Equilibria of biological aggregations with nonlocal repulsive-attractive interactions

We consider the aggregation equation $\rho_{t}-\nabla\cdot(\rho\nabla K\ast\rho) =0$ in $\mathbb{R}^{n}$, where the interaction potential $K$ incorporates short-range Newtonian repulsion and long-range power-law attraction. We study the global well-posedness of solutions and investigate analytically and numerically the equilibrium solutions. We show that there exist unique equilibria supported on a ball of $\mathbb{R}^n$. By using the method of moving planes we prove that such equilibria are radially symmetric and monotone in the radial coordinate. We perform asymptotic studies for the limiting cases when the exponent of the power-law attraction approaches infinity and a Newtonian singularity, respectively. Numerical simulations suggest that equilibria studied here are global attractors for the dynamics of the aggregation model

Thermoelastic Equation of State of Boron Suboxide B6O up to 6 GPa and 2700 K: Simplified Anderson-Gr\"uneisen Model and Thermodynamic Consistency

p-V-T equation of state of superhard boron suboxide B6O has been measured up to 6 GPa and 2700 K using multianvil technique and synchrotron X-ray diffraction. To fit the experimental data, the theoretical p-V-T equation of state has been derived in approximation of the constant value of the Anderson-Gr\"uneisen parameter {\delta}T. The model includes bulk modulus B0 =181 GPa and its first pressure derivative B0' = 6 at 300 K; two parameters describing thermal expansion at 0.1 MPa, i.e. a = 1.4x10-5 K-1 and b = 5x10-9 K-2, as well as {\delta}T = 6. The good agreement between fitted and experimental isobars has been achieved to the absolute volume changes up to 5% as compared to volume at standard conditions, V0. The fitted thermal expansion at 0.1 MPa is well consistent with the experimental data, as well as with ambient-pressure heat capacity cp, bulk modulus B0 and {\delta}T describing its evolution with volume and temperature. The fitted value of Gr\"uneisen parameter {\gamma} = 0.85 is in agreement with previous empiric estimations for B6O and experimental values for other boron-rich solids

Renormalized Equilibria of a Schloegl Model Lattice Gas

A lattice gas model for Schloegl's second chemical reaction is described and analyzed. Because the lattice gas does not obey a semi-detailed-balance condition, the equilibria are non-Gibbsian. In spite of this, a self-consistent set of equations for the exact homogeneous equilibria are described, using a generalized cluster-expansion scheme. These equations are solved in the two-particle BBGKY approximation, and the results are compared to numerical experiment. It is found that this approximation describes the equilibria far more accurately than the Boltzmann approximation. It is also found, however, that spurious solutions to the equilibrium equations appear which can only be removed by including effects due to three-particle correlations.Comment: 21 pages, REVTe

Ancient multiple-layer solutions to the Allen-Cahn equation

We consider the parabolic one-dimensional Allen-Cahn equation $$u_t= u_{xx}+ u(1-u^2)\quad (x,t)\in \mathbb{R}\times (-\infty, 0].$$ The steady state $w(x) =\tanh (x/\sqrt{2})$, connects, as a "transition layer" the stable phases $-1$ and $+1$. We construct a solution $u$ with any given number $k$ of transition layers between $-1$ and $+1$. At main order they consist of $k$ time-traveling copies of $w$ with interfaces diverging one to each other as $t\to -\infty$. More precisely, we find $$u(x,t) \approx \sum_{j=1}^k (-1)^{j-1}w(x-\xi_j(t)) + \frac 12 ((-1)^{k-1}- 1)\quad \hbox{as} t\to -\infty,$$ where the functions $\xi_j(t)$ satisfy a first order Toda-type system. They are given by $$\xi_j(t)=\frac{1}{\sqrt{2}}\left(j-\frac{k+1}{2}\right)\log(-t)+\gamma_{jk},\quad j=1,...,k,$$ for certain explicit constants $\gamma_{jk}.

An Empirical Study of Finding Approximate Equilibria in Bimatrix Games

While there have been a number of studies about the efficacy of methods to find exact Nash equilibria in bimatrix games, there has been little empirical work on finding approximate Nash equilibria. Here we provide such a study that compares a number of approximation methods and exact methods. In particular, we explore the trade-off between the quality of approximate equilibrium and the required running time to find one. We found that the existing library GAMUT, which has been the de facto standard that has been used to test exact methods, is insufficient as a test bed for approximation methods since many of its games have pure equilibria or other easy-to-find good approximate equilibria. We extend the breadth and depth of our study by including new interesting families of bimatrix games, and studying bimatrix games upto size $2000 \times 2000$. Finally, we provide new close-to-worst-case examples for the best-performing algorithms for finding approximate Nash equilibria

Two-population replicator dynamics and number of Nash equilibria in random matrix games

We study the connection between the evolutionary replicator dynamics and the number of Nash equilibria in large random bi-matrix games. Using techniques of disordered systems theory we compute the statistical properties of both, the fixed points of the dynamics and the Nash equilibria. Except for the special case of zero-sum games one finds a transition as a function of the so-called co-operation pressure between a phase in which there is a unique stable fixed point of the dynamics coinciding with a unique Nash equilibrium, and an unstable phase in which there are exponentially many Nash equilibria with statistical properties different from the stationary state of the replicator equations. Our analytical results are confirmed by numerical simulations of the replicator dynamics, and by explicit enumeration of Nash equilibria.Comment: 9 pages, 2x2 figure