Supersymmetric Theory of Stochastics:Demystification of Self-Organized Criticality

Garcin, Matthieu

doi:10.1201/b20232-17

Cited by 5 publications

(9 citation statements)

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“…Here, we show that TFTs of another type, specifically the gauge-field-less Witten-type TFTs known as topological sigma models [4], describe the recently proposed digital memcomputing machines (DMMs) [5,6] -engineered dynamical systems with point attractors being the solutions of the corresponding logic circuit that solves a specific task. This result derives from the recent finding that any stochastic differential equation possesses a topological supersymmetry [7,8], and the realization that the solution search by a DMM proceeds via an instantonic phase. Certain TFT correlators in DMMs then reveal the presence of a transient long-range order both in space and time, associated with the effective breakdown of the topological supersymmetry by instantons.…”

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confidence: 94%

“…On the other hand, using methods of dynamical systems theory [11] and stochastic quantization [12], it was recently shown that all stochastic differential equations (SDEs), that describe all natural and engineered dynamical systems, including all non-quantum computational devices, possess a topological supersymmetry [7,8]. The supercharge of this supersymmetry is the exterior derivative on the phase space, and its ubiquitousness in dynamical systems is the algebraic reflection of the preservation of the continuity of the phase space by a continuous-time dynamics.…”

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confidence: 99%

“…( 3) is the standard path-integral representation of a Witten-type (cohomological) TFT -more specifically a TSM (due to the absence of gauge fields) -with the action consisting only of a gauge-fixing term [2]. It is also of a topological character since it depends only on the topology of the phase space, and physically it represents, up to a topological factor, the partition function of the noise [8]. Now that we have established the TSM representation of DMMs, let us try to understand where their computational power originates from.…”

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confidence: 99%

“…Indeed, at the initial point of the evolution, the system has unstable variables (instanton modulii) and thus the trajectory is highly sensitive to the external influence and/or initial conditions. This temporal long-range order can be interpreted as the sort of dynamical long-range order in ordinary chaotic dynamical systems, that have topological supersymmetry broken spontaneously [8].…”

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Topological Field Theory and Computing with Instantons

2017

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It is well known that dynamical systems may be employed as computing machines. However, not all dynamical systems offer particular advantages compared to the standard paradigm of computation, in regard to efficiency and scalability. Recently, it was suggested that a new type of machines, named digital -hence scalable-memcomputing machines (DMMs), that employ non-linear dynamical systems with memory, can solve complex Boolean problems efficiently. This result was derived using functional analysis without, however, providing a clear understanding of which physical features make DMMs such an efficient computational tool. Here, we show, using recently proposed topological field theory of dynamical systems, that the solution search by DMMs is a composite instanton. This process effectively breaks the topological supersymmetry common to all dynamical systems, including DMMs. The emergent long-range order -a collective dynamical behaviorallows logic gates of the machines to correlate arbitrarily far away from each other, despite their non-quantum character. We exemplify these results with the solution of prime factorization, but the conclusions generalize to DMMs applied to any other Boolean problem.Unconventional computing paradigms that employ topological features have considerable advantages over standard ones. The prototypical example is topological quantum computation [1,2] that exploits, in an essential way, the topological character of the ground states of some strongly-correlated quantum electron systems, such as p-wave superconductors and fractional quantum Hall systems, to realize computation unencumbered by decoherence and noise. [3] Recently, a new computing paradigm has been advanced, named memcomputing [4][5][6] that employs non-linear (nonquantum) dynamical systems to compute in and with memory. Implemented in digital (hence scalable) form, memcomputing machines (DMMs) are a collection of logic gates networked

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confidence: 94%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Topological Field Theory and Computing with Instantons

2017

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“…[8] This theory is an approximation-free and coordinate-free theory of SDEs that can be dubbed the supersymmetric theory of stochastics (STS) and that has provided a rigorous stochastic generalization of the concept of deterministic dynamical chaos. [9] Another important result from the STS, [10] which is directly relevant to this paper, is revealing the theoretical essence of the stochastic dynamical behavior known as noise-induced chaos, [11] self-organized criticality, [7] intermittency [12,13] etc. and that we call for brevity the N-phase.…”

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confidence: 97%

Stochastic dynamics and combinatorial optimization

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Wang

2017

Mod. Phys. Lett. B

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Natural dynamics is often dominated by sudden nonlinear processes such as neuroavalanches, gamma-ray bursts, solar flares 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, selforganized criticality, dynamical complexity 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.

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Topological supersymmetry breaking: The definition and stochastic generalization of chaos and the limit of applicability of statistics

2016

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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., can not 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. * iovchinnikov@ucla.edu

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Supersymmetric Theory of Stochastics:Demystification of Self-Organized Criticality

Cited by 5 publications

References 0 publications

Topological Field Theory and Computing with Instantons

Topological Field Theory and Computing with Instantons

Stochastic dynamics and combinatorial optimization

Topological supersymmetry breaking: The definition and stochastic generalization of chaos and the limit of applicability of statistics

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