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We focus on the derivation and analysis of a model for multi-component phase separation occurring on biological membranes, inspired by observations of lipid raft formation. The model integrates local membrane composition with local membrane curvature, describing the membrane's geometry through a perturbation method represented as a graph over an undeformed Helfrich minimising surface, such as a sphere. The resulting energy consists of a small deformation functional coupled to a Cahn-Hilliard functional. By applying Onsager's variational principle, we obtain a multi-component Cahn-Hilliard equation for the vector $\boldsymbol\varphi$ of protein concentrations coupled to an evolution equation for the small deformation $u$ along the normal direction to the reference membrane. Then, in the case of a constant mobility matrix, we consider the Cauchy problem and we prove that it is (globally) well posed in a weak setting. We also demonstrate that any weak solution regularises in finite time and satisfies the so-called "strict separation property". This property allows usto show that any weak solution converges to a single stationary state by a suitable version of the Lojasiewicz-Simon inequality.

Quantum networks will provide unconditional security for communication, computation and distributed sensing tasks. We report on an experimental demonstration of private parameter estimation, which allows a global phase to be evaluated without revealing the constituent local phase values. This is achieved by sharing a Greenberger-Horne-Zeilinger (GHZ) state among three users who first verify the shared state before performing the sensing task. We implement the verification protocol, based on stabilizer measurements, and measure an average failure rate of 0.038(5) which we use to establish the security and privacy parameters. We validate the privacy conditions established by the protocol by evaluating the quantum Fisher information of the experimentally prepared GHZ states., Comment: 14 pages, 8 figures

As an introduction to a Special Issue of International Journal of Theoretical and Applied Finance in Honour of the Memory of Thomas Robert Hurd we present a brief synopsis of Tom Hurd's scientific career and a bibliography of his scientific publications., Comment: 9 pages, 1 photograph, to appear in International Journal of Theoretical and Applied Finance (2024) Vol. 27 (3,4), Special Issue in Honour of the Memory of Thomas Robert Hurd (1956 - 2022), edited by T. R. Bielecki, M. R. Grasselli and L. P. Hughston

The widespread application of machine learning (ML) to the chemical sciences is making it very important to understand how the ML models learn to correlate chemical structures with their properties, and what can be done to improve the training efficiency whilst guaranteeing interpretability and transferability. In this work, we demonstrate the wide utility of prediction rigidities, a family of metrics derived from the loss function, in understanding the robustness of ML model predictions. We show that the prediction rigidities allow the assessment of the model not only at the global level, but also on the local or the component-wise level at which the intermediate (e.g. atomic, body-ordered, or range-separated) predictions are made. We leverage these metrics to understand the learning behavior of different ML models, and to guide efficient dataset construction for model training. We finally implement the formalism for a ML model targeting a coarse-grained system to demonstrate the applicability of the prediction rigidities to an even broader class of atomistic modeling problems., Comment: 15 pages, 9 figures, submitted to Faraday Discussions: "Data-driven discovery in the chemical sciences", 2024

The electronic density of states (DOS) provides information regarding the distribution of electronic states in a material, and can be used to approximate its optical and electronic properties and therefore guide computational material design. Given its usefulness and relative simplicity, it has been one of the first electronic properties used as target for machine-learning approaches going beyond interatomic potentials. A subtle but important point, well-appreciated in the condensed matter community but usually overlooked in the construction of data-driven models, is that for bulk configurations the absolute energy reference of single-particle energy levels is ill-defined. Only energy differences matter, and quantities derived from the DOS are typically independent on the absolute alignment. We introduce an adaptive scheme that optimizes the energy reference of each structure as part of training, and show that it consistently improves the quality of ML models compared to traditional choices of energy reference, for different classes of materials and different model architectures. On a practical level, we trace the improved performance to the ability of this self-aligning scheme to match the most prominent features in the DOS. More broadly, we believe that this work highlights the importance of incorporating insights into the nature of the physical target into the definition of the architecture and of the appropriate figures of merit for machine-learning models, that translate in better transferability and overall performance.

We study the number $N_{<0}(H_s)$ of negative eigenvalues, counting multiplicities, of the fractional Schr\"odinger operator $H_s=(-\Delta)^s-V(x)$ on $L^2(\mathbb{R}^d)$, for any $d\ge1$ and $s\ge d/2$. We prove a bound on $N_{<0}(H_s)$ which depends on $s-d/2$ being either an integer or not, the critical case $s=d/2$ requiring a further analysis. Our proof relies on a splitting of the Birman-Schwinger operator associated to this spectral problem into low- and high-energies parts, a projection of the low-energies part onto a suitable subspace, and, in the critical case $s=d/2$, a Cwikel-type estimate in the weak trace ideal $\mathcal{L}^{2,\infty}$ to handle the high-energies part.

We consider local and nonlocal Cahn-Hilliard equations with constant mobility and singular potentials including, e.g., the Flory-Huggins potential, subject to no-flux (or periodic) boundary conditions. The main goal is to show that the presence of a suitable class of reaction terms allows to establish the existence of a weak solution to the corresponding initial and boundary value problem even though the initial condition is a pure state. In other words, the separation process takes place even in presence of a pure phase, provided that it is triggered by a convenient reaction term. This fact was already observed by the authors in a previous contribution devoted to a specific biological model. In this context, we generalize the previously-mentioned concept by examining the essential assumptions required for the reaction term to apply the new strategy. Also, we explore the scenario involving the nonlocal Cahn-Hilliard equation and provide illustrative examples that contextualize within our abstract framework., Comment: 32 pages

We consider an evolution system modeling a flow of colloidal particles which are suspended in an incompressible fluid and accounts for colloidal crystallization. The system consists of the Navier-Stokes equations for the volume averaged velocity coupled with the so-called Phase-Field Crystal equation for the density deviation. Considering this system in a periodic domain and assuming that the viscosity as well as the mobility depend on the density deviation, we first prove the existence of a weak solution in dimension three. Then, in dimension two, we establish the existence of a (unique) strong solution.

We consider the Schr{\"o}dinger operator --$\Delta$ + V on the Euclidean space with potential in the Lorentz space L^{n/2,1} and we find necessary and sufficient conditions for zero to be a resonance or an eigenvalue. We consider functions with gradient in L^2 and that verify the equation (--$\Delta$ + V)$\psi$ = 0, namely the kernel of (--$\Delta$ + V) in the homogeneous Sobolev space of order one. We prove that a function in this set is either in a weak Lebesgue space or in L^2 , in the latter case we have a zero eigenfunction. The set of eigenfunctions is the hyperplane of functions that are orthogonal to V, furthermore we show that under some classic orthogonality conditions a zero eigenfunction belongs to the weak Lebesgue space of order one or to L^1. We study dimensions n $\ge$ 3 and in dimension three we generalize a result proved by Beceanu.

Regression methods are fundamental for scientific and technological applications. However, fitted models can be highly unreliable outside of their training domain, and hence the quantification of their uncertainty is crucial in many of their applications. Based on the solution of a constrained optimization problem, we propose "prediction rigidities" as a method to obtain uncertainties of arbitrary pre-trained regressors. We establish a strong connection between our framework and Bayesian inference, and we develop a last-layer approximation that allows the new method to be applied to neural networks. This extension affords cheap uncertainties without any modification to the neural network itself or its training procedure. We show the effectiveness of our method on a wide range of regression tasks, ranging from simple toy models to applications in chemistry and meteorology.

Accessing the thermal transport properties of glasses is a major issue for the design of production strategies of glass industry, as well as for the plethora of applications and devices where glasses are employed. From the computational standpoint, the chemical and morphological complexity of glasses calls for atomistic simulations where the interatomic potentials are able to capture the variety of local environments, composition, and (dis)order that typically characterize glassy phases. Machine-learning potentials (MLPs) are emerging as a valid alternative to computationally expensive ab initio simulations, inevitably run on very small samples which cannot account for disorder at different scales, as well as to empirical force fields, fast but often reliable only in a narrow portion of the thermodynamic and composition phase diagrams. In this article, we make the point on the use of MLPs to compute the thermal conductivity of glasses, through a review of recent theoretical and computational tools and a series of numerical applications on vitreous silica and vitreous silicon, both pure and intercalated with lithium., Comment: 11 pages, 6 figures. Supplementary Material as ancillary file

The vast amount of computational studies on electrical conduction in solid-state electrolytes is not mirrored by comparable efforts addressing thermal conduction, which has been scarcely investigated despite its relevance to thermal management and (over)heating of batteries. The reason for this lies in the complexity of the calculations: on one hand, the diffusion of ionic charge carriers makes lattice methods formally unsuitable, due to the lack of equilibrium atomic positions needed for normal-mode expansion. On the other hand, the prohibitive cost of large-scale molecular dynamics (MD) simulations of heat transport in large systems at ab initio levels has hindered the use of MD-based methods. In this paper, we leverage recently developed machine-learning potentials targeting different ab initio functionals (PBEsol, r$^2$SCAN, PBE0) and a state-of-the-art formulation of the Green-Kubo theory of heat transport in multicomponent systems to compute the thermal conductivity of a promising solid-state electrolyte, Li$_3$PS$_4$, in all its polymorphs ($\alpha$, $\beta$, and $\gamma$). By comparing MD estimates with lattice methods on the low-temperature, nondiffusive $\gamma$-Li$_3$PS$_4$, we highlight strong anharmonicities and negligible nuclear quantum effects, hence further justifying MD-based methods even for nondiffusive phases. Finally, for the ion-conducting $\alpha$ and $\beta$ phases, where the multicomponent Green-Kubo MD approach is mandatory, our simulations indicate a weak temperature dependence of the thermal conductivity, a glass-like behavior due to the effective local disorder characterizing these Li-diffusing phases., Comment: 11 pages, 5 figures

Anonymous quantum conference key agreement (AQCKA) allows a group of users within a network to establish a shared cryptographic key without revealing their participation. Although this can be achieved using bi-partite primitives alone, it is costly in the number of network rounds required. By allowing the use of multi-partite entanglement, there is a substantial efficiency improvement. We experimentally implement the AQCKA task in a six-user quantum network using Greenberger-Horne-Zeilinger (GHZ)-state entanglement and obtain a significant resource cost reduction in line with theory when compared to a bi-partite-only approach. We also demonstrate that the protocol retains an advantage in a four-user scenario with finite key effects taken into account.

We develop methods for the solution of inhomogeneous Robin type boundary value problems (BVPs) that arise for certain linear parabolic Partial Differential Equations (PDEs) on a half line, as well as a second order generalisation. We are able to obtain non-standard solutions to equations arising in a range of areas, including mathematical finance, stochastic analysis, hyperbolic geometry and mathematical physics. Our approach uses the odd and even Hilbert transforms. The solutions we obtain and the method itself seem to be new.

Lithium ortho-thiophosphate (Li$_3$PS$_4$) has emerged as a promising candidate for solid-state-electrolyte batteries, thanks to its highly conductive phases, cheap components, and large electrochemical stability range. Nonetheless, the microscopic mechanisms of Li-ion transport in Li$_3$PS$_4$ are far to be fully understood, the role of PS$_4$ dynamics in charge transport still being controversial. In this work, we build machine learning potentials targeting state-of-the-art DFT references (PBEsol, r$^2$SCAN, and PBE0) to tackle this problem in all known phases of Li$_3$PS$_4$ ($\alpha$, $\beta$ and $\gamma$), for large system sizes and timescales. We discuss the physical origin of the observed superionic behavior of Li$_3$PS$_4$: the activation of PS$_4$ flipping drives a structural transition to a highly conductive phase, characterized by an increase of Li-site availability and by a drastic reduction in the activation energy of Li-ion diffusion. We also rule out any paddle-wheel effects of PS$_4$ tetrahedra in the superionic phases -- previously claimed to enhance Li-ion diffusion -- due to the orders-of-magnitude difference between the rate of PS$_4$ flips and Li-ion hops at all temperatures below melting. We finally elucidate the role of inter-ionic dynamical correlations in charge transport, by highlighting the failure of the Nernst-Einstein approximation to estimate the electrical conductivity. Our results show a strong dependence on the target DFT reference, with PBE0 yielding the best quantitative agreement with experimental measurements not only for the electronic band-gap but also for the electrical conductivity of $\beta$- and $\alpha$-Li$_3$PS$_4$., Comment: Main: 18 pages, 9 figures; Supporting Information: 13 pages, 13 figures

We propose an optimal iterative scheme for federated transfer learning, where a central planner has access to datasets ${\cal D}_1,\dots,{\cal D}_N$ for the same learning model $f_{\theta}$. Our objective is to minimize the cumulative deviation of the generated parameters $\{\theta_i(t)\}_{t=0}^T$ across all $T$ iterations from the specialized parameters $\theta^\star_{1},\ldots,\theta^\star_N$ obtained for each dataset, while respecting the loss function for the model $f_{\theta(T)}$ produced by the algorithm upon halting. We only allow for continual communication between each of the specialized models (nodes/agents) and the central planner (server), at each iteration (round). For the case where the model $f_{\theta}$ is a finite-rank kernel regression, we derive explicit updates for the regret-optimal algorithm. By leveraging symmetries within the regret-optimal algorithm, we further develop a nearly regret-optimal heuristic that runs with $\mathcal{O}(Np^2)$ fewer elementary operations, where $p$ is the dimension of the parameter space. Additionally, we investigate the adversarial robustness of the regret-optimal algorithm showing that an adversary which perturbs $q$ training pairs by at-most $\varepsilon>0$, across all training sets, cannot reduce the regret-optimal algorithm's regret by more than $\mathcal{O}(\varepsilon q \bar{N}^{1/2})$, where $\bar{N}$ is the aggregate number of training pairs. To validate our theoretical findings, we conduct numerical experiments in the context of American option pricing, utilizing a randomly generated finite-rank kernel., Comment: 51 pages, 2 figures

We investigate the Cahn-Hilliard and the conserved Allen-Cahn equations with logarithmic type potential and conservative noise in a periodic domain. These features ensure that the order parameter takes its values in the physical range and, albeit the stochastic nature of the problems, that the total mass is conserved almost surely in time. For the Cahn-Hilliard equation, existence and uniqueness of probabilistically-strong solutions is shown up to the three-dimensional case. For the conserved Allen-Cahn equation, under a restriction on the noise magnitude, existence of martingale solutions is proved even in dimension three, while existence and uniqueness of probabilistically-strong solutions holds in dimension one. The analysis is carried out by studying the Cahn-Hilliard/conserved Allen-Cahn equations jointly, that is a linear combination of both the equations, which has an independent interest., Comment: 44 pages

We study a Cahn-Hilliard-Darcy system with mass sources, which can be considered as a basic, though simplified, diffuse interface model for the evolution of tumor growth. This system is equipped with an impermeability condition for the (volume) averaged velocity $\mathbf{u}$ as well as homogeneous Neumann boundary conditions for the phase function $\varphi$ and the chemical potential $\mu$. The source term in the convective Cahn-Hilliard equation contains a control $R$ that can be thought, for instance, as a drug or a nutrient in applications. Our goal is to study a distributed optimal control problem in the two dimensional setting with a cost functional of tracking-type. In the physically relevant case with unmatched viscosities for the binary fluid mixtures and a singular potential, we first prove the existence and uniqueness of a global strong solution with $\varphi$ being strictly separated from the pure phases $\pm 1$. This well-posedness result enables us to characterize the control-to-state mapping $\mathcal{S}:R \mapsto \varphi$. Then we obtain the existence of an optimal control, the Fr\'{e}chet differentiability of $\mathcal{S}$ and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we prove the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality., Comment: Minor revision has been made. This is a full length preprint version with detailed computations

Machine learning (ML) models for molecules and materials commonly rely on a decomposition of the global target quantity into local, atom-centered contributions. This approach is convenient from a computational perspective, enabling large-scale ML-driven simulations with a linear-scaling cost, and also allow for the identification and post-hoc interpretation of contributions from individual chemical environments and motifs to complicated macroscopic properties. However, even though there exist practical justifications for these decompositions, only the global quantity is rigorously defined, and thus it is unclear to what extent the atomistic terms predicted by the model can be trusted. Here, we introduce a quantitative metric, which we call the local prediction rigidity (LPR), that allows one to assess how robust the locally decomposed predictions of ML models are. We investigate the dependence of LPR on the aspects of model training, particularly the composition of training dataset, for a range of different problems from simple toy models to real chemical systems. We present strategies to systematically enhance the LPR, which can be used to improve the robustness, interpretability, and transferability of atomistic ML models.

We propose a new theoretical framework that exploits convolution kernels to transform a Volterra path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. This transformation is achieved by embedding a Markovian "memory process" within the dynamics of the non-Markovian process. We discuss existence and path-wise regularity of solutions for the stochastic Volterra equations introduced and we provide a financial application to volatility modeling. We also propose a numerical scheme for simulating the processes. The numerical scheme exhibits a strong convergence rate of 1/2, which is independent of the roughness parameter of the volatility process. This is a significant improvement compared to Euler schemes used in similar models. We propose a new theoretical framework that exploits convolution kernels to transform a Volterra path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. This transformation is achieved by embedding a Markovian "memory process" (the goldfish) within the dynamics of the non-Markovian process (the elephant). Most notably, it is also possible to go back, i.e., the transformation is reversible. We discuss existence and path-wise regularity of solutions for the stochastic Volterra equations introduced and we propose a numerical scheme for simulating the processes, which exhibits a remarkable convergence rate of $1/2$. In particular, in the fractional kernel case, the strong convergence rate is independent of the roughness parameter, which is a positive novelty in contrast with what happens in the available Euler schemes in the literature in rough volatility models.

The dynamics of (few) electrons dissolved in an ionic fluid--as when a small amount of metal is added to a solution while upholding its electronic insulation--manifests interesting properties that can be ascribed to nontrivial topological features of particle transport (e.g., Thouless' pumps). In the adiabatic regime, the charge distribution and the dynamics of these dissolved electrons are uniquely determined by the nuclear configuration. Yet, their localization into effective potential wells and their diffusivity are dictated by how the self-interaction is modeled. In this article, we investigate the role of self-interaction in the description of localization and transport properties of dissolved electrons in non-stoichiometric molten salts. Although the account for the exact (Fock) exchange strongly localizes the dissolved electrons, decreasing their tunneling probability and diffusivity, we show that the dynamics of the ions and of the dissolved electrons are largely uncorrelated, irrespective of the degree to which the electron self-interaction is treated, and in accordance with topological arguments., Comment: Accepted version. 7 pages, 6 figures

We consider a multi-component version of the conserved Allen-Cahn equation proposed by J. Rubinstein and P. Sternberg in 1992 as an alternative model for phase separation. In our case, the free energy is characterized by a mixing entropy density which belongs to a large class of physically relevant entropies like, e.g., the Boltzmann-Gibbs entropy. We establish the well-posedness of the Cauchy-Neumann problem with respect to a natural notion of (finite) energy solution which is more regular under appropriate assumptions and is strictly separated from pure phases if the initial datum is. We then prove that the energy solution becomes more regular and strictly separated instantaneously. Also, we show that any finite energy solution converges to a unique equilibrium. The validity of a dissipative inequality (identity for strong solutions) allows us to analyze the problem within the theory of infinite-dimensional dissipative dynamical systems. On account of the obtained results, we can associate to our problem a dissipative dynamical system and we can prove that it has a global attractor as well as an exponential attractor.

We consider a phase-field model which describes the interactions between the blood flow and the thrombus. The latter is supposed to be a viscoelastic material. The potential describing the cohesive energy of the mixture is assumed to be of Flory-Huggins type (i.e. logarithmic). This ensures the boundedness from below of the dissipation energy. In the two dimensional case, we prove the local (in time) existence and uniqueness of a strong solution, provided that the two viscosities of the pure fluid phases are close enough. We also show that the order parameter remains strictly separated from the pure phases if it is so at the initial time.