549 research outputs found
Complexity, parallel computation and statistical physics
The intuition that a long history is required for the emergence of complexity
in natural systems is formalized using the notion of depth. The depth of a
system is defined in terms of the number of parallel computational steps needed
to simulate it. Depth provides an objective, irreducible measure of history
applicable to systems of the kind studied in statistical physics. It is argued
that physical complexity cannot occur in the absence of substantial depth and
that depth is a useful proxy for physical complexity. The ideas are illustrated
for a variety of systems in statistical physics.Comment: 21 pages, 7 figure
Parallel dynamics and computational complexity of the Bak-Sneppen model
The parallel computational complexity of the Bak-Sneppen evolution model is
studied. It is shown that Bak-Sneppen histories can be generated by a massively
parallel computer in a time that is polylogarithmic in the length of the
history. In this parallel dynamics, histories are built up via a nested
hierarchy of avalanches. Stated in another way, the main result is that the
logical depth of producing a Bak-Sneppen history is exponentially less than the
length of the history. This finding is surprising because the self-organized
critical state of the Bak-Sneppen model has long range correlations in time and
space that appear to imply that the dynamics is sequential and history
dependent. The parallel dynamics for generating Bak-Sneppen histories is
contrasted to standard Bak-Sneppen dynamics. Standard dynamics and an alternate
method for generating histories, conditional dynamics, are both shown to be
related to P-complete natural decision problems implying that they cannot be
efficiently implemented in parallel.Comment: 37 pages, 12 figure
Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity
The computational complexity of internal diffusion-limited aggregation (DLA)
is examined from both a theoretical and a practical point of view. We show that
for two or more dimensions, the problem of predicting the cluster from a given
set of paths is complete for the complexity class CC, the subset of P
characterized by circuits composed of comparator gates. CC-completeness is
believed to imply that, in the worst case, growing a cluster of size n requires
polynomial time in n even on a parallel computer.
A parallel relaxation algorithm is presented that uses the fact that clusters
are nearly spherical to guess the cluster from a given set of paths, and then
corrects defects in the guessed cluster through a non-local annihilation
process. The parallel running time of the relaxation algorithm for
two-dimensional internal DLA is studied by simulating it on a serial computer.
The numerical results are compatible with a running time that is either
polylogarithmic in n or a small power of n. Thus the computational resources
needed to grow large clusters are significantly less on average than the
worst-case analysis would suggest.
For a parallel machine with k processors, we show that random clusters in d
dimensions can be generated in O((n/k + log k) n^{2/d}) steps. This is a
significant speedup over explicit sequential simulation, which takes
O(n^{1+2/d}) time on average.
Finally, we show that in one dimension internal DLA can be predicted in O(log
n) parallel time, and so is in the complexity class NC
Stationary states and energy cascades in inelastic gases
We find a general class of nontrivial stationary states in inelastic gases
where, due to dissipation, energy is transfered from large velocity scales to
small velocity scales. These steady-states exist for arbitrary collision rules
and arbitrary dimension. Their signature is a stationary velocity distribution
f(v) with an algebraic high-energy tail, f(v) ~ v^{-sigma}. The exponent sigma
is obtained analytically and it varies continuously with the spatial dimension,
the homogeneity index characterizing the collision rate, and the restitution
coefficient. We observe these stationary states in numerical simulations in
which energy is injected into the system by infrequently boosting particles to
high velocities. We propose that these states may be realized experimentally in
driven granular systems.Comment: 4 pages, 4 figure
Evidence against a mean field description of short-range spin glasses revealed through thermal boundary conditions
A theoretical description of the low-temperature phase of short-range spin
glasses has remained elusive for decades. In particular, it is unclear if
theories that assert a single pair of pure states, or theories that are based
infinitely many pure states-such as replica symmetry breaking-best describe
realistic short-range systems. To resolve this controversy, the
three-dimensional Edwards-Anderson Ising spin glass in thermal boundary
conditions is studied numerically using population annealing Monte Carlo. In
thermal boundary conditions all eight combinations of periodic vs antiperiodic
boundary conditions in the three spatial directions appear in the ensemble with
their respective Boltzmann weights, thus minimizing finite-size corrections due
to domain walls. From the relative weighting of the eight boundary conditions
for each disorder instance a sample stiffness is defined, and its typical value
is shown to grow with system size according to a stiffness exponent. An
extrapolation to the large-system-size limit is in agreement with a description
that supports the droplet picture and other theories that assert a single pair
of pure states. The results are, however, incompatible with the mean-field
replica symmetry breaking picture, thus highlighting the need to go beyond
mean-field descriptions to accurately describe short-range spin-glass systems.Comment: 13 pages, 11 figures, 3 table
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