Box 3.3 Relaxation of systems
Denoting the upward and downward relaxation probabilities by Wap and Wpa (with Wap ^ Wpa), the rate of change of Na is given by:
At thermal equilibrium dNa /dt = 0, and denoting the equilibrium population by N0a and N0p we see that:
The populations follow from Boltzmann's law and so the ratio of the two transition probabilities must also be equal to exp(-AE/kT). Expressing Na and Np in terms of N and n (n = Na - Np) we obtain:
This may be rewritten as:
in which n0, the population difference at thermal equilibrium, is equal to:
Wfia - Wap
Wpa +Wap and 1/Tj is expressed by:
T1 thus has the dimensions of time and is called the "spin-lattice relaxation time". It is a measure of the time taken for energy to be transferred to other degrees of freedom, i.e., for the spin system to approach thermal equilibrium: Large values of T1 (minutes or even hours for some nuclei) indicate very slow relaxation (Carrington and McLachlan: Introduction to magnetic resonance).
It is now possible to say something about the width and shape of the resonance absorption line, which certainly cannot be represented by a Dirac 5 function.
First, it is clear that, because of the spin relaxation, the spin states have a finite lifetime. The resulting line broadening can be estimated from the uncertainty relation:
and thus we find that the line width due to spin-lattice relaxation will be of the order of 1/T1.
Given the remarks made at the start of this section, one may indeed start to wonder and speculate about the relations of these physical systems that obey universal laws when involved at the level of chemistry and biology and how or if these affect living systems at all. This is exactly what the physicist Walter M. Elsasser did and it may be worthwhile to spend a few moments studying his work and conclusions.
What really differs between physics and biology: four principles of Elsasser The one contributor from Table 3.1 that literally takes the step from physics into biology was Walter M. Elsasser who's "roaming" life is quite impressive. The details of his life are described in a biography1 by Rubin (1995), who was acquainted with Elsasser in the last 10 years of his life. Most of the information on Elsasser's below is based on this biography and Elsasser's own autobiography (Elsasser, 1978). From these works, one can almost sense that Elsasser's contributions were sparked by ontic openness on his own "body and soul" throughout his career. Rubin (1995) summarized Elsasser's (1987) four basic principles of organisms: (A) ordered heterogeneity, (B) creative selection, (C) holistic memory, and (D) operative symbolism. The first principle is the key reference to ontic openness, while the other points address how this order arises in this "messy" world of immense numbers. In other words, the latter three seem more to be ad hoc inventions necessary to elaborate and explain the first.
According to Rubin, Theophile Khan influenced Elsasser's understanding of the overwhelming complexity dominating biological systems as compared with the relative simplicity of physics. Probably, he was also influenced by Wigner from whom he is likely to have picked up group or set theory.
These studies, together with periodical influence from von Neumann, caused him to realize a fundamental difference between physical systems on one side and living systems on the other. Due to his early life education in atomic physics, he considered physical systems as homogenous sets—all atoms and molecules of a kind basically possess the same properties and behavior. At this level, and always near to equilibrium conditions, the world is deterministic and reversible processes dominate.
'This excellent biography is available on the Internet in several forms. Philosophy of Science students will be provided with a deep insight in how production of a scientist may not necessarily depend on skill or education, but may rather be determined by political and sociological regimens throughout his life.
As opposed to this view, he considered living systems to differ in this fundamental aspect of the homogenous sets. Living systems, he argued, are highly heterogeneous and far more complex than physical systems. Their behavior as opposed to physical systems is non-deterministic and irreversible. This is what we today would designate as far from equilibrium systems or dissipative structures.
The views of Elsasser are at this point derived from studies and knowledge about biological systems at cellular and sub-cellular level, i.e., the boarder between the "dead" physico-chemical systems and the living systems. The "distinction" falls somewhere between the pure chemical oscillations, like in the Beluzov-Zhabotinsky reaction and the establishing of biochemical cycles (autocatalytic cycles or hypercycles of Eigen and Schuster) together with chirality and the coupling to asymmetries introduced by separation of elements and processes by membranes. Part of the living systems indeterminacy is caused by an intrinsic and fundamental (ontic) property of the systems—(ontic) openness.
Around the late 1960s, Elsasser directed his attention to the question of what possibly could have happened since the beginning of the universe, i.e., since the Big Bang—the thinking is much along the same line as Jorgensen formulated some decades later where Heisenberg's uncertainty relation is transferred2 to ecosystems (see later this Section).
Elsasser's starting point was to calculate, roughly at least, how many quantum-level events could have taken place since the Big Bang. Since events at quantum level happens within one billionth of a second he calculates a number to be in order of 1025. Then considering that the number of particles in the form of simple protons that may have been involved in these events to be approximately 1085 he calculates the number of possible events to be 10110. Any number beyond this "simply loses its meaning with respect to physical reality" (Ulanowicz, 2006a). Elsasser puts a limit at around 10100 (a number known as Googol). Any number beyond this is referred to as an immense number. In Elsasser's terminology an immense number is a number whose logarithm itself is large. We claim that such numbers make no sense. And yet, as we saw with the examples from music, any simple everyday event, such as a piece of music, breaks this limit of physical events easily—almost before it is started.
But where does the relevance to ecosystems come in one may ask? Good question—and for once—a very simple answer. The point is that any ecosystem easily goes to a level of complexity where the number of possible events that may occur reaches or exceeds immense numbers. Again, Ulanowicz points out that "One doesn't need Avogadro's number of particles (1023) to produce combinations in excess of 10110, a system with merely 80 or so distinguishable components will suffice" (Ulanowicz, 2006a) as 80! is on the order of 7 X 10118.
Now, as the vast majority of ecosystems, if not all, exceed this number of components it means that far more possibilities could have been realized, so that out of the phase space of possibilities on a few combinations have been realized. Any state that has occurred is also likely to occur only once—and is picked out of super-astronomical number of
2This transfer would in the context of philosophy of science be designated as a theoretical reduction—indeed with large epistemic consequences. This is opposed to Elsasser's approach that we here consider within the normal paradigm of physics.
possibilities. The other side of the story, as the title indicates, is that we are also left with a large number of possibilities that have never been and are never going to be realized. In other words, almost all events we may observe around us are literally unique. There are simple, repeatable events in nature within the domain of classical probability, but they are sets of a measure zero in comparison with unique events.
Meanwhile, we cannot foretell the possibilities of the next upcoming events. If we consider any particular situation, we face a world of unpredictability—a world that is totally ontic open. In fact, taken together, the above means that we should forget about making predictions about ecosystem development or even trying to do this. Luckily, as we shall see later, Karl Popper (1990) advocated a "milder" version of ontic openness.
Whereas up till now we have dealt with heterogeneity at the level of probabilities the following points from Elsasser try to explain how nature copes with this situation.
This point addresses the problems that arise from the immense heterogeneity. How do living systems "decide" among the extraordinary large number of possibilities that exist? Elsasser was precisely aware that living systems were non-deterministic, non-mechanist systems, as opposed to the physical systems that are always identical. As Rubin (1995) states, they "repeat themselves over and over again. . . but each organism is unique".
Elsasser gives agency to the organisms, although judging from this point alone it is not very easy to see where or how the "creativity" arises. Therefore, this point cannot be viewed as isolated from the two additional points below. Selection mechanisms are not ignored in this view that just stresses the intrinsic causes of evolution.
With memory Elsasser addresses part of what is missing from agency. Again, according to Rubin, the criterion for living system to choose is information stability. Some memory system has to be introduced, as the living systems have to ensure the stability. This point, in addition to agency, also involves history and the ability to convey this history, i.e., heredity to living, organic systems. Although again a part misses on how this information is physically going to be stored, preserved, and conveyed.
Lastly, symbolism provides the mechanism for storing this information by introducing DNA as "material carrier of this information". This cannot be seen as isolated from the history of science in the area of genetics. Much of the Elsasser's philosophical work has been written when the material structure and organization of our hereditary material, the chromosomes, was revealed.
The above arguments could be taken as if Elsasser was still basically a true reductionist as we have now got everything reduced into "simple" mechanisms for the conveyance of history. Elsasser was indeed aware of this point and saw the process in a dualistic (not to say dialectic) manner as he stated this mechanism to be holistic in the sense that it had to "involve the entire cell or organism" (see Section 3.6).
According to Jorgensen (1995) "some of the principles of quantum mechanics are (silently and slowly) introduced in ecology" during the last 15 years (this was probably written significantly earlier than 1995!). This is stated to be valid in particular to the area of modeling with the following remarks: "An ecosystem is too complex to allow us to make the number of observations needed to set up a very detailed model—even if we still consider models with a complexity far from that of nature. The number of components (state variables) in an ecosystem is enormous". Taking this argument there is a clear correlation to the ontic openness of Elsasser, for instance through the presentation by Ulanowicz quoted above. Again, the number of components in an ecosystem alone is enough to form a system that is ontic open.
To the empiricist, this means that we have to use our limited resources in time and in particular money in the best possible manner. Who wants to spend unnecessary efforts? Who does not want to be as economically efficient as possible given that research money is always a limiting constraint? Meanwhile, the calculations made by Jorgensen imply a theorem of intrinsic empirical incompleteness. The argument goes as follows (see Box 3.4).
According to Jorgensen the Heisenberg uncertainty principle may now be reformulated, so that it refers to two other measures: uncertainty in time and energy (note the product of the two is consistent with Planck's constant, namely energy times time). The analogous formula reduces to:
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