Emergent Spatiotemporal Complexity in Field Theory
Essay written by Marcelo Gleiser. Originally published in Complexity and the Arrow of Time, edited by Charles H. Lineweaver, Paul C. W. Davies, and Michael Ruse, Cambridge University Press, 2013.
The origin of spatiotemporal order in physical and biological systems is a key scientific question of our time. How does microscopic matter self-organize to create living and non-living macroscopic structures? Do systems capable of generating spatiotemporal complexity obey certain universal principles? We propose that progress along these questions may be made by searching for fundamental properties of non-linear field models which are common to several areas of physics, from elementary particle physics to condensed matter and biological physics. In particular, we've begun exploring what models that support localized coherent (soliton-like) solutions - both timedependent and time-independent - can tell us about the emergence of spatiotemporal order. Of interest to us is the non-equilibrium dynamics of such systems and how it differs when they are allowed to interact with external environments. It is argued that the emergence of spatiotemporal order delays energy equipartition and that growing complexity correlates with growing departure from equipartition. We further argue that the emergence of complexity is related to the existence of attractors in field configuration space and propose a new entropic measure to quantify the degree of ordering of localized energy configurations.
6. I SOLITONS AND SELF-ORGANIZATION
A key question across the natural sciences is how simple material entities self-organize to create coherent structures capable of complex behavior. As an example, phenomena as diverse as water waves and symmetry-breaking during phase transitions can give rise to solitons, topologically or non-topologically stable spatially-localized structures ("'energy lumps") that keep their profiles as they move across space. They beautifully illustrate cooperative behavior in Nature, that is, how interacting discrete entities work in tandem to generate complex structures that minimize energy and other physical quantities (Infeld & Rowlands, 2000; Walgraef, 1997; Cross & Hohenberg, 1993; Rajamaran, 1987; Lee & Pang, 1992). For these to exist, a dynamic compromise must be achieved between what could be called gathering and dispersive tendencies: while attractive interactions tend to collect particles in small volumes, gradient energies want to spread them out. The study of such structures started in earnest in August 1834, when the Scottish engineer John Scott Russell was conducting experiments to improve the design of canals for boats. He observed, to his amazement, that sometimes, when the boat stopped suddenly, a "mass of water" kept travelling ahead of the boat for miles without losing its spatial shape. He called this "singular and beautiful phenomenon" a "Wave of Translation" (Scott, 2007). Today we call it a solitary wave or soliton.
In elementary particle physics, solitons owe their stability to a conserved charge, which can be either topological (that is, related to the non-trivial structure of the vacuum, i.e., the set of minima of the potential energy describing the system) or non-topological (that is, related to the conservation of certain charge-like quantities such as particle number). In both cases, solitons are time-independent solutions of the non-linear partial differential equations describing the models, which are robust against perturbations. They can be thought of as being composed of a superposition of many different momentum modes. Semi-classically, they can be thought of as collections of particles trapped in a bag-like configuration. It turns out that the same, or qualitatively similar, PDEs appear in many different areas of physics, albeit often in different spatial dimensionalities and in non-relativistic and/or high-viscosity approximations. This is due to the fact that many of the equations describing the interactions of fundamental matter fields are essentially non-linear wave equations with amplitude-dependent non-linearities determined by the particular interactions of the model. For example, the famous sineGordon equation (when sin x is truncated to second order, as in sin x = x - x'/3!) is related to the 1d Klein-Gordon equation with a double-well potential widely used in describing symmetry breaking in particle physics and condensed matter theories (Rajamaran, 1987). Since the 1970s, a large amount of literature exploring the properties of such solitons has been produced. The focus is invariably the same: a given model is described by a given PDE or coupled PDEs. Static solutions to these equations describe localized, coherent spatial structures such as kinks, vortices, or monopoles. In some cases, these structures maintain their profiles after scattering (what are known as "real" solitons), while in others they may suffer perturbations but still approximately maintain their spatial coherence or combine into more complex hybrid objects.
One of the key points I'd like to make is that focusing only on static (time-independent) solutions, as has mostly been the case for the past four decades, adds an unnecessary and very stringent constraint on the classes of models that can exhibit interesting spatiotemporal coherent behavior. Once this constraint is relaxed and we look for time-dependent but still spatially-localized structures, a whole new world opens up, with a plethora of possibilities. As I have shown over the past 15 years with various collaborators, oscillating solitonlike structures appear in a wide class of fundamental models describing symmetry breaking in elementary particle physics, cosmology, and in condensed matter systems. I called them oscillons in 1994 (Gleiser, 1994; Copeland, Gleiser, & Muller, 1995). Since then, their remarkable properties have attracted much attention in high energy physics and cosmology (Gleiser & Sornborger, 2000; Honda & Choptuik, 2002; Adib, Gleiser, & Almeida, 2002; Graham & Stamatopoulos, 2006; Farhi et al., 2008; Fodor et al., 2006; Gleiser & Howell, 2005; Gleiser & Sicilia, 2008, 2009; Hertzberg, 2010; Amin & Shirokoff, 2010; Amin et al., 2012). Interestingly, also in 1994, spatiotemporal patterns emerging in vibrating grains have been discovered and also given the name oscillons (Melo et al., 1994; Umbanhowar et al., 1996; Tsimring & Aranson, 1997; Jeong & Moon, 1999). Similar localized oscillating structures have been found in the plasma of stellar interiors (Umurhan et al., 1998), in extensions of the Swift-Hohenberg model (Crawford & Riecke, 1999), in deep water vibrations (Shats, Xia, & Punzmann, 2012), and in the non-linear Schrödinger equation (Stenflo & Yu, 2007), to list but a few examples. They all seem to have very similar qualitative properties to fundamental oscillons, although a more detailed analysis should be done to establish their correspondence.
6.2 EMERGENT SPATIOTEMPORAL COMPLEXITY IN FIELD THEORY
I will use the name oscillons to characterize any long-living, oscillating coherent field configuration found in non-linear field models in arbitrary spatial dimensions. These can involve a single real scalar field (Bogolubsky & Makhankov, 1976; Gleiser, 1994; Copeland, Gleiser, & Muller, 1995; Gleiser & Sornborger, 2000; Honda & Choptuik, 2002; Adib, Gleiser, & Almeida, 2002; Graham & StamatopouLos, 2006; Farhi et al., 2008; Fodor et al., 2006; Gleiser & Howell, 2005; Gleiser & Sicilia, 2008, 2009; Hertzberg, 2010; Amin & Shirokoff, 2010, Amin et al., 2012) or several interacting fields (Gleiser & Thorarinson, 2007, 2009). In discrete lattice models, qualitatively similar configurations are known by the name of discrete breathers (Flach & Willis, 1998). The name breathers also describes bound states of 1d kink-antikink pairs, which are localized in space and oscillate in time, somewhat like a drumhead that wouldn't stop vibrating: the non-linear interactions restrict the radiation of energy to spatial infinity (Gleiser & Sicilia, 2008).
In a sense, oscillons are the answer to Derrick's theorem (Rajaraman, 1987), which forbids the existence of time-independent solitons in more than one spatial dimension for models featuring only a single real scalar field. This disappointing result can be evaded in two (….)
