56 research outputs found
Topological Supersymmetry Breaking as the Origin of the Butterfly Effect
Previously, there existed no clear explanation why chaotic dynamics is always
accompanied by the infinitely long memory of perturbations (and/or initial
conditions) known as the butterfly effect (BE). In this paper, it is shown that
within the recently proposed approximation-free supersymmetric theory of
stochastic (partial) differential equations (SDE), the BE is a derivable
consequence of (stochastic) chaos, a rigorous definition of which is the
spontaneous breakdown of topological supersymmetry that all SDEs possess. It is
also discussed that the concept of ergodicy must be refined under the condition
of the spontaneous breakdown of pseudo-time-reversal symmetry when the model
has "physical" states of multiple eigenvalues that survive the physical limit
of the infinitely long temporal propagation.Comment: 26 pages, 4 figure
Topological field theory of dynamical systems
Here, it is shown that the path-integral representation of any stochastic or
deterministic continuous-time dynamical model is a cohomological or Witten-type
topological field theory, i.e., a model with global topological supersymmetry
(Q-symmetry). As many other supersymmetries, Q-symmetry must be perturbatively
stable due to what is generically known as non-renormalization theorems. As a
result, all (equilibrium) dynamical models are divided into three major
categories: Markovian models with unbroken Q-symmetry, chaotic models with
Q-symmetry spontaneously broken on the mean-field level by, e.g., fractal
invariant sets (e.g., strange attractors), and intermittent or self-organized
critical (SOC) models with Q-symmetry dynamically broken by the condensation of
instanton-antiinstanton configurations (earthquakes, avalanches etc.) SOC is a
full-dimensional phase separating chaos and Markovian dynamics. In the
deterministic limit, however, antiinstantons disappear and SOC collapses into
the "edge of chaos". Goldstone theorem stands behind spatio-temporal
self-similarity of Q-broken phases known under such names as algebraic
statistics of avalanches, 1/f noise, sensitivity to initial conditions etc.
Other fundamental differences of Q-broken phases is that they can be
effectively viewed as quantum dynamics and that they must also have
time-reversal symmetry spontaneously broken. Q-symmetry breaking in
non-equilibrium situations (quenches, Barkhausen effect, etc.) is also briefly
discussed.Comment: 18 pages, 4 figures, published versio
Transfer operators and topological field theory
The transfer operator (TO) formalism of the dynamical systems (DS) theory is
reformulated here in terms of the recently proposed supersymetric theory of
stochastic differential equations (SDE). It turns out that the stochastically
generalized TO (GTO) of the DS theory is the finite-time Fokker-Planck
evolution operator. As a result comes the supersymmetric trivialization of the
so-called sharp trace and sharp determinant of the GTO, with the former being
the Witten index, which is also the stochastic generalization of the Lefschetz
index so that it equals the Euler characteristic of the (closed) phase space
for any flow vector field, noise metric, and temperature. The enabled
possibility to apply the spectral theorems of the DS theory to the
Fokker-Planck operators allows to extend the previous picture of the
spontaneous topological supersymmetry (Q-symmetry) breaking onto the situations
with negative ground state's attenuation rate. The later signifies the
exponential growth of the number of periodic solutions/orbits in the large time
limit, which is the unique feature of chaotic behavior proving that the
spontaneous breakdown of Q-symmetry is indeed the field-theoretic definition
and stochastic generalization of the concept of deterministic chaos. In
addition, the previously proposed low-temperature classification of SDEs, i.e.,
thermodynamic equilibrium / noise-induced chaos ((anti)instanton condensation,
intermittent) / ordinary chaos (non-integrability of the flow vector field), is
complemented by the discussion of the high-temperature regime where the sharp
boundary between the noise-induced and ordinary chaotic phases must smear out
into a crossover, and at even higher temperatures the Q-symmetry is restored.
The Weyl quantization is discussed in the context of the Ito-Stratonovich
dilemma.Comment: 51 pages, 3 figure
Bogoliubov-like mode in the Tonks-Girardeau Gas
We reformulate 1D boson-fermion duality in path-integral terms. The result is
a 1D counterpart of the boson-fermion duality in the 2D Chern-Simons gauge
theory. The theory is consistent and enables, using standard resummation
techniques, to obtain the long-wave-length asymptotics of the collective mode
in 1D boson systems at the Tonks-Girardeau regime. The collective mode has the
dispersion of Bogoliubov phonons: , where
is the bosons density and is a Fourier component of the
two-body potential.Comment: 4 pages, 1 figur
Stochastic Dynamics and Combinatorial Optimization
Natural dynamics is often dominated by sudden nonlinear processes such as
neuroavalanches, gamma-ray bursts, solar flares \emph{etc}. that exhibit
scale-free statistics much in the spirit of the logarithmic Ritcher scale for
earthquake magnitudes. On phase diagrams, stochastic dynamical systems (DSs)
exhibiting this type of dynamics belong to the finite-width phase (N-phase for
brevity) that precedes ordinary chaotic behavior and that is known under such
names as noise-induced chaos, self-organized criticality, dynamical complexity
\emph{etc.} Within the recently formulated approximation-free supersymemtric
theory of stochastics, the N-phase can be roughly interpreted as the
noise-induced "overlap" between integrable and chaotic deterministic dynamics.
As a result, the N-phase dynamics inherits the properties of the both. Here, we
analyze this unique set of properties and conclude that the N-phase DSs must
naturally be the most efficient optimizers: on one hand, N-phase DSs have
integrable flows with well-defined attractors that can be associated with
candidate solutions and, on the other hand, the noise-induced
attractor-to-attractor dynamics in the N-phase is effectively chaotic or
a-periodic so that a DS must avoid revisiting solutions/attractors thus
accelerating the search for the best solution. Based on this understanding, we
propose a method for stochastic dynamical optimization using the N-phase DSs.
This method can be viewed as a hybrid of the simulated and chaotic annealing
methods. Our proposition can result in a new generation of hardware devices for
efficient solution of various search and/or combinatorial optimization
problems.Comment: revtex4-
Digital Memcomputing: from Logic to Dynamics to Topology
Digital memcomputing machines (DMMs) are a class of computational machines
designed to solve combinatorial optimization problems. A practical realization
of DMMs can be accomplished via electrical circuits of highly non-linear,
point-dissipative dynamical systems engineered so that periodic orbits and
chaos can be avoided. A given logic problem is first mapped into this type of
dynamical system whose point attractors represent the solutions of the original
problem. A DMM then finds the solution via a succession of elementary
instantons whose role is to eliminate solitonic configurations of logical
inconsistency ("logical defects") from the circuit. By employing a
supersymmetric theory of dynamics, a DMM can be described by a cohomological
field theory that allows for computation of certain topological matrix elements
on instantons that have the mathematical meaning of intersection numbers on
instantons. We discuss the "dynamical" meaning of these matrix elements, and
argue that the number of elementary instantons needed to reach the solution
cannot exceed the number of state variables of DMMs, which in turn can only
grow at most polynomially with the size of the problem. These results shed
further light on the relation between logic, dynamics and topology in digital
memcomputing
Introduction to Supersymmetric Theory of Stochastics
Many natural and engineered dynamical systems, including all living objects,
exhibit signatures of what can be called spontaneous dynamical long-range order
(DLRO). This order's omnipresence has long been recognized by the scientific
community, as evidenced by a myriad of related concepts, theoretical and
phenomenological frameworks, and experimental phenomena such as turbulence,
noise, dynamical complexity, chaos and the butterfly effect, the Richter
scale for earthquakes and the scale-free statistics of other sudden processes,
self-organization and pattern formation, self-organized criticality, etc.
Although several successful approaches to various realizations of DLRO have
been established, the universal theoretical understanding of this phenomenon
remained elusive. The possibility of constructing a unified theory of DLRO has
emerged recently within the approximation-free supersymmetric theory of
stochastics (STS). There, DLRO is the spontaneous breakdown of the topological
or de Rham supersymmetry that all stochastic differential equations (SDEs)
possess. This theory may be interesting to researchers with very different
backgrounds because the ubiquitous DLRO is a truly interdisciplinary entity.
The STS is also an interdisciplinary construction. This theory is based on
dynamical systems theory, cohomological field theories, the theory of
pseudo-Hermitian operators, and the conventional theory of SDEs. Reviewing the
literature on all these mathematical disciplines can be time-consuming. As
such, a concise and self-contained introduction to the STS, the goal of this
paper, may be useful.Comment: 44 pages; 13 figures; revtex 4-1; improved format, typos, ref
Hydrodynamic Tensor-DFT with correct susceptibility
In a previous work we developed a family of orbital-free tensor equations for
DFT [J. Chem. Phys. 124, 024105 (2006)]. The theory is a combination of the
coupled hydrodynamic moment equations hierarchy with a cumulant truncation of
the one-body electron density matrix. A basic ingredient in the theory is how
to truncate the series of equation of motion for the moments. In the original
work we assumed that the cumulants vanish above a certain order (N). Here we
show how to modify this assumption to obtain the correct susceptibilities. This
is done for N=3, a level above the previous study. At the desired truncation
level a few relevant terms are added, which, with the right combination of
coefficients, lead to excellent agreement with the Kohn-Sham Lindhard
susceptibilities for an uninteracting system. The approach is also powerful
away from linear response, as demonstrated in a non-perturbative study of a
jellium with a repulsive core, where excellent matching with Kohn-Sham
simulations is obtained while the Thomas Fermi and von-Weiszacker methods show
significant deviations. In addition, time-dependent linear response studies at
the new N=3 level demonstrate our previous assertion that as the order of the
theory is increased, new additional transverse sound modes appear mimicking the
RPA transverse dispersion region.Comment: 10 pages, 3 figure
Topological supersymmetry breaking: Definition and stochastic generalization of chaos and the limit of applicability of statistics
The concept of deterministic dynamical chaos has a long history and is well
established by now. Nevertheless, its field theoretic essence and its
stochastic generalization have been revealed only very recently. Within the
newly found supersymmetric theory of stochastics (STS), all stochastic
differential equations (SDEs) possess topological or de Rahm supersymmetry and
stochastic chaos is the phenomenon of its spontaneous breakdown. Even though
the STS is free of approximations and thus is technically solid, it is still
missing a firm interpretational basis in order to be physically sound. Here, we
make a few important steps toward the construction of the interpretational
foundation for the STS. In particular, we discuss that one way to understand
why the ground states of chaotic SDEs are conditional (not total) probability
distributions, is that some of the variables have infinite memory of initial
conditions and thus are not "thermalized", i.e., cannot be described by the
initial-conditions-independent probability distributions. As a result, the
definitive assumption of physical statistics that the ground state is a
steady-state total probability distribution is not valid for chaotic SDEs.Comment: 20 pages, 1 figure, revtex 4-
A Liouville equation for systems which exchange particles with reservoirs: transport through a nano-device
A Redfield-like Liouville equation for an open system that couples to one or
more leads and exchanges particles with them is derived. The equation is
presented for a general case. A case study of time-dependent transport through
a single quantum level for varying electrostatic and chemical potentials in the
leads is presented. For the case of varying electrostatic potentials the
proposed equation yields, for the model study, the results of an exact
solution.Comment: 8 pages, 2 figure
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