The Mathematics of Chinese Checkers

Our goal for this project was to expand and improve upon the findings of George I. Bell and Nicholas Fonseca, who have both written papers on optimization in Chinese Checkers. While their work focuses mainly on cooperative games between one, two, and three players, we have considered games for six players. While doing this, we have redefined the playing board in a more intuitive manner, while developing and proving its associated distance formula. As well, we have found the shortest game for six players, and are working to generalize a formula for the number of moves required to finish a six player game as fast as possible. This could further incite research to generalize a lower bound for any number of players

Tracer diffusion in active suspensions

We study the diffusion of a Brownian probe particle of size $R$ in a dilute dispersion of active Brownian particles (ABPs) of size $a$, characteristic swim speed $U_0$, reorientation time $\tau_R$, and mechanical energy $k_s T_s = \zeta_a U_0^2 \tau_R /6$, where $\zeta_a$ is the Stokes drag coefficient of a swimmer. The probe has a thermal diffusivity $D_P = k_B T/\zeta_P$, where $k_B T$ is the thermal energy of the solvent and $\zeta_P$ is the Stokes drag coefficient for the probe. When the swimmers are inactive, collisions between the probe and the swimmers sterically hinder the probe's diffusive motion. In competition with this steric hindrance is an enhancement driven by the activity of the swimmers. The strength of swimming relative to thermal diffusion is set by $Pe_s = U_0 a /D_P$. The active contribution to the diffusivity scales as $Pe_s^2$ for weak swimming and $Pe_s$ for strong swimming, but the transition between these two regimes is nonmonotonic. When fluctuations in the probe motion decay on the time scale $\tau_R$, the active diffusivity scales as $k_s T_s /\zeta_P$: the probe moves as if it were immersed in a solvent with energy $k_s T_s$ rather than $k_B T$.Comment: 5 pages, 3 figures, submitted for publication. Please contact authors regarding supplemental informatio

Fluctuation-dissipation in active matter

In a colloidal suspension at equilibrium, the diffusive motion of a tracer particle due to random thermal fluctuations from the solvent is related to the particle’s response to an applied external force, provided this force is weak compared to the thermal restoring forces in the solvent. This is known as the fluctuation-dissipation theorem (FDT) and is expressed via the Stokes-Einstein-Sutherland (SES) relation D = k_BT/ζ, where D is the particle’s self-diffusivity (fluctuation), ζ is the drag on the particle (dissipation), and k_BT is the thermal Boltzmann energy. Active suspensions are widely studied precisely because they are far from equilibrium—they can generate significant nonthermal internal stresses, which can break the detailed balance and time-reversal symmetry—and thus cannot be assumed to obey the FDT a priori. We derive a general relationship between diffusivity and mobility in generic colloidal suspensions (not restricted to near equilibrium) using generalized Taylor dispersion theory and derive specific conditions on particle motion required for the FDT to hold. Even in the simplest system of active Brownian particles (ABPs), these conditions may not be satisfied. Nevertheless, it is still possible to quantify deviations from the FDT and express them in terms of an effective SES relation that accounts for the ABPs conversion of chemical into kinetic energy

Fluctuation-dissipation in active matter

In a colloidal suspension at equilibrium, the diffusive motion of a tracer particle due to random thermal fluctuations from the solvent is related to the particle’s response to an applied external force, provided this force is weak compared to the thermal restoring forces in the solvent. This is known as the fluctuation-dissipation theorem (FDT) and is expressed via the Stokes-Einstein-Sutherland (SES) relation D = k_BT/ζ, where D is the particle’s self-diffusivity (fluctuation), ζ is the drag on the particle (dissipation), and k_BT is the thermal Boltzmann energy. Active suspensions are widely studied precisely because they are far from equilibrium—they can generate significant nonthermal internal stresses, which can break the detailed balance and time-reversal symmetry—and thus cannot be assumed to obey the FDT a priori. We derive a general relationship between diffusivity and mobility in generic colloidal suspensions (not restricted to near equilibrium) using generalized Taylor dispersion theory and derive specific conditions on particle motion required for the FDT to hold. Even in the simplest system of active Brownian particles (ABPs), these conditions may not be satisfied. Nevertheless, it is still possible to quantify deviations from the FDT and express them in terms of an effective SES relation that accounts for the ABPs conversion of chemical into kinetic energy

The Mathematics of Chinese Checkers

Our goal for this project was to expand and improve upon the findings of George I. Bell and Nicholas Fonseca, who have both written papers on optimization in Chinese Checkers. While their work focuses mainly on cooperative games between one, two, and three players, we have considered games for six players. While doing this, we have redefined the playing board in a more intuitive manner, while developing and proving its associated distance formula. As well, we have found the shortest game for six players, and are working to generalize a formula for the number of moves required to finish a six player game as fast as possible. This could further incite research to generalize a lower bound for any number of players

Determining the Winner in a Graph Theory Game

We are investigating who has the winning strategy in a game in which two players take turns drawing arrows trying to complete cycle cells. The game boards are graphs, objects with dots and lines between them. A cycle cell looks like a polygon (triangle, square, pentagon, etc.). We examined game boards where the winning strategy was previously unknown. Starting with a pentagon and a heptagon glued by two sides, we worked to solve multiple classes of graphs involving stacked polygons. We also explored variations of the game where cycles, as defined in graph theory, are used in place of cycle cells, which opens the game up to non-planar graphs, such as complete graphs and gives the game a graph theory twist on top of topology. The original game was described by Francis Su in his book Mathematics for Human Flourishing

Single Particle Motion in Active Matter

"Active matter" refers to a broad class of materials in which the constituent particles or organisms are able to self-propel (swim) by some internal physicochemical mechanism. Though the origin of this self-propulsive motion is a rich area of study, we are primarily interested in the collective effects of this motion on the physical properties &#8212; and in particular, the rheology &#8212; of the active material as a whole. As such we model self-propulsive motion using the minimal active Brownian particle (ABP) model: a particle of size a , swims in a direction q with a speed U0, and the direction of its motion changes randomly over some time scale τR. On a macroscopic scale, active motion leads to unique hydrodynamic and mechanical stresses exerted by the particles on their embedding medium. These stresses arise from the microscopic force associated with particle locomotion &#8212; the swim force F swim. Though the idea of the swim force is widely recognized in the abstract, little attention has been given to the characterization and mechanical consequences of this force. In this work we are particularly interested the role of the swim force in the effective motion of passive constituents in active environments, and how the swim force affects long-ranged hydrodynamic interactions (HI) in active suspensions. We examine these issues through the lens of microrheology: tracking the motion of a colloidal probe particle through an active medium, and using its motions to infer the effective viscoelastic properties of the suspension. Using generalized Taylor dispersion theory, we find an activity-driven enhancement to the diffusion of the probe in an active medium. This first-principles theory unites many experimental observations of tracer diffusion, and provides simple physical descriptions of the problem that do not rely on the specific self-propulsion mechanism of the swimmer. This same framework is then used to compute the suspension microviscosity (as measured by the drag on the probe particle), and the fluctuation-dissipation relation in an active system. We find that activity reduces the drag on the probe, but the drag is still larger than it would be in a Newtonian fluid; this stands in contrast to experimental measurements of reduced shear viscosities. We show that the microviscosity of a suspension is reduced &#8212; and may even become negative! &#8212; due to HI, and that this effect is not due to the fluid velocity disturbance associated with the swimmers' self-propulsion.</p