Langevin equation with multi-Poissonian noise

A general master equation is shown to be equivalent to a Langevin equation whose noise is expressed as a linear superposition of Poissonian random variables (multi-Poissonian noise). As typical examples, a birth and death process and a Boltzmann-Langevin equation are given.     

Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule

By introducing fractional Gaussian noise into the generalized Langevin equation, the subdiffusion of a particle can be described as a stationary Gaussian process with analytical tractability. This model is capable of explaining the equilibrium fluctuation of the distance between an electron transfer donor and acceptor pair within a protein that spans a broad range of time scales, and is in excellent agreement with a single-molecule experiment.     

Exact solution of a Langevin equation with nonlinear periodic noise

An exact stochastic average of a Langevin equation with a multiplicative nonlinear periodic noise is performed. The noise is described by an arbitrary periodic function of the diffusion Wiener-Lévy stochastic process. The solution of this stochastic equation is given by periodic solutions of the Hill equation.     

Non-Markovian Diffusion Over a Saddle with a Generalized Langevin Equation

The diffusion over a simple parabolic barrier is exactly solved with a non-Markovian Generalized Langevin Equation. For a short relaxation time, the problem is shown to be similar to a Markovian one, with a smaller effective friction. But for longer relaxation time, the average trajectory starts to oscillate and the system can have a very fast first passage over the barrier. For very long relaxation times, the solution tends to a zero-friction limit. PACS: 02.50.EY, 05.40.−a, 25.70.Jj     

Generalized Langevin equations with time-dependent temperature

A generalized Langevin equation describing the evolution of a particle in a heat bath with a time-dependent temperature is derived for a simple model. The temperature is controlled by introducing dissipative terms in the dynamical equations of the heat bath particles. The Langevin equation contains a term that is specifically associated with the variation of the temperature.     

Random integral equation formulation of a generalized Langevin equation

In this paper, the generalized Langevin equation introduced by Kubo and Mori is formulated as a random integral equation. We consider (1) the existence and uniqueness of the solution, (2) moments of the solution process, (3) a comparison theorem for solution processes, and (4) the Cauchy polygonal approximation to the solution.     

Langevin equation with two fractional orders

A new type of fractional Langevin equation of two different orders is introduced. The solutions for this equation, known as the fractional Ornstein-Uhlenbeck processes, based on Weyl and Riemann-Liouville fractional derivatives are obtained. The basic properties of these processes are studied. An example of the spectral density of ocean wind speed which has similar spectral density as that of Weyl fractional Ornstein-Uhlenbeck process is given.     

Generalized Langevin dynamics simulation: numerical integration and application of the generalized Langevin equation with an exponential model for the friction kernel

An efficient procedure is introduced for a generalized Langevin dynamics simulation when the exponential model is taken for the friction kernel. The leap frog algorithm is used for numerical integration of the generalized Langevin equation. Simulation with this model has been performed on a cyclic undecapeptide, cyclosporin A (CPA). By comparison with the results obtained from previous simulations, the method proves to be reliable and efficient in the simulation of CPA.     

Langevin particle in Gaussian noise

The problem of memory of past is discussed for a Langevin particle subject to Gaussian noise of a general type. The most probable trajectories of a particle are found and a generalisation of the Girsanov formula is proposed.     

Generalized Langevin Equations for Systems with Local Interactions

Journal of Statistical Physics - We present a new method to approximate the Mori–Zwanzig (MZ) memory integral in generalized Langevin equations describing the evolution of smooth observables...     

On the complex Langevin equation

We discuss the Langevin equation for a complex Boltzmann distribution, allowing for modifications of the process. The relation between the two different time development operators involved is analyzed, with emphasis on their spectra.     

Langevin equation in field theory

A new approach to quantum field theory is developed based on the Langevin equation (stochastic quantization). Applications to conventional and gauge theories are discussed, as well as various extensions; the Langevin difference equation, the complex Langevin equation in Minkowski space, etc.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 66–76, March, 1986.     

Localized solution of a simple nonlinear quantum Langevin equation

A simple nonlinear quantum Langevin equation is introduced as phenomenological equation for quantum brownian motion. Easy calculations yield a unique localized wave function in the stationary regime. The given example may encourage more general use of nonlinear quantum Langevin equations for damped quantum systems, e.g. in measurement theory, in heavy ion physics, etc.     

A generalized Langevin equation for dealing with nonadditive fluctuations

A suitable extension of the Mori memory-function formalism to the non-Hermitian case allows a multiplicative process to be described by a Langevin equation of non-Markoffian nature. This generalized Langevin equation is then shown to provide for the variable of interest the same autocorrelation function as the well-known theoretical approach developed by Kubo, the stochastic Liouville equation (SLE) theory. It is shown, furthermore, that the present approach does not disregard the influence of the variable of interest on the time evolution of its thermal bath. The stochastic process under study can also be described by a Fokker-Planck-like equation, which results in a Gaussian equilibrium distribution for the variable of interest. The main flaw of the SLE theory, that resulting in an uncorrect equilibrium distribution, is therefore completely eliminated.     

Solving Langevin equation with the bicolour rooted tree method

Stochastic differential equations, especially the one called Langevin equation, play an important role in many fields of modern science. In this paper, we use the bicolour rooted tree method, which is based on the stochastic Taylor expansion, to get the systematic pattern of the high order algorithm for Langevin equation. We propose a popular test problem, which is related to the energy relaxation in the double well, to test the validity of our algorithm and compare our algorithm with other usually used algorithms in simulations. And we also consider the time-dependent Langevin equation with the Ornstein-Uhlenbeck noise as our second example to demonstrate the versatility of our method.     

Homogenization for a Class of Generalized Langevin Equations with an Application to Thermophoresis

We study a class of systems whose dynamics are described by generalized Langevin equations with state-dependent coefficients. We find that in the limit, in which all the characteristic time scales vanish at the same rate, the position variable of the system converges to a homogenized process, described by an equation containing additional drift terms induced by the noise. The convergence results are obtained using the main result in Hottovy et al. (Commun Math Phys 336(3):1259–1283, 2015), whose version is proven here under a weaker spectral assumption on the damping matrix. We apply our results to study thermophoresis of a Brownian particle in a non-equilibrium heat bath.