Out-of-equilibrium

The average chain length at time t after the system is moved out of equilibrium is given by L t) = [Pg.540]

C. Misbah, Y. Saito. Vicinal surfaces at, and out of, equilibrium. Phys Rev E (to appear). 

Address questions such as How do two sides of a conflict co-evolve with one another and Can one side exploit what it knows of this co-evolutionary process to compel the other side to remain out of equilibrium ... 

Molecular view of a gasnsolution equilibrium, (a) At equilibrium, the rate of escape of gas molecules from the solution equals the rate of capture of gas molecules by the solution, (b) An increase in gas pressure causes more gas molecules to dissolve, throwing the system out of equilibrium, (c) The concentration of solute increases until the rates of escape and capture once again balance. 

As we all know from thermodynamics, closed systems in equilibrium have minimum free energy and maximum entropy. If such a system were brought out of equilibrium, i.e. to a state with lower entropy and higher free energy, it would automatically decay to the state of equilibrium, and it would lose all information about its previous states. A system s tendency to return to equilibrium is given by its free energy. An example is a batch reaction that is run to completion. 

One of the major advances in the application of Jarzynski s identity to the calculation of free energies came from coupling it to path sampling [46, 47]. In a typical application with fast switching, the system is rapidly driven out of equilibrium as the coupling parameter is changed, and nearly all trajectories are essentially... 

A final observation is in order the quantitative application of the equilibrium thermodynamical formalism to living systems and especially to ecosystems is generally inadequate since they are complex in their organisation, involving many interactions and feedback loops, several hierarchical levels may have to be considered, and the sources and types of energy involved can be multiple. Furthermore, they are out-of-equilibrium open flow systems and need to be maintained in such condition since equilibrium is death. Leaving aside very simple cases, in the present state of the art we are, therefore, limited to general semiquantitative statements or descriptions (e.g. ecosystem narratives ). 

In this chapter we consider the problem of the kinetics of the heterogeneous reactions by which minerals dissolve and precipitate. This topic has received a considerable amount of attention in geochemistry, primarily because of the slow rates at which many minerals react and the resulting tendency of waters, especially at low temperature, to be out of equilibrium with the minerals they contact. We first discuss how rate laws for heterogeneous reactions can be integrated into reaction models and then calculate some simple kinetic reaction paths. In Chapter 26, we explore a number of examples in which we apply heterogeneous kinetics to problems of geochemical interest. 

Here we are talking about evaporation under thermodynamic equilibrium. We can also have evaporation under nonequilibrium conditions. For example, if the pressure of a liquid is suddenly dropped below its saturation pressure, flash evaporation will occur. The resulting vapor will be at the boiling point or saturation temperature corresponding to the new pressure, but the bulk of the original liquid will remain (out of equilibrium) at the former higher temperature. Eventually, all of the liquid will become vapor at the lower pressure. The distinction between flash evaporation and equilibrium evaporation is illustrated in Figure 6.6 for water. 

But the entire conception here is that of equilibrium solvation of the transition state by the Debye ionic atmosphere, and closer inspection [51] indicates that this assumption can hardly be justified indeed, time scale considerations reveal that it will nearly always be violated. The characteristic time for the system to cross the reaction barrier is cot, 0.1 ps say. On the other hand, the time required for equilibration of the atmosphere is something like the time for an ion to diffuse over the atmosphere dimension, the Debye length K- this time is = 1 ns for a salt concentration C= 0.1M and only drops to lOps for C 1M. Thus the ionic atmosphere is perforce out of equilibrium during the barrier passage, and in analogy with ionic transport problems, there should be an ionic atmosphere friction operative on the reaction coordinate which can influence the reaction rate. 

In words, s describes the interaction of the solute charge distribution component p, with the arbitrary solvent orientational polarization mediated by the cavity surface. The arbitrary weights p,, previously defined by (2.11), enter accordingly the definition of the solvent coordinates, and reduce, in the equilibrium solvation regime, to the weights tv,, such that the solvent coordinates are no longer arbitrary, but instead depend on the solute nuclear geometry and assume the form se<> = lor. weq. In equilibrium, the solvent coordinates are correlated to the actual electronic structure of the solute, while out of equilibrium they are not. 

One of the drawbacks of crew-cut micelles is that they systematically require the use of a nonselective solvent for their preparation and they definitely represent out-of-equilibrium micelles once they have been transferred... 

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