From Van der Waals' equation to the scaling laws

Two order parameters for the Kuramoto model on complex networks

We investigate the behavior of two different order parameters for the Kuramoto model in the desynchronized phase. Since the primary role of the order parameter is to distinguish different phases, we focus on the ability to discern the desynchronized phase from the synchronized one on complex networks with the size N. From the exact derivation of the difference between two order parameters, Δ, on a star network, we find that these order parameters disagree in the desynchronized phase. We also show that the hub plays an important role and provide an analytic conjecture on the condition that the two order parameters agree with each other as N→∞. The conjecture is numerically confirmed.

Master crossover functions for one-component fluids

By introducing three well-defined dimensionless numbers, we establish the link between the scale dilatation method able to estimate master (i.e., unique) singular behaviors of the one-component fluid subclass and the universal crossover functions recently estimated [Garrabos and Bervillier, Phys. Rev. E 74, 021113 (2006)] from the bounded results of the massive renormalization scheme applied to the Phi(d)(4)(n) model of scalar order parameter (n=1) and three dimensions (d=3), representative of the Ising-like universality class. The master (i.e., rescaled) crossover functions are then able to fit the singular behaviors of any one-component fluid without adjustable parameter, using only one critical energy scale factor, one critical length scale factor, and two dimensionless asymptotic scale factors, which characterize the fluid critical interaction cell at its liquid-gas critical point. An additional adjustable parameter accounts for quantum effects in light fluids at the critical temperature. The effective extension of the thermal field range along the critical isochore where the master crossover functions seems to be valid corresponds to a correlation length greater than three times the effective range of the microscopic short-range molecular interaction.

Nature of vapor-liquid asymmetry in fluid criticality

We have investigated the nature and experimental consequences of vapor-liquid asymmetry in near-critical fluids within the framework of "complete scaling" [M. E. Fisher and G. Orkoulas, Phys. Rev. Lett. 85, 696 (2000); Y. C. Kim, Phys. Rev. E 67, 061506 (2003)]. We used the thermodynamic freedom for a choice of the critical-entropy value to simplify "complete scaling" to a form with only two independent parameters, responsible for two different sources of the asymmetry. We then developed a procedure to obtain these two parameters from mean-field equations of state. By combining accurate liquid-vapor coexistence and heat-capacity data, we have unambiguously separated two nonanalytic contributions from the two sources of vapor-liquid asymmetry and proved the validity of "complete scaling." Since the nonanalytic asymmetry effects in the critical region are fully determined by the Ising critical exponents for the symmetric lattice-gas model, there is no need for a special renormalization-group theoretical treatment of "non-Ising" asymmetry in fluid criticality.

Generic van der Waals Equation of State, Modified Free Volume Theory of Diffusion, and Viscosity of Simple Liquids

The shear viscosity formula derived by the density fluctuation theory in previous papers is computed for argon, krypton, and methane by using the self-diffusion coefficients derived in the modified free volume theory with the help of the generic van der Waals equation of state. In the temperature regime near or above the critical temperature, the density dependence of the shear viscosity can be accounted for by ab initio calculations with the self-diffusion coefficients provided by the modified free volume theory if the minimum (critical) free volume is set equal to the molecular volume and the volume overlap parameter (alpha) is taken about unity in the expression for the self-diffusion coefficient. In the subcritical temperature regime, if the density fluctuation range parameter is chosen appropriately at a temperature, then the resulting expression for the shear viscosity can well account for its density and temperature dependence over the ranges of density and temperature experimentally studied. In the sense that once the density fluctuation range is fixed at a temperature, the theory can account for the experimental data at other subcritical temperatures on the basis of the intermolecular force only; the theory is predictive even in the subcritical regime of temperature. Theory is successfully tested in comparison with experimental data for self-diffusion coefficients and shear viscosity for argon, krypton, and methane.

The theory of everything

We discuss recent developments in our understanding of matter, broadly construed, and their implications for contemporary research in fundamental physics.