Large density expansion of a hydrodynamic theory for self-propelled particles

Recently, an Enskog-type kinetic theory for Vicsek-type models for self-propelled particles has been proposed [T. Ihle, Phys. Rev. E 83, 030901 (2011)]. This theory is based on an exact equation for a Markov chain in phase space and is not limited to small density. Previously, the hydrodynamic equations were derived from this theory and its transport coefficients were given in terms of infinite series. Here, I show that the transport coefficients take a simple form in the large density limit. This allows me to analytically evaluate the well-known density instability of the polarly ordered phase near the flocking threshold at moderate and large densities. The growth rate of a longitudinal perturbation is calculated and several scaling regimes, including three different power laws, are identified. It is shown that at large densities, the restabilization of the ordered phase at smaller noise is analytically accessible within the range of validity of the hydrodynamic theory. Analytical predictions for the width of the unstable band, the maximum growth rate and for the wave number below which the instability occurs are given. In particular, the system size below which spatial perturbations of the homogeneous ordered state are stable is predicted to scale with $\sqrt{M}$ where $M$ is the average number of collision partners. The typical time scale until the instability becomes visible is calculated and is proportional to M

Reply to comment on "Towards a quantitative kinetic theory of polar active matter" by Bertin et al

A reply on the comment of Bertin, Chate, Ginelli, Gregoire, Leonard and Peshkov, arxiv:1404.3950v1, in this special issue

Discussion on Peshkov et al., "Boltzmann-Ginzburg-Landau approach for continuous descriptions of generic Vicsek-like models"

A discussion on the contribution of Peshkov, Bertin, Ginelli and Chate, arxiv:1404.3275v1, in this special issue

Transport coefficients of multi-particle collision algorithms with velocity-dependent collision rules

Detailed calculations of the transport coefficients of a recently introduced particle-based model for fluid dynamics with a non-ideal equation of state are presented. Excluded volume interactions are modeled by means of biased stochastic multiparticle collisions which depend on the local velocities and densities. Momentum and energy are exactly conserved locally. A general scheme to derive transport coefficients for such biased, velocity dependent collision rules is developed. Analytic expressions for the self-diffusion coefficient and the shear viscosity are obtained, and very good agreement is found with numerical results at small and large mean free paths. The viscosity turns out to be proportional to the square root of temperature, as in a real gas. In addition, the theoretical framework is applied to a two-component version of the model, and expressions for the viscosity and the difference in diffusion of the two species are given.Comment: 31 pages, 8 figures, accepted by J. Phys. Cond. Matte

Active matter beyond mean-field: Ring-kinetic theory for self-propelled particles

A ring-kinetic theory for Vicsek-style models of self-propelled agents is derived from the exact N-particle evolution equation in phase space. The theory goes beyond mean-field and does not rely on Boltzmann's approximation of molecular chaos. It can handle pre-collisional correlations and cluster formation which both seem important to understand the phase transition to collective motion. We propose a diagrammatic technique to perform a small density expansion of the collision operator and derive the first two equations of the BBGKY-hierarchy. An algorithm is presented that numerically solves the evolution equation for the two-particle correlations on a lattice. Agent-based simulations are performed and informative quantities such as orientational and density correlation functions are compared with those obtained by ring-kinetic theory. Excellent quantitative agreement between simulations and theory is found at not too small noises and mean free paths. This shows that there is parameter ranges in Vicsek-like models where the correlated closure of the BBGKY-hierarchy gives correct and nontrivial results. We calculate the dependence of the orientational correlations on distance in the disordered phase and find that it seems to be consistent with a power law with exponent around -1.8, followed by an exponential decay. General limitations of the kinetic theory and its numerical solution are discussed

Spin susceptibility and magnetic short-range order in the Hubbard model

The uniform static spin susceptibility in the paraphase of the one-band Hubbard model is calculated within a theory of magnetic short--range order (SRO) which extends the four-field slave-boson functional-integral approach by the trans- formation to an effective Ising model and the self-consistent incorporation of SRO at the saddle point. This theory describes a transition from the paraphase without SRO for hole dopings $\delta > \delta_{c_2}$ to a paraphase with anti- ferromagnetic SRO for $\delta_{c_1} < \delta < \delta_{c_2}$. In this region the susceptibility consists of interrelated `itinerant' and `local' parts and increases upon doping. The zero--temperature susceptibility exhibits a cusp at $\delta_{c_2}$ and reduces to the usual slave-boson result for larger dopings. Using the realistic value of the on--site Coulomb repulsion $U=8t$ for LSCO, the peak position ($\delta_{c_2} = 0.26$) as well as the doping dependence reasonably agree with low--temperature susceptibility experiments showing a maximum at a hole doping of about 25\%.Comment: 4 pages, 1 Postscript figure, revtex-style, accepted for publishing: Phys. Rev. B, 54, ... (1996

Seasonal Asymmetric Price Transmission in Ghanaian Tomato Markets: Adapting Johansen’s Estimation Method

We assess market integration and price transmission of perishable agricultural produce in Sub-Saharan Africa by studying Ghanaian tomato markets which are characterized by pronounced seasonality in production and trade flows. We analyse the tomato markets of Ghana by simultaneously regarding its five most important markets, Navrongo, Techiman, Kumasi, Tamale and Accra, in a multivariate asymmetric price transmission framework. The estimation of the model is based on a unique dataset and on a modified version of the Johansen estimation procedure which is suitable for estimating such multivariate models. We estimate the price transmission parameters for four regimes which are a combination of the seasonal patterns in trade flows and asymmetries in the longrun price equilibrium between the most important production region (Techiman) and the most important consumption centre for tomatoes (Accra). We find strong evidence for integration of the five markets. In general, price transmission appears to be fast. Disequilibria mainly trigger price responses in the two production regions of Navrongo and Techiman. The regimes are found to matter for the whole system of tomato markets. Disequilibrium is shown to spillover between the price relationships. Consequently, tomato markets in Ghana appear to be integrated and function very well since price signals are rapidly passed through the country.asymmetric price transmission, cointegration, Ghana, regime-dependent model, seasonality, tomato, vector error-correction model, Crop Production/Industries, Demand and Price Analysis, C32, Q11, Q13, F14, F15,