Time-dependent properties of run-and-tumble particles. II.: Current fluctuations

We investigate steady-state current fluctuations in two models of run-and-tumble particles (RTPs) on a ring of $L$ sites, for \textit{arbitrary} tumbling rate $\gamma=\tau_p^{-1}$ and density $\rho$; model I consists of standard hardcore RTPs, while model II is an analytically tractable variant of model I, called long-ranged lattice gas (LLG). We show that, in the limit of $L$ large, the fluctuation of cumulative current $Q_i(T, L)$ across $i$th bond in a time interval $T \gg 1/D$ grows first {\it subdiffusively} and then {\it diffusively} (linearly) with $T$, where $D$ is the bulk diffusion coefficient. Remarkably, regardless of the model details, the scaled bond-current fluctuations $D \langle Q_i^2(T, L) \rangle/2 \chi L \equiv {\cal W}(y)$ as a function of scaled variable $y=DT/L^2$ collapse onto a {\it universal} scaling curve ${\cal W}(y)$, where $\chi(\rho,\gamma)$ is the collective particle {\it mobility}. In the limit of small density and tumbling rate $\rho, \gamma \rightarrow 0$ with $\psi=\rho/\gamma$ fixed, there exists a scaling law: The scaled mobility $\gamma^{a} \chi(\rho, \gamma)/\chi^{(0)} \equiv {\cal H} (\psi)$ as a function of $\psi$ collapse onto a scaling curve ${\cal H}(\psi)$, where $a=1$ and $2$ in models I and II, respectively, and $\chi^{(0)}$ is the mobility in the limiting case of symmetric simple exclusion process (SSEP). For model II (LLG), we calculate exactly, within a truncation scheme, both the scaling functions, ${\cal W}(y)$ and ${\cal H}(\psi)$. We also calculate spatial correlation functions for the current, and compare our theory with simulation results of model I; for both models, the correlation functions decay exponentially, with correlation length $\xi \sim \tau_p^{1/2}$ diverging with persistence time $\tau_p \gg 1$. Overall our theory is in excellent agreement with simulations and complements the findings of Ref. {\it arXiv:2209.11995}

A Brownian particle having a fluctuating mass

We focus on the dynamics of a Brownian particle whose mass fluctuates. First we show that the behaviour is similar to that of a Brownian particle moving in a fluctuating medium, as studied by Beck [Phys. Rev. Lett. 87 (2001) 180601]. By performing numerical simulations of the Langevin equation, we check the theoretical predictions derived in the adiabatic limit, and study deviations outside this limit. We compare the mass velocity distribution with truncated Tsallis distributions [J. Stat. Phys. 52 (1988) 479] and find excellent agreement if the masses are chi- squared distributed. We also consider the diffusion of the Brownian particle by studying a Bernoulli random walk with fluctuating walk length in one dimension. We observe the time dependence of the position distribution kurtosis and find interesting behaviours. We point out a few physical cases where the mass fluctuation problem could be encountered as a first approximation for agglomeration- fracture non equilibrium processes.Comment: submitted to PR

Linear Stochastic Models of Nonlinear Dynamical Systems

We investigate in this work the validity of linear stochastic models for nonlinear dynamical systems. We exploit as our basic tool a previously proposed Rayleigh-Ritz approximation for the effective action of nonlinear dynamical systems started from random initial conditions. The present paper discusses only the case where the PDF-Ansatz employed in the variational calculation is ``Markovian'', i.e. is determined completely by the present values of the moment-averages. In this case we show that the Rayleigh-Ritz effective action of the complete set of moment-functions that are employed in the closure has a quadratic part which is always formally an Onsager-Machlup action. Thus, subject to satisfaction of the requisite realizability conditions on the noise covariance, a linear Langevin model will exist which reproduces exactly the joint 2-time correlations of the moment-functions. We compare our method with the closely related formalism of principal oscillation patterns (POP), which, in the approach of C. Penland, is a method to derive such a linear Langevin model empirically from time-series data for the moment-functions. The predictive capability of the POP analysis, compared with the Rayleigh-Ritz result, is limited to the regime of small fluctuations around the most probable future pattern. Finally, we shall discuss a thermodynamics of statistical moments which should hold for all dynamical systems with stable invariant probability measures and which follows within the Rayleigh-Ritz formalism.Comment: 36 pages, 5 figures, seceq.sty for sequential numbering of equations by sectio

Application of Stationary Wavelet Support Vector Machines for the Prediction of Economic Recessions

This paper examines the efficiency of various approaches on the classification and prediction of economic expansion and recession periods in United Kingdom. Four approaches are applied. The first is discrete choice models using Logit and Probit regressions, while the second approach is a Markov Switching Regime (MSR) Model with Time-Varying Transition Probabilities. The third approach refers on Support Vector Machines (SVM), while the fourth approach proposed in this study is a Stationary Wavelet SVM modelling. The findings show that SW-SVM and MSR present the best forecasting performance, in the out-of sample period. In addition, the forecasts for period 2012-2015 are provided using all approaches