Functional Fokker-Planck Equation Research Articles

Exact solution of the functional Fokker–Planck equation for cell growth with asymmetric cell division

In this communication, by means of the operator algebra we have succeeded to derive the exact solution of the functional Fokker–Planck Equation for a stochastic cell growth model with asymmetric cell division in closed form and investigated the time evolution of the derived density function. It is found that the density function exhibits an interesting “multi-peak” structure generated by cell division as time evolves. The knowledge of the exact closed-form solution not only provides a basis for understanding the concepts and properties behind the functional Fokker–Planck Equation, but it can also be useful as a benchmark to test approximate and analytical procedures because the existence of exact closed-form solutions is rare for the functional Fokker–Planck Equations.

Stochastic motion of vortex filaments in He II under random force

Langevin dynamics are applied to describe the stochastic motion of vortex filaments in He II under random force. The article describes a functional formalism, which is a modification of the method developed earlier by Migdal to deal with the stochastic dynamics of classical vortex filaments. In particular, starting with the Langevin-type equation, the functional Fokker–Planck equation for the characteristic functional was obtained. Based on this equation, and under the assumption that the random force correlator satisfies the fluctuation-dissipation theorem, thermodynamic equilibrium in a system of chaotic quantized vortices was investigated. Additionally, the case of stationary helium and the case of counterflow with the constant relative velocity of the normal and superfluid components were considered. Some physical consequences of the results obtained are also discussed.

Grassmann phase space methods for fermions. II. Field theory

In both quantum optics and cold atom physics, the behaviour of bosonic photons and atoms is often treated using phase space methods, where mode annihilation and creation operators are represented by c-number phase space variables, with the density operator equivalent to a distribution function of these variables. The anti-commutation rules for fermion annihilation, creation operators suggests the possibility of using anti-commuting Grassmann variables to represent these operators. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of Grassmann phase space methods in quantum-atom optics to treat fermionic systems is rather rare, though fermion coherent states using Grassmann variables are widely used in particle physics.This paper presents a phase space theory for fermion systems based on distribution functionals, which replace the density operator and involve Grassmann fields representing anti-commuting fermion field annihilation, creation operators. It is an extension of a previous phase space theory paper for fermions (Paper I) based on separate modes, in which the density operator is replaced by a distribution function depending on Grassmann phase space variables which represent the mode annihilation and creation operators. This further development of the theory is important for the situation when large numbers of fermions are involved, resulting in too many modes to treat separately. Here Grassmann fields, distribution functionals, functional Fokker–Planck equations and Ito stochastic field equations are involved. Typical applications to a trapped Fermi gas of interacting spin 1/2 fermionic atoms and to multi-component Fermi gases with non-zero range interactions are presented, showing that the Ito stochastic field equations are local in these cases. For the spin 1/2 case we also show how simple solutions can be obtained both for the untrapped case and for an optical lattice trapping potential.

Potential and flux field landscape theory. I. Global stability and dynamics of spatially dependent non-equilibrium systems

We established a potential and flux field landscape theory to quantify the global stability and dynamics of general spatially dependent non-equilibrium deterministic and stochastic systems. We extended our potential and flux landscape theory for spatially independent non-equilibrium stochastic systems described by Fokker-Planck equations to spatially dependent stochastic systems governed by general functional Fokker-Planck equations as well as functional Kramers-Moyal equations derived from master equations. Our general theory is applied to reaction-diffusion systems. For equilibrium spatially dependent systems with detailed balance, the potential field landscape alone, defined in terms of the steady state probability distribution functional, determines the global stability and dynamics of the system. The global stability of the system is closely related to the topography of the potential field landscape in terms of the basins of attraction and barrier heights in the field configuration state space. The effective driving force of the system is generated by the functional gradient of the potential field alone. For non-equilibrium spatially dependent systems, the curl probability flux field is indispensable in breaking detailed balance and creating non-equilibrium condition for the system. A complete characterization of the non-equilibrium dynamics of the spatially dependent system requires both the potential field and the curl probability flux field. While the non-equilibrium potential field landscape attracts the system down along the functional gradient similar to an electron moving in an electric field, the non-equilibrium flux field drives the system in a curly way similar to an electron moving in a magnetic field. In the small fluctuation limit, the intrinsic potential field as the small fluctuation limit of the potential field for spatially dependent non-equilibrium systems, which is closely related to the steady state probability distribution functional, is found to be a Lyapunov functional of the deterministic spatially dependent system. Therefore, the intrinsic potential landscape can characterize the global stability of the deterministic system. The relative entropy functional of the stochastic spatially dependent non-equilibrium system is found to be the Lyapunov functional of the stochastic dynamics of the system. Therefore, the relative entropy functional quantifies the global stability of the stochastic system with finite fluctuations. Our theory offers an alternative general approach to other field-theoretic techniques, to study the global stability and dynamics of spatially dependent non-equilibrium field systems. It can be applied to many physical, chemical, and biological spatially dependent non-equilibrium systems.

Landscape Framework and Global Stability for Stochastic Reaction Diffusion and General Spatially Extended Systems with Intrinsic Fluctuations

Spatially extended systems are widely encountered in physics, chemistry, and biology for studying many important natural phenomena. In this work, we established a landscape framework for studying general stochastic spatially extended systems with intrinsic statistical fluctuations that can be applied to both equilibrium systems with detailed balance and nonequilibrium systems without detailed balance. We set up the master equation for general stochastic spatially extended systems (functional master equation) from two fundamental dynamical ingredients that characterize the elementary state transitions of spatially extended systems. We explored the entire spectrum of the various approximations of the functional master equation under certain conditions, from the functional Kramers-Moyal equation to the functional Fokker-Planck equation and its equivalent, the functional Langevin equation, to the macroscopic deterministic equation. We uncovered the Lyapunov functionals which are required to quantify the global stability and function of the system for both the deterministic and stochastic spatially extended systems. The global potential landscape functional for stochastic spatially extended systems can be quantified by the steady state probability distribution functional. In the small fluctuation limit, the potential landscape functional becomes the Lyapunov functional for the corresponding deterministic spatially extended system. The relative entropy functional (or free energy functional) proves to be the Lyapunov functional for the stochastic spatially extended systems. The potential landscape functional and the relative entropy functional quantify the global stability and function of the deterministic and stochastic spatially extended systems for both equilibrium and nonequilibrium conditions. The chemical reaction-diffusion systems as a typical and important class of spatially extended systems is explored on its own terms as well as used as a direct application of the general framework to derive more specific results for reaction-diffusion systems. A biological system, the bicoid protein concentration distribution in fruit fly embryo development, which can be modeled as a specific type of reaction-diffusion dynamics, is studied using the proposed framework. We found both the bicoid concentration and its fluctuation decay from anterior to posterior when the source producing the bicoid protein is located at the anterior point. The corresponding local funneled landscape basin of the bicoid concentration becomes narrower and steeper from anterior to posterior in such a case.

Phase space theory of Bose–Einstein condensates and time-dependent modes

A phase space theory approach for treating dynamical behaviour of Bose–Einstein condensates applicable to situations such as interferometry with BEC in time-dependent double well potentials is presented. Time-dependent mode functions are used, chosen so that one, two,…highly occupied modes describe well the physics of interacting condensate bosons in time dependent potentials at well below the transition temperature. Time dependent mode annihilation, creation operators are represented by time dependent phase variables, but time independent total field annihilation, creation operators are represented by time independent field functions. Two situations are treated, one (mode theory) is where specific mode annihilation, creation operators and their related phase variables and distribution functions are dealt with, the other (field theory) is where only field creation, annihilation operators and their related field functions and distribution functionals are involved. The field theory treatment is more suitable when large boson numbers are involved. The paper focuses on the hybrid approach, where the modes are divided up between condensate (highly occupied) modes and non-condensate (sparsely occupied) modes. It is found that there are extra terms in the Ito stochastic equations both for the stochastic phases and stochastic fields, involving coupling coefficients defined via overlap integrals between mode functions and their time derivatives. For the hybrid approach both the Fokker–Planck and functional Fokker–Planck equations differ from those derived via the correspondence rules, the drift vectors are unchanged but the diffusion matrices contain additional terms involving the coupling coefficients.Results are also presented for the combined approach where all the modes are treated as one set. Here both the Fokker–Planck and functional Fokker–Planck equations are exactly the same as those derived via the correspondence rules. However, although the Ito stochastic field equations are also unchanged, the Ito equations for the stochastic phases contain an extra classical term involving the coupling coefficients.

Decoherence effects in Bose–Einstein condensate interferometry I. General theory

The present paper outlines a basic theoretical treatment of decoherence and dephasing effects in interferometry based on single component Bose–Einstein condensates in double potential wells, where two condensate modes may be involved. Results for both two mode condensates and the simpler single mode condensate case are presented. The approach involves a hybrid phase space distribution functional method where the condensate modes are described via a truncated Wigner representation, whilst the basically unoccupied non-condensate modes are described via a positive P representation. The Hamiltonian for the system is described in terms of quantum field operators for the condensate and non-condensate modes. The functional Fokker–Planck equation for the double phase space distribution functional is derived. Equivalent Ito stochastic equations for the condensate and non-condensate fields that replace the field operators are obtained, and stochastic averages of products of these fields give the quantum correlation functions that can be used to interpret interferometry experiments. The stochastic field equations are the sum of a deterministic term obtained from the drift vector in the functional Fokker–Planck equation, and a noise field whose stochastic properties are determined from the diffusion matrix in the functional Fokker–Planck equation. The stochastic properties of the noise field terms are similar to those for Gaussian–Markov processes in that the stochastic averages of odd numbers of noise fields are zero and those for even numbers of noise field terms are the sums of products of stochastic averages associated with pairs of noise fields. However each pair is represented by an element of the diffusion matrix rather than products of the noise fields themselves, as in the case of Gaussian–Markov processes. The treatment starts from a generalised mean field theory for two condensate modes, where generalised coupled Gross–Pitaevskii equations are obtained for the modes and matrix mechanics equations are derived for the amplitudes describing possible fragmentations of the condensate between the two modes. These self-consistent sets of equations are derived via the Dirac–Frenkel variational principle. Numerical studies for interferometry experiments would involve using the solutions from the generalised mean field theory in calculations for the stochastic fields from the Ito stochastic field equations.

Stochastic Gompertz model of tumour cell growth

In this communication, based upon the deterministic Gompertz law of cell growth, a stochastic model in tumour growth is proposed. This model takes account of both cell fission and mortality too. The corresponding density function of the size of the tumour cells obeys a functional Fokker–Planck equation which can be solved analytically. It is found that the density function exhibits an interesting “multi-peak” structure generated by cell fission as time evolves. Within this framework the action of therapy is also examined by simply incorporating a therapy term into the deterministic cell growth term.

Theory of decoherence in Bose-Einstein condensate interferometry

A full treatment of decoherence and dephasing effects in BEC interferometry has been developed based on using quantum correlation functions for treating interferometric effects. The BEC is described via a phase space distribution functional of the Wigner type for the condensate modes and the positive P type for the non-condensate modes. Ito equations for stochastic condensate and non-condensate field functions replace the functional Fokker-Planck equation for the distribution functional and stochastic averages of field function products determine the quantum correlation functions.

Stochastic formulation of the renormalization group: supersymmetric structure and topology of the space of couplings

The exact or Wilson renormalization group equations can be formulated as a functional Fokker–Planck equation in the infinite-dimensional configuration space of a field theory, suggesting a stochastic process in the space of couplings. Indeed, the ordinary renormalization group differential equations can be supplemented with noise, making them stochastic Langevin equations. Furthermore, if the renormalization group is a gradient flow, the space of couplings can be endowed with a supersymmetric structure a la Parisi–Sourlas. The formulation of the renormalization group as supersymmetric quantum mechanics is useful for analysing the topology of the space of couplings by means of Morse theory. We present simple examples with one or two couplings.

Nonlinear gluon evolution in the color glass condensate: I

We consider a nonlinear evolution equation recently proposed to describe the small- x hadronic physics in the regime of very high gluon density. This is a functional Fokker–Planck equation in terms of a classical random color source, which represents the color charge density of the partons with large x. In the saturation regime of interest, the coefficients of this equation must be known to all orders in the source strength. In this first paper of a series of two, we carefully derive the evolution equation, via a matching between classical and quantum correlations, and set up the framework for the exact background source calculation of its coefficients. We address and clarify many of the subtleties and ambiguities which have plagued past attempts at an explicit construction of this equation. We also introduce the physical interpretation of the saturation regime at small x as a Color Glass Condensate. In the second paper we shall evaluate the expressions derived here, and compare them to known results in various limits.

Saturation and universality in QCD at small x

We find approximate solutions to the renormalization group equation which governs the quantum evolution of the effective theory for the Color Glass Condensate. This is a functional Fokker-Planck equation which generates in particular the non-linear evolution equations previously derived by Balitsky and Kovchegov within perturbative QCD. In the limit where the transverse momentum of the external probe is large compared to the saturation momentum, our approximations yield the Gaussian Ansatz for the effective action of the McLerran-Venugopalan model. In the opposite limit, of a small external momentum, we find that the effective theory is governed by a scale-invariant universal action which has the correct properties to describe gluon saturation.

The renormalization group equation for the color glass condensate

We present an explicit and simple form of the renormalization group equation which governs the quantum evolution of the effective theory for the Color Glass Condensate (CGC). This is a functional Fokker–Planck equation for the probability density of the color field which describes the CGC in the covariant gauge. It is equivalent to the Euclidean time evolution equation for a second quantized current–current Hamiltonian in two spatial dimensions. The quantum corrections are included in the leading log approximation, but the equation is fully non-linear with respect to the generally strong background field. In the weak field limit, it reduces to the BFKL equation, while in the general non-linear case it generates the evolution equations for Wilson-line operators previously derived by Balitsky and Kovchegov within perturbative QCD.

A Modified Stochastic Gompertz Model for Tumour Cell Growth

Based upon the deterministic Gompertz law of cell growth, we have proposed a stochastic model of tumour cell growth, in which the size of the tumour cells is bounded. The model takes account of both cell fission (which is an ‘action at a distance’ effect) and mortality too. Accordingly, the density function of the size of the tumour cells obeys a functional Fokker–Planck Equation (FPE) associated with the bounded stochastic process. We apply the Lie‐algebraic method to derive the exact analytical solution via an iterative approach. It is found that the density function exhibits an interesting kink‐like structure generated by cell fission as time evolves.

Thermally activated depinning of flux lines from columnar defects.

The dynamics of a fluctuating vortex in the presence of a columnar defect is described with a functional Fokker-Planck equation. Within this picture thermally activated depinning due to an external electric current perpendicular to the defect axis corresponds to a nonzero probability current over a saddle point ${\mathbf{u}}_{1}$ of the energy in configuration space. If the energy barrier is large enough this probability current may be calculated summing over all fluctuations orthogonal to the direction of steepest descent at ${\mathbf{u}}_{1}$. This gives the prefactor in the Arrhenius law to be proportional to ${\mathit{j}}^{5/2}$${\mathit{T}}^{\mathrm{\ensuremath{-}}1/2}$ for the escape rate of a pinned vortex, and to ${\mathit{j}}^{3/2}$${\mathit{T}}^{\mathrm{\ensuremath{-}}1/2}$ for the resistivity \ensuremath{\rho}.

Statistical model of the thermal ignition of a distributed system

The present work is concerned with the instability of combustible mixtures against finite-amplitude nucleus-like fluctuations of temperature that lead to a stochastic thermal ignition. The phenomenon is associated with the presence of a potential barrier that must be crossed by nuclei to initiate the stochastic ignition, which turns out to be possible even under the deterministic explosion limit given by the well-known Frank-Kamenetskii model. A stochastic analysis of such phenomena has been given using a functional Fokker-Planck equation. The following discussion of the ignition is restricted to a one-dimensional case when the consumption of reactant is not taken into account. The results obtained here may be applied only to weak fluctuations.

Properties of confined states in phase dynamics.

In this paper we point out that for the confined states discussed recently for the phase associated with a stationary pattern, a conservation law for both the phase variation and the local wave vector exists. We discuss the physical implications of this observation as a function of the imposed boundary conditions. We investigate the influence of noise and we find that for the phase change and for the local wave vector, the stationary solution of the associated functional differential Fokker-Planck equation can be explicitly given.

Quantization and phase-space methods for slowly varying optical fields in a dispersive nonlinear medium.

We consider the quantization of slowly varying optical fields in a dispersive nonlinear medium and the application of phase-space methods to the resulting quantum field equations. A pragmatic approach to the quantization of the electromagnetic field is adopted whereby we apply canonical quantization to the Hamiltonian expressed in terms of the slowly varying electric field envelope, all approximations (quasimonochromatic and paraxial) having been made at the classical level. This approach allows us to include material dispersion, diffraction, and nonlinearity. Using phase-space methods we then develop a c-number functional Fokker-Planck equation from which the quantum statistical properties of propagating optical fields can be deduced.

Dynamical interpretation of a classical complex free energy

Using a functional Fokker-Planck equation for a scalar phi 4 model, it is shown that the initial decay rate out of a metastable state after an instantaneous quench to an unstable situation is proportional to the corresponding imaginary part of the free energy, as calculated by Langer (1969).

Field-theoretic study of the nonlinear Fokker-Planck equation

A new field-theoretic formulation of the Fokker-Planck approach to non-equilibrium statistical mechanics is presented. Starting with the nonlinear functional Fokker-Planck equation, a new generating functional is derived. No use of auxiliary conjugate fields or response functions is needed. The Feynman rules are deduced, and the renormalisation of the theory is carried out. Finally, the renormalisation group equation is solved, and scaling laws and critical exponents are calculated, which are in good agreement with previous results obtained through different formalisms.