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 system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s, 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
Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called '' ...
and
topological field theories. Besides these and related disciplines such as
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
and
supersymmetric field theories, STS is also connected with the traditional theory of
stochastic differential equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
s and the theory of pseudo-Hermitian operators.
The theory began with the application of
BRST gauge fixing procedure to Langevin SDEs,
that was later adapted to
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
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 operator
Transfer may refer to:
Arts and media
* ''Transfer'' (2010 film), a German science-fiction movie directed by Damir Lukacevic and starring Zana Marjanović
* ''Transfer'' (1966 film), a short film
* ''Transfer'' (journal), in management studies
...
s 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 form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s. This evolution has an intrinsic BRST or topological supersymmetry representing the preservation of topology and/or the concept of proximity in the
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
by continuous time dynamics. The theory identifies a model as
chaotic
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
, 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
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
and
self-organized criticality
Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. Their macroscopic behavior thus displays the spatial or temporal scale-invariance characteristic of the critical point of a phase ...
can be understood as a consequence of the
Goldstone theorem.
History and relation to other theories
The first relation between supersymmetry and stochastic dynamics was established in two papers in 1979 and 1982 by
Giorgio Parisi
Giorgio Parisi (born 4 August 1948) is an Italian theoretical physicist, whose research has focused on quantum field theory, statistical mechanics and complex systems. His best known contributions are the QCD evolution equations for parton densit ...
and Nicolas Sourlas,
who demonstrated that the application of the
BRST gauge fixing procedure to Langevin SDEs, i.e., to SDEs with linear phase spaces, gradient flow vector fields, and additive noises, results in N=2 supersymmetric models. The original goal of their work was
dimensional reduction
Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero. In physics, a theory in ''D'' spacetime dimensions can be redefined in a lower number of dimensions ''d'', by taking all the fields ...
, i.e., a specific cancellation of divergences in Feynman diagrams proposed a few years earlier by
Amnon Aharony
Amnon Aharony (Hebrew: אמנון אהרוני; born: 7 January 1943) is an Israeli Professor (Emeritus) of Physics in the School of Physics and Astronomy at Tel Aviv University, Israel and in the Physics Department of Ben Gurion University of t ...
,
Yoseph Imry
Yoseph Imry (Hebrew: יוסף אמרי; born 23 February 1939 – 29 May 2018) was an Israeli physicist.
He was best known for taking part in the foundation of mesoscopic physics, a relatively new branch of condensed matter physics. It is conce ...
, and
Shang-keng Ma
Shang-keng Ma (September 24, 1940, Chongqing, Sichuan, China – November 24, 1983, La Jolla, California, ) was a Chinese theoretical physicist, known for his work on the theory of critical phenomena and random systems. He is known as the co-auth ...
.
Since then, relation between so-emerged supersymmetry of Langevin SDEs and a few physical concepts
have been established including the
fluctuation dissipation theorems,
Jarzynski equality
The Jarzynski equality (JE) is an equation in statistical mechanics that relates free energy differences between two states and the irreversible work along an ensemble of trajectories joining the same states. It is named after the physicist Chr ...
,
Onsager principle of microscopic reversibility,
solutions of Fokker–Planck equations,
self-organization
Self-organization, also called spontaneous order in the social sciences, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process can be spontaneous when suffi ...
, etc.
A similar approach was used to establish that
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
,
its stochastic generalization,
and higher-order Langevin SDEs
also have supersymmetric representations. Real dynamical systems, however, are never purely Langevin or classical mechanical. In addition, physically meaningful Langevin SDEs never break supersymmetry spontaneously. Therefore, for the purpose of the identification of the spontaneous supersymmetry breaking as
dynamical chaos, the generalization of the Parisi–Sourlas approach to SDEs of general form is needed. This generalization could come only after a rigorous formulation of the theory of pseudo-Hermitian operators
because the stochastic evolution operator is pseudo-Hermitian in the general case. Such generalization
showed that all SDEs possess N=1 BRST or topological supersymmetry (TS) and this finding completes the story of relation between supersymmetry and SDEs.
In parallel to the BRST procedure approach to SDEs, mathematicians working in the
dynamical systems theory
Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called '' ...
introduced and studied the concept of generalized transfer operator defined for random dynamical systems.
This concept underlies the most important object of the STS, the stochastic evolution operator, and provides it with a solid mathematical meaning.
STS has a close relation with algebraic topology and its topological sector belongs to the class of models known as
Witten-type topological or cohomological field theory.
[
]
As a supersymmetric theory, BRST procedure approach to SDEs can be viewed as one of the realizations of the concept of Nicolai map.
Parisi–Sourlas approach to Langevin SDEs
In the context of supersymmetric approach to stochastic dynamics, the term Langevin SDEs denotes SDEs with Euclidean phase space,
, gradient flow vector field, and additive
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
,
where
,
is the noise variable,
is the noise intensity, and
, which in coordinates
and
, is the gradient flow vector field with
being the Langevin function often interpreted as the energy of the purely dissipative stochastic dynamical system.
The Parisi–Sourlas method is a way of construction of the
path integral representation of the Langevin SDE. It can be thought of as a
BRST gauge fixing procedure that uses the Langevin SDE as a gauge condition. Namely, one considers the following functional integral,
where
denotes the r.h.s. of the Langevin SDE,
is the operation of stochastic averaging with
being the normalized distribution of noise configurations,
is the Jacobian of the corresponding functional derivative, and the path integration is over all closed paths,
, where
and
are the initial and final moments of temporal evolution.
Dimensional reduction
The Parisi–Sourlas construction originally aimed at "dimensional reduction" proposed in 1976 by
Amnon Aharony
Amnon Aharony (Hebrew: אמנון אהרוני; born: 7 January 1943) is an Israeli Professor (Emeritus) of Physics in the School of Physics and Astronomy at Tel Aviv University, Israel and in the Physics Department of Ben Gurion University of t ...
,
Yoseph Imry
Yoseph Imry (Hebrew: יוסף אמרי; born 23 February 1939 – 29 May 2018) was an Israeli physicist.
He was best known for taking part in the foundation of mesoscopic physics, a relatively new branch of condensed matter physics. It is conce ...
, and
Shang-keng Ma
Shang-keng Ma (September 24, 1940, Chongqing, Sichuan, China – November 24, 1983, La Jolla, California, ) was a Chinese theoretical physicist, known for his work on the theory of critical phenomena and random systems. He is known as the co-auth ...
[ who proved that ''to all orders in perturbation expansion, the critical exponents in a ''d''-dimensional (4 < ''d'' < 6) system with short-range exchange and a random quenched field are the same as those of a (''d''–2)-dimensional pure system.''] Their arguments indicated that the "Feynman diagrams which give the leading singular behavior for the random case are identically equal, apart from combinatorial factors, to the corresponding Feynman diagrams for the pure case in two fewer dimensions."[
]
Topological interpretation
Topological aspects of the Parisi–Sourlas construction can be briefly outlined in the following manner. The delta-functional, i.e., the collection of the infinite number of delta-functions, ensures that only solutions of the Langevin SDE contribute to . In the context of BRST procedure, these solutions can be viewed as Gribov copies. Each solution contributes either positive or negative unity: with being the index of the so-called Nicolai map, , which in this case is the map from the space of closed paths in to the space of noise configurations, a map that provides a noise configuration at which a given closed path is a solution of the Langevin SDE. can be viewed as a realization of Poincaré–Hopf theorem
In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincar ...
on the infinite-dimensional space of close paths with the Langevin SDE playing the role of the vector field and with the solutions of Langevin SDE playing the role of the critical points with index . is independent of the noise configuration because it is of topological character. The same it true for its stochastic average, , which is not the partition function of the model but, instead, its Witten index
In quantum field theory and statistical mechanics, the Witten index at the inverse temperature β is defined as a modification of the standard partition function:
:\textrm -1)^F e^/math>
Note the (-1)F operator, where F is the fermion numbe ...
.
Path integral representation
With the help of a standard field theoretic technique that involves introduction of additional field called Lagrange multiplier, , and a pair of fermionic fields called Faddeev–Popov ghost
In physics, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral formulat ...
s, , the Witten index can be given the following form,
where denotes collection of all the fields, p.b.c. stands for periodic boundary conditions, the so-called gauge fermion, , with and , and the BRST symmetry defined via its action on arbitrary functional as . In the BRST formalism, the Q-exact pieces like, , serve as gauge fixing tools. Therefore, the path integral expression for can be interpreted as a model whose action contains nothing else but the gauge fixing term. This is a definitive feature of Witten-type topological field theories and in this particular case of BRST procedure approach to SDEs, the BRST symmetry can be also recognized as the topological supersymmetry.
A common way to explain the BRST procedure is to say that the BRST symmetry generates the fermionic version of the gauge transformations, whereas its overall effect on the path integral is to limit the integration only to configurations that satisfy a specified gauge condition. This interpretation also applies to Parisi–Sourlas approach with the deformations of the trajectory and the Langevin SDE playing the roles of the gauge transformations and the gauge condition respectively.
Operator representation
Physical fermions in the high-energy physics and condensed matter models have antiperiodic boundary conditions in time. The unconventional periodic boundary conditions for fermions in the path integral expression for the Witten index is the origin of the topological character of this object. These boundary conditions reveal themselves in the operator representation of the Witten index as the alternating sign operator, where is the operator of the number of ghosts/fermions and the finite-time stochastic evolution operator (SEO), , where, is the infinitesimal SEO with being the Lie derivative along the subscript vector field, being the Laplacian, being the exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
, which is the operator representative of the topological supersymmetry (TS), and , where and are bosonic and fermionic momenta, and with square brackets denoting bi-graded commutator, i.e., it is an anticommutator if both operators are fermionic (contain odd total number of 's and 's) and a commutator otherwise. The exterior derivative and are supercharges. They are nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the class ...
, e.g., , and commutative with the SEO. In other words, Langevin SDEs possess N=2 supersymmetry. The fact that is a supercharge is accidental. For SDEs of arbitrary form, this is not true.
Hilbert space
The wavefunctions are functions not only of the bosonic variables, , but also of the Grassmann number
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as ...
s or fermions, , from the tangent space of . The wavefunctions can be viewed as differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s on with the fermions playing the role of the differentials . The concept of infinitesimal SEO generalizes the Fokker–Planck operator, which is essentially the SEO acting on top differential forms that have the meaning of the total probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s. Differential forms of lesser degree can be interpreted, at least locally on , as conditional probability distribution
In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the ...
s. Viewing the spaces of differential forms of all degrees as wavefunctions of the model is a mathematical necessity. Without it, the Witten index representing the most fundamental object of the model—the partition function of the noise—would not exist and the dynamical partition function would not represent the number of fixed points of the SDE ( see below). The most general understanding of the wavefunctions is the coordinate-free objects that contain information not only on trajectories but also on the evolution of the differentials and/or Lyapunov exponents.
Relation to nonlinear sigma model and algebraic topology
In Ref., a model has been introduced that can be viewed as a 1D prototype of the topological nonlinear sigma models (TNSM), a subclass of the Witten-type topological field theories. The 1D TNSM is defined for Riemannian phase spaces while for Euclidean phase spaces it reduces to the Parisi–Sourlas model. Its key difference from STS is the diffusion operator which is the Hodge Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
for 1D TNSM and for STS . This difference is unimportant in the context of relation between STS and algebraic topology, the relation established by the theory of 1D TNSM (see, e.g., Refs.).
The model is defined by the following evolution operator