Supersymmetric theory of stochastic dynamics | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-29

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Supersymmetric_theory_of_stochastic_dynamics


00:02:39 1 History and relation to other theories
00:05:41 2 Parisi–Sourlas approach to Langevin SDEs
00:10:50 2.1 Topological interpretation
00:14:05 2.2 Path integral representation
00:19:07 2.3 Operator representation
00:24:08 2.4 Hilbert space
00:26:17 3 Relation to nonlinear sigma model and algebraic topology
00:37:53 4 BRST procedure for SDEs of arbitrary form
00:40:15 4.1 Ambiguity of path integral representation and Ito–Stratonovich dilemma
00:47:10 4.2 Eigensystem of stochastic evolution operator
00:49:27 5 STS without BRST procedure
00:49:39 5.1 The mathematical meaning of stochastic evolution operator
00:57:05 5.2 The meaning of supersymmetry and the butterfly effect
00:59:58 6 Spontaneous supersymmetry breaking and dynamical chaos
01:05:43 7 STS as a topological field theory
01:11:32 8 Applications
01:13:05 8.1 Classification of stochastic dynamics
01:14:42 8.2 Demystification of self-organized criticality
01:16:48 8.3 Kinematic dynamo theory
01:17:38 8.4 Transient dynamics
01:19:14 8.5 Low energy effective theories for dynamical chaos



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SUMMARY
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Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches. From the mathematical point of view, STS is interesting because it bridges the two major parts of mathematical physics – the dynamical systems theory and topological field theories. Besides these and related disciplines such as algebraic topology and supersymmetric field theories, STS is also connected with the traditional theory of stochastic differential equations and the theory of pseudo-Hermitian operators.
The theory begun with the application of BRST gauge fixing procedure to Langevin SDEs, that was later adapted to classical mechanics and its stochastic generalization, higher-order Langevin SDEs, and, more recently, to SDEs of arbitrary form, which allowed to link BRST formalism to the concept of transfer operators and recognize spontaneous breakdown of BRST supersymmetry as a stochastic generalization of dynamical chaos.
The main idea of the theory is to study, instead of trajectories, the SDE-defined temporal evolution of differential forms. This evolution has an intrinsic BRST or topological supersymmetry representing the preservation of topology and/or the concept of proximity in the phase space by continuous time dynamics. The theory identifies a model as chaotic, in the generalized, stochastic sense, if its ground state is not supersymmetric, i.e., if the supersymmetry is broken spontaneously. Accordingly, the emergent long-range behavior that always accompanies dynamical chaos and its derivatives such as turbulence and self-organized criticality can be understood as a consequence of the Goldstone theorem.