Square-Lattice Ising Model in a Weak Uniform Magnetic Field: Renormalization Group Analysis

For the two-dimensional ferromagnetic Ising critical point, I show that the known values of the critical exponents imply the absence of logarithms of the reduced temperature in the leading contributions to any field derivative of the free energy at zero magnetic field. For the square-lattice Ising antiferromagnet in a weak magnetic field, I compute the critical line Tc(H)=Tc0(1-0.038 023 259H2) and the leading contribution to the susceptibility χ=0.014 718 006 6H2ln(1/ǁtǁ), where t is the reduced temperature

Equilibrium Polymerization on the Equivalent-Neighbor Lattice

The equilibrium polymerization problem is solved exactly on the equivalent-neighbor lattice. The Flory-Huggins entropy of mixing is exact for this lattice. We verify the discrete version of the n-vector model when n→0 is equivalent to the equal reactivity polymerization process in the whole parameter space, including the polymerized phase. The polymerization processes for polymers satisfying the Schulz distribution exhibit nonuniversal critical behavior. A close analogy is found between the polymerization problem of index r and the Bose-Einstein ideal gas in d=-2r dimensions, with the critical polymerization corresponding to the Bose-Einstein condensation

Entropy Driven Phase Transition in Polymer Gels: Mean Field Theory

We present a mean field model of a gel consisting of P polymers, each of length L and Nz polyfunctional monomers. Each polyfunctional monomer forms z covalent bonds with the 2P bifunctional monomers at the ends of the linear polymers. We find that the entropy dependence on the number of polyfunctional monomers exhibits an abrupt change at Nz = 2P/z due to the saturation of possible crosslinks. This non-analytical dependence of entropy on the number of polyfunctionals generates a first-order phase transition between two gel phases: one poor and the other rich in poly-functional molecules

Entropy Driven Phase Transition in Polymer Gels: Mean Field Theory

We present a mean field model of a gel consisting of P polymers, each of length L and Nz polyfunctional monomers. Each polyfunctional monomer forms z covalent bonds with the 2P bifunctional monomers at the ends of the linear polymers. We find that the entropy dependence on the number of polyfunctional monomers exhibits an abrupt change at Nz = 2P/z due to the saturation of possible crosslinks. This non-analytical dependence of entropy on the number of polyfunctionals generates a first-order phase transition between two gel phases: one poor and the other rich in poly-functional molecules

Equation of state from the Potts-percolation model of a solid

We expand the Potts-percolation model of a solid to include stress and strain. Neighboring atoms are connected by bonds. We set the energy of a bond to be given by the Lennard-Jones potential. If the energy is larger than a threshold the bond is more likely to fail, whereas if the energy is lower than the threshold, the bond is more likely to be alive. In two dimensions we compute the equation of state: stress as a function of interatomic distance and temperature by using renormalization-group and Monte Carlo simulations. The phase diagram, the equation of state, and the isothermal modulus are determined. When the Potts heat capacity is divergent the continuous transition is replaced by a weak first-order transition through the van der Waals loop mechanism. When the Potts transition is first order the stress exhibits a large discontinuity as a function of the interatomic distance

Dynamic Firm Location Network Model with Anticipatory Scenarios for the Northeast Ohio Region

Public policy and planning decisions require glimpses into the future, to assess how the social-ecological systems we plan for might evolve with or without policy intervention. To do so, one approach gaining currency is using anticipatory tools rather than predictions. Anticipation entails generating a range of possible systems futures (scenarios), instead of attempting to predict the one that will prevail. We use here a scenario-generating model, to anticipate where in a region businesses are likely to locate in time. Using data for Northeast Ohio, including the Cleveland–Akron–Lorain– Elyria, Ohio Combined Statistical Area, we estimate the model parameters. We evaluate its prediction accuracy against 2001–2015 regional data. To illustrate how policymakers could use the model, we generate three scenarios to explore what might happen to the spatial configuration of businesses if policies were implemented to attract businesses at specific locations or discourage them from locating in parts of the region

Radial Motion in A Central Potential for Singular Mass Densities

We study the radial motion of an object in the gravitational field produced by an isotropic mass density that is singular at the origin. This problem applies to elliptical galaxies and can be used to illustrate motion in a central field appropriate for an intermediate-level mechanics course

Multicritical Susceptibility Sum Rules

Asymptotically close to the Nth-order multicritical point of an N-phase system, there are N-1 sum rules involving the mean-field susceptibilities measured in each of the coexisting phases. These sum rules provide the experimentalist a convenient and stringent test of the theory. In particular, they facilitate the detection of nonclassical effects, especially valuable for N\u3e3 where the fluctuation effect is dominated by the classical contribution for three-dimensional systems

Random-Field Blume-Capel Model: Mean-Field Theory

The global phase diagram of the Blume-Capel model in a random field obeying the bimodal symmetric distribution is determined by using the mean-field method. The phase diagram includes an isolated ordered critical end point and two lines of tricritical points. A new phase emerges for strong enough random fields: the ferromagnetic-nonmagnetic phase. It is argued that such a phase occurs in three dimensions