arXiv subjects

Karl Saunders

Publications and source records attributed to Karl Saunders.

17 recordsLinked to original sources

arXiv2025

Use of a genetic algorithm in university scheduling for equitable and efficient determination of teaching assignments

Here a genetic algorithm (GA) is presented that creates a teaching schedule for a university physics department by algorithmically assigning ${\sim}200$ classes to ${\sim}50$ professors for each of three academic terms per year. The algorithm is driven by chromosomes of the GA that encode proposed pairings between enumerated lists of professors and classes. The fitness of the pairings is measured by considering both contractual work constraints and individual teaching preferences. The algorithm uses standard crossover and mutation operations to seek ever more optimal schedules over many generations. Here we detail the implementation and performance of the algorithm, including some interpretability findings. Overall, we are very pleased with the algorithm, as it is typically able to converge within minutes, with over $90\%$ of needed classes assigned. A metric is used to assign each professor's schedule a score, which measures how well their preferences were satisfied. These scores can be used to ensure longitudinal equity in the assignment of classes among professors.

cs.NE↗

arXiv2019

A New Twist on the Electroclinic Critical Point: Type I and Type II Smectic $C^*$ Systems

This analysis of the electroclinic effect in ferroelectric liquid crystals with a first order Smectic-$A^*$--Smectic-$C^*$ (Sm-$A^*$--Sm-$C^*$) transition, shows they can be either Type I or Type II. In temperature--field parameter space Type I systems exhibit a macroscopically achiral (in which the Sm-$C^*$ helical superstructure is expelled) low-tilt (LT) Sm-$C$--high-tilt (HT) Sm-$C$ critical point, which terminates a LT Sm-$C$--HT Sm-$C$ first order boundary. This boundary extends to an achiral-chiral triple point where the achiral LT Sm-$C$ and HT Sm-$C$ phases coexist with the chiral Sm-$C^*$ phase. In Type II systems the critical point, triple point, and first order boundary are replaced by a Sm-$C^*$ region, between LT and HT achiral Sm-$C$ phases, at low and high fields respectively. When the field is ramped up, the Type II system displays a reentrant Sm-$C$--Sm-$C^*$-Sm-$C$ phase sequence. A discontinuity in the tilt of the optical axis at each of the two phase transitions means the Type II system is tristable. The system is Type I or Type II depending on the ratio of two length scales, one is the zero-field Sm-$C^*$ helical pitch, the other depends on the latent heat at the zero-field first order Sm-$A^*$--Sm-$C^*$ transition. A system could be experimentally tuned by varying enantiomeric excess, between Type I and Type II behavior. We also show this Type I vs Type II behavior is the Ising universality class analog of Type I vs Type II behavior in XY universality class systems. Lastly, we make a complete mapping of the phase boundaries in temperature--field--enantiomeric excess parameter space (not just near the critical point) which shows a variety of interesting features, including a multicritical point, tricritical points and a doubly reentrant Sm-$C$--Sm-$C^*$-Sm-$C$--Sm-$C^*$ phase sequence.

cond-mat.soft↗

arXiv2013

Modeling the Field Control of the Surface Electroclinic Effect near Continuous and First Order Smectic-A-Smectic-C Transitions

We present and analyze a model for the combination of bulk and surface electroclinic effects in the smectic-A* (Sm-A) phase near a Sm-A*--Sm-C* transition. As part of our analysis we calculate the dependence of the surface tilt on external electric field and show that it can be eliminated, or even reversed from its zero-field value. This is in good agreement with previous experimental work on a system (W415) with a continuous Sm-A*--Sm-C* transition. We also analyze, for the first time, the combination of bulk and surface electroclinic effects in systems with a first order Sm-A*--Sm-C* transition. The variation of surface tilt with electric field in this case is much more dramatic, with discontinuities and hysteresis. Near each type of Sm-A*--Sm-C* transition we obtain the temperature dependence of the field required to eliminate surface tilt. Additionally, we analyze the effect of varying the system's enantiomeric excess, showing that it strongly affects the field dependence of surface tilt, in particular, near a first order Sm-A*--Sm-C* transition. In this case, increasing enantiomeric excess can change the field dependence of surface tilt from continuous to discontinuous. Our model also allows us to calculate the variation of layer spacing in going from surface to bulk, which in turn allows us to estimate the strain resulting from the difference between the surface and bulk layer spacing. We show that for certain ranges of applied electric field, this strain can result in layer buckling which reduces the overall quality of the liquid crystal cell. For de Vries materials, with small tilt-induced change in layer spacing, the induced strain for a given surface tilt should be smaller. However, we argue that this may be offset by the fact that de Vries materials, which typically have Sm-A*--Sm-C* transitions near a tricritical point, will generally have larger surface tilt.

cond-mat.soft↗

arXiv2011

de Vries behavior of the electroclinic effect in the smectic-A* phase near a biaxiality-induced smectic-A* -- smectic-C* tricritical point

Using a generalized Landau theory involving orientational, layering, tilt, and biaxial order parameters we analyze the smectic-A* and smectic-C* (Sm-A* -- Sm-C*) transition, showing that a combination of small orientational order and large layering order leads to Sm-A* -- Sm-C* transitions that are either continuous and close to tricriticality or first order. The model predicts that in such systems the increase in birefringence upon entry to the Sm-C* phase will be especially rapid. It also predicts that the change in layer spacing at the Sm-A* -- Sm-C* transition will be proportional to the orientational order. These are two hallmarks of Sm-A* -- Sm-C* transitions in de Vries materials. We analyze the electroclinic effect in the Sm-A* phase and show that as a result of the zero-field Sm-A* -- Sm-C* transition being either continuous and close to tricriticality or first order (i.e for systems with a combination of weak orientational order and strong layering order) the electroclinic response of the tilt will be unusually strong. Additionally, we investigate the associated electrically induced change in birefringence and layer spacing, demonstrating de Vries behavior for each, i.e. an unusually large increase in birefringence and an unusually small layer contraction. Both the induced change in birefringence and layer spacing are shown to scale quadratically with the induced tilt angle.

cond-mat.soft↗

arXiv2011

The Surface Electroclinic Effect near the First Order Smectic-A--Smectic-C transition

We analyze the surface electroclinic effect (SECE) in a material that exhibits a first order bulk smectic-$A^*$ (Sm-$A^*$) -- smectic-$C^*$ (Sm-$C^*$) transition. The effect of a continuously varying degree of enantiomeric excess on the SECE is also investigated. We show that due to the first order nature of the bulk Sm-$A^*$ -- Sm-$C^*$ transition, the SECE can be unusually strong and that as enantiomeric excess is varied, a jump in surface induced tilt is expected. A theoretical state map, in enantiomeric excess - temperature space, features a critical point which terminates a line of first order discontinuities in the surface induced tilt. This critical point is analogous to that found for the phase diagram (in electric field - temperature space) for the bulk electroclinic effect. Analysis of the decay of the surface induced tilt, as one moves from surface into bulk shows that for sufficiently high surface tilt the decay will exhibit a well defined spatial kink within which it becomes especially rapid. We also propose that the SECE is additionally enhanced by the de Vries nature (i.e. small layer shrinkage at the bulk Sm-A* -- Sm-C* transition) of the material. As such the SECE provides a new means to characterize the de Vries nature of a material. We discuss the implications for using these materials in device applications and propose ways to investigate the predicted features experimentally.

cond-mat.soft↗

arXiv2011

The Biaxial Smectic-A* Phase -- A New Phase, Already But Unknowingly Discovered?

The biaxial smectic-A* (Sm-A_B*) phase, appearing in the phase sequence Sm-A*--Sm-A*_B--Sm-C*, is analyzed using Landau theory. It is found to possess a helical superstructure with a pitch that is significantly shorter than the pitch of the Sm-C* helical superstructure. The Sm-A_B*--Sm-C* transition can be either 1st or 2nd order, and correspondingly there will be either a jump or continuous variation in the pitch. The behaviors of the birefringence and electroclinic effect are analyzed and found to be similar to those of a Sm-C*_alpha phase. As such, it is possible that the Sm-A_B* phase could be misidentified as a Sm-C*alpha phase. Ways to distinguish the two phases are discussed.

cond-mat.soft↗

arXiv2008

De Vries Behavior in Smectics near a Biaxiality Induced Smectic A - Smectic C Tricritical Point

We show that a generalized Landau theory for the smectic A and C phases exhibits a biaxiality induced AC tricritical point. Proximity to this tricritical point depends on the degree of orientational order in the system; for sufficiently large orientational order the AC transition is 3D XY-like, while for sufficiently small orientational order, it is either tricritical or 1st order. We investigate each of the three types of AC transitions near tricriticality and show that for each type of transition, small orientational order implies de Vries behavior in the layer spacing, an unusually small layer contraction. This result is consistent with, and can be understood in terms of, the "diffuse cone" model of de Vries. Additionally, we show that birefringence grows upon entry to the C phase. For a continuous transition, this growth is more rapid the closer the transition is to tricriticality. Our model also predicts the possibility of a nonmontonic temperature dependence of birefringence.

cond-mat.soft↗

arXiv2006

Disordering to Order: de Vries behavior from a Landau theory for smectics

We show that Landau theory for the isotropic, nematic, smectic A, and smectic C phases generically, but not ubiquitously, implies de Vries behavior. I.e., a continuous AC transition can occur with little layer contraction; the birefringence decreases as temperature T is lowered above this transition, and increases again below the transition. This de Vries behavior occurs in models with unusually small orientational order, and is preceded by a first order I − A transition. A first order AC transition with elements of de Vries behavior can also occur. These results correspond well with experimental work to date.

cond-mat.soft↗

arXiv2006

Hydrodynamics of polar liquid crystals

Starting from a microscopic definition of an alignment vector proportional to the polarization, we discuss the hydrodynamics of polar liquid crystals with local $C_{\infty v}$-symmetry. The free energy for polar liquid crystals differs from that of nematic liquid crystals ($D_{\infty h}$) in that it contains terms violating the ${\bf{n}}\to -{\bf{n}}$ symmetry. First we show that these $\mathcal{Z}_2$-odd terms induce a general splay instability of a uniform polarized state in a range of parameters. Next we use the general Poisson-bracket formalism to derive the hydrodynamic equations of the system in the polarized state. The structure of the linear hydrodynamic modes confirms the existence of the splay instability.

cond-mat.soft↗

arXiv2006

Pattern stabilization through parameter alternation in a nonlinear optical system

We report the first experimental realization of pattern formation in a spatially extended nonlinear system when the system is alternated between two states, neither of which exhibits patterning. Dynamical equations modeling the system are used for both numerical simulations and a weakly nonlinear analysis of the patterned states. The simulations show excellent agreement with the experiment. The nonlinear analysis provides an explanation of the patterning under alternation and accurately predicts both the observed dependence of the patterning on the frequency of alternation, and the measured spatial frequencies of the patterns.

nlin.PS↗

arXiv2005

Elasticity, fluctuations and vortex pinning in ferromagnetic superconductors: A "columnar elastic glass"

We study the elasticity, fluctuations and pinning of a putative spontaneous vortex solid in ferromagnetic superconductors. Using a rigorous thermodynamic argument, we show that in the idealized case of vanishing crystalline pinning anisotropy the long-wavelength tilt modulus of such a vortex solid vanishes identically, as guaranteed by the underlying rotational invariance. The vanishing of the tilt modulus means that, to lowest order, the associated tension elasticity is replaced by the softer, curvature elasticity. The effect of this is to make the spontaneous vortex solid qualitatively more susceptible to the disordering effects of thermal fluctuations and random pinning. We study these effects, taking into account the nonlinear elasticity, that, in three dimensions, is important at sufficiently long length scales, and showing that a ``columnar elastic glass'' phase of vortices results. This phase is controlled by a previously unstudied zero-temperature fixed point and it is characterized by elastic moduli that have universal strong wave-vector dependence out to arbitrarily long length scales, leading to non-Hookean elasticity. We argue that, although translationally disordered for weak disorder, the columnar elastic glass is stable against the proliferation of dislocations and is therefore a topologically ordered {\em elastic} glass. As a result, the phenomenology of the spontaneous vortex state of isotropic magnetic superconductors differs qualitatively from a conventional, external-field-induced mixed state. For example, for weak external fields $H$, the magnetic induction scales {\em universally} like $B(H)\sim B(0)+ c H^α$, with $α\approx 0.72$.

cond-mat.supr-con↗

arXiv2003

Mean Field Theory of Collective Transport with Phase Slips

The driven transport of plastic systems in various disordered backgrounds is studied within mean field theory. Plasticity is modeled using non-convex interparticle potentials that allow for phase slips. This theory most naturally describes sliding charge density waves; other applications include flow of colloidal particles or driven magnetic flux vortices in disordered backgrounds. The phase diagrams exhibit generic phases and phase boundaries, though the shapes of the phase boundaries depend on the shape of the disorder potential. The phases are distinguished by their velocity and coherence: the moving phase generically has finite coherence, while pinned states can be coherent or incoherent. The coherent and incoherent static phases can coexist in parameter space, in contrast with previous results for exactly sinusoidal pinning potentials. Transitions between the moving and static states can also be hysteretic. The depinning transition from the static to sliding states can be determined analytically, while the repinning transition from the moving to the pinned phases is computed by direct simulation.

cond-mat.dis-nn↗

arXiv2003

Driven depinning of strongly disordered media and anisotropic mean-field limits

Extended systems driven through strong disorder are modeled generically using coarse-grained degrees of freedom that interact elastically in the directions parallel to the driving force and that slip along at least one of the directions transverse to the motion. A realization of such a model is a collection of elastic channels with transverse viscous couplings. In the infinite range limit this model has a tricritical point separating a region where the depinning is continuous, in the universality class of elastic depinning, from a region where depinning is hysteretic. Many of the collective transport models discussed in the literature are special cases of the generic model.

cond-mat.dis-nn↗

arXiv2002

Viscoelasticity from a Microscopic Model of Dislocation Dynamics

It is shown that the dynamics of a two-dimensional crystal with a finite concentration of dislocations, as well as vacancy and interstitial defects, is governed by the hydrodynamic equations of a viscoelastic medium. At the longest length scales the viscoelasticity is described by the simplest Maxwell model, whose shear and compressional relaxation times are obtained in terms of microscopic quantities, including the density of free dislocations. At short length scales, bond orientational order effects become important and lead to wavevector dependent corrections to the relaxation times.

cond-mat.soft↗

arXiv2001

"Soft" Anharmonic Vortex Glass in Ferromagnetic Superconductors

Ferromagnetic order in superconductors can induce a {\em spontaneous} vortex (SV) state. For external field ${\bf H}=0$, rotational symmetry guarantees a vanishing tilt modulus of the SV solid, leading to drastically different behavior than that of a conventional, external-field-induced vortex solid. We show that quenched disorder and anharmonic effects lead to elastic moduli that are wavevector-dependent out to arbitrarily long length scales, and non-Hookean elasticity. The latter implies that for weak external fields $H$, the magnetic induction scales {\em universally} like $B(H)\sim B(0)+ c H^α$, with $α\approx 0.72$. For weak disorder, we predict the SV solid is a topologically ordered vortex glass, in the ``columnar elastic glass'' universality class.

cond-mat.soft↗

arXiv2000

A Discotic Disguised as a Smectic: A Hybrid Columnar Bragg Glass

We show that discotics, lying deep in the columnar phase, can exhibit an x-ray scattering pattern which mimics that of a somewhat unusual smectic liquid crystal. This exotic, new glassy phase of columnar liquid crystals, which we call a ``hybrid columnar Bragg glass'', can be achieved by confining a columnar liquid crystal in an anisotropic random environment of e.g., strained aerogel. Long-ranged orientational order in this phase makes {\em single domain} x-ray scattering possible, from which a wealth of information could be extracted. We give detailed quantitative predictions for the scattering pattern in addition to exponents characterizing anomalous elasticity of the system.

cond-mat.soft↗

arXiv1999

Two new topologically ordered glass phases of smectics confined in anisotropic random media

We show that smectic liquid crystals confined in_anisotropic_ porous structures such as e.g.,_strained_ aerogel or aerosil exhibit two new glassy phases. The strain both ensures the stability of these phases and determines their nature. One type of strain induces an ``XY Bragg glass'', while the other creates a novel, triaxially anisotropic ``m=1 Bragg glass''. The latter exhibits anomalous elasticity, characterized by exponents that we calculate to high precision. We predict the phase diagram for the system, and numerous other experimental observables.

cond-mat.soft↗