Framework Of Stochastic Differential Equations Research Articles

Fluctuation theorem as a special case of Girsanov theorem

Stochastic thermodynamics is an important development in the direction of finding general thermodynamic principles for non-equilibrium systems. We believe stochastic thermodynamics has the potential to benefit from the measure-theoretic framework of stochastic differential equations (SDEs). Toward this, in this work, we show that fluctuation theorem (FT) is a special case of the Girsanov theorem, which is an important result in the theory of SDEs. We report that by employing Girsanov transformation of measures between the forward and the reversed dynamics of a general class of Langevin dynamic systems, we arrive at the integral fluctuation relation. Following the same approach, we derive the FT also for the overdamped case. Our derivation is applicable to both transient and steady state conditions and can also incorporate diffusion coefficients varying as a function of state and time, e.g. in the context of multiplicative noise. We expect that the proposed method will be an easy route towards deriving the FT irrespective of the complexity and non-linearity of the system.

Weakly convergent stochastic simulation of electron collisions in plasmas

Collisions between charged particles can be described and solved using Monte Carlo methods in the framework of stochastic differential equations (SDEs). In this paper, we start from an SDE including the extended Lorentz collision operator, which can recover the collisions between a sampling electron and background ions and electrons. On this basis, we construct a second order weakly convergent algorithm (WCA2) to simulate collisional effects of electrons in plasmas. Superseding the Weiner process by a three-point distribution, WCA2 possesses high weakly convergent accuracy as well as low computational costs. The definition and properties of weak convergence are discussed in detail. The weakly convergent order of WCA2 is verified both theoretically and numerically. Through two trial moment functions, we carefully analyze the numerical solutions of the SDE using rigorous statistical tests in the sense of weak convergence. The criteria and practical operations of finding the benchmark solution of SDEs are introduced at length. In order to illustrate the power of WCA2, we apply it to simulate the backward runaways in plasmas, which is a dramatic physical phenomenon. By comparison with the Euler-Maruyama method and the Cadjan-Ivanov method, the advantage and efficiency of WCA2 is exhibited. The backward runaway probability and its dependence on initial conditions are accurately studied using WCA2.

An Almost-Solvable Model of Complex Network Dynamics

We discuss a specific model, which we refer to as RandLOE, of a large multi-agent network whose dynamic is prescribed via a combination of deterministic local laws and random exogenous factors. The RandLOE approach lies outside the framework of Stochastic Differential Equations, but lends itself to analytic examination as well as to stable simulation even for relatively large networks. RandLOE is based on the logistic operator equation (LOE), which is a multidimensional dynamical system extending the classical logistic equation via an operator-algebraic interaction term. The network is defined by interpreting the LOE variable as an adjacency matrix of a complete graph. Depending on the choice of parameters, it can display a number of essentially distinct dynamical characteristics: e.g. cycles of expansion and contraction.

Microscopic derivation of particle-based coarse-grained dynamics

In this paper we revisit the derivation of equations of motion for coarse-grained (CG) particles from the microscopic Hamiltonian dynamics of the underlying atomistic system in equilibrium. The derivation is based on the projection operator method and time-convolution equation. We demonstrate that due to the energy exchange between CG and intraparticle phase space coordinates in the microscopic system, the choice of projection operator is not unique, leading to different CG equations of motion that have the form of the nonlinear generalized Langevin equation (GLE). We derive the idempotence properties for the projection operators along the system trajectories and show that these properties result in streaming terms of the respective GLEs that are conservative forces and allow the expression of the non-conservative forces explicitly through thermodynamic averages, which can be measured from the microscopic simulations. The difference between GLEs that are presented herein lies in how the non-conservative forces are partitioned into dissipative and projected contributions. We compute the projected force and analyze conditions under which the projected (stochastic) force is orthogonal to (uncorrelated with) the momenta of CG particles, therefore justifying a transition to a framework of stochastic differential equations. We show that a position- and momentum-independent memory function appears only if the projected force is fully decoupled from the past CG positions and momenta, respectively. In the case of non-vanishing correlations between the projected force and the CG coordinates in past times, we derive explicitly the position- and momentum-dependent memory function in a form of projection onto a space spanned by N-order Hermite polynomials. The expressions presented herein can be used to construct a hierarchy of thermodynamically consistent CG models with momentum-dependent memory functions. They can also be used to design computational schemes for obtaining the parameters for GLEs and their variants such as dissipative particle dynamics equations from the microscopic data. We illustrate these applications by presenting the GLE with a memory function that is quadratic in the particle momenta.

Scaling laws and dispersion equations for Lévy particles in one-dimensional fractal porous media

Over some range of scales natural porous media often display a fractal Eulerian velocity or conductivity field. If one assumes the fractal conductivity gives rise to fractal drift velocities, then particle paths may be studied in the framework of stochastic differential equations (SODEs). On the microscale, trajectories are modeled as solutions to a SODE with Markovian, stationary, ergodic drift subject to a fluctuating Lévy force. The Lévy force allows for self-motile particles such as flagellated microbes. On the mesoscale the trajectories are modeled as solutions to a SODE with Lévy (fractal) drift and diffusion arising from the microscale asymptotics. On the macroscale the process is driven by the asymptotics of the mesoscale drift without diffusion. Asymptotic scaling laws and dispersion equations are presented.

Boltzmann–Gibbs distribution of fortune and broken time reversible symmetry in econodynamics

Within the framework of stochastic differential equations it is demonstrated that the existence of Boltzmann–Gibbs type distribution in economy is independent of the time reversal symmetry in econodynamics. Both power law and exponential distributions can be accommodated naturally. The demonstration is based on a mathematical structure discovered during a study in gene regulatory network dynamics. Further possible analogy between equilibrium economy and thermodynamics is explored, suggesting that statistical physics methods can indeed play an important role in the study of complex systems.

Expansion of a bounded plasma with dust particles

The expansion of a bounded plasma with dust particles is investigated by means of computer modelling, taking into account the dynamics of the dust particle charge as well as the Coulomb collisions of electrons and ions with dust particles. The PIC method is used for the computer modelling. The collection of electrons and ions by dust particles is described in a way similar to orbit-limited probe theory. Coulomb interactions are described in the framework of stochastic differential equations. It is shown that the mean distribution functions of electrons and ions are influenced by the dust particles during plasma expansion. The evolution of the ion distribution function leads to a strong deviation from equilibrium. Dust particles also influence the temporal behaviour of the plasma parameters.

Elements of a systematic search in animal behavior and model simulations

Starting from a position paper by Hans Bremermann, general aspects of search behavior are envisaged: locomotion itself, the capability of orientation and possible storage of information about the ‘goals’ of a search. The importance of stochasticity in these processes is briefly discussed. In particular, experimental findings and theoretical concepts on the homing search of desert isopods (Hemilepistus reaumuri) are presented. Using the general framework of stochastic differential equations for the angular turning rate of a migrating individual, search paths with characteristic loops and meanders can be modelled and simulated. Search success is quantified by measuring the degree of path overlap and by computing an index of area search intensity. Quantities are plotted versus path length, both for observed isopod data and for typical simulated search paths. Certain elements of a systematic search are described and explained by a hypothesis about temporal locomotion control, based on the isopod's ability for path integration and directional compensation. Finally, possible effects of orientational cues are mentioned.

QUANTUM STOCHASTIC EQUATIONS FOR A NON-LINEAR DAMPED OSCILLATOR

A unified framework of stochastic differential equations for quantum systems, formulated within Non-Equilibrium Thermo Field Dynamics (NETFD), is applied to a model of non-linear damped oscillator. The quantum stochastic Liouville equation and the quantum Langevin equations (both of Ito and Stratonovich type), which are consistent with the corresponding master equation, are written down explicitly in the case of a non-conventional treatment. This solves Kubo's third problem: how one can obtain the correlations of random force operators for the Langevin equation compatible with the master equation derived by the non-conventional treatment of the damping theory.

A unified framework of stochastic differential equations for quantum systems

A unified framework of stochastic differential equations for quantum systems

QUANTUM STOCHASTIC HEISENBERG EQUATION

By extending the quantum stochastic Liouville equation formulated by one of the authors (T. A.) within Non-Equilibrium Thermo Field Dynamics (NETFD), it is shown that, by means of stochastic Heisenberg equation, one can construct a unified framework of stochastic differential equations for quantum systems with the quantum white noise of non-thermal ensemble. A new aspect is given to deal with the difficulties attending quantum Langevin equations.