Out-of-equilibrium systems

This notion of bifurcation point, connected with those of instability and fluctuations, is the basis of this branch of science of self-organization in out-of-equilibrium systems. The story begins with Alan Turing, who, in search of the chemical basis of... 

Out-of-equilibrium systems (non-linear, dynamic processes), such as the Zabotinski-Belousov reaction, and other oscillating reactions bifurcation, and order out of chaos convection phenomena tornadoes, vortexes... 

Such a quantity, denoted as 7 eff(co), and parameterized by the age of the system, has been defined, for real to, via an extension of both the Einstein relation and the Nyquist formula. It has been argued in Refs. 5 and 6 that the effective temperature defined in this way plays in out-of-equilibrium systems the same role as does the thermodynamic temperature in systems at equilibrium (namely, the effective temperature controls the direction of heat flow and acts as a criterion for thermalization). 

With this chapter we enter the peculiar world of instability. A world considered by many as imperfect because the symmetry of time (meaning that an inversion of sign of the time variable is without effect on processes) is broken, conferring an arrow to the time. This is the domain of out-of-equilibrium systems and of irreversible processes, as opposed to equilibrium resulting from reversible processes. 

The proper representation of the thermodynamic properties of out-of-equilibrium systems such as glassy polymers is still an open question. Reliable correlations and predictive expressions for the mass transport properties in polymeric glasses as a function of temperature and concentration are also lacking. 

J. Kurchan. Recent theories of glasses as out of equilibrium systems. Comptes Rendus De L Academic Des Sciences Serie Iv Physique Astrophysique, 2 239-247, 2001. 

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

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 ). 

Fig. 2.1. Schematic diagram of a reaction model. The heart of the model is the equilibrium system, which contains an aqueous fluid and, optionally, one or more minerals. The system s constituents remain in chemical equilibrium throughout the calculation. Transfer of mass into or out of the system and variation in temperature drive the system to a series of new equilibria over the course of the reaction path. The system s composition may be buffered by equilibrium with an external gas reservoir, such as the atmosphere.

More complicated models account for the transport of mass or heat into or out of the system, so that its composition or temperature, or both, vary over the course of the calculation. The system s initial equilibrium state provides the starting point... 

In a second example of a flow-through path, we model the evaporation of seawater (Fig. 2.6). The equilibrium system in this case is a unit mass of seawater. Water is titrated out of the system over the course of the path, concentrating the seawater and causing minerals to precipitate. The minerals sink to the sea floor as they... 

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. 

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