The dynamic behaviour of the Pople and Karasz model

doi:10.1016/0022-3697(84)90139-2

Journal of Physics and Chemistry of Solids

Volume 45, Issues 8–9, 1984, Pages 955-962

https://doi.org/10.1016/0022-3697(84)90139-2 Get rights and content

Abstract

Using the Pople and Karasz model for the solidification of plastic crystals, we construct two different sets of dynamic equations for the translational and rotational order parameters. The first is a straight generalization of the Pople and Karasz model, whereby one coupling parameter is a function of the other, and vice versa. The second generalization is based on the most probable path method of Kikuchi. In order to accomplish this we start with an appropriate transformation of the parameters. It is then shown that it is necessary to incorporate the spacial-angular correlation in order to apply this method. Computations for both systems of equations are given to demonstrate the behaviour of the long range order parameter if it is initially far from equilibrium.

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Cited by (21)

Dynamic phase diagrams of the Blume-Capel model in an oscillating field by the path probability method
2014, Physica A: Statistical Mechanics and its Applications
We calculate the dynamic phase transition (DPT) temperatures and present the dynamic phase diagrams in the Blume–Capel model under the presence of a time-dependent oscillating external magnetic field by using the path probability method. We study the time variation of the average order parameters to obtain the phases in the system and the paramagnetic (P), ferromagnetic (F) and the F+P mixed phases are found. We also investigate the thermal behavior of the dynamic order parameters to analyze the nature (continuous and discontinuous) of transitions and to obtain the DPT points. We present the dynamic phase diagrams in three planes, namely (T, h), (d, T) and (k₂/k₁, T), where T is the reduced temperature, h the reduced magnetic field amplitude, d the reduced crystal-field interaction and the k₂, k₁ rate constants. The phase diagrams exhibit dynamic tricritical and reentrant behaviors as well as a double critical end point and triple point, strongly depending on the values of the interaction parameters and the rate constants. We compare and discuss the dynamic phase diagrams with dynamic phase diagrams that are obtained within the Glauber-type stochastic dynamics based on the mean-field theory and the effective field theory.
Multicritical phase diagrams of the ferromagnetic spin-frac(3, 2) Blume-Emery-Griffiths model with repulsive biquadratic coupling including metastable phases: The cluster variation method and the path probability method with the point distribution
2008, Journal of Magnetism and Magnetic Materials
We study the thermal variations of the ferromagnetic spin- $\frac{3}{2}$ Blume–Emery–Griffiths (BEG) model with repulsive biquadratic coupling by using the lowest approximation of the cluster variation method (LACVM) in the absence and presence of the external magnetic field. We obtain metastable and unstable branches of the order parameters besides the stable branches and phase transitions of these branches are investigated extensively. The classification of the stable, metastable and unstable states is made by comparing the free energy values of these states. We also study the dynamics of the model by using the path probability method (PPM) with the point distribution in order to make sure that we find and define the metastable and unstable branches of the order parameters completely and correctly. We present the metastable phase diagrams in addition to the equilibrium phase diagrams in the (kT/J, K/J) and (kT/J, D/J) planes. It is found that the metastable phase diagrams always exist at the low temperatures, which are consistent with experimental and theoretical works.
Solid-plastic and plastic-isotropic liquid phase transitions on molecular crystals
2005, Physica B: Condensed Matter
Crystalline solid–plastic solid and plastic solid–isotropic liquid phase transitions are studied using a modified Pople and Karasz model. The thermodynamic properties of the model are evaluated by the cluster variation method. The variation of positional and orientational order parameters with temperature is studied. We have also obtained theoretical isotherms, volume and entropy changes of transitions at zero pressure and theoretical phase diagrams at high pressures. The results are compared with the original Pople and Karasz theory and the other by a modified Pople and Karasz theory which was done by Chandrasekhar et al.
Study of melting of molecular crystals by a modified Pople-Karasz model
2005, Physica A: Statistical Mechanics and its Applications
A new modified model that combines the modified models of Chandrasekhar et al. with those of Keskin and Özgan, which are based on the Pople–Karasz theory, is applied to study the thermodynamics of melting and solid–solid transitions of molecular crystals. The thermodynamic properties of the disordered system are evaluated relative to those of the perfectly ordered one using the lowest approximation of the cluster-variation method, which is identical to the mean-field approximation. A good agreement is found between the present modified theory and the available experimental data. For melting transitions the agreement is excellent and much better than with the calculations of the Pople–Karasz theory and its previous modified theories. Approximate agreement is obtained for the solid–solid transitions. However, for these transition the experimental agreement with the present modified theory is still better than previous modified theories except at zero and low pressures.
Dynamics of voltage-gated ion channels in cell membranes by the path probability method
2004, Physica A: Statistical Mechanics and its Applications
Dynamics of voltage-gated ion channels in the excitable cell membranes is formulated by the path probability method of nonequilibrium statistical physics and approaches of the system toward the steady or equilibrium states are presented. For a single-particle noninteractive two-state model, a first-order rate equation or dynamic equation is derived by introducing the path probability rate coefficients which satisfy the detailed balancing relation. Using known parameters for the batrachotoxin (BTX)-modified sodium channels in giand squid axon as an example, the rate equation is solved and voltage dependence of the time constant (τ) and its temperature effect are investigated. An increase in voltage caused a shift in τ towards shorter durations while increasing temperature caused a shift in time distribution towards longer durations. Results are compared with the kinetic model for the squid axon BTX-modified sodium channels by the cut-open axon technique and a very good agreement is found.
Dynamic behaviour of the modified Pople-Karasz model
2003, Physica A: Statistical Mechanics and its Applications
Citation Excerpt :
On the other hand, it is a very good method for studying the metastable behaviour in Ising systems [21,22,39,41–43].
The dynamic behaviour of the modified Pople–Karasz model is studied by the direct relaxation method and the path probability method. First the equilibrium behaviour of the model is given briefly in order to understand the dynamic behaviour. Then the direct relaxation method, which is based on the detailed balance conditions, and the path probability method are applied to the model and the dynamic equations or the rate equations are obtained. Dynamic equations are solved either by means of the flow diagram or by using the Runge–Kutta method, or both. The stable, metastable and unstable solutions are shown in the flow diagrams explicitly, and the role of the unstable state as separators between the stable and metastable is described. Moreover, the “flatness” property of the metastable state and the “overshooting” phenomenon are seen explicitly from the relaxation curves of the order parameters.

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Journal of Physics and Chemistry of Solids

The dynamic behaviour of the Pople and Karasz model

Abstract

Physica A

J. Phys. Chem. Solids

Proc. Roy. Soc. A

Dynamic phase diagrams of the Blume-Capel model in an oscillating field by the path probability method

Multicritical phase diagrams of the ferromagnetic spin-frac(3, 2) Blume-Emery-Griffiths model with repulsive biquadratic coupling including metastable phases: The cluster variation method and the path probability method with the point distribution

Solid-plastic and plastic-isotropic liquid phase transitions on molecular crystals

Study of melting of molecular crystals by a modified Pople-Karasz model

Dynamics of voltage-gated ion channels in cell membranes by the path probability method

Dynamic behaviour of the modified Pople-Karasz model