Microscopic rules

Fig. 12.6. Schematic representation of the first three levels of a dynamical hierarchy, or, to use Baas term ([baas94] see below), hyperstructure. Level-1 is described by microscopic rules level-2 by second-order rules and so on.

It will appear especially from Wolfram s work that a priori prediction of macroscopic behavior even for many particle systems that follow simple interaction rules is often not possible. The behavior can be sensitive to initial conditions and disturbances. These are, of course, conditions that are optimum for a learning system, where microscopic rules have to be adapted to macroscopic requirements. The optimum condition for emergence of life-like multiplication appears close to conditions where the system behavior becomes unpredictable. An important feature of a living cell is its finite lifetime. Sustained existence... 

Thermodynamics was developed mostly in the nineteenth century. This was after the acceptance of the modern atomic theory of Dalton but before the ideas of quantum mechanics (which imply that the microscopic universe of atoms and electrons follow different rules than the macroscopic world of large masses). Therefore, thermodynamics mostly deals with large collections of atoms and molecules. The laws of thermodynamics are macroscopic rules. Later in the text, we will cover microscopic rules (that is, quantum mechanics), but for now remember that thermodynamics deals with systems we can see, feel, weigh, and manipulate with our own hands. 

Lattice gases are micro-level rule-based simulations of macro-level fluid behavior. Lattice-gas models provide a powerful new tool in modeling real fluid behavior ([doolenQO], [doolenQl]). The idea is to reproduce the desired macroscopic behavior of a fluid by modeling the underlying microscopic dynamics. 

Apart from their pedagogical value, reversible rules may be used to explore possible relationships between discrete dynamical systems and the dynamics of real mechanical systems, for which the microscopic laws are known to be time-reversal invariant. What sets such systems apart from continuous idealizations is their exact reversibility, discreteness assures us that computer simulations run for arbitrarily long times will never suffer from roundoff or truncation errors. As Toffoli points out, ...the results that one obtains have thus the force of theorems [toff84a]. ... 

A schematic representation of emergence is given in figure 12.6, which depicts the first three levels of a dynamical hierarchy and the rules or laws describing their behavior. The first, or lowest, level might be thought of as the level on which a CA system is usually defined. It consists of the lattice sites and values that define the microscopic dynamics. 

Lorentz-Invariance on a Lattice One of the most obvious shortcomings of a CA-based microphysics has to do with the lack of conventional symmetries. A lattice, by definition, has preferred directions and so is structurally anisotropic. How can we hope to generate symmetries where none fundamentally exist A strong hint comes from our discussion of lattice gases in chapter 9, where we saw that symmetries that do not exist on the microscopic lattice level often emerge on the macroscopic dyneimical level. For example, an appropriate set of microscopic LG rules can spawn circular wavefronts on anisotropic lattices. 

Because of our inability to analyze the interaction of microscopic QM systems and macroscopic measuring devices to a sufficient degree, we make use of a set of empirical rules that are known as measurement theory. Some day, measurement theory will become a proven set of theorems in QM,, as the proponents of the decoherence theory, among others, claim. Until such time, it is beneficial to introduce the measurement process, and the principles associated with it, separately from the dynamics described by the Schrbdinger equation. 

The microscopic rate constant is derived from the quantum mechanical transition probability by considering the system to be initially present in one of the vibronic levels on the initial potential surface. The initial level is coupled by spin-orbit interaction to the manifold of vibronic levels belonging to the final potential surface. The microscopic rate constant is then obtained, following the Fermi-Golden rule, as ... 

If the test is positive, the urine is examined microscopically for red blood cells. If no red blood cells are found, a tentative diagnosis of myoglobinuria is made, serum chemistries are obtained, and the patient is held to rule out rhabdomyolysis. If the uric acid and creatinine kinase (CK) values are normal, and the patient is asymptomatic, he/she is discharged from the hospital. Routine toxicology tests include urinary PCP, serum alcohol, and hypnotic screen. 

Urinalysis to rule out infection as a cause of the patient s voiding symptoms also check urinalysis for microscopic hematuria, which typically accompanies benign prostatic hyperplasia. 

Ninety-five percent patients monitored for recurrence show CA 125 concentrations greater than 35 U/ml and residual ovarian carcinoma. However, a negative result is not conclusive because half of the patients with negative results have microscopic residual carcinoma. Therefore, it is essential to have a second-look procedure in order to rule out residual carcinoma. Postsurgical monitoring of patients... 

The principle of a lattice gas is to reproduce macroscopic behavior by modeling the underlying microscopic dynamics. In order to successfully predict the macro-level behavior of a fluid from micro-level rules, three requirements must be satisfied. First, the number of particles must be conserved and, in most cases, so is the particle momentum. States of all the cells in the neighborhood depend on the states of all the others, but neighborhoods do not overlap. This makes application of conservation laws simple because if they apply to one neighborhood they apply to the whole lattice. 

Recently2 it has been asserted that the very existence of dissociative electron transfer reactions is ruled out by application of the principle of microscopic reversibility. The line of argument was as follows. In the reaction of the cleaving substrate RX, say, with an electron donor D (the same argument could be developed for an oxidative cleavage triggered by an electron acceptor),... 

The nanoscale world is exciting because it is governed by rules differing from those in the macroscopic, or even microscopic, realm. It is a world where quantum mechanics dominates the scene, and events on the single-molecule scale are critical. What we know about the behavior of material on our scale is no longer true on the nanometer scale, and our formularies must be re-written. In order to study this quantum world, a quantum-mechanical probe is essential. Electron tunneling provides that quantum-mechanical tool. 

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