Well-defined Continuum Limit Research Articles

Heterarchical modelling of comminution for rotary mills: part I—particle crushing along streamlines

Rotary mills aim to effectively reduce the size of particles through a process called comminution. Modelling comminution in rotary mills is a challenging task due to substantial material deformation and the intricate interplay of particle kinematics of segregation, mixing, crushing, and abrasion. Existing particle-based simulations tend to provide predictions that cannot cope with the large number of particles within rotary mills, their wide range of sizes, and the physics dictating the crushing of individual particles. Similarly, there is currently no deterministic modelling means to determine the evolving population of particle sizes at any point in time and space within the mill. The aim of this two-part contribution is to address these gaps by advancing a framework for a novel stochastic comminution model for rotary mills, which has a well-defined deterministic continuum limit and can cope with arbitrarily large numbers of particles. This work describes the basic physics and structure of the new model within a heterarchical framework for ball and autogenous grinding mills. The primary focus of this Part I paper is to develop a computational model for the integration of motion of material along streamlines inside a mill. Coupled to this process is the kinetic physics dictating particle crushing. In a subsequent work, Part II, segregation and mixing will be added to this model such that realistic behaviour from the mill can be observed.Graphical

Continuum Limit of the Green Function in Scaled Affine φ44 Quantum Euclidean Covariant Relativistic Field Theory

Through path integral Monte Carlo computer experiments, we prove that the affine quantization of the φ44-scaled Euclidean covariant relativistic scalar field theory is a valid quantum field theory with a well-defined continuum limit of the one- and two-point functions. Affine quantization leads to a completely satisfactory quantization of field theories in situations involving scaled behavior, leading to an unexpected term, ℏ2/φ2, which arises only in the quantum aspects.

Lattice regularizations of θ vacua: Anomalies and qubit models

Anomalies are a powerful way to gain insight into possible lattice regularizations of a quantum field theory. In this work, we argue that the continuum anomaly for a given symmetry can be matched by a manifestly symmetric, local, lattice regularization in the same spacetime dimensionality only if (i) the symmetry action is offsite, or (ii) if the continuum anomaly is reproduced exactly on the lattice. We consider lattice regularizations of a class of prototype models of QCD: the ($1+1$)-dimensional asymptotically free Grassmannian nonlinear sigma model ($\mathrm{NL}\ensuremath{\sigma}\mathrm{Ms}$) with a $\ensuremath{\theta}$ term. Using the Grassmannian $\mathrm{NL}\ensuremath{\sigma}\mathrm{Ms}$ as a case study, we provide examples of lattice regularizations in which both possibilities are realized. For possibility (i), we argue that Grassmannian $\mathrm{NL}\ensuremath{\sigma}\mathrm{Ms}$ can be obtained from $\mathrm{SU}(N)$ antiferromagnets with a well-defined continuum limit, reproducing both the infrared physics of $\ensuremath{\theta}$ vacua and the ultraviolet physics of asymptotic freedom. These results enable the application of new classical algorithms to lattice Monte Carlo studies of these quantum field theories, and provide a viable realization suited for their quantum simulation. On the other hand, we show that, perhaps surprisingly, the conventional lattice regularization of $\ensuremath{\theta}$ vacua due to Berg and L\"uscher reproduces the anomaly exactly on the lattice, providing a realization of the second possibility.

Extending the average spectrum method: Grid point sampling and density averaging

Analytic continuation of imaginary time or frequency data to the real axis is a crucial step in extracting dynamical properties from quantum Monte Carlo simulations. The average spectrum method provides an elegant solution by integrating over all non-negative spectra weighted by how well they fit the data. In a recent paper, we found that discretizing the functional integral as in Feynman's path-integrals, does not have a well-defined continuum limit. Instead, the limit depends on the discretization grid whose choice may strongly bias the results. In this paper, we demonstrate that sampling the grid points, instead of keeping them fixed, also changes the functional integral limit and rather helps to overcome the bias considerably. We provide an efficient algorithm for doing the sampling and show how the density of the grid points acts now as a default model with a significantly reduced biasing effect. The remaining bias depends mainly on the width of the grid density, so we go one step further and average also over densities of different widths. For a certain class of densities, including Gaussian and exponential ones, this width averaging can be done analytically, eliminating the need to specify this parameter without introducing any computational overhead.

Entanglement dynamics in short- and long-range harmonic oscillators

We study the time evolution of the entanglement entropy in the short and long-range coupled harmonic oscillators that have well-defined continuum limit field theories. We first introduce a method to calculate the entanglement evolution in generic coupled harmonic oscillators after quantum quench. Then we study the entanglement evolution after quantum quench in harmonic systems that the couplings decay effectively as $1/r^{d+\alpha}$ with the distance $r$. After quenching the mass from non-zero value to zero we calculate numerically the time evolution of von Neumann and R\'enyi entropies. We show that for $1<\alpha<2$ we have a linear growth of entanglement and then saturation independent of the initial state. For $0<\alpha<1$ depending on the initial state we can have logarithmic growth or just fluctuation of entanglement. We also calculate the mutual information dynamics of two separated individual harmonic oscillators. Our findings suggest that in our system there is no particular connection between having a linear growth of entanglement after quantum quench and having a maximum group velocity or generalized Lieb-Robinson bound.

Spectral sums of the Dirac-Wilson operator and their relation to the Polyakov loop

We investigate and compute spectral sums of the Wilson lattice Dirac operator for quenched $SU(3)$ gauge theory. It is demonstrated that there exist sums which serve as order parameters for the confinement-deconfinement phase transition and get their main contribution from the IR end of the spectrum. They are approximately proportional to the Polyakov loop. In contrast to earlier studied spectral sums, some of them are expected to have a well-defined continuum limit.

Symmetropy of earthquake patterns: asymmetry and rotation in a disordered seismic source

We demonstrate that the idea of symmetropy can be used for quantification of earthquake patterns. The symmetropy can be considered as a measure of asymmetry. A pattern is richer in asymmetry when the symmetropy is smaller. The specific results of its applications are obtained as follows. In a discrete model of a seismic source with self-organized criticality, the spatial patterns of earthquakes during critical states and sub-critical states are distinguished by the behaviour of the symmetropy: sub-critical patterns show that the symmetropy is approximately a constant but this has various values during critical states. The critical patterns show asymmetric property without any asymmetric force from the outside and without asymmetric intracellular rule. We show that the emergence of asymmetric patterns is a generic feature of dynamic ruptures in our model. Such a generic asymmetry results from the model which is an inherently discrete system consisting of finite-sized cells. These cells may represent geometrical disordered fault zones. We further discuss rotational motions that generate seismic rotational waves. In micromorphic continuum theory, such rotations are attributed to dynamic ruptures in disordered systems. We note that the concept of disorder in this theory is expressed by a set of finite-sized microstructures and is consistent with the concept of disorder modelled in the present study. Thus, we suggest that the spatially asymmetric patterns of earthquakes might be related to the rotational motions, because both come from dynamic ruptures in a discrete fault zone without a well-defined continuum limit.

Regge calculus from discontinuous metrics

Regge calculus is considered as a particular case of the more general system where the linklengths of any two neighbouring 4-tetrahedra do not necessarily coincide on their common face. This system is treated as that one described by metric discontinuous on the faces. In the superspace of all discontinuous metrics the Regge calculus metrics form some hypersurface defined by continuity conditions. Quantum theory of the discontinuous metric system is assumed to be fixed somehow in the form of quantum measure on (the space of functionals on) the superspace. The problem of reducing this measure to the Regge hypersurface is addressed. The quantum Regge calculus measure is defined from a discontinuous metric measure by inserting the δ-function-like phase factor. The requirement that continuity conditions be imposed in a “face-independent” way fixes this factor uniquely. The term “face-independent” means that this factor depends only on the (hyper)plane spanned by the face, not on it's form and size. This requirement seems to be natural from the viewpoint of existence of the well-defined continuum limit maximally free of lattice artefacts.

A long range spherical model and exact solutions of an energy enstrophy theory for two-dimensional turbulence

The equilibrium statistical mechanics of the energy–enstrophy theory for the two-dimensional (2D) Euler equations is solved exactly. A family of lattice vortex gas models for the Euler equations is derived and shown to have a well-defined nonextensive continuum limit. This family of continuous-spin lattice Hamiltonians is shown to be nondegenerate under different point vortex discretizations of the Euler equations. Under the assumptions that the energy, total circulation and the enstrophy (mean squared vorticity) are conserved, this lattice vortex gas model is equivalent to a long range version of Kac’s exactly solvable spherical model with logarithmic interaction. The spherical model formulation is based on the fundamental observation that the conservation of enstrophy is mathematically equivalent to Kac’s spherical constraint. This spherical model is shown to have a free energy that is analytic in the properly scaled inverse temperatures β̃ in the range 0=β̃*&amp;lt;β̃&amp;lt;β̃c=4π/3. Phase transitions occur at the positive value β̃c and β̃*=0. Spin–spin correlations are calculated giving two-point vorticity correlations that are important to the study of turbulence. There are exactly three distinct phases in the energy-enstrophy theory for 2D flows, namely (a) an uncorrelated high positive temperature phase, (b) an antiferromagnetic checkerboard vorticity pattern at low positive temperature, and (c) a highly correlated physical domain scale vorticity pattern (for instance, a large positive vorticity region surrounded by a sea of negative vorticity) at negative temperatures. The boundary β̃*=0 agrees with the known numerical and analytical results on the occurrence of coherent or ordered structures at negative temperatures. The critical temperature β̃c&amp;gt;0 is new, as is the corresponding checkerboard low positive temperature phase. Physical interpretations of the results in this paper are obtained.

Exact solutions of an energy-enstrophy theory for the barotropic vorticity equation on a rotating sphere

The equilibrium statistical mechanics of the energy-enstrophy theory for the barotropic vorticity equation is solved exactly in the sense that a explicitly nonGaussian configurational integral is calculated in closed form. A family of lattice vortex gas models for the barotropic vorticity equation (BVE) is derived and shown to have a well-defined nonextensive continuum limit as the coarse graining is refined. This family of continuous-spin lattice Hamiltonians is shown to be nondegenerate under different point vortex discretizations of the BVE. Under the assumption that the energy and the enstrophy (mean-squared absolute vorticity) are conserved, a long-range version of Kac's spherical model with logarithmic interaction is derived and solved exactly in the zero total circulation or neutral vortex gas case by the method of steepest descent. The spherical model formulation is based on the fundamental observation that the conservation of enstrophy is mathematically equivalent to Kac's spherical constraint. Two new features of this spherical model are (i) it allows negative temperatures, and (ii) a nonextensive thermodynamic limit where the strength of the interaction scales with the number of lattice sites but where the size of the physical domain remains fixed; novel interpretations of the saddle point criterion for negative temperatures will be formulated. This spherical model is shown to have a free energy that is analytic in the properly scaled inverse temperatures β ̃ in the range 0= β ̃ ∗< β ̃ < β ̃ c=N ∗ 2 π 2/2K in the nonextensive continuum limit, with K being the fixed value of the enstrophy. The boundary β ̃ ∗=0 agrees with the known numerical and analytical results on the occurrence of coherent or ordered structures at negative temperatures. Spin–spin correlations are calculated giving two-point vorticity correlations that are important to the study of turbulence. Physical interpretations of the results in this paper are obtained and applied to planetary atmospheres.

Embedded topological defects in hot electroweak theory: a lattice study

We study the properties of Nambu monopoles and Z-vortices in the 3 D lattice SU(2) Higgs theory which represents the Standard Model at high temperature. We show that the densities of the Nambu monopoles and the Z-vortices are O(1) in the symmetric phase and generically small in the Higgs phase. Near to the critical Higgs mass and in the vicinity of the phase transition the densities are no more negligible in the broken phase. The percolation probability of the Z-vortex lines is found as a new disorder parameter for this phase transition. We conclude that the transition to the symmetric phase is accompanied by Z-vortex condensation. Simulations comparing elementary and extended vortices and monopoles at different β G values, aiming to show that the density of vortices and monopoles of fixed physical size might have a well-defined continuum limit, gives encouraging but so far inconclusive results.

Thermodynamics of gauge-invariant U(1) vortices from lattice Monte Carlo simulations

We study non-perturbatively and from first principles the thermodynamics of vortices in 3d U(1) gauge+Higgs theory, or the Ginzburg-Landau model, which has frequently been used as a model for cosmological topological defect formation. We discretize the system and introduce a gauge-invariant definition of a vortex passing through a loop on the lattice. We then study with Monte Carlo simulations the total vortex density, extract the physically meaningful part thereof, and demonstrate that it has a well-defined continuum limit. The total vortex density behaves as a pseudo order parameter, having a discontinuity in the regime of first order transitions and behaving continuously in the regime of second order transitions. Finally, we discuss further gauge-invariant observables to be measured.

Dynamic friction and the origin of the complexity of earthquake sources.

We study a simple antiplane fault of finite length embedded in a homogeneous isotropic elastic solid to understand the origin of seismic source heterogeneity in the presence of nonlinear rate- and state-dependent friction. All the mechanical properties of the medium and friction are assumed homogeneous. Friction includes a characteristic length that is longer than the grid size so that our models have a well-defined continuum limit. Starting from a heterogeneous initial stress distribution, we apply a slowly increasing uniform stress load far from the fault and we simulate the seismicity for a few 1000 events. The style of seismicity produced by this model is determined by a control parameter associated with the degree of rate dependence of friction. For classical friction models with rate-independent friction, no complexity appears and seismicity is perfectly periodic. For weakly rate-dependent friction, large ruptures are still periodic, but small seismicity becomes increasingly nonstationary. When friction is highly rate-dependent, seismicity becomes nonperiodic and ruptures of all sizes occur inside the fault. Highly rate-dependent friction destabilizes the healing process producing premature healing of slip and partial stress drop. Partial stress drop produces large variations in the state of stress that in turn produce earthquakes of different sizes. Similar results have been found by other authors using the Burridge and Knopoff model. We conjecture that all models in which static stress drop is only a fraction of the dynamic stress drop produce stress heterogeneity.

Conformal matrix models as an alternative to conventional multi-matrix models

We introduce conformal multi-matrix models (CMMs) as an alternative to conventional multi-matrix model description of two-dimensional gravity interacting with c < 1 matter. We define CMMs as solutions to (discrete) extended Virasoro constraints. We argue that the so defined alternatives of multi-matrix models represent the same universality classes in the continuum limit, while at the discrete level they provide explicit solutions to the multi-component KP-hierarchy and by definition satisfy the discrete W-constraints. We prove that discrete CMMs coincide with the ( p, q)-series of 2d gravity models in a well-defined continuum limit, thus demonstrating that they provided a proper generalization of the hermitian one-matrix model.

The Schrödinger functional — a renormalizable probe for non-abelian gauge theories

Following Symanzik we argue that the Schrödinger functional in lattice gauge theories without matter fields has a well-defined continuum limit. Due to gauge invariance no extra counterterms are required. The Schrödinger functional is, moreover, accessible to numerical simulations. It may hence be used to study the scaling properties of the theory and in particular the evolution of the renormalized gauge coupling from low to high energies. A concrete proposition along this line is made and the necessary perturbative analysis of the Schrödinger functional is carried through to one-loop order.

The solution space of the unitary matrix model string equation and the Sato Grassmannian

The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on points $V_1$ and $V_2$ in the big cell $\Gr$ of the Sato Grassmannian $Gr$. This is a consequence of a well-defined continuum limit in which the string equation has the simple form $\lb \cp ,\cq_- \rb =\hbox{\rm 1}$, with $\cp$ and $\cq_-$ $2\times 2$ matrices of differential operators. These conditions on $V_1$ and $V_2$ yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraints $\L_n\,(n\geq 0)$, where $\L_n$ annihilate the two modified-KdV $\t$-functions whose product gives the partition function of the Unitary Matrix Model.

Critical properties of a model of planar random surfaces

We describe results for a model of planar random surfaces which suggest that the model has a well-defined continuum limit from which a relativistic quantum theory of strings without tachyons can be reconstructed. The mean-field theory of that model is developed, and Osterwalder-Schrader positivity is exhibited. Some scaling relations and mean-field bounds for critical exponents and correlations are established. It is argued that, in dimension <6, our model may serve to calculate the leading O( N −2) correction to the two-glue-ball amplitude and that in high dimensions the Regge slope is infinite.

On the geometric lattice approximation to a realistic model of QCD

We suggest a model of QCD with four flavour degress of freedom on the lattice. This model has a well-defined continuum limit and no spurious quark degrees of freedom. The formulation is realistic in so far as the different quarks may have different bare masses. For Monte Carlo calculations on finite lattices our suggestion should be superior to comparable other models. The model is formulated within the Dirac-Kahler description of fermions which we repeat in a short glossary.