3 research outputs found
Zero Temperature Dynamics of 2D and 3D Ising Ferromagnets
We consider zero-temperature, stochastic Ising models with nearest-neighbor
interactions in two and three dimensions. Using both symmetric and asymmetric
initial configurations, we study the evolution of the system with time. We
examine the issue of convergence of the dynamics and discuss the nature of the
final state of the system. By determining a relation between the median number
of spin flips per site, the probability p that a spin in the initial spin
configuration takes the value +1, and lattice size, we conclude that in two and
three dimensions, the system converges to a frozen (but not necessarily
uniform) state when p is not equal to 1/2. Results for p=1/2 in three
dimensions are consistent with the conjecture that the system does not evolve
towards a fully frozen limiting state. Our simulations also uncover `striped'
and `blinker' states first discussed by Spirin et al., and their statistical
properties are investigated.Comment: 17 pages, 12 figure
The metastate approach to thermodynamic chaos
In realistic disordered systems, such as the Edwards-Anderson (EA) spin
glass, no order parameter, such as the Parisi overlap distribution, can be both
translation-invariant and non-self-averaging. The standard mean-field picture
of the EA spin glass phase can therefore not be valid in any dimension and at
any temperature. Further analysis shows that, in general, when systems have
many competing (pure) thermodynamic states, a single state which is a mixture
of many of them (as in the standard mean-field picture) contains insufficient
information to reveal the full thermodynamic structure. We propose a different
approach, in which an appropriate thermodynamic description of such a system is
instead based on a metastate, which is an ensemble of (possibly mixed)
thermodynamic states. This approach, modelled on chaotic dynamical systems, is
needed when chaotic size dependence (of finite volume correlations) is present.
Here replicas arise in a natural way, when a metastate is specified by its
(meta)correlations. The metastate approach explains, connects, and unifies such
concepts as replica symmetry breaking, chaotic size dependence and replica
non-independence. Furthermore, it replaces the older idea of non-self-averaging
as dependence on the bulk couplings with the concept of dependence on the state
within the metastate at fixed coupling realization. We use these ideas to
classify possible metastates for the EA model, and discuss two scenarios
introduced by us earlier --- a nonstandard mean-field picture and a picture
intermediate between that and the usual scaling/droplet picture.Comment: LaTeX file, 49 page
Metastable States in Spin Glasses and Disordered Ferromagnets
We study analytically M-spin-flip stable states in disordered short-ranged
Ising models (spin glasses and ferromagnets) in all dimensions and for all M.
Our approach is primarily dynamical and is based on the convergence of a
zero-temperature dynamical process with flips of lattice animals up to size M
and starting from a deep quench, to a metastable limit. The results (rigorous
and nonrigorous, in infinite and finite volumes) concern many aspects of
metastable states: their numbers, basins of attraction, energy densities,
overlaps, remanent magnetizations and relations to thermodynamic states. For
example, we show that their overlap distribution is a delta-function at zero.
We also define a dynamics for M=infinity, which provides a potential tool for
investigating ground state structure.Comment: 34 pages (LaTeX); to appear in Physical Review