Dark matter density profiles: A comparison of nonextensive theory with N-body simulations

Density profiles of simulated galaxy cluster-sized dark matter haloes are analysed in the context of a recently introduced nonextensive theory of dark matter and gas density distributions. Nonextensive statistics accounts for long-range interactions in gravitationally coupled systems and is derived from the fundamental concept of entropy generalisation. The simulated profiles are determined down to radii of ~1% of R_200. The general trend of the relaxed, spherically averaged profiles is accurately reproduced by the theory. For the main free parameter kappa, measuring the degree of coupling within the system, and linked to physical quantities as the heat capacity and the polytropic index of the self-gravitating ensembles, we find a value of -15. The significant advantage over empirical fitting functions is provided by the physical content of the nonextensive approach.Comment: 6 pages, 3 figures, accepted for publication in A&

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

Statistics of reversible bond dynamics observed in force-clamp spectroscopy

We present a detailed analysis of two-state trajectories obtained from force-clamp spectroscopy (FCS) of reversibly bonded systems. FCS offers the unique possibility to vary the equilibrium constant in two-state kinetics, for instance the unfolding and refolding of biomolecules, over many orders of magnitude due to the force dependency of the respective rates. We discuss two different kinds of counting statistics, the event-counting usually employed in the statistical analysis of two-state kinetics and additionally the so-called cycle-counting. While in the former case all transitions are counted, cycle-counting means that we focus on one type of transitions. This might be advantageous in particular if the equilibrium constant is much larger or much smaller than unity because in these situations the temporal resolution of the experimental setup might not allow to capture all transitions of an event-counting analysis. We discuss how an analysis of FCS data for complex systems exhibiting dynamic disorder might be performed yielding information about the detailed force-dependence of the transition rates and about the time scale of the dynamic disorder. In addition, the question as to which extent the kinetic scheme can be viewed as a Markovian two-state model is discussed.Comment: 25 pages, 10 figures, Phys. Rev. E, in pres

Field theoretic formulation of a mode-coupling equation for colloids

The only available quantitative description of the slowing down of the dynamics upon approaching the glass transition has been, so far, the mode-coupling theory, developed in the 80's by G\"otze and collaborators. The standard derivation of this theory does not result from a systematic expansion. We present a field theoretic formulation that arrives at very similar mode-coupling equation but which is based on a variational principle and on a controlled expansion in a small dimensioneless parameter. Our approach applies to such physical systems as colloids interacting via a mildly repulsive potential. It can in principle, with moderate efforts, be extended to higher orders and to multipoint correlation functions

Non-equilibrium dynamics of gene expression and the Jarzynski equality

In order to express specific genes at the right time, the transcription of genes is regulated by the presence and absence of transcription factor molecules. With transcription factor concentrations undergoing constant changes, gene transcription takes place out of equilibrium. In this paper we discuss a simple mapping between dynamic models of gene expression and stochastic systems driven out of equilibrium. Using this mapping, results of nonequilibrium statistical mechanics such as the Jarzynski equality and the fluctuation theorem are demonstrated for gene expression dynamics. Applications of this approach include the determination of regulatory interactions between genes from experimental gene expression data

Semi-Analytic Estimates of Lyapunov Exponents in Lower-Dimensional Systems

Recent work has shown that statistical arguments, seemingly well-justified in higher dimensions, can also be used to derive reasonable, albeit less accurate, estimates of the largest Lyapunov exponent ${\chi}$ in lower-dimensional Hamiltonian systems. This letter explores the detailed assumptions incorporated into these arguments. The predicted values of ${\chi}$ are insensitive to most of these details, which can in any event be relaxed straightforwardly, but {\em can} depend sensitively on the nongeneric form of the auto-correlation function characterising the time-dependence of an orbit. This dependence on dynamics implies a fundamental limitation to the application of thermodynamic arguments to such lower-dimensional systems.Comment: 6 pages, 3 PostScript figure