Transient rectification of Brownian diffusion with asymmetric initial distribution

In an ensemble of non-interacting Brownian particles, a finite systematic average velocity may temporarily develop, even if it is zero initially. The effect originates from a small nonlinear correction to the dissipative force, causing the equation for the first moment of velocity to couple to moments of higher order. The effect may be relevant when a complex system dissociates in a viscous medium with conservation of momentum

Fidelity of holonomic quantum computations in the case of random errors in the values of control parameters

We investigate the influence of random errors in external control parameters on the stability of holonomic quantum computation in the case of arbitrary loops and adiabatic connections. A simple expression is obtained for the case of small random uncorrelated errors. Due to universality of mathematical description our results are valid for any physical system which can be described in terms of holonomies. Theoretical results are confirmed by numerical simulations.Comment: 7 pages, 3 figure

Generalized Fokker-Planck equation, Brownian motion, and ergodicity

Microscopic theory of Brownian motion of a particle of mass $M$ in a bath of molecules of mass $m\ll M$ is considered beyond lowest order in the mass ratio $m/M$. The corresponding Langevin equation contains nonlinear corrections to the dissipative force, and the generalized Fokker-Planck equation involves derivatives of order higher than two. These equations are derived from first principles with coefficients expressed in terms of correlation functions of microscopic force on the particle. The coefficients are evaluated explicitly for a generalized Rayleigh model with a finite time of molecule-particle collisions. In the limit of a low-density bath, we recover the results obtained previously for a model with instantaneous binary collisions. In general case, the equations contain additional corrections, quadratic in bath density, originating from a finite collision time. These corrections survive to order $(m/M)^2$ and are found to make the stationary distribution non-Maxwellian. Some relevant numerical simulations are also presented

How accurate are the non-linear chemical Fokker-Planck and chemical Langevin equations?

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order $\Omega^{-3/2}$ for reaction systems which do not obey detailed balance and at least accurate to order $\Omega^{-2}$ for systems obeying detailed balance, where $\Omega$ is the characteristic size of the system. Hence the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order $\Omega^{-1/2}$ and variance estimates accurate to order $\Omega^{-3/2}$. This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.Comment: 39 pages, 3 figures, accepted for publication in J. Chem. Phy

Stochastic dynamics beyond the weak coupling limit: thermalization

We discuss the structure and asymptotic long-time properties of coupled equations for the moments of a Brownian particle's momentum derived microscopically beyond the lowest approximation in the weak coupling parameter. Generalized fluctuation-dissipation relations are derived and shown to ensure convergence to thermal equilibrium at any order of perturbation theory.Comment: 6+ page

On Nonlinear Diffusion with Multiplicative Noise

Nonlinear diffusion is studied in the presence of multiplicative noise. The nonlinearity can be viewed as a ``wall'' limiting the motion of the diffusing field. A dynamic phase transition occurs when the system ``unbinds'' from the wall. Two different universality classes, corresponding to the cases of an ``upper'' and a ``lower'' wall, are identified and their critical properties are characterized. While the lower wall problem can be understood by applying the knowledge of linear diffusion with multiplicative noise, the upper wall problem exhibits an anomaly due to nontrivial dynamics in the vicinity of the wall. Broad power-law distribution is obtained throughout the bound phase.Comment: 4 pages, LaTeX, text and figures also available at http://matisse.ucsd.edu/~hw

Demographic growth and the distribution of language sizes

It is argued that the present log-normal distribution of language sizes is, to a large extent, a consequence of demographic dynamics within the population of speakers of each language. A two-parameter stochastic multiplicative process is proposed as a model for the population dynamics of individual languages, and applied over a period spanning the last ten centuries. The model disregards language birth and death. A straightforward fitting of the two parameters, which statistically characterize the population growth rate, predicts a distribution of language sizes in excellent agreement with empirical data. Numerical simulations, and the study of the size distribution within language families, validate the assumptions at the basis of the model.Comment: To appear in Int. J. Mod. Phys. C (2008

Exact Distributions for Stochastic Gene Expression Models with Bursting and Feedback

Stochasticity in gene expression can give rise to fluctuations in protein levels and lead to phenotypic variation across a population of genetically identical cells. Recent experiments indicate that bursting and feedback mechanisms play important roles in controlling noise in gene expression and phenotypic variation. A quantitative understanding of the impact of these factors requires analysis of the corresponding stochastic models. However, for stochastic models of gene expression with feedback and bursting, exact analytical results for protein distributions have not been obtained so far. Here, we analyze a model of gene expression with bursting and feedback regulation and obtain exact results for the corresponding protein steady-state distribution. The results obtained provide new insights into the role of bursting and feedback in noise regulation and optimization. Furthermore, for a specific choice of parameters, the system studied maps on to a two-state biochemical switch driven by a bursty input noise source. The analytical results derived thus provide quantitative insights into diverse cellular processes involving noise in gene expression and biochemical switching.Comment: Accepted in Phys. Rev. Let

A new method of quantization of classical solutions

Using stochastic quantization method we derive equations for correlators of quantum fluctuations around the classical solution in the massless phi^4 theory. The obtained equations are then solved in the lowest orders of perturbation theory, and the first correction to the free propagator of a quantum fluctuation is calculated.Comment: 8 page

Spin-torque shot noise in magnetic tunnel junctions

Spin polarized current may transfer angular momentum to a ferromagnet, resulting in a spin-torque phenomenon. At the same time the shot noise, associated with the current, leads to a non-equilibrium stochastic force acting on the ferromagnet. We derive stochastic version of Landau-Lifshitz-Gilbert equation for a magnetization of a ''free'' ferromagnetic layer in contact with a ''fixed'' ferromagnet. We solve the corresponding Fokker-Planck equation and show that the non-equilibrium noise yields to a non-monotonous dependence of the precession spectrum linewidth on the current.Comment: 5 pages, 2 figure