Physical Kinetics (Book): Difference between revisions

Line 15:

We of course aim to describe specific physical processes, but not without also introducing the general mathematical principles. As is covered in stochastic quantization within the [[Statistical_Physics_(Book)#Applications|statistical physics applications]] and in statistical field theory, random processes must be integrated through a Riemann-sum-like discrete collection of random samples. This is because continuous random processes are not smooth, so the naive idea of differentiating them and writing differential equations with random source terms does not make sense. Another basic introduction to stochastic differential equations (SDEs) is also within the open quantum systems book, in the context of quantum particles affected by a large classical system.

SDEs associated to random classical fields also have a fundamental relationship to quantum supersymmetric gauge theories, where this time the supersymmetry originates algebraically from the algebra of differential forms and the exterior derivative as opposed to speculative fundamental particle physics. This perspective was initiated by Parisi, and is continued in statistical physics. A kinetic application of this idea is to model the electromagnetic fields of the brain and neuronal processes like neuroavalanches and long range order with SDEs and with this gauge-geometry in mind. This is new, so there are no books on the topic, but we recommend starting with the paper by Igor V. Ovchinnikov and Skirmantas Janusonis: [https://arxiv.org/abs/2102.03849 Toward an Effective Theory of Neurodynamics: Topological Supersymmetry Breaking, Network Coarse-Graining, and Instanton Interaction] or with Ovchinnikov's introductory paper on the mathematical ideas: [https://arxiv.org/abs/1511.03393 Introduction to Supersymmetric Theory of Stochastics] which contains many helpful references including the original writing by Parisi himself.

SDEs associated to random classical fields also have a fundamental relationship to quantum supersymmetric gauge theories, where this time the supersymmetry originates algebraically from the algebra of differential forms and the exterior derivative as opposed to speculative fundamental particle physics. Partially, this analogy is evident through the use of Feynman-like diagrams in kinetic theory independently. This perspective was initiated by Parisi, and is continued in statistical physics. A kinetic application of this idea is to model the electromagnetic fields of the brain and neuronal processes like neuroavalanches and long range order with SDEs and with this gauge-geometry in mind. This is new, so there are no books on the topic, but we recommend starting with the paper by Igor V. Ovchinnikov and Skirmantas Janusonis: [https://arxiv.org/abs/2102.03849 Toward an Effective Theory of Neurodynamics: Topological Supersymmetry Breaking, Network Coarse-Graining, and Instanton Interaction] or with Ovchinnikov's introductory paper on the mathematical ideas: [https://arxiv.org/abs/1511.03393 Introduction to Supersymmetric Theory of Stochastics] which contains many helpful references including the original writing by Parisi himself, e.g. [https://www.sciencedirect.com/science/article/abs/pii/0550321382905387 Supersymmetric field theories and stochastic differential equations].

=== Applications ===