## The Schrodinger Ratio

The Schrodinger ratio was defined by Howard Odum as the ratio of the supporting energy flow to the structure of an open (living) system. He derived it from Schrodinger's concept that maintenance of a low-entropy structure depends on continuous inflow of low-entropy energy (or exergy inflow) and on exportation of entropy. It is the ratio of entropy-generation rate to entropy embodied in structures under given environmental conditions, conceived as an indicator of the capacity of self-adaptive dissipative structures to self-maintain by dissipating entropy.

Open living systems feed on low-entropy energy inflows to achieve a state of minimum entropy and maintain themselves far from thermodynamic equilibrium (which is a state of maximum entropy); they maintain a steady state (of minimum entropy) in time. The higher the ratio the lower the capacity of the system to convert incoming exergy into internal organization. Nicolis and Prigogine observed that since internal entropy increases due to processes within a system, the entropy content of an open system can decrease by virtue of low-entropy energy inflows from the external environment or other systems with different conditions of temperature and pressure. If Se is the incoming entropy flow and Si the internal entropy production rate, a system can reduce and maintain its entropy content if Se is negative (Schrodinger's negen-tropy) and greater than Si. The entropy variation of the system is therefore given by Si + Se and may be negative due to the following condition:

Sprod = Si + Se < 0 if Se < 0 and - Se > Si 

In the case of ecosystems, which are self-organizing living systems, the Schrodinger ratio is conceived as the ratio of biological entropy production to free energy stored; in other words, the exergy stored in the living biomass by biological components. The ratio is also known as the specific entropy production or specific dissipation of a system. According to the concept of exergy, temperature T, pressure p, and chemical potentials y = (yb..., yn) of the system under study are supposed to differ from those of the external environment: T0, p0, and y° = (u!°, ..., yn0). Since, the internal energy, volume, and number of particles of the environment are so large that processes of the system do not produce any significant change in the temperature, pressure, or potentials of the environment, these constant values may considered to be those of the reference system. The Schrodinger ratio is given by the following expression (in units of time _ *):

(s° - s ) = Ex/T° = w where Sprod is the entropy production of the system and (S° -S) is the difference between the maximum entropy at thermodynamic equilibrium with the environment (S°) and the entropy content of the system (S) at the absolute temperature T0 corresponding to the temperature of the external environment (as the reference system). This difference is also a measure of the internal order (organization) achieved by a system with respect to the environment. Since the entropy content under none-quilibrium conditions cannot be greater than that at equilibrium, (S° -S) is always positive.

Sprod is the sum of the contributions from abiotic (mechanical dissipation of energy, photophysical processes, chemical reactions, etc.) and biotic processes (anabolic and catabolic reactions).

The exergy of an ecosystem, Ex = T0 (S0 - S), is its thermodynamic distance from equilibrium. Ex = 0 indicates a condition of equilibrium. 