Thermodynamics nonlinear

Evans D J 1983 Computer experiment for nonlinear thermodynamics of Couette flow J. Chem. Phys. 78 3297-302... 

C. M. Marques, M. E. Cates. Nonlinear thermodynamic relaxation in living polymer systems. J Phys II (France) 7 489-492, 1991. 

Even if we make the stringent assumption that errors in the measurement of each variable ( >,. , M.2,...,N, j=l,2,...,R) are independently and identically distributed (i.i.d.) normally with zero mean and constant variance, it is rather difficult to establish the exact distribution of the error term e, in Equation 2.35. This is particularly true when the expression is highly nonlinear. For example, this situation arises in the estimation of parameters for nonlinear thermodynamic models and in the treatment of potentiometric titration data (Sutton and MacGregor. 1977 Sachs. 1976 Englezos et al., 1990a, 1990b). 

Nonlinear Hamiltonian system, geometric transition state theory, 200-201 Nonlinear thermodynamics coefficients linear limit, 36 entropy production rate, 39 parity, 28-29... 

The apparent dispersion coefficient in Equation 10.8 describes the zone spreading observed in linear chromatography. This phenomenon is mainly governed by axial dispersion in the mobile phase and by nonequilibrium effects (i.e., the consequence of a finite rate of mass transfer kinetics). The band spreading observed in preparative chromatography is far more extensive than it is in linear chromatography. It is predominantly caused by the consequences of the nonlinear thermodynamics, i.e., the concentration dependence of the velocity associated to each concentration. When the mass transfer kinetics is fast, the influence of the apparent axial dispersion is small or moderate and results in a mere correction to the band profile predicted by thermodynamics alone. 

It is essential to note that nonlinear thermodynamics of nonequilib rium processes critically change the status of the Second Law of thermody namics. It appeared that in open systems that are far from equilibrium, this law determines not only the inevitability of the destruction of equilibrium... 

The Onsager reciprocal relations are not satisfied in open strongly non equilibrium systems. As a result, the assumption on minimization of the entropy production rate is not substantiated. Therefore, the universal criterion of the system that is evolution far from equilibrium should be a generalization of the principle of the minimized entropy production rate in specific terms of nonlinear thermodynamics. 

A fundamental corollary of the Glansdorf Prigogine criterion (3.2) is a potentiality of the formation of ordered structures at the occurrence of irreversible processes in the region of nonlinear thermodynamics in open systems that are far from their equilibrium. Prigogine created the term dissipative structures to describe the structures that arise when some controlling parameters exceed certain critical values and are classified as spatial, temporal, or spatial temporal. Some typical dissipative structures are discussed in Sections 3.5 and 4.6. 

The stationary states that faU into fragment 1 of the curve (Figure 3.5) are stable at minor deviations of a from (1q in virtue of the theorem on the minimal rates of entropy production in these states. On further run ning away from the point a = aq, we may faU outside the region of applicability of nonlinear thermodynamics while remaining in the thermodynamic branch, which is described, for example, by a station ary state functional as a kind of positively defined Lyapunov function... 

The relative importance of the contributions of the nonlinear thermodynamics of phase equilibrium and of the finite mass transfer kinetics to the band profile is evidenced and quantified by the numerical value of e. When e = 0, Cg = cjsiCm), and Eqs. 10.59a and 10.59b reduce to Eq. 10.58 by summing them up, which gives the equation of ideal chromatography. This asymptotic result can be justified in a few cases. On the other hand, the slower the mass transfer kinetics, the larger will be the value of e. 

Equation 10.115 has a considerable fundamental and practical importance. It combines parameters of fimdamentally different origins, the plate number at infinite dilution, N, which characterizes the intensity of axial dispersion taking place in the column and two parameters of thermod5mamic origin, the retention factor at infinite dilution, ICg, related to the initial slope of the isotherm, and the loading factor, proportional to the sample size and related to the saturation capacity of the isotherm. Accordingly, Eq. 10.115 indicates the extent to which the self-sharpening effect on the band profile due to the nonlinear thermodynamics is balanced by the dispersive effect of axial and eddy diffusion and of the mass transfer resistances. 

Another important relationship between the kinetic coefficients is the so-called principle of symmetry , as formulated by P. Curie and introduced to nonlinear thermodynamics by Kondepudi and Prigogine [37]. As applied to thermodynamics, this postulates that a scalar quantity could not evoke a vector effect. For example, a scalar thermodynamic force - chemical affinity (driving the process of chemical reaction) that has very high isotropy symmetry - could not cause heat flow, which has a particular direction and is therefore anisotropic. Taking into account the reciprocal relationships, this can be formulated as... 

Parameter Estimation in Nonlinear Thermodynamic Modeis Cubic Equations of State... 

The two described effects origin from nonlinear thermodynamic interactions between the components. If in a concrete separation the peak resolution is not high... 

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