Introduction to Supersymmetric Theory of Stochastics

Овчинников, И. В.

doi:10.3390/e18040108

Cited by 47 publications

(91 citation statements)

References 126 publications

(180 reference statements)

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“…[6] The TE state is (one of) the ground state(s) of the DS with the Langevin potential playing the role of the Lyapunov function,…”

Section: Chaotic and Thermal Annealing Methodsmentioning

confidence: 99%

“…The recently found approximation-free STS [6] lays at the heart of the proposition in this paper and we would like to begin with a brief discussion of this theory. The easiest way to address the STS is to consider the following class of SDEs that covers most of the models in the literature,ẋ…”

Section: General Phase Diagram Of Stochastic Dynamicsmentioning

confidence: 99%

“…[6] and Refs. therein) as a result of the conjecture that the mathematical essence of the so-called selforganized criticality [7] is the instanton-mediated spontaneous breakdown of the topological supersymmetry that all stochastic differential equations (SDEs) possess.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic dynamics and combinatorial optimization

Овчинников

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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“…[6] The TE state is (one of) the ground state(s) of the DS with the Langevin potential playing the role of the Lyapunov function,…”

Section: Chaotic and Thermal Annealing Methodsmentioning

confidence: 99%

Section: General Phase Diagram Of Stochastic Dynamicsmentioning

confidence: 99%

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Stochastic dynamics and combinatorial optimization

Овчинников

Wang

2017

Mod. Phys. Lett. B

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“…75 This analogy is not far-fetched since it has been shown that any classical dynamical system (with and without noise) can be expressed within a topological field theory. 72 The topological field theory that emerges is of a Witten type. 76 Therefore, any transient dynamics can be described within the same instantonic formalism, where, of course, the critical points, and the topological sector of the theory change according to the physical system considered.…”

Section: Instantonsmentioning

confidence: 99%

“…(3) in a path-integral representation, 71 the partition function of such representation has an action functional, S Eucl , which is of topological character. 72 As any physical system, the trajectory chosen by the system is the one that renders such action functional stationary. 73 Instantons turn out to be those trajectories that render the topological S Eucl stationary 69,70 dS Eucl ðx cl ðt; rÞÞ ¼ 0:…”

Section: Instantonsmentioning

confidence: 99%

Perspective: Memcomputing: Leveraging memory and physics to compute efficiently

Ventra

Traversa

2018

Journal of Applied Physics

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It is well known that physical phenomena may be of great help in computing some difficult problems efficiently. A typical example is prime factorization that may be solved in polynomial time by exploiting quantum entanglement on a quantum computer. There are, however, other types of (non-quantum) physical properties that one may leverage to compute efficiently a wide range of hard problems. In this perspective we discuss how to employ one such property, memory (time non-locality), in a novel physics-based approach to computation: Memcomputing. In particular, we focus on digital memcomputing machines (DMMs) that are scalable. DMMs can be realized with non-linear dynamical systems with memory. The latter property allows the realization of a new type of Boolean logic, one that is self-organizing. Self-organizing logic gates are "terminal-agnostic", namely they do not distinguish between input and output terminals. When appropriately assembled to represent a given combinatorial/optimization problem, the corresponding self-organizing circuit converges to the equilibrium points that express the solutions of the problem at hand. In doing so, DMMs take advantage of the long-range order that develops during the transient dynamics. This collective dynamical behavior, reminiscent of a phase transition, or even the "edge of chaos", is mediated by families of classical trajectories (instantons) that connect critical points of increasing stability in the system's phase space. The topological character of the solution search renders DMMs robust against noise and structural disorder. Since DMMs are non-quantum systems described by ordinary differential equations, not only can they be built in hardware with available technology, they can also be simulated efficiently on modern classical computers. As an example, we will show the polynomial-time solution of the subset-sum problem for the worst cases, and point to other types of hard problems where simulations of DMMs' equations of motion on classical computers have already demonstrated substantial advantages over traditional approaches. We conclude this article by outlining further directions of study.

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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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Introduction to Supersymmetric Theory of Stochastics

Cited by 47 publications

References 126 publications

Stochastic dynamics and combinatorial optimization

Stochastic dynamics and combinatorial optimization

Perspective: Memcomputing: Leveraging memory and physics to compute efficiently

Topological Field Theory and Computing with Instantons

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