Onsager's Theory Of Irreversible Thermodynamics Research Articles

Dynamic dipolar and quadrupolar susceptibilities for the spin-1 Blume-Emery-Griffiths model based on Onsager theory of irreversible thermodynamics

Abstract The expressions for dynamic (complex) dipolar and quadrupolar susceptibilities are obtained within the framework of the linear theory of irreversible thermodynamics in the Blume-Emery-Griffiths model. Frequency, temperature as well as crystal field dependences of the dispersion relations and absorption factors are investigated for two different phase diagram topologies which take place for K/J = 3 and K/J = 5.0. Their behaviors near the second- and first-order transition points as well as multi-critical points such as tricritical, triple and critical endpoint are presented.

The crystal-field dependency of sound attenuation in the spin-3/2 ising model

The effects of crystal-field (D) on sound attenuation are considered for the spin-3/2 Ising model by using Onsager theory of irreversible thermodynamics. It is assumed that the sound wave couples to the order-parameter fluctuations, therefore, it decays mainly via order-parameter relaxation process. The order-parameters, magnetization and quadrupole moment, are defined in terms of exact recursion relations (ERR) on Bethe lattice (BL). After our analysis, two relaxation times are obtained and they are used to calculate the sound attenuation coefficient (α). Consequently, the critical behaviors of sound attenuation coefficient are investigated in terms of frequency (w) and Onsager coefficient (γ) for the coordination numbers q=3, 4 ad 6 near the second-order (Tc) and first-order (Tt) phase transition temperatures in the ferromagnetic phase regions for the negative and positive D values and the results are presented on the (kBT/J, α) planes. It is found that the peaks about Tt’s are observed at the same temperature, but the peaks about Tc’s are observed shifted to lower and higher temperatures in increasing (w)’s and (γ)’s, respectively. In addition, the peaks are also obtained near the tricritical points for all q.

The crystal field effects on sound attenuation for a spin-1 Ising model on the Bethe lattice

The sound attenuation phenomenon is investigated by using the Onsager theory of irreversible thermodynamics for a spin-1 Ising model with the inclusion of the crystal field effects on the Bethe lattice. The recursion relations are calculated in a transcendental form to obtain the order-parameters and then the sound attenuation is analyzed. The relationships of sound attenuation with temperature, frequency and Onsager coefficient are examined near the second- and first-order phase transition temperatures, Tc and Tt respectively, for given negative crystal field values and coordination numbers on the Bethe lattice. The sound wave couples to the order-parameter fluctuations which decay mainly via the order-parameter relaxation process. Thus, two relaxation times are obtained which are used to find an expression for the sound attenuation coefficient. The attenuation maxima are found near the second- and first-order phase transition temperatures in the ferromagnetic and quadrupole phase regions, respectively, for the coordination numbers q = 3,4 and 6. The attenuation peaks are observed at the same temperature before Tt and are found to be shifted to lower (higher) temperatures with increasing value of frequency (Onsager coefficient) before Tc for any crystal field values. The attenuation peaks are found at lower values and at higher temperatures with negatively increasing crystal field values in the quadrupolar phase regions. In addition, the sound attenuation peaks are also studied at some tricritical points for q = 3,4 and 6 for some critical values of the crystal field.

The Bethe lattice treatment of sound attenuation for a spin- 3/2 Ising model

The sound attenuation phenomena is investigated for a spin- 3/2 Ising model on the Bethe lattice in terms of the recursion relations by using the Onsager theory of irreversible thermodynamics. The dependencies of sound attenuation on the temperature (T), frequency (w), Onsager coefficient (γ) and external magnetic field (H) near the second-order (Tc) and first-order (Tt) phase transition temperatures are examined for given coordination numbers q on the Bethe lattice. It is assumed that the sound wave couples to the order-parameter fluctuations which decay mainly via the order-parameter relaxation process, thus two relaxation times are obtained and which are used to obtain an expression for the sound attenuation coefficient (α). Our investigations revealed that only one peak is obtained near Tt and three peaks are found near Tc when the Onsager coefficient is varied at a given constant frequency for q=3. Fixing the Onsager coefficient and varying the frequency always leads to two peaks for q=3,4 and 6 near Tc. The sound attenuation peaks are observed near Tt at lower values of external magnetic field, but as it increases the sound attenuation peaks decrease and eventually disappear.

The Sound Attenuation for the Spin-1 Ising Model on the Bethe Lattice

The temperature ( T ), frequency ( w ), Onsager coefficient (γ) and external magnetic field ( H ) dependencies of the sound attenuation near the phase transition temperature ( T c ) is studied for the spin-1 Ising model on the Bethe lattice for given coordination numbers q in terms of the recursion relations by using Onsager theory of irreversible thermodynamics. It was assumed that the sound wave couples to the order-parameter fluctuations which decay mainly via the order-parameter relaxation process. Two relaxation times are obtained and which are used to find an expression for the sound attenuation coefficient (α). It was found that α presents only one peak for all frequencies when q =6, and two peaks for q =3 and 4 at low frequencies. In addition, α increases with the increasing values of frequency and with the decreasing values of external magnetic field.

Critical behaviors of the sound attenuation in a spin-1 Ising model

Critical behaviors of the sound attenuation in a spin-1 Ising system with bilinear (J) and biquadratic (K) interactions are investigated within the framework of cluster variation method and Onsager theory of irreversible thermodynamics. The sound wave is assumed to couple mainly to the order parameter fluctuations which decay via order parameter relaxation processes. Two relaxation times are obtained and an expression is found for the sound attenuation coefficient (α) in terms of these relaxation times. The temperature behavior of the sound attenuation near the phase transition temperatures (Tc) is analyzed according to various values of Onsager coefficients (γij) and sound frequency (ω). For T<Tc it is found that the maxima of the attenuation shifted to lower temperatures with increasing ω and γij (i≠j) values. For T>Tc the data give evidence that there is no relaxational contribution to sound attenuation coming from order parameter fluctuations. On the other hand, a convergence is found in attenuation just below the critical and the tricritical points as (Tc−T), while a jump-discontinuity is observed for the first-order behavior. The frequency variation of the sound attenuation is also investigated and in addition to ω2-attenuation dependence observed in the hydrodynamic regime it is observed that in the high frequency region the attenuation is independent of ω and the ratio of two interaction parameters (J/K).

Sound dispersion in a spin-1 Ising system near the second-order phase transition point

Sound dispersion relation is derived for a spin-1 Ising system and its behaviour near the second-order phase transition point or the critical point is analyzed. The method used is a combination of molecular field approximation and Onsager theory of irreversible thermodynamics. If we assume a linear coupling of sound wave with the order parameter fluctuations in the system, we find that the dispersion which is the relative sound velocity change with frequency behaves as ω 0 ε 0, where ω is the sound frequency and ε the temperature distance from the critical point. In the ordered region, one also observes a frequency-dependent velocity or dispersion minimum which is shifted from the corresponding attenuation maxima. These phenomena are in good agreement with the calculations of sound velocity in other magnetic systems such as magnetic metals, magnetic insulators, and magnetic semiconductors.

Comparison of Macro- and Microscopic Theories Describing Multicomponent Mass Transport in Microporous Media

A detailed discussion is presented of three well-known macroscopic theories for describing the mass transport behavior of multicomponent mixtures; these include Fick's law, the Onsager theory of irreversible thermodynamics, and the Maxwell−Stefan theory. The merits and drawbacks of these theories are discussed, together with their interrelation. These macroscopic theories and an entirely microscopic theory are applied to describe the transport of a mixture of mobile species in a microporous material obeying ideal Langmuir sorption behavior, which is an issue of considerable interest in the field of membrane technology. If a unary system, i.e., a single mobile species, is considered, the chemical diffusion coefficient appears to be independent of the concentration. In this case, mass transport can simply be described using Fick's law. For a multicomponent mixture, the chemical diffusion coefficients become a function of composition and Fick's law can no longer be used in a straightforward way. The two remaining macroscopic theories and the microscopic theory can adequately describe the mass transport behavior, assuming that mechanical interactions between the mobile species are negligible. When mechanical interactions are not negligible, only the Maxwell−Stefan theory yields a proper description. Applying the Onsager theory of irreversible thermodynamics in that case would lead to off-diagonal transport coefficients that are nonzero. Also, the microscopic theory used in this study is then no longer applicable, since it simply neglects interactions between mobile species. It is demonstrated that the use of virtual chemical potentials for the mobile species, by contrast with real chemical potentials of building units, complicates the description of mass transport.

Variational properties of irreversible processes

The thermodynamic integral principle, equivalent to the Onsager theory of irreversible thermodynamics, is analyzed in detail for a purely dissipative system. Different reformulations of the principle are also given together with the derivation of the corresponding Euler—Lagrange equations. One of them, the dual field formulation, is of special interest: It is an exact variational principle in terms of the intensive parameters and their dual fields introduced in place of the thermodynamic current densities. Finally, the possibility of deducing variational statements in terms of volume and surface dissipation functionals is discussed.

On the characterization of fluxes in nonlinear irreversible thermodynamics

The linear Onsager theory of irreversible thermodynamics is extended to include nonlinear phenomenological relations by means of Onsager fluxes. Such fluxes satisfy a full system of reciprocity relations, vanish in thermodynamic equilibrium, and give a non-negative production of entropy. A complete characterization of Onsager fluxes is obtained in terms of non-negative scalar valued functions which vanish in thermodynamic equilibrium. These same functions are also shown to characterize all C 2 fluxes which satisfy the second law of thermodynamics. Each system of Onsager fluxes is shown to derive from a dissipation function which attains its absolute minimum in thermodynamic equilibrium. The reaction rates given by reaction kinetics are shown to be Onsager fluxes and their dissipation functions are explicitly calculated.