Nonequilibrium thermodynamics approaches

The next three chapters deal with the most widely used classes of methods free energy perturbation (FEP) [3], methods based on probability distributions and histograms, and thermodynamic integration (TI) [1, 2], These chapters represent a mix of traditional material that has already been well covered, as well as the description of new techniques that have been developed only recendy. The common thread followed here is that different methods share the same underlying principles. Chapter 5 is dedicated to a relatively new class of methods, based on calculating free energies from nonequilibrium dynamics. In Chap. 6, we discuss an important topic that has not received, so far, sufficient attention - the analysis of errors in free energy calculations, especially those based on perturbative and nonequilibrium approaches. 

The review is organized as follows In Section 2 we present the multiscale equilibrium thermodynamics in the setting of contact geometry. The time evolution (multiscale nonequilibrium thermodynamics) representing approach of a mesoscopic level LmeSoi to the level of equilibrium thermodynamics Leth is discussed in Section 3. A generalization in which the level Leth is replaced by another mesoscopic level LmesoZ is considered in Section 4. The notion of multiscale thermodynamics of systems arises in the analysis of this type of time evolution. 

The important argument in favor of fractal approach application is the usage of two order parameter values, which are necessary for correct description of polymer mediums structure and properties features. As it is known, solid phase polymers are thermodynamically nonequilibrium mediums, for which Prigogine-Defay criterion is not fulfilled, and therefore, two order parameters are required, as a minimum, for their structure description. In its turn, one order parameter is required for Euclidean object characterization (its Euclidean dimension d). In general case three parameters (dimensions) are necessary for fractal object correct description dimension of Euclidean space d, fractal (Hausdorff) object dimension d and its spectral (fraction)... 

In this review we put less emphasis on the physics and chemistry of surface processes, for which we refer the reader to recent reviews of adsorption-desorption kinetics which are contained in two books [2,3] with chapters by the present authors where further references to earher work can be found. These articles also discuss relevant experimental techniques employed in the study of surface kinetics and appropriate methods of data analysis. Here we give details of how to set up models under basically two different kinetic conditions, namely (/) when the adsorbate remains in quasi-equihbrium during the relevant processes, in which case nonequilibrium thermodynamics provides the needed framework, and (n) when surface nonequilibrium effects become important and nonequilibrium statistical mechanics becomes the appropriate vehicle. For both approaches we will restrict ourselves to systems for which appropriate lattice gas models can be set up. Further associated theoretical reviews are by Lombardo and Bell [4] with emphasis on Monte Carlo simulations, by Brivio and Grimley [5] on dynamics, and by Persson [6] on the lattice gas model. 

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

Because the focus is on a single, albeit rather general, theory, only a limited historical review of the nonequilibrium field is given (see Section IA). That is not to say that other work is not mentioned in context in other parts of this chapter. An effort has been made to identify where results of the present theory have been obtained by others, and in these cases some discussion of the similarities and differences is made, using the nomenclature and perspective of the present author. In particular, the notion and notation of constraints and exchange with a reservoir that form the basis of the author s approach to equilibrium thermodynamics and statistical mechanics [9] are used as well for the present nonequilibrium theory. 

Arguably a more practical approach to higher-order nonequilibrium states lies in statistical mechanics rather than in thermodynamics. The time correlation function gives the linear response to a time-varying field, and this appears in computational terms the most useful methodology, even if it may lack the... 

Two major approaches to describing combined convection and diffusion have been used by pharmaceutical scientists. These are the convective diffusion approach and the nonequilibrium thermodynamics approach, described in Sections IV.A.l and IV.A.2, respectively. 

The convective diffusion equations presented above have been used to model tablet dissolution in flowing fluids and the penetration of targeted macro-molecular drugs into solid tumors [5], In comparison with the nonequilibrium thermodynamics approach described below, the convective diffusion equations have the advantage of theoretical rigor. However, their mathematical complexity dictates a numerical solution in all but the simplest cases. 

Nonequilibrium thermodynamics provides a second approach to combined convection and diffusion problems. The Kedem-Katchalsky equations, originally developed to describe combined convection and diffusion in membranes, form the basis of this approach [6,7] ... 

In the pharmaceutical sciences, the nonequilibrium thermodynamics approach has been particularly important in the design of osmotic drug delivery devices, as discussed in Chapter 11. It has also been used to describe the convective transport of a binding antibody in an in vitro model of a solid tumor [8], As our appreciation of the roles of convection and osmosis in drug delivery increases, the nonequilibrium thermodynamics approach may find wider appeal. 

Under those conditions P behaves as a Lagrangian in mechanics. Furthermore, as P is a nonnegative function for any positive value of the concentrations X,, by a theorem due to Lyapounov, the asymptotic stability of nonequilibrium steady states is ensured (theorem of minimum entropy production.1-23 These steady states are thus characterized by a minimum level of the dissipation in the linear domain of nonequilibrium thermodynamics the systems tend to states approaching equilibrium as much as their constraints permit. Although entropy may be lower than at equilibrium, the equilibrium type of order still prevails. The steady states belong to what has been called the thermodynamic branch, as it contains the equilibrium state as a particular case. 

The question of the efficiency of biological transport systems was examined extensively in the 1960s on the basis of linear nonequilibrium thermodynamics. I think it would be appropriate to give a brief account of the treatment here, especially since Professor Prigogine s early work was the source of most of our ideas at the time. The formal approach of... 

In the bottom-up approach, a large variety of ordered nano-, micro-and macrostructures may be obtained by changing the balance of all the attractive and repulsive forces between the structure-forming molecules or particles. This can be achieved by altering the environmental conditions (temperature, pH, ionic strength, presence of specific substances or ions) and the concentration of molecules/particles in the system (Min et al., 2008). As this takes place, the interrelated processes of formation and stabilization are both important considerations in the production of nanoparticles. In addition, as particles grow in size a number of intrinsic properties change, some qualitatively, others quantitatively some affect the equilibrium (thermodynamic) properties, and others affect the nonequilibrium (dynamic) properties such as relaxation times. 

To obtain a local quantification of entropy in a nonequilibrium material, consider a continuous system that has gradients in temperature, chemical potential, and other intensive thermodynamic quantities. Fluxes of heat, mass, and other extensive quantities will develop as the system approaches equilibrium. Assume that... 

In natural waters organisms and their abiotic environment are interrelated and interact upon each other. Such ecological systems are never in equilibrium because of the continuous input of solar energy (photosynthesis) necessary to maintain life. Free energy concepts can only describe the thermodynamically stable state and characterize the direction and extent of processes that are approaching equilibrium. Discrepancies between predicted equilibrium calculations and the available data of the real systems give valuable insight into those cases where chemical reactions are not understood sufficiently, where nonequilibrium conditions prevail, or where the analytical data are not sufficiently accurate or specific. Such discrepancies thus provide an incentive for future research and the development of more refined models. 

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