Abstract We study a class of radially symmetric Coulomb gas ensembles at inverse temperature $$\beta =2$$ β = 2 , for which the droplet consists of a number of concentric annuli, having at least one bounded “gap” G , i.e., a connected component of the complement of the droplet, which disconnects the droplet. Let n be the total number of particles. Among other things, we deduce fine asymptotics as $$n \rightarrow \infty $$ n → ∞ for the edge density and the correlation kernel near the gap, as well as for the cumulant generating function of fluctuations of smooth linear statistics. We typically find an oscillatory behaviour in the distribution of particles which fall near the edge of the gap. These oscillations are given explicitly in terms of a discrete Gaussian distribution, weighted Szegő kernels, and the Jacobi theta function, which depend on the parameter n .

Abstract We reconsider the fundamental problem of coarse-graining infinite-dimensional Hamiltonian dynamics to obtain a macroscopic system which includes dissipative mechanisms. In particular, we study the thermodynamical implications concerning Hamiltonians, energy, and entropy and the induced geometric structures such as Poisson and Onsager brackets (symplectic and dissipative brackets). We start from a general finite-dimensional Hamiltonian system that is coupled linearly to an infinite-dimensional heat bath with linear dynamics. The latter is assumed to admit a compression to a finite-dimensional dissipative semigroup (i.e., the heat bath is a dilation of the semigroup) describing the dissipative evolution of new macroscopic variables. Already in the finite-energy case (zero-temperature heat bath) we obtain the so-called GENERIC structure (General Equation for Non-Equilibrium Reversible Irreversible Coupling), with conserved energy, nondecreasing entropy, a new Poisson structure, and an Onsager operator describing the dissipation. However, their origin is not obvious at this stage. After extending the system in a natural way to the case of positive temperature, giving a heat bath with infinite energy, the compression property leads to an exact multivariate Ornstein-Uhlenbeck process that drives the rest of the system. Thus, we are able to identify a conserved energy, an entropy, and an Onsager operator (involving the Green-Kubo formalism) which indeed provide a GENERIC structure for the macroscopic system.

We demonstrate the existence of topologically nontrivial energy minimizing maps of a given positive degree from bounded domains in the plane to S2 in a variational model describing magnetizations in ultrathin ferromagnetic films with Dzyaloshinskii–Moriya interaction. Our strategy is to insert tiny truncated Belavin–Polyakov profiles in carefully chosen locations of lower degree objects such that the total energy increase lies strictly below the expected Dirichlet energy contribution, ruling out loss of degree in the limits of minimizing sequences. The argument requires that the domain be either sufficiently large or sufficiently slender to accommodate a prescribed degree. We also show that these higher degree minimizers concentrate on point-like skyrmionic configurations in a suitable parameter regime.

Abstract We consider the $$\phi ^4_1$$ ϕ 1 4 measure in an interval of length $$\ell $$ ℓ , defined by a symmetric double-well potential W and inverse temperature $$\beta $$ β . Our results concern its asymptotic behavior in the joint limit $$\beta , \ell \rightarrow \infty $$ β , ℓ → ∞ , both in the subcritical regime $$\ell \ll \textrm{e}^{\beta C_W}$$ ℓ ≪ e β C W and in the supercritical regime $$\ell \gg \textrm{e}^{\beta C_W}$$ ℓ ≫ e β C W , where $$C_W$$ C W denotes the surface tension. In the former case, in which the measure concentrates on the pure phases, we prove the corresponding large deviation principle. The associated rate function is the Modica–Mortola functional modified to take into account the entropy of the locations of the interfaces. Furthermore, we provide the sharp asymptotics of the probability of having a given number of transitions between the two pure phases. In the supercritical regime, the measure no longer concentrates and we show that the interfaces are asymptotically distributed according to a Poisson point process.

Abstract A static variational model for shape formation in heteroepitaxial crystal growth is considered. The energy functional takes into account surface energy, elastic misfit-energy and nucleation energy of dislocations. A scaling law for the infimal energy is proven. The results quantify the expectation that in certain parameter regimes, island formation or topological defects are favorable. This generalizes results in the purely elastic setting from [23]. To handle dislocations in the lower bound, a new variant of a ball-construction combined with thorough local estimates is presented.

Abstract Most theories and applications of elasticity rely on an energy function that depends on the strains from which the stresses can be derived. This is the traditional setting of Green elasticity, also known as hyper-elasticity. However, in its original form the theory of elasticity does not assume the existence of a strain energy function. In this case, called Cauchy elasticity, stresses are directly related to the strains. Since the emergence of modern elasticity in the 1940s, research on Cauchy elasticity has been relatively limited. One possible reason for this is that for Cauchy materials, the net work performed by stress along a closed path in the strain space may be nonzero. Therefore, such materials may require access to both energy sources and sinks. This characteristic has led some mechanicians to question the viability of Cauchy elasticity as a physically plausible theory of elasticity. In this paper, motivated by its relevance to recent applications, such as the modeling of active solids, we revisit Cauchy elasticity in a modern form. First, we show that in the general theory of anisotropic Cauchy elasticity, stress can be expressed in terms of six functions, that we call Edelen-Darboux potentials. For isotropic Cauchy materials, this number reduces to three, while for incompressible isotropic Cauchy elasticity, only two such potentials are required. Second, we show that in Cauchy elasticity, the link between balance laws and symmetries is lost, in general, since Noether’s theorem does not apply. In particular, we show that, unlike hyperleasticity, objectivity is not equivalent to the balance of angular momentum. Third, we formulate the balance laws of Cauchy elasticity covariantly and derive a generalized Doyle–Ericksen formula. Fourth, the material symmetry and work theorems of Cauchy elasticity are revisited, based on the stress-work 1-form that emerges as a fundamental quantity in Cauchy elasticity. The stress-work 1-form allows for a classification via Darboux’s theorem that leads to a classification of Cauchy elastic solids based on their generalized energy functions. Fifth, we discuss the relevance of Carathéodory’s theorem on accessibility property of Pfaffian equations. Sixth, we show that Cauchy elasticity has an intrinsic geometric hystresis, which is the net work of stress in cyclic deformations. If the orientation of a cyclic deformation is reversed, the sign of the net work of stress changes, from which we conclude that stress in Cauchy elasticity is neither dissipative nor conservative. Seventh, we establish connections between Cauchy elasticity and the existing constitutive equations for active solids. Eighth, linear anisotropic Cauchy elasticity is examined in detail, and simple displacement-control loadings are proposed for each symmetry class to characterize the corresponding antisymmetric elastic constants. Ninth, we discuss both isotropic and anisotropic Cauchy anelasticity and show that the existing solutions for stress fields of distributed eigenstrains (and particularly defects) in hyperelastic solids can be readily extended to Cauchy elasticity. Tenth, we introduce Cosserat–Cauchy materials and demonstrate that an anisotropic three-dimensional Cosserat–Cauchy elastic solid has at most twenty four generalized energy functions.

Abstract We develop a Birman–Schwinger principle for the spherically symmetric, asymptotically flat Einstein–Vlasov system. The principle characterizes the stability properties of steady states such as the positive definiteness of an Antonov-type operator or the existence of exponentially growing modes in terms of a one-dimensional variational problem for a Hilbert–Schmidt operator. This requires a refined analysis of the operators arising from linearizing the system, which uses action-angle type variables. For the latter, a single-well structure of the effective potential for the particle flow of the steady state is required. This natural property can be verified for a broad class of singularity-free steady states. As a particular example for the application of our Birman–Schwinger principle we consider steady states where a Schwarzschild black hole is surrounded by a shell of Vlasov matter. We prove the existence of such steady states and derive linear stability if the mass of the Vlasov shell is small compared to the mass of the black hole.