Thermodynamic depth

Related to Bennett s logical depth (see above), the thermodynamic depth, T) S) of a system S in state S measures how much information must be processed in order for S to evolve to S. 

Lloyd and Pagels show that these three requirements lead uniquely to an average complexity of a state proportional to the Shannon entropy of the set of (experimentally determined) trajectories leading to the given state (= EiPi oSzPi)- The thermodynamic depth of a state S to which the system S has evolved via the possible trajectory is equal to the amount of information required to specify that trajectory, or Djj S) x log2Pi. In the case of Hamiltonian systems, Lloyd and 

Pagels show that thermodynamic depth is proportional to the difference between the state s thermodynamic entropy (i.e. its coarse grained entropy) and its finegrained entropy, given by fcex volume of points in phase space corresponding to the system s trajectory, where k], is Boltzman s constant. 

Lloyd and Pagels [IloydSS] prove that the only function that is consistent with their requirements (l)-(3), given above, is 

Central to the general science of complexity, and to the study of artificial life in particular, is the notion of emergence. Emergence refers to the appearance of higher-level properties and behaviors of a system that - while obviously originating from the collective dynamics of that system s components - cannot be directly deduced from the lower-level properties or behaviors of that system. 

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Thermodynamics is a deductive science built on the foundation of two fundamental laws that circumscribe the behavior of macroscopic systems the first law of thermodynamics affirms the principle of energy conservation the second law states the principle of entropy increase. In-depth treatments of thermodynamics may be found in References 1—7. 

This reaction is catalyzed by iron, and extensive research, including surface science experiments, has led to an understanding of many of the details (72). The adsorption of H2 on iron is fast, and the adsorption of N2 is slow and characterized by a substantial activation energy. N2 and H2 are both dis so datively adsorbed. Adsorption of N2 leads to reconstmction of the iron surface and formation of stmctures called iron nitrides that have depths of several atomic layers with compositions of approximately Fe N. There is a bulk compound Fe N, but it is thermodynamically unstable when the surface stmcture is stable. Adsorbed species such as the intermediates NH and NH2 have been identified spectroscopically. 

The lower the value of the more likely it is that S is positive indicating a thermodynamic tendency for the process to occur. Longitudinal wave theory has been appHed to the defoamer spreading process as in equation 5 where P is the penetration depth of a spreading droplet of initial radius R, viscosity Tj, and density p. 

The scheme of commercial methane synthesis includes a multistage reaction system and recycle of product gas. Adiabatic reactors connected with waste heat boilers are used to remove the heat in the form of high pressure steam. In designing the pilot plants, major emphasis was placed on the design of the catalytic reactor system. Thermodynamic parameters (composition of feed gas, temperature, temperature rise, pressure, etc.) as well as hydrodynamic parameters (bed depth, linear velocity, catalyst pellet size, etc.) are identical to those in a commercial methana-tion plant. This permits direct upscaling of test results to commercial size reactors because radial gradients are not present in an adiabatic shift reactor. 

There are two principal chemical concepts we will cover that are important for studying the natural environment. The first is thermodynamics, which describes whether a system is at equilibrium or if it can spontaneously change by undergoing chemical reaction. We review the main first principles and extend the discussion to electrochemistry. The second main concept is how fast chemical reactions take place if they start. This study of the rate of chemical change is called chemical kinetics. We examine selected natural systems in which the rate of change helps determine the state of the system. Finally, we briefly go over some natural examples where both thermodynamic and kinetic factors are important. This brief chapter cannot provide the depth of treatment found in a textbook fully devoted to these physical chemical subjects. Those who wish a more detailed discussion of these concepts might turn to one of the following texts Atkins (1994), Levine (1995), Alberty and Silbey (1997). 

E. L. Shock (1990) provides a different interpretation of these results he criticizes that the redox state of the reaction mixture was not checked in the Miller/Bada experiments. Shock also states that simple thermodynamic calculations show that the Miller/Bada theory does not stand up. To use terms like instability and decomposition is not correct when chemical compounds (here amino acids) are present in aqueous solution under extreme conditions and are aiming at a metastable equilibrium. Shock considers that oxidized and metastable carbon and nitrogen compounds are of greater importance in hydrothermal systems than are reduced compounds. In the interior of the Earth, CO2 and N2 are in stable redox equilibrium with substances such as amino acids and carboxylic acids, while reduced compounds such as CH4 and NH3 are not. The explanation lies in the oxidation state of the lithosphere. Shock considers the two mineral systems FMQ and PPM discussed above as particularly important for the system seawater/basalt rock. The FMQ system acts as a buffer in the oceanic crust. At depths of around 1.3 km, the PPM system probably becomes active, i.e., N2 and CO2 are the dominant species in stable equilibrium conditions at temperatures above 548 K. When the temperature of hydrothermal solutions falls (below about 548 K), they probably pass through a stability field in which CH4 and NII3 predominate. If kinetic factors block the achievement of equilibrium, metastable compounds such as alkanes, carboxylic acids, alkyl benzenes and amino acids are formed between 423 and 293 K. 

A very large number of synthetic, as well as many natural, macrocycles have now been studied in considerable depth. A major thrust of many of these studies has been to investigate the unusual properties frequently associated with cyclic ligand complexes. In particular, the investigation of spectral, electrochemical, structural, kinetic, and thermodynamic aspects of macrocyclic complex formation have all received considerable attention. 

The complexation of the alkaline earth metals is reminiscent of the behaviour of several of the naturally occurring antibiotics and, like the latter, the crown often exhibits remarkable selectivity for particular ions. The thermodynamic factors underlying the selectivity of many of the crowns have been studied in some depth and the results related to such parameters as cavity size, number of donor atoms present, possible ring conformations on complex formation and the solvation energies of the various species involved. 

From the chemical point of view, the solvent in which the CL experiment is carried out can have a dramatic influence on the efficiency of the CL reaction as solvation can alter the shapes, the depths, and the densities of the vibrational states of the potential surfaces representing the ground states of products and reactants and the lowest excited singlet state of the potential fluorophore. The alteration of the intersections of these potential energy surfaces can affect the enthalpies of reaction and the enthalpies of activation for dark and lumigenic reactions. In some cases, these changes will favor CL (if AH decreases relative to AHa) and in some cases, they will make it thermodynamically unfavorable for CL to occur. 

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