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Seismicity, Statistical Physics Approaches to

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

A fundamental challenge in many scientific disciplines concerns upscaling, that is, of determining the regularities and laws of evolution atsome large scale from those known at a lower scale: biology (from molecules to cells, from cells to organs); neurobiology (from neurons to brainfunction), psychology (from brain to emotions, from evolution to understanding), ecology (from species to the global web of ecological interactions),condensed matter physics (from atoms and molecules to organized phases such as solid, liquid, gas, and intermediate structures), social sciences (fromindividual humans to social groups and to society), economics (from producers and consumers to the whole economy), finance (from investors to the globalfinancial markets), Internet (from e-pages to the world wide web 2.0), semantics (from letters and words to sentences and meaning), and so on. Earthquakephysics is no exception, with the challenge of understanding the transition from the...

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Abbreviations

Chaos:

Chaos occurs in dynamical systems with two ingredients: (i) nonlinear recurrent re-injection of the dynamics into a finite domain in phase space and (ii) exponential sensitivity of the trajectories in phase space to initial conditions.

Continuous phase transitions:

If there is a finite discontinuity in the first derivative of the thermodynamic potential, then the phase transition is termed first-order. During such a transition, a system either absorbs or releases a fixed amount of latent heat (e.?g. the freezing/melting of water/ice). If the first derivative is continuous but higher derivatives are discontinuous or infinite, then the phase transition is called continuous, of the second kind, or critical. Examples include the critical point of the liquid–gas transition, the Curie point of the ferromagnetic transition, or the superfluid transition [127,235].

Critical exponents:

Near the critical point, various thermodynamic quantities diverge as power laws with associated critical exponents. In equilibrium systems, there are scaling relations that connect some of the critical exponents of different thermodynamic quantities [32,127,203,216,235].

Critical phenomena:

Phenomena observed in systems that undergo a continuous phase transition. They are characterized by scale invariance: the statistical properties of a system at one scale are related to those at another scale only through the ratio of the two scales and not through any one of the two scales individually. The scale invariance is a result of fluctuations and correlations at all scales, which prevents the system from being separable in the large scale limit at the critical point [32,203,235].

Declustering:

In studies of seismicity, declustering traditionally refers to the deterministic identification of fore-, main- and aftershocks in sequences (or clusters) of earthquakes clustered in time and space. Recent, more sophisticated techniques, e.?g. stochastic declustering, assign to earthquakes probabilities of being triggered or spontaneous.

Dynamical scaling and exponents:

Non-equilibrium critical phase transitions are also characterized by scale invariance, scaling functions and critical exponents. Furthermore, some evidence supports the claim that universality classes also exist for non-equilibrium phase transitions (e.?g. the directed percolation and the Manna universality class in sandpile models), although a complete classification of classes is lacking and may in fact not exist at all. Much interest has recently focused on directed percolation, which, as the most common universality class of absorbing state phase transitions, is expected to occur in many physical, chemical and biological systems [85,135,203].

Finite size scaling:

If a thermodynamic or other quantity is investigated at the critical point under a change of the system size, the scaling behavior of the quantity with respect to the system size is known as finite size scaling [32]. The quantity may refer to a thermodynamic quantity such as the free energy or it may refer to an entire probability distribution function. At criticality, the sole length scale in a finite system is the upper cut-off s c, which diverges in the thermodynamic limit \( { L \to \infty } \). Assuming a lower cut-off \( { s_0 \ll s_\text{c},s } \), a finite size scaling ansatz for the distribution \( { P(s;s_\text{c}) } \) of the observable variable s, which depends on the upper cut-off s c is then given by:

$$ P(s;s_\text{c})=a s^{-\tau} G(s/s_\text{c}) \quad \text{for}\quad s, s_\text{c} \gg s_0\:, $$
(1)

where the parameter a is a non-universal metric factor, t is a universal (critical) exponent, and G is a universal scaling function that decays sufficiently fast for \( { s \gg s_\text{c} } \) [32,36]. Pruessner [163] provides a simple yet instructive and concise introduction to scaling theory and how to find associated exponents. System-specific corrections appear to sub-leading order.

Fractal:

A deterministic or stochastic mathematical object that is defined by its exact or statistical self-similarity at all scales. Informally, it often refers to a rough or fragmented geometrical shape which can be subdivided into parts which look approximately the same as the original shape. A fractal is too irregular to be described by Euclidean geometry and has a fractal dimension that is larger than its topological dimension but less than the dimension of the space it occupies.

Mean-Field:

An effective or average interaction field designed to approximately replace the interactions from many bodies by one effective interaction which is constant in time and space, neglecting fluctuations.

Mechanisms for power laws:

Power laws may be the hallmark of critical phenomena, but there are a host of other mechanisms that can lead to power laws (see Chapter 14 of [203] for a list of power law mechanisms as well as [37,143]). Observations of scale invariant statistics therefore do not necessarily imply SOC, of course. Power laws express the existence of a symmetry (scale invariance) and there are many mechanisms by which a symmetry can be obtained or restored.

Non-equilibrium phase transitions:

In contrast to systems at equilibrium, non-equilibrium phase transitions involve dynamics, energy input and dissipation. Detailed balance is violated and no known equivalent of the partition function exists, from which all thermodynamic quantities of interest derive in equilibrium. Examples of non-equilibrium phase transitions include absorbing state phase transitions, reaction-diffusion models, and morphological transitions of growing surfaces [85,135].

Phase transitions:

In (equilibrium) statistical mechanics, a phase transition occurs when there is a singularity in the free energy or one of its derivatives. Examples include the freezing of water, the transition from ferromagnetic to paramagnetic behavior in magnets, and the transition from a normal conductor to a superconductor [127,235].

Renormalization group theory:

A mathematical theory built on the idea that the critical point can be mapped onto a fixed point of a suitably chosen transformation on the system's Hamiltonian. It provides a foundation for understanding scaling and universality and provides tools for calculating exponents and scaling functions. Renormalization group theory provides the basis for our understanding of critical phenomena [32,216,235]. It has been extended to non-Hamiltonian systems and provides a general framework for constructing theories of the macro-world from the microscopic description.

Self-organized criticality (SOC):

Despite two decades of research since its inception by [13] and the ambitious claim by [11] that, as a mechanism for the ubiquitous power laws in Nature, SOC was “How Nature Works”, a commonly accepted definition along with necessary and sufficient conditions for SOC is still lacking [93,163,203]. A less rigorous definition may be the following: Self-organized criticality refers to a non-equilibrium, critical and marginally stable steady-state, which is attained spontaneously and without (explicit) tuning of parameters. It is characterized by power law event distributions and fractal geometry (in some cases) and may be expected in slowly driven, interaction-dominated threshold systems [93]. Some authors additionally require that temporal and/or spatial correlations decay algebraically (e.?g. [84], but see [163]). Definitions in the literature range from broad (simply the absence of characteristic length scales in non-equilibrium systems) to narrow (the criticality is due to an underlying continuous phase transition with all of its expected properties) (see, e.?g., [162] for evidence that precipitation is an instance of the latter definition of SOC in which a non-linear feedback of the order parameter on the control parameter turns a critical phase transition into a self-organized one attracting the dynamics [198]).

Spinodal decomposition:

In contrast to the slow process of phase separation via nucleation and slow growth of a new phase in a material inside the metasstable region near a first-order phase transition, spinodal decomposition is a non-equilibrium, rapid and critical-like dynamical process of phase separation that occurs quickly and throughout the material. It needs to be induced by rapidly quenching the material to reach a sub-area (sometimes a line) of the unstable region of the phase diagram which is characterized by a negative derivative of the free energy.

Statistical physics:

is the set of concepts and mathematical techniques allowing one to derive the large-scale laws of a physical system from the specification of the relevant microscopic elements and of their interactions.

Turbulence:

In fluid mechanics, turbulence refers to a regime in which the dynamics of the flow involves many interacting degrees of freedom, and is very complex with intermittent velocity bursts leading to anomalous scaling laws describing the energy transfer from injection at large scales to dissipation at small scales.

Universality:

In systems with little or no frozen disorder, equilibrium continuous phase transitions fall into a small set of universality classes that are characterized by the same critical exponents and by certain scaling functions that become identical near the critical point. The class depends only on the dimension of the space and the dimension of the order parameter. For instance, the critical point of the liquid–gas transition falls into the same universality class as the 3D Ising model. Even some phase transitions occurring in high-energy physics are expected to belong to the Ising class. Universality justifies the development and study of extremely simplified models (caricatures) of Nature, since the behavior of the system at the critical point can nevertheless be captured (in some cases exactly). However, non-universal features remain even at the critical point but are less important, e.?g. amplitudes of fluctuations or system-specific corrections to scaling that appear at sub-leading order [32,216,235,239].

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  1. Department of Management, Technology and Economics, ETH Zurich, Switzerland

    Didier Sornette

  2. Swiss Seismological Service, Institute of Geophysics, ETH Zurich, Switzerland

    Maximilian J. Werner

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Sornette, D., Werner, M.J. (2009). Seismicity, Statistical Physics Approaches to. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_467

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