The change in entropy for an open system, dSsystem, consists of an external, exogenous contribution from the environment, deS = Sfa- Sout, and an internal, endogenous contribution due to system state, diS, which must always be positive by the second law of thermodynamics (Prigogine, 1980). Prigogine uses the concept of entropy and the second law of thermodynamics far from thermodynamic equilibrium, which is outside the framework of classical thermodynamics, but he uses the concepts only locally. There are three possibilities for the entropy balance:
The system loses order in the first case. Gaining order (case 2), is only possible if -deS > diS > 0. Creation of order in a system must be associated with a greater flux of entropy out of the system than into the system. This implies that the system must be open or at least non-isolated.
Case 3, Equation 2.11, corresponds to a stationary situation, for which Ebeling et al. (1990) used the following two equations for the energy (U ) balance and the entropy (S ) balance :
and dSsystem /dt = 0 or deS /dt = - diS /dt = 0 (2.13)
Usually the thermodynamic processes are isothermal and isobaric. This implies that we can interpret the third case (Equations 2.11-2.13) by use of the free energy:
It means that a "status quo" situation for an ecosystem requires input of free energy or exergy to compensate for the loss of free energy and corresponding formation of heat due to maintenance processes, i.e. respiration and evapotranspiration. If the system is not receiving a sufficient amount of free energy, the entropy will increase. If the entropy of the system will continue to increase, thus, the system will approach thermodynamic equilibrium—the system will die; see Section 2.2. This is in accordance with Ostwald (1931): life without the input of free energy is not possible.
An average energy flow of approximately 1017 W by solar radiation ensures the maintenance of life on earth. The surface temperature of the sun is 5800K and of the earth on average approximately 280 K. This implies that the following export of entropy per unit of time takes place from the earth to the open space:
corresponding to 1 W/m2 K.
Prigogine uses the term dissipative structure to denote self-organizing systems, thereby indicating that such systems dissipate energy (produce entropy) for the maintenance of their organization (order). The following conclusions are appropriate:
All living systems, because they are subject to the second law of thermodynamics, are inherently dissipative structures. The anabolism combats and compensates for the cata-bolic deterioration of structure; the two processes operate against one another. Note that the equilibrium "attractor" represents a resting or refractory state, one that is passively devolved to if system openness or non-isolation are compromised (Jorgensen et al., 1999). The term is also commonly used to express the situation when a system is actively pushed or "forced" toward a steady state. Though widespread, we do not subscribe to this usage and make a distinction between steady states and equilibria for two reasons:
(1) The state-space system theory we outlined in the conservation chapter of Ecosystems Emerging (Patten et al., 1997) precludes anything in system dynamics but a unique input-state-output relationship. Therefore, given an initial state, state-space theory asserts that there exists one and only one sequence of inputs that will put an open system in a given state at a specified final time. For this terminal state to be an "attractor", many input sequences would have to be able to place the system in it, and from many initial states—the attractor would be hard to avoid. This is inconsistent with dynamical state theory.
(2) As observed above, a steady state is a forced (non-zero input) condition; there is nothing "attractive" about it. Without a proper forcing function it will never be reached or maintained. A steady state that is constant may appear equilibrial, but it is really far from equilibrium and maintained by a steady input of energy or matter. We regard equilibrium as a zero-input or resting condition. What are often recognized as local attractors in mathematical models really have no counterparts in nature. Steady states are forced conditions, not to be confused with unforced equilibria which represent states to which systems settle when they are devoid of inputs. The only true natural attractor in reality, and it is global, is the unforced thermodynamic equilibrium.
As an ecosystem is non-isolated, the entropy changes during a time interval, dt can be decomposed into the entropy flux due to exchanges with the environment, and the entropy production due to the irreversible processes inside the system such as diffusion, heat conduction, and chemical reactions. This can also be expressed by use of exergy:
where deEx/dt represents the exergy input to the system and diEx/dt is the exergy consumed (is negative) by the system for maintenance, etc. Equation 2.16—an exergy version of Equations 2.9 and 2.10—shows among other things that systems can only maintain a non-equilibrium steady state by compensating the internal exergy consumption with a positive exergy influx (deEx/dt > 0). Such an influx induces order into the system. In ecosystems the ultimate exergy influx comes from solar radiation, and the order induced is, e.g. biochemical molecular order. If deEx > -diEx (the exergy consumption in the system), the system has surplus exergy input, which may be utilized to construct further order in the system, or as Prigogine (1980) calls it: dissipative structure. The system will thereby move further away from thermodynamic equilibrium. Evolution shows that this situation has been valid for the ecosphere on a long-term basis. In spring and summer ecosystems are in the typical situation that deEx exceeds -diEx. If deEx < -diEx, the system cannot maintain the order already achieved, but will move closer to the thermody-namic equilibrium, i.e. it will lose order. This may be the situation for ecosystems during fall and winter or due to environmental disturbances.
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