Thermodynamic Response Functions Research Articles

Instabilities of hydrogen bond network in liquid water

A new model of the hydrogen bond network in liquid water is developed. Using the technique of polymer science, the grand canonical partition function of water is expressed by the functional integral of the field variables. The method is applied to anomalies of liquid water. Using the mean-field lattice-gas approximation it is shown that there are two instability lines, one in the supercooled region and the other in the stretched (P<0) region. Approaching the instability line in the supercooled region, the fluctuation of ice-likeness grows anomalously large and liquid water becomes unstable to be ice. Approaching the instability line in the stretched region, the fluctuation of the concentration of non-hydrogen-bonded sites grows large and liquid water becomes unstable to be gas. Divergence of the thermodynamic response functions and the mean-field values of exponents are discussed.

Isothermal molecular-dynamics ensembles.

S. Nose [Mol. Phys. 52, 255 (1984); J. Chem. Phys. 81, 511 (1984)] has developed a form of molecular dynamics that generates the canonical ensemble. Since molecular-dynamics calculations are carried out at constant total linear momentum, we investigate the necessary modifications in Nose’s theory when the constraint of constant total linear momentum is imposed. We show that Nose’s theory can be modified so that it generates phase-space trajectories corresponding to a constant-momentum canonical ensemble with zero total momentum. We also present an exact treatment of the statistical mechanics of this constant-momentum canonical ensemble, including expressions relating averages to thermodynamic variables and formulas involving thermodynamic response functions which are often used in molecular-dynamics calculations.

Fundamental treatment of molecular-dynamics ensembles.

In microcanonical molecular dynamics the conservation of total momentum implies that the trajectories generate an ensemble with constant total momentum. We present the exact statistical mechanics of this microcanonical molecular dynamics ensemble. Using the adiabatic invariant phase volume to define the entropy, exact formulas, containing thermodynamic response functions, are derived for the microcanonical molecular dynamics ensemble and a numerical comparison of these formulas is made with fluctuation formulas which are valid in the thermodynamic limit and often used in molecular dynamics. No significant difference is found between the results calculated using the different formulas for a system of 432 particles; however, for simulations with small numbers of particles or small-cluster studies, the differences could be important.

Exact solution of the impure one-dimensional n-vector model with bilinear and biquadratic exchange interactions

Abstract We demonstrate explicitly that the impure one-dimensional n-vector model with bilinear and biquadratic exchange interactions can be solved exactly. Results are provided for the free energy, equal-time spin-spin correlation functions and the initial susceptibility of the corresponding bond and site model in the annealed and quenched limit. A numerical analysis of the thermodynamic response functions in the n = 3 case is performed, and therein some interesting consequences of the simultaneous influence of the biquadratic interactions and impurities are established. The presence of the biquadratic interaction is interpreted as an effect of the lattice compressibility.

Symmetry properties and the two-phase coexistence curve of a two-dimensional bonded lattice fluid

By examining the symmetry properties of a two-dimensional bonded lattice fluid we are able to obtain in a simple way the two-phase coexistence curve and the line of symmetry. The authors show that, in the density-temperature plane, the line of symmetry, the analytically continued two-phase diameter and the curve on which the chemical potential is equal to the analytic continuation of its values along the coexistence curve are all distinct. These curves intersect at the critical point in nonzero angles. They examine the behaviour of the thermodynamic response functions on the coexistence curve and show that on the high-density branch of the curve, the isothermal compressibility and the coefficient of isobaric thermal expansion exhibit anomalies similar to those shown by the water system. They show also that the discontinuous change in the constant pressure heat capacity as the system changes from the high-density to the low-density phase becomes negative below a certain temperature.

Instantaneous and equilibrium Grüneisen parameters for a nonlinear viscoelastic polymer

In studying the propagation of shock waves in nonlinear viscoelastic solids, it is useful to characterize the mechanical response of the material in terms of instantaneous and equilibrium stress-strain functions. A natural extension of this concept to include thermodynamic influences leads to the consideration of instantaneous and equilibrium thermodynamic response functions for the stress and internal energy. An important factor in the thermoviscoelastic behavior concerns the stress/energy response as characterized by the Grüneisen parameter, or in the present context, an instantaneous and equilibrium parameter. An example of these parameters and their temperature dependence is given for the solid polymer, polymethyl methacrylate, based on a thermoviscoelastic constitutive model developed for predicting steady shock wave propagation.

Response functions, superconductivity and ferromagnetism in the many fermion system

At the phase transition, the anomalies of thermodynamic and static response functions coincide due to the fact that the free energy, expressed in terms of response functions at imaginary frequency including zero, has its pole at zero frequency. Previous work has considered the singularities which arise from the polarisation of the medium. In this paper, a study is made of the divergences arising from the modification of the vertex function. The diamagnetic current is related to the free energy via the Ward Identity so that the specific heat anomaly inevitably gives rise to the Meissner effect. Below the transition the system adjusts to the singularity by going into the BCS condensed state. The similarity between BCS theory and ferromagnetism is shown both through a Weiss field approach and diagrammatically by including special correlations between k and - k. Its importance for an understanding of the broken symmetry problem is discussed. Finally, the parallel behaviour of the susceptibility and specific heat of a Bloch ferromagnet is demonstrated.

Time-Dependent Correlations in a Solvable Ferromagnetic Model

The exact equation of state, two-time correlation function, and linear response function, are calculated in the limit of infinite $N$ for a classical $N$-spin system with a ferromagnetic phase transition, in a particular nonuniform magnetic field. The correlation function can be analytically continued in temperature (or magneticfield strength) from the nonferromagnetic to the ferromagnetic region of the $T\ensuremath{-}H$ plane; the result of such continuation is not, however, the correlation function for the ferromagnetic region, but a function which grows exponentially in time. The frequency-dependent linear response function has a pole at zero frequency throughout the ferromagnetic region due to a broken symmetry; the corresponding function in the nonferromagnetic region developes a pole at zero frequency as the ferromagnetic region is approached, but when the function is continued in temperature (or field strength) into the ferromagnetic region, the pole detaches itself from the origin and moves up into the complex frequency plane, signifying an exponential growth in time of the linear response. The purpose of the model is to demonstrate that this kind of behavior does not contradict any general structural properties of equilibrium thermodynamic correlation or response functions. The possible general significance of such behavior for a theory of metastable states is discussed.