrasts sharply with the effective quantum dimer model prediction which favors the ``R'' state, and demonstrates the breakdown of the perturbative mechanism at the Heisenberg point. These results provide a nonperturbative characterization of field-induced phases in pyrochlore magnets and predictive guidance for spin-1 and spin-$\tfrac{3}{2}$ materials.

The nearly scale-invariant primordial power spectrum provides the standard initial conditions for cosmological perturbations. However, the largest scales remain only weakly constrained by CMB observations, leaving room for deviations such as an infrared (IR) cut-off. This possibility is further motivated by the persistence of large-scale CMB anomalies, most notably the low quadrupole power. In this work, we revisit several broad classes of phenomenologically motivated IR cut-off scenarios using parametrised functional forms of the primordial power spectrum. We confront these models with the latest CMB, BAO, and supernova data and derive updated constraints on the cut-off scale and associated features. Our results remain consistent with earlier studies, showing that although such models suppress power at low multipoles, the improvement in fit is marginal and does not overcome the associated parameter penalties. We therefore find no statistically significant evidence favouring IR cut-off models over the standard power-law spectrum with current data. We further explore the interplay between IR cut-off features and a possible increase in the reionisation optical depth, motivated by the recent CMB-BAO tension highlighted by DESI DR2 within the $Λ$CDM framework. We show that the additional freedom introduced by large-scale suppression is generally insufficient to support a substantial increase in optical depth, owing to the weak statistical preference for suppressed large-scale temperature power. Finally, we examine the implications of IR cut-off models for large-scale CMB anomalies by analysing the corresponding anomaly statistics within a Bayesian framework.

We study the mean-field dynamics of a system of $N$ interacting bosons starting from an initially condensated state. For a broad class of mean-field Hamiltonians, including models with arbitrary bounded interactions and models with unbounded interaction potentials, we prove that the probability of having $n$ particles outside the condensate decays exponentially in $n$ for any finite evolution time. Our results strengthen previously known bounds that provide only polynomial control on the probability of having $n$ excitations.

We consider the phenomenon of string breaking in the context of the spatial Wilson loops using the gauge/string duality. In particular, we discuss the impact of light flavors on the pseudopotential. We also introduce the notion of the spatial string breaking distance and estimate it for $SU(3)$ gauge theory in the temperature range $0\,\text{-}\,3\,T_c$.

We consider the amplification of bosonic interactions through parametric control that implements squeezing along orthogonal quadratures. We show that bosonic interactions described by certain classes of quadratic and quartic Hamiltonians can be enhanced in this way while simultaneously overcoming noise and decoherence. In general, the amplification method enhances both desired and undesired interactions present in the system. Depending on the case, however, detrimental processes can be less amplified than the desired couplings. We leverage this observation to improve the fidelity for preparing Bell-type entangled states between two bosonic modes in the presence of noise and losses. We also investigate noise models for which the protocol either fails or partially achieves a loss-tolerant state preparation speedup. Our work facilitates faster preparation of complex quantum states and implementation of entangling gates in the presence of decoherence mechanisms.

We extend Stein's maximal theorem to the bilinear setting. Let $M$ be a homogeneous space with a transitive action of a compact abelian group, and let $1 \le p,q \le 2$ and $1/2 \le r \le 1$ satisfy $1/p + 1/q = 1/r$. For a family of translation-invariant bilinear operators \[ T_m : L^p(M) \times L^q(M) \to L^r(M), \qquad m \in \mathbb{N}, \] that converge almost everywhere, we prove that the associated maximal operator \[ T^*(f,g) = \sup_m |T_m(f,g)| \] is of weak type $L^p(M) \times L^q(M) \to L^{r,\infty}(M)$. The proof relies on probabilistic methods and a bilinear extension of Stein's lemma for double Rademacher series. We also establish a bilinear analogue of Sawyer's extension of Stein's theorem for positive bilinear operators commuting with a mixing family of measure-preserving transformations. Applications include strong-type boundedness of maximal bilinear tail operators associated with ergodic transformations in the natural exponent range $r = (1/p + 1/q)^{-1}$ for $p,q > 1$, as well as almost everywhere convergence results for bilinear Bochner--Riesz means and other bilinear ergodic averages on the torus.

Turbulence in the magnetized plasma is well understood to be the consequence of wave interactions. When the Hall effect is added to the minimum magnetohydrodynamics (MHD), the MHD waves become dispersive and different nonlinear interactions are expected. The emergent turbulent state will thus be expected to be different. For incompressible Hall MHD we develop a reduced model for wave-wave interactions concentrating on those processes that will lead to phase coherent modifications to the linear dispersion of a given wave. We show that these special interactions provide an amplitude-dependent contribution to the linear dispersion relation, which yields nonlinear frequency shifts. The resonance-driven frequency shifts are dominant and add damping or growth to the linear dispersion. The damping/growth rates represent the nonlinear time scales for energy redistribution and can be used in conjunction with a conjecture like the "critical balance" to estimate the energy spectral content.

The redshifted 21-cm signal emitted by neutral Hydrogen (HI) is a promising probe to understand the evolution of the topology of ionized regions during the Epoch of Reionization (EoR). The topology of ionized regions allows us to infer the nature and properties of ionizing sources, i.e., early galaxies and AGNs. Traditional Fourier statistics, such as the power spectrum, help us quantify the strength of fluctuations in this field at different length scales but do not preserve its phase information. Analyzing the 21-cm brightness temperature field in the image domain retains its non-Gaussian characteristics and morphological information. One such approach is to track the coalescence of multiple ionized regions to form one contiguous ionized region spanning the universe. This is referred to as percolation, and its onset is quantified by a sharp rise in the value of the Largest Cluster Statistic (LCS) approaching unity. In this work, we carry out a percolation analysis of 21-cm brightness temperature fields by studying the redshift evolution of the LCS along a lightcone to distinguish between several simulated reionization scenarios. We have extended previous results on reionization model comparison from the analysis of coeval 21-cm maps to understand how the lightcone effect biases the observed percolation behavior and affects the distinguishability of the source models. We estimate the LCS of subvolumes of different sizes in the 21-cm lightcone maps and study their redshift evolution for different reionization scenarios using a moving volume approach. We find that the percolation transition inferred from a lightcone approaches that from the coeval box as we increase the bandwidth of the moving volume in all but one reionization scenario.

Fluid molecular ferroelectrics are a new class of organic materials where ferroelectricity is found in conjunction with 3D fluidity whilst still retaining spontaneous polarization values comparable to their traditional solid state counterparts. One of the major challenges for soft condensed matter physics is predicting whether a fluid molecular material will form ferroelectric phase with nematic or smectic order. Through the synthesis of forty five systematically varied molecules, and by analogy to solid molecular ferroelectrics, is it shown that subtle hydrogen fluorine substitution allows for tuneable syn-parallel pairing motifs resulting in either specific pairings leading too geometrically constrained lamellar order or diversified pairings stabilising nematic ordering. Large-scale, fully atomistic molecular dynamics simulations reveal that smectic ferroelectricity emerges from discrete lateral pairing modes, whereas nematic phases arise from a multiplicity of equivalent polar configurations. Together, these findings establish experimentally validated design principles for fluid molecular ferroelectrics and provide a predictive framework for engineering functional polar fluids.

We propose that the superconducting diode (SD) efficiency can be significantly enhanced near the transition between two superconducting states by choosing parameters where, before the system goes normal with increasing supercurrent, it switches into a different superconducting order for one direction of the current but not for the other. This mechanism for producing high SD efficiency relies on the expectation that the critical current depends sensitively on the superconducting order. We demonstrate this explicitly by performing detailed calculations for a bilayer superconductor with an in-plane magnetic field, which admits the standard Bardeen-Cooper-Schrieffer (BCS) and the orbital Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) orders as a function of the strength of the magnetic field. We predict a sharp peak in the SD efficiency in the FFLO state close to the transition, which arises from a complex interplay between the two superconducting orders. An implication of our study is that the measurement of the SD efficiency can provide fundamental insight into the nature of the BCS-FFLO transition both as a function of the magnetic field and the supercurrent.

Context. Characterization of warm giants is crucial to constrain giant planet formation and evolution. Measuring the mass and radius of these planets, combined with their moderated irradiation, allows us to estimate their planetary bulk composition, which is a key quantity to comprehend giant planet formation and structure. Aims. We present the discovery of two transiting warm giant planets orbiting solar-type stars from the Transiting Exoplanet Survey Satellite (TESS), which were characterized by further spectroscopic and photometric ground-based observations. Methods. We performed a joint analysis of photometric data with radial velocities to confirm and characterize TOI-883 b and TOI-899 b, two sub-Saturns orbiting solar-like stars. Results. TOI-883 b and TOI-899 b have masses of $0.123 \pm 0.012$ $M_J$ and $0.213 \pm 0.024$ $M_J$, radius of $0.604 \pm 0.028$ $R_J$ and $0.991 \pm 0.044$ $R_J$, periods of $10.06$ d and $12.85$ d and equilibrium temperature of $1086 \pm 19$ K and $1040 \pm 19$ K, respectively. Conclusions. While having similar masses, orbital periods and stellar host properties, these planets seem to have different internal compositions, which could point to distinct formation histories. Both planets are suitable targets for atmospheric studies to further constrain formation scenarios of planets in the Neptune-Saturn mass range

Most cold atoms experiments in microgravity platforms or in Space are achieved using atom chips, leading to limitations in terms of optical access and inhomogeneous magnetic fields. Optical dipole traps do not have these drawbacks but have difficulties producing atomic samples with a large number of atoms at ultra low temperature in the absence of gravity. Here, we report on an efficient evaporative cooling in two-crossed laser beams during parabolic flights. Time-averaged potentials combine the advantages of large capture volume and trap compression, increasing the initial phase space density and collision rate to favor the evaporative process. With this technique we demonstrate the production of an ultra cold gas of $2.5\times 10^4$ rubidium atoms at a temperature below 100 nK in less than 4 seconds. Our experiment paves the way for the development of quantum sensors applied to fundamental physics and geodesy as well as the study of ultracold atomic physics in Space.

We outline an extension of paraproduct decompositions for compositions of the form $A(f)$ where $A \in C^{d}(\mathbb{R}), f \in Λ_α([0,1]^d)$ developed in [arXiv:2503.12629] and [arXiv:2508.13322] to settings where $(A \in C^1(\mathbb{R}),f \in Λ_α(X))$ and $ (A \in C^2(\mathbb{R}),f \in Λ_α(X \times Y))$. To do so, we construct partition trees on $X$ and $X \times Y$ such that analysis with respect to scale is sensible. We obtain results resembling those of [arXiv:2503.12629] and [arXiv:2508.13322], but with the finite sets $X$ and $X \times Y $ as support. In particular we construct the paraproduct $Π_{A',A''}^{L,S}: f \to \tilde{A}_{L,S}(f) + Δ_{L,S}(A,f)$ such that $Δ_{L,S}(A,f) \in Λ_{2α}(X \times Y)$ and $\lVert Δ_{L,S}(A,f) \rVert_{Λ_{2α}(X \times Y)} \leq C_A \lVert f \rVert_{Λ_α(X \times Y)}$. Analogous results are obtained when the support is just one finite set, $X$. This extension is motivated by situations where one wishes to separate the singular and smooth components of such compositions in graph signal processing environments.

We present a randomized algorithm for the single-source shortest paths (SSSP) problem on directed graphs with arbitrary real-valued edge weights that runs in $n^{2+o(1)}$ time with high probability. This result yields the first almost linear-time algorithm for the problem on dense graphs ($m = Θ(n^2)$) and improves upon the best previously known bounds for moderately dense graphs ($m = ω(n^{1.306})$). Our approach builds on the hop-reduction via shortcutting framework introduced by Li, Li, Rao, and Zhang (2025), which iteratively augments the graph with shortcut edges to reduce the negative hop count of shortest paths. The central computational bottleneck in prior work is the cost of explicitly constructing these shortcuts in dense regions. We overcome this by introducing a new compression technique using auxiliary Steiner vertices. Specifically, we construct these vertices to represent large neighborhoods compactly in a structured manner, allowing us to efficiently generate and propagate shortcuts while strictly controlling the growth of vertex degrees and graph size.

We present the RIS-VSign system, an active reconfigurable intelligent surface (RIS)-assisted multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) framework for vital signs extraction under an integrated sensing and communication (ISAC) model. The system consists of two stages: the phase selector of RIS and the extraction of respiration rate. To mitigate synchronization-induced common phase drifts, the difference of Möbius transformation (DMT) is integrated into the deep learning framework, named DMTNet, to jointly configure multiple active RIS elements. Notably, the training data are generated in simulation without collecting real-world measurements, and the resulting phase selector is validated experimentally. For sensing, multi-antenna measurements are fused by the DC-offset calibration and the DeepMining-MMV processing with CA-CFAR detection and Newton's refinements. Prototype experiments indicate that active RIS deployment improves respiration detectability while simultaneously enabling higher-order modulation; without RIS, respiration detection is unreliable and only lower-order modulation is supported.

Understanding mechanisms of ion transport in bulk materials is central to designing next-generation ion conductors for energy storage devices, yet studies employing all-atom molecular dynamics (MD) have largely been limited to reporting overall transport coefficients without a quantitative, spatiotemporally resolved breakdown of \emph{how} charge is carried. We present a computational framework that analyzes MD trajectories to quantitatively interpret macroscopic transport by decomposing it into additive contributions from physically motivated events. They are defined either through heuristically identified microscopic transitions, capturing events such as single-ion hops, multi-ion hops, and vehicular motion, or through transitions between chemically interpretable coordination macrostates. The construction guarantees that attributed contributions sum exactly to the Onsager transport coefficients estimated via the Green-Kubo/Einstein formalism, while scanning the sampling window exposes characteristic temporal scales at which distinct transport mechanisms emerge and dominate. Applied across three prototypical electrolytes-inorganic crystals, liquids, and polymers-the framework quantitatively resolves long-standing debates (e.g., the role of concerted motion and exchange), identifies dominant mechanisms and rate-limiting steps, quantifies their frequencies and effectiveness, and extracts activation energies for distinct transport modes, thereby distilling design rules for fast conduction. This general and reproducible analysis tool turns MD trajectories into quantitative mechanism maps, enabling the ion-conductor community to adjudicate mechanistic hypotheses and accelerate discovery.

Background: Human behavior shapes infectious disease dynamics, yet its integration into transmission models remains fragmented. Recent epidemics, particularly COVID-19, highlight the need for models capturing adaptation to perceived risk, social influence, and policy signals. This review synthesizes post-2020 models incorporating behavioral adaptation, examines their theoretical grounding, and evaluates how behavioral constructs modify transmission, vaccination, and compliance. Methods: Following PRISMA guidelines, we searched Scopus and PubMed (2020-2025), screening 1,274 records with citation chaining. We extracted data on disease context, country, modeling framework, behavioral mechanisms (prevalence-dependent, policy/media, imitation/social learning), and psychosocial constructs (personal threat, coping appraisal, barriers, social norms, cues to action). A total of 216 studies met inclusion criteria. Results: COVID-19 accounted for 73% of studies. Most used compartmental ODE models (81%) and focused on theoretical or U.S. settings. Behavioral change was mainly reactive: 47% applied prevalence-dependent feedback, 25% included awareness/media dynamics, and 19% relied on exogenous policy triggers. Game-theoretic or social learning approaches were rare (less or equal than 5%). Behavioral effects primarily modified contact or transmission rates (91%). Psychosocial constructs were unevenly represented: cues to action (n=159) and personal threat (n=145) dominated, whereas coping appraisal (n=82), barriers (n=36), and social norms (n=25) were less common. Conclusions: We propose a taxonomy structured by behavioral drivers, social scale, and memory to clarify dominant paradigms and their empirical basis. Mapping models to psychosocial constructs provides guidance for more theory-informed and data grounded-integration of behavioral processes in epidemiological modeling.

Twinning is a primary deformation mechanism in Mg alloys. This study focuses on tension twins during uniaxial compression of Mg-Y alloys, with three key aspects: the orientation specificity of twin grains, the relative evolution of CRSS with increasing Y content, and the local stress and strain evolution at twin sites. Experimental characterization and crystal plasticity modeling were performed. In Mg-7wt.%Y, TT2-{112-1} tension twins were observed in addition to the common TT1-{101-2} twins. Increasing Y suppressed TT1 formation while promoting TT2 activity. A previously unreported group of crystallographic orientations with a higher global Schmid factor for <c+a> slip was identified, which exhibited TT1 twinning with increasing compression strain. To elucidate Y effects on twin activity and local mechanical fields, both TT1 and TT2 tension twin modes were incorporated into PRISMS-Plasticity, an open-source, finite element-based crystal plasticity solver. Four binary Mg-Y alloys were modeled under compression, and statistical analysis was conducted to correlate initial orientations, stress-strain distributions, and twin activities as functions of Y concentration. The plasticity analysis revealed that increasing Y decreases the CRSS ratio of prismatic and pyramidal slip relative to TT1 twinning, while the slip-to-twin CRSS ratio for TT2 increases, thereby serving as a potential indicator of differential twin activity with Y addition in Mg alloys. Additionally, despite their small volume fraction, TT2 twin sites were predicted higher local strain accumulation locally, relative to the representative volume element and TT1 twins, suggesting their potential influence on localized phenomena such as recrystallization or twin nucleation. These findings provide insight into local mechanical behavior in Mg alloys and support alloy design for advanced engineering applications.

In this paper, we investigate the symmetry properties of positive solutions $u$ to a semilinear elliptic equation under mixed Dirichlet-Neumann boundary conditions in symmetric domains. First, we establish a maximum principle tailored to mixed-boundary problems in domains of either small volume or narrow width, thereby enabling the application of the moving plane method. Secondly, in contrast to the purely Dirichlet case, a key challenge is to establish the non-vanishing of the tangential derivative of $u$ along the Neumann boundary. To address this, we employ local analysis techniques of angular derivatives, as introduced by Hartman and Wintner [Amer. J. Math., 1953]. Thirdly, we identify the signs of directional derivatives of $u$ along sections of the moving line. Using a planar sub-spherical sector as an example, we illustrate how these new innovative techniques and the moving plane method can be combined to derive symmetry and monotonicity results, particularly when the amplitude is less than or equal to $2π/3$.

Cerium hydride has a variety of interesting properties, including a known lattice contraction and densification with increasing hydrogen content. However, precise stoichiometric control is not experimentally straightforward and {\it ab initio} approaches are not computationally feasible for many properties such as melting and low temperature diffusion. Therefore, we develop a machine-learned interatomic potential for cerium hydride that is valid for H to Ce ratios from 2.0 to 3.0. A query-by-committee active learning approach is used to develop the training set. Leveraging classical molecular dynamics simulations, we assess a range of properties and provide fundamental mechanisms for the trends with stoichiometry. A majority of the properties follow the trend of lattice contraction, being governed by the stronger lattice binding induced by adding octahedral atoms.

We develop the theoretical model that describes dynamic non-equilibrium effects of external inertial and axion fields in a system of particles with spin. The possibility of using the spin density and the current density of non-relativistic quantum particle systems for the detection of the hypothetical axion-like dark matter is discussed. The resulting closed system of dynamic equations encompasses the continuity equation, the momentum balance equation, and the spin density evolution equation, accounting for the influence of the spin-rotation coupling and the external axion fields. The new formalism opens up new perspectives for an experimental search of dark matter axions.

The (strong) Bruhat order for permutations provides a partial ordering defined as follows: two permutations are comparable if one can be obtained from the other by a sequence of adjacent transpositions that each increase the number of inversions by $1$. Given two random permutations, what is the probability that they are comparable in the Bruhat order? This problem was first considered in a 2006 work of Hammett and Pittel, which showed an exponential lower bound and a polynomial upper bound. The lower bound was very recently improved to the subexponential bound of $\exp(-n^{1/2 + o(1)})$ by Boretsky, Cornejo, Hodges, Horn, Lesnevich, and McAllister. Hammett and Pittel predicted that the probability should decrease polynomially. We show that the probability decreases faster than any polynomial and is on the order of $\exp(-Θ(\log^2 n))$.

The self-modulation (SM) instability transforms a long charged particle bunch traveling in plasma into a train of microbunches that resonantly drives large-amplitude wakefields. We present the first determination of the saturation length of SM using experimental and numerical results. The saturation length is the distance over which wakefields reach their maximum amplitude along the plasma. By varying the plasma length and measuring the radius of the transverse distribution of the bunch, we find that the saturation length of SM decreases with plasma density and initial field amplitude, e.g., when seeding. The saturation length is a fundamental parameter of the instability, and these results are key for understanding SM and designing plasma wakefield accelerators driven by long bunches, such as AWAKE, or by long laser pulses for radiation production.

Particle-stabilised emulsions are a cornerstone of soft matter science due to their broad application and fundamental relevance. Computer simulations provide key insights into the formation and behaviour of these emulsions, yet current methods are limited by the spatiotemporal scales accessible for study. The principal issue is that particles are resolved individually. In this work, an alternative strategy is introduced based on phase-field theory, for which we establish the framework. By evolving continuous fields, large-scale dynamics can be simulated in a computationally efficient manner. Our approach is then applied to model the complex formation of a bicontinuous interfacially jammed emulsion gel (bijel) via solvent-transfer induced phase separation (STrIPS). By resolving the coupled dynamics of liquid phase separation and nanoparticle adsorption, the model allows for the characterisation of the influence of nanoparticles on the morphology. Higher concentrations of nanoparticles are found to reduce the average domain size of STrIPS bijels, in line with previous experimental evidence. The presented phase-field model thus represents a promising approach for the morphological investigation of complex particle-stabilised emulsions.