We give a comprehensive presentation of methods for calculating the Casimir force to arbitrary accuracy, for any number of objects, arbitrary shapes, susceptibility functions, and separations. The technique is applicable to objects immersed in media other than vacuum, nonzero temperatures, and spatial arrangements in which one object is enclosed in another. Our method combines each object's classical electromagnetic scattering amplitude with universal translation matrices, which convert between the bases used to calculate scattering for each object, but are otherwise independent of the details of the individual objects. The method is illustrated by re-deriving the Lifshitz formula for infinite half spaces, by demonstrating the Casimir-Polder to van der Waals cross-over, and by computing the Casimir interaction energy of two infinite, parallel, perfect metal cylinders either inside or outside one another. Furthermore, it is used to obtain new results, namely the Casimir energies of a sphere or a cylinder opposite a plate, all with finite permittivity and permeability, to leading order at large separation.

Throughout cell cycle progression, the expression of multiple transcripts oscillate, and whether these are under the centralized control of the CDK-APC/C proteins or can be driven by a de-centralized transcription factor (TF) cascade is a fundamental question for understanding cell cycle regulation. In budding yeast, we find that the transcription of nearly all genes, as assessed by RNA-seq or fluorescence microscopy in single cells, is dictated by CDK-APC/C. Three exceptional genes are transcribed in a pulsatile pattern in a variety of CDK-APC/C arrests. Pursuing one of these transcripts, the SIC1 inhibitor of B-type cyclins, we use a combination of mathematical modeling and experimentation to provide evidence that, counter-intuitively, Sic1 provides a failsafe mechanism promoting nuclear division when levels of mitotic cyclins are low.

We examine whether fluctuation-induced forces can lead to stable levitation. First, we analyze a collection of classical objects at finite temperature that contain fixed and mobile charges, and show that any arrangement in space is unstable to small perturbations in position. This extends Earnshaw's theorem for electrostatics by including thermal fluctuations of internal charges. Quantum fluctuations of the electromagnetic field are responsible for Casimir/van der Waals interactions. Neglecting permeabilities, we find that any equilibrium position of items subject to such forces is also unstable if the permittivities of all objects are higher or lower than that of the enveloping medium; the former being the generic case for ordinary materials in vacuum.Earnshaw's theorem [1] states that a collection of charges cannot be held in stable equilibrium solely by electrostatic forces. The charges can attract or repel, but cannot be stably levitated. While the stability of matter (due to quantum phenomena), and dramatic demonstrations of levitating frogs [2], are vivid reminders of the caveats to this theorem, it remains a powerful indicator of the constraints to stability in electrostatics. There is much current interest in forces induced by fluctuating charges (e.g., mobile ions in solution), or fluctuating electromagnetic (EM) fields (e.g., the Casimir force between metal plates). The former (due to thermal fluctuations) may lead to unexpected phenomena such as attraction of like-charged macroions, and is thought to be relevant to interactions of biological molecules. The latter (due mainly to quantum fluctuations) is important to the attraction (and stiction) of components of microelectromechanical (MEM) devices. Here, we extend Earnshaw's theorem to some fluctuation-induced forces, thus placing strong constraints on the possibility of obtaining stable equilibria, and repulsion between neutral objects.An extension of Earnshaw's theorem [1] to polarizable objects by Braunbek [3,4] establishes that dielectric and paramagnetic ( > 1 and µ > 1) matter cannot be stably levitated by electrostatic forces, while diamagnetic (µ < 1) matter can. This is impressively demonstrated by superconductors and frogs that fly freely above magnets [2]. If the enveloping medium is not vacuum, the criteria for stability are modified by substituting the static electric permittivity M and magnetic permeability µ M of the medium in place of the vacuum value of 1 in the respective inequalities. In fact, if the medium itself has a dielectric constant higher than the objects ( < M ), stable levitation is possible, as demonstrated for bubbles in liquids (see Ref. [5], and references therein). For dynamic fields the restrictions of electrostatics do not apply; for example, lasers can lift and hold dielectric beads with index of refraction n = √ µ > 1 [6]. In addition to the force which keeps the bead in the center of the laser beam, there is radiation pressure which pushes the bead along the direction of the Poynting vector. Ashkin and Gordon hav...

We study collective interaction effects that result from the change of free quantum electrodynamic field fluctuations by one-and two-dimensional perfect metal structures. The Casimir interactions in geometries containing plates and cylinders is explicitly computed using partial wave expansions of constrained path integrals. We generalize previously obtained results and provide a more detailed description of the technical aspects of the approach [1]. We find that the interactions involving cylinders have a weak logarithmic dependence on the cylinder radius, reflecting that one-dimensional perturbations are marginally relevant in 4D space-time. For geometries containing two cylinders and one or two plates, we confirm a previously found non-monotonic dependence of the interaction on the object's separations which does not follow from pair-wise summation of two-body forces. Qualitatively, this effect is explained in terms of fluctuating charges and currents and their mirror images.

The Casimir force has been computed exactly for only a few simple geometries, such as infinite plates, cylinders, and spheres. We show that a parabolic cylinder, for which analytic solutions to the Helmholtz equation are available, is another case where such a calculation is possible. We compute the interaction energy of a parabolic cylinder and an infinite plate (both perfect mirrors), as a function of their separation and inclination, H and θ, and the cylinder's parabolic radius R. As H/R → 0, the proximity force approximation becomes exact. The opposite limit of R/H → 0 corresponds to a semi-infinite plate, where the effects of edge and inclination can be probed.PACS numbers: 42.25. Fx, 03.70.+k, Casimir's computation of the force between two parallel metallic plates [1] originally inspired much theoretical interest as a macroscopic manifestation of quantum fluctuations of the electromagnetic field in vacuum. Following its experimental confirmation in the past decade [2], however, it is now an important force to reckon with in the design of microelectromechanical systems [3]. Potential practical applications have motivated the development of numerical methods to compute Casimir forces for objects of any shape [4]. The simplest and most commonly used methods for dealing with complex shapes rely on pairwise summations, as in the proximity force approximation (PFA), which limits their applicability.Recently we have developed a formalism [5,6] that relates the Casimir interaction among several objects to the scattering of the electromagnetic field from the objects individually. (For additional perspectives on the scattering formalism, see references in [6].) This approach simplifies the problem, since scattering is a well-developed subject. In particular, the availability of scattering formulae for simple objects, such as spheres and cylinders, has enabled us to compute the Casimir force between two spheres [5], a sphere and a plate [7], multiple cylinders [8], etc. In this work we show that parabolic cylinders provide another example where the scattering amplitudes can be computed exactly. We then use the exact results for scattering from perfect mirrors to compute the Casimir force between a parabolic cylinder and a plate. In the limiting case when the radius of curvature at its tip vanishes, the parabolic cylinder becomes a semi-infinite plate (a knife's edge), and we can consider how the energy depends on the boundary condition it imposes and the angle it makes to the plane.The surface of a parabolic cylinder in Cartesian coordinates is described by y = (x 2 − R 2 )/2R for all z, as shown in Fig. 1, where R is the radius of curvature at the tip. In parabolic cylinder coordinates [9], defined through x = µλ, y = (λ 2 − µ 2 )/2, z = z, the surface is simply µ = µ 0 = √ R for −∞ < λ, z < ∞. One advantage of the latter coordinate system is that the Helmholtz equationwhich we consider for imaginary wavenumber k = iκ, admits separable solutions. Since sending λ → −λ and µ → −µ returns us to the same point, we restr...

We analyze the Casimir force between two parallel infinite metal cylinders, with nearby metal plates (sidewalls), using complementary methods for mutual confirmation. The attractive force between cylinders is shown to have a nonmonotonic dependence on the separation to the plates. This intrinsically multi-body phenomenon, which occurs with either one or two sidewalls (generalizing an earlier result for squares between two sidewalls), does not follow from any simple two-body force description. We can, however, explain the nonmonotonicity by considering the screening (enhancement) of the interactions by the fluctuating charges (currents) on the two cylinders, and their images on the nearby plate(s). Furthermore, we show that this effect also implies a nonmonotonic dependence of the cylinder-plate force on the cylinder-cylinder separation.PACS numbers: 12.20. Ds, 42.50.Ct, 42.50.Lc Casimir forces arise from quantum vacuum fluctuations, and have been the subject of considerable theoretical and experimental interest [1,2,3,4]. We consider here the force between metallic cylinders with one or two parallel metal sidewalls ( Fig. 1) using two independent exact computational methods, and find an unusual nonmonotonic dependence of the force on the sidewall separation. These nonmonotonic effects cannot be predicted by commonly used two-body Casimir-force estimates, such as the proximity-force approximation (PFA) [5,6] that is based on the parallel-plate limit, or by addition of Casimir-Polder 'atomic' interactions (CPI) [6,7,8].In previous work, we demonstrated a similar nonmonotonic force between two metal squares in proximity to two parallel metal sidewalls, for either perfect or realistic metals [9]. This work, with perfect-metal cylinders [10], demonstrates that the effect is not limited to squares (i.e., it does not arise from sharp corners or parallel flat surfaces), nor does it require two sidewalls. The nonmonotonicity stems from a competition between forces from transverse electric (TE) and transverse magnetic (TM) field polarizations: In the latter case, the interaction between fluctuating charges on the cylinders is screened by opposing image charges, in the former case it is enhanced by analogous fluctuating image currents. Furthermore, we show that a related nonmonotonic variation arises for the force between the cylinders and a sidewall as a function of separation between the cylinders, a geometry potentially amenable to experiment.Casimir forces are not two-body interactions: quantum fluctuations in one object induce fluctuations throughout the system which in turn act back on the first object. However, both the PFA and CPI view Casimir forces as the result of attractive two-body ("pairwise") interactions. They are reasonable approximations only in certain limits (e.g., low curvature for PFA), and can fail qualitatively as well as quantitatively otherwise. Pairwise estimates fail to account for two important aspects of the Casimir forces in the geometry we consider [11]. First, a monotonic pairwise attrac...

The identification of cell borders (‘segmentation’) in microscopy images constitutes a bottleneck for large-scale experiments. For the model organism Saccharomyces cerevisiae, current segmentation methods face challenges when cells bud, crowd, or exhibit irregular features. We present a convolutional neural network (CNN) named YeaZ, the underlying training set of high-quality segmented yeast images (>10 000 cells) including mutants, stressed cells, and time courses, as well as a graphical user interface and a web application (www.quantsysbio.com/data-and-software) to efficiently employ, test, and expand the system. A key feature is a cell-cell boundary test which avoids the need for fluorescent markers. Our CNN is highly accurate, including for buds, and outperforms existing methods on benchmark images, indicating it transfers well to other conditions. To demonstrate how efficient large-scale image processing uncovers new biology, we analyze the geometries of ≈2200 wild-type and cyclin mutant cells and find that morphogenesis control occurs unexpectedly early and gradually.

Casimir forces between conductors at the submicron scale are paramount to the design and operation of microelectromechanical devices. However, these forces depend nontrivially on geometry, and existing analytical formulae and approximations cannot deal with realistic micromachinery components with sharp edges and tips. Here, we employ a novel approach to electromagnetic scattering, appropriate to perfect conductors with sharp edges and tips, specifically wedges and cones. The Casimir interaction of these objects with a metal plate (and among themselves) is then computed systematically by a multiple-scattering series. For the wedge, we obtain analytical expressions for the interaction with a plate, as functions of opening angle and tilt, which should provide a particularly useful tool for the design of microelectromechanical devices. Our result for the Casimir interactions between conducting cones and plates applies directly to the force on the tip of a scanning tunneling probe. We find an unexpectedly large temperature dependence of the force in the cone tip which is of immediate relevance to experiments.fluctuations | quantum electrodynamics T he inherent appeal of the Casimir force as a macroscopic manifestation of quantum "zero-point" fluctuations has inspired many studies over the decades that followed its discovery (1). Casimir's original result (2) for the force between perfectly reflecting mirrors separated by vacuum was quickly extended to include slabs of material with specified (frequency-dependent) dielectric response (3). Quantitative experimental confirmation, however, had to await the advent of high-precision scanning probes in the 1990s (4-7). Recent studies have aimed to reduce or reverse the attractive Casimir force for practical applications in micron-sized mechanical machines, where Casimir forces may cause components to stick and the machine to fail. In the presence of an intervening fluid, experiments have indeed observed repulsion due to quantum (8) or critical thermal (9) fluctuations. Metamaterials, fabricated designs of microcircuitry, have also been proposed as candidates for Casimir repulsion across vacuum (10).Although there have been many studies of the role of materials (dielectrics, conductors, etc.), the treatment of shapes and geometry has remained comparatively less investigated. Interactions between nonplanar shapes are typically calculated via the proximity force approximation (PFA), which sums over infinitesimal segments treated as locally parallel plates (11). This approximation represents a serious limitation because the majority of experiments measure the force between a sphere and a plate with precision that is now sufficient to probe deviations from PFA in this and other geometries (12, 13). Practical applications are likely to explore geometries further removed from parallel plates. Several numerical schemes (14, -16), and even an analog computer (17), have recently been developed for computing Casimir forces in general geometries. However, analytical formulae for quick...