*Stefan Thurner, Rudolf Hanel, and Peter Klimekl*

- Published in print:
- 2018
- Published Online:
- November 2018
- ISBN:
- 9780198821939
- eISBN:
- 9780191861062
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198821939.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Evolutionary processes combine many features of complex systems: they are algorithmic; states co-evolve with interactions; they show power law statistics; they are selforganized critical; and they ...
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Evolutionary processes combine many features of complex systems: they are algorithmic; states co-evolve with interactions; they show power law statistics; they are selforganized critical; and they are driven non-equilibrium systems. Evolution is a dynamical process that changes the composition of large sets of interconnected elements, entities, or species over time. The essence of evolutionary processes is that, through the interaction of existing entities with each other and with their environment, they give rise to an open-ended process of creation and destruction of new entities. Evolutionary processes are critical, co-evolutionary, and combinatorial, meaning that thew entities are created from combinations of existing ones. We review the concepts of the replicator equation, fitness landscapes, cascading events, the adjacent possible. We review several classical quantitative approaches to evolutionary dynamics such as the NK model and the Bak–Snappen model. We propose a general and universal framework for evolutionary dynamics that is critical, co-evolutionary, and combinatorial.Less

Evolutionary processes combine many features of complex systems: they are algorithmic; states co-evolve with interactions; they show power law statistics; they are selforganized critical; and they are driven non-equilibrium systems. Evolution is a dynamical process that changes the composition of large sets of interconnected elements, entities, or species over time. The essence of evolutionary processes is that, through the interaction of existing entities with each other and with their environment, they give rise to an open-ended process of creation and destruction of new entities. Evolutionary processes are critical, co-evolutionary, and combinatorial, meaning that thew entities are created from combinations of existing ones. We review the concepts of the replicator equation, fitness landscapes, cascading events, the adjacent possible. We review several classical quantitative approaches to evolutionary dynamics such as the NK model and the Bak–Snappen model. We propose a general and universal framework for evolutionary dynamics that is critical, co-evolutionary, and combinatorial.

*M. E. J. Newman and R. G. Palmer*

- Published in print:
- 2003
- Published Online:
- November 2020
- ISBN:
- 9780195159455
- eISBN:
- 9780197562000
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195159455.003.0008
- Subject:
- Archaeology, Prehistoric Archaeology

The models discussed in the last chapter are intriguing, but present a number of problems. In particular, most of the results about them come from computer ...
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The models discussed in the last chapter are intriguing, but present a number of problems. In particular, most of the results about them come from computer simulations, and little is known analytically about their properties. Results such as the power-law distribution of extinction sizes and the system's evolution to the "edge of chaos" are only as accurate as the simulations in which they are observed. Moreover, it is not even clear what the mechanisms responsible for these results are, beyond the rather general arguments that we have already given. In order to address these shortcomings, Bak and Sneppen (1993; Sneppen et al. 1995; Sneppen 1995; Bak 1996) have taken Kauffman's ideas, with some modification, and used them to create a considerably simpler model of large-scale coevolution which also shows a power-law distribution of avalanche sizes and which is simple enough that its properties can, to some extent, be understood analytically. Although the model does not directly address the question of extinction, a number of authors have interpreted it, using arguments similar to those of section 1.2.2.5, as a possible model for extinction by biotic causes. The Bak-Sneppen model is one of a class of models that show "self-organized criticality," which means that regardless of the state in which they start, they always tune themselves to a critical point of the type discussed in section 2.4, where power-law behavior is seen. We describe self-organized criticality in more detail in section 3.2. First, however, we describe the Bak-Sneppen model itself. In the model of Bak and Sneppen there are no explicit fitness landscapes, as there are in NK models. Instead the model attempts to mimic the effects of landscapes in terms of "fitness barriers." Consider figure 3.1, which is a toy representation of a fitness landscape in which there is only one dimension in the genotype (or phenotype) space. If the mutation rate is low compared with the time scale on which selection takes place (as Kauffman assumed), then a population will spend most of its time localized around a peak in the landscape (labeled P in the figure).
Less

The models discussed in the last chapter are intriguing, but present a number of problems. In particular, most of the results about them come from computer simulations, and little is known analytically about their properties. Results such as the power-law distribution of extinction sizes and the system's evolution to the "edge of chaos" are only as accurate as the simulations in which they are observed. Moreover, it is not even clear what the mechanisms responsible for these results are, beyond the rather general arguments that we have already given. In order to address these shortcomings, Bak and Sneppen (1993; Sneppen et al. 1995; Sneppen 1995; Bak 1996) have taken Kauffman's ideas, with some modification, and used them to create a considerably simpler model of large-scale coevolution which also shows a power-law distribution of avalanche sizes and which is simple enough that its properties can, to some extent, be understood analytically. Although the model does not directly address the question of extinction, a number of authors have interpreted it, using arguments similar to those of section 1.2.2.5, as a possible model for extinction by biotic causes. The Bak-Sneppen model is one of a class of models that show "self-organized criticality," which means that regardless of the state in which they start, they always tune themselves to a critical point of the type discussed in section 2.4, where power-law behavior is seen. We describe self-organized criticality in more detail in section 3.2. First, however, we describe the Bak-Sneppen model itself. In the model of Bak and Sneppen there are no explicit fitness landscapes, as there are in NK models. Instead the model attempts to mimic the effects of landscapes in terms of "fitness barriers." Consider figure 3.1, which is a toy representation of a fitness landscape in which there is only one dimension in the genotype (or phenotype) space. If the mutation rate is low compared with the time scale on which selection takes place (as Kauffman assumed), then a population will spend most of its time localized around a peak in the landscape (labeled P in the figure).

*Constantino Tsallis*

- Published in print:
- 2004
- Published Online:
- November 2020
- ISBN:
- 9780195159769
- eISBN:
- 9780197562024
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195159769.003.0006
- Subject:
- Earth Sciences and Geography, Atmospheric Sciences

Statistical mechanics is clearly mechanics (classical, quantum, special or general relativistic, or any other) plus the theory of probabilities, as is well known. It ...
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Statistical mechanics is clearly mechanics (classical, quantum, special or general relativistic, or any other) plus the theory of probabilities, as is well known. It is our understanding, however, that it is more than that. It is also the adoption of a specific entropic functional, which will, in some sense, adequately shortcut the vast, and for most practical purposes useless, detailed microscopic mechanical information on the system. It is, in particular, through this functional that the connection with thermodynamics and its macroscopic laws will be established. This particular functional is determined by the specific type (or geometry) of occupation of the phase space (or Hilbert space or analogous space). This geometrical structure depends in turn not only on the microscopic dynamics that the system obeys, but also on the initial conditions at which the system is placed at t = 0. In colloquial terms, we could say that the microscopic dynamics determine where the system is allowed to live, whereas the initial conditions determin where it likes to live within the allowed region. This viewpoint is consistent with Einstein's perspective on classical statistical mechanics, and especially with his criticism [82, 92] of the celebrated Boltzmann principle However, the problem is that, up to now, no systematic manner exists for univocally determining the entropic functional to be used, given the dynamics and the initial conditions. The optimization of this entropy under the physically appropriate constraints is expected to provide the correct probability distribution for the microscopic states of the macroscopic stationary state of the system. Boltzmann, then complemented by Gibbs, proposed the celebrated form which is the foundation of standard statistical mechanics.
Less

Statistical mechanics is clearly mechanics (classical, quantum, special or general relativistic, or any other) plus the theory of probabilities, as is well known. It is our understanding, however, that it is more than that. It is also the adoption of a specific entropic functional, which will, in some sense, adequately shortcut the vast, and for most practical purposes useless, detailed microscopic mechanical information on the system. It is, in particular, through this functional that the connection with thermodynamics and its macroscopic laws will be established. This particular functional is determined by the specific type (or geometry) of occupation of the phase space (or Hilbert space or analogous space). This geometrical structure depends in turn not only on the microscopic dynamics that the system obeys, but also on the initial conditions at which the system is placed at t = 0. In colloquial terms, we could say that the microscopic dynamics determine where the system is allowed to live, whereas the initial conditions determin where it likes to live within the allowed region. This viewpoint is consistent with Einstein's perspective on classical statistical mechanics, and especially with his criticism [82, 92] of the celebrated Boltzmann principle However, the problem is that, up to now, no systematic manner exists for univocally determining the entropic functional to be used, given the dynamics and the initial conditions. The optimization of this entropy under the physically appropriate constraints is expected to provide the correct probability distribution for the microscopic states of the macroscopic stationary state of the system. Boltzmann, then complemented by Gibbs, proposed the celebrated form which is the foundation of standard statistical mechanics.