Dissipative structure states

In the time independent case the array is stationary and does not dissipate any energy in the time dependent case it moves along the barrier with an arbitrary velocity < V. The moving array of vortices appears, as in the Anderson-Dayem bridges, when the current through the junction exceeds a certain critical current 1. In that case the system is in a dissipative structure state. 

As we have seen, the Josephson effects appear to occupy an intermediate position between phase transitions and bifurcations to dissipative structure states. One of the reasons for this special behavior of superconductors is that they can carry electric currents without dissipating any energy. There is another reason for the Josephson effects to occupy this intermediate position. In the case of ordinary dissipative structures, the structures are macroscopic and can be described by classical physics. 

These patterns are an example of what are sometimes called dissipative structures, which arise in many complex systems. Dissipative structures are dynamical patterns that retain their organized state by persistently dissipating matter and energy into an otherwise thermodynamically open environment. 

Although these systems involve two variables, their steady-state solutions can be calculated in general and a more complete mathematical analysis of dissipative structures is possible. From a practical point of view it is interesting to note that systems obeying equations of the form (2) may be found in artificial membrane reactors.22 Examples are presented by D. Thomas in this volume. 

Fig. 11. Bifurcation diagram for an even wave number and

Fig. 15. Spatiotemporal dissipative structure in a one-dimensional system with fixed boundary conditions, dashed line unstable homogeneous steady state. Full line stable periodic regime. A = 2, B = 5.45 D, = 8 10 3 D2 = 4 10 3.

Fig. 16. Bifurcation diagram of temporal dissipative structures, c (maximal amplitude of the oscillation minus the homogeneous steady-state value) is sketched versus B for a two-dimensional system with zero flux boundary conditions. The first bifurcation occurs at B = Bn and corresponds to a stable homogeneous oscillation. At B, two space-dependent unstable solutions bifurcate simultaneously. They become stable at B a and Bfb. Notice that as it is generally the case Bfa Bfb.

Fig. 17. Temporal dissipative structure after various time intervals during the period of oscillation. The reaction medium is a circle with zero flux boundary conditions. The lines correspond to isoconentrations. A =2, B = 5.4, Dt = 8 10 3, D2 = 4- 1G"3. Curves of equal concentration for Y are represented by full or broken lines when the concentration is, respectively, larger or smaller than the unstable steady state. The radius of the circle r0 = 0.5861. 

Experimentally, in chemical systems a variety of dynamic states can be observed resulting from thermodynamic and kinetic conditions as defined as a prerequisite for the evolution of dissipative structures. Today we distinguish the following states (1) the maintenance of multiple steady states with transitions from one to another, (2) rotation on a limit cycle... 

Boundary conditions in the systems I have considered primarily ensure the maintenance of a state of nonequilibrium with respect to the environment their relation with the size and form of the reaction medium, with the chemical and transport processes, determines the nature and properties of the dissipative structures that occur. These boundary conditions are imposed once and for all and do not couple with surface effects or electrostatic interactions. Such a coupling is likely to be the source of self-organization processes also, but was not the object of my talk. 

At the phenomenological thermodynamic level, when we go far from equilibrium, the striking new feature is that new dynamical states of matter arise. We may call these states dissipative structures as they present both structure and coherence and their maintenance requires dissipation of energy.6 Dissipative processes that destroy structure at and near equilibrium may create these structures when sufficiently far from equilibrium. 

The steady-state solution that is an extension of the equilibrium state, called the thermodynamic branch, is stable until the parameter A reaches the critical value A,. For values larger than A<, there appear two new branches (61) and (62). Each of the new branches is stable, but the extrapolation of the thermodynamic branch (a ) is unstable. Using the mathematical methods of bifurcation theory, one can determine the point A, and also obtain the new solution, (i.e., the dissipative structures) in the vicinity of A, as a function of (A - A,.). One must emphasize that... 

All work that is introduced per unit time into the heat pump is ultimately lost work, but with the advantage that the nonequilibrium state is maintained. Given the structured nature of the nonequilibrium state as discussed above, given the dissipation of the work input for maintenance, it will come as no surprise that Ilya Prigogine spoke of dissipative structures. Thinking about it,... 

D. Kondepudi and I. Prigogine, Modern Thermodynamics, From Heat Engines to Dissipative Structures, Wiley, New York (1999). N.W. Tschoegl, Fundamentals of Equilibrium and Steady-State Thermodynamics, Elsevier, Amsterdam (2000). 

Systems that exchange entropy with their surroundings may undergo spontaneous transformation to dissipative structures and self-organization. The forces that exist in irreversible processes create these organized states, which range from convection patterns of Benard cells to biological cycles. 

Example 12.10 Multiple steady states Multiple steady states and dissipative structures may play an important role in nerve excitations. Consider the following simple set of reactions ... 

There are two types of macroscopic structures equilibrium and dissipative ones. A perfect crystal, for example, represents an equilibrium structure, which is stable and does not exchange matter and energy with the environment. On the other hand, dissipative structures maintain their state by exchanging energy and matter constantly with environment. This continuous interaction enables the system to establish an ordered structure with lower entropy than that of equilibrium structure. For some time, it is believed that thermodynamics precludes the appearance of dissipative structures, such as spontaneous rhythms. However, thermodynamics can describe the possible state of a structure through the study of instabilities in nonequilibrium stationary states. 

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