Information content of spontaneous symmetry breaking

Marcelo Gleiser and Nikitas Stamatopoulos

Phys. Rev. D 86, 045004 – Published 1 August 2012

Abstract

We propose a measure of order in the context of nonequilibrium field theory and argue that this measure, which we call relative configurational entropy (RCE), may be used to quantify the emergence of coherent low-entropy configurations, such as time-dependent or time-independent topological and nontopological spatially extended structures. As an illustration, we investigate the nonequilibrium dynamics of spontaneous symmetry breaking in three spatial dimensions. In particular, we focus on a model where a real scalar field, prepared initially in a symmetric thermal state, is quenched to a broken-symmetric state. For a certain range of initial temperatures, spatially localized, long-lived structures known as oscillons emerge in synchrony and remain until the field reaches equilibrium again. We show that the RCE correlates with the number density of oscillons, thus offering a quantitative measure of the emergence of nonperturbative spatiotemporal patterns that can be generalized to a variety of physical systems.

Received 14 May 2012

DOI:https://doi.org/10.1103/PhysRevD.86.045004

Authors & Affiliations

Marcelo Gleiser^* and Nikitas Stamatopoulos^†

Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA

^*mgleiser@dartmouth.edu
^†nstamato@dartmouth.edu

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 86, Iss. 4 — 15 August 2012

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Evolution of the field’s zero mode (volume-averaged field) after the quench. The initial temperature was set to $T = 0.25$ . Note the damping of the amplitude as energy is transferred to higher $k$ modes.Reuse & Permissions
Figure 2
Radially averaged two-point correlation functions for the field $ϕ$ at $T = 0.25$ . On the left, we show the simulation data at equilibrium and the analytical approximation $⟨ | ϕ_{eq}^{latt} (k) |^{2} ⟩$ (solid line) of Eq. (10). On the right, the same data is plotted along with the two-point correlation function data at time $44 m^{- 1}$ after the quench (diamonds). Modes with $k ≲ 1.5 m$ get significantly amplified while the rest remain in thermal equilibrium throughout the simulation.Reuse & Permissions
Figure 3
Snapshot of the field $ϕ$ at $t = 44 m^{- 1}$ after the system is quenched from a single to a symmetric double well. The simulation size shown here is $L^{3} = 128^{3}$ . The field was initially thermalized at $T = 0.25$ . Oscillons (spatial ordering) appear as spikes about the zero mode of the field. Three isosurfaces are shown at $ϕ = 0.5$ , 1.3, 1.6 in purple, cyan, and red, respectively. In black and white, darker regions correspond to larger values of the field. As the system evolves towards its new equilibrium, the spatiotemporal ordering is gradually lost and oscillons subsequently disappear. A full visualization of the process can be seen at 29, where it is also clear that for early times oscillons emerge in synchrony (time ordering).Reuse & Permissions
Figure 4
Relative entropy density $S_{f} (| k |, t)$ and relative entropy $S_{f} (t)$ as a function of time for a simulation of initial temperature $T = 0.25$ . Initially the relative entropy is zero, corresponding to the time just after the quench when the system is still in equilibrium. As parametric resonance takes place, spikes in the low $k$ part of the relative entropy density emerge, signaling the formation of oscillons. The amplitude of the spikes correlate with the number of oscillons formed, as explained below. As the system evolves back to equilibrium, the oscillons disappear and the relative configurational entropy goes back to zero. The wave vector magnitude $k$ has units of m and time has units of $m^{- 1}$ .Reuse & Permissions
Figure 5
Relative configurational entropy $S_{f} (t)$ (continuous line) and number density of oscillons $n_{osc} (t)$ (dashed line) as a function of time for a simulation of initial temperature $T = 0.25$ . Initially, both the relative entropy and the number density of oscillons are zero, since the system starts in equilibrium. After the quench, a clear correlation is seen between the spikes in $S_{f} (t)$ and the maxima of $n_{osc}$ . The wave vector magnitude $| k |$ has units of m and time has units of $m^{- 1}$ .Reuse & Permissions
Figure 6
Ensemble-averaged values of the maxima of the RCE and of the oscillon number density for a range of different initial temperatures. For low temperatures $T ≲ 0.15$ , no oscillon formation is possible and the RCE is zero. As the temperature increases, the maxima of the RCE increase in amplitude at the same rate as the maxima of the oscillon number density. For temperatures $T \sim 0.28 - 0.29$ the rates begin to diverge as the Hartree approximation is no longer valid.Reuse & Permissions

Physical Review D

covering particles, fields, gravitation, and cosmology