Derivation of mean-field equations for stochastic particle systems

We study stochastic particle systems on a complete graph and derive effective mean-field rate equations in the limit of diverging system size, which are also known from cluster aggregation models. We establish the propagation of chaos under generic growth conditions on particle jump rates, and the limit provides a master equation for the single site dynamics of the particle system, which is a non-linear birth death chain. Conservation of mass in the particle system leads to conservation of the first moment for the limit dynamics, and to non-uniqueness of stationary distributions. Our findings are consistent with recent results on exchange driven growth, and provide a connection between the well studied phenomena of gelation and condensation.Comment: 26 page

Existence results for mean field equations

Let $\Omega$ be an annulus. We prove that the mean field equation -\Delta\psi=\frac{e\sp{-\beta\psi}}{\int\sb{\Omega}e\sp{-\beta\psi}} admits a solution with zero boundary for $\beta\in (-16\pi,-8\pi)$. This is a supercritical case for the Moser-Trudinger inequality.Comment: Filling a gap in the argument and adding 2 referrence

Semiclassical Analysis of Extended Dynamical Mean Field Equations

The extended Dynamical Mean Field Equations (EDMFT) are analyzed using semiclassical methods for a model describing an interacting fermi-bose system. We compare the semiclassical approach with the exact QMC (Quantum Montecarlo) method. We found the transition to an ordered state to be of the first order for any dimension below four.Comment: RevTex, 39 pages, 16 figures; Appendix C added, typos correcte

Singular mean field equations on compact Riemann surfaces

For a general class of elliptic PDE's in mean field form on compact Riemann surfaces with exponential nonlinearity, we address the question of the existence of solutions with concentrated nonlinear term, which, in view of the applications, are physically of definite interest. In the model, we also include the possible presence of singular sources in the form of Dirac masses, which makes the problem more degenerate and difficult to attack

Mean field equations, hyperelliptic curves and modular forms: II

A pre-modular form $Z_n(\sigma; \tau)$ of weight $\tfrac{1}{2} n(n + 1)$ is introduced for each $n \in \Bbb N$, where $(\sigma, \tau) \in \Bbb C \times \Bbb H$, such that for $E_\tau = \Bbb C/(\Bbb Z + \Bbb Z \tau)$, every non-trivial zero of $Z_n(\sigma; \tau)$, namely $\sigma \not\in E_\tau[2]$, corresponds to a (scaling family of) solution to the mean field equation \begin{equation} \tag{MFE} \triangle u + e^u = \rho \, \delta_0 \end{equation} on the flat torus $E_\tau$ with singular strength $\rho = 8\pi n$. In Part I (Cambridge J. Math. 3, 2015), a hyperelliptic curve $\bar X_n(\tau) \subset {\rm Sym}^n E_\tau$, the Lam\'e curve, associated to the MFE was constructed. Our construction of $Z_n(\sigma; \tau)$ relies on a detailed study on the correspondence $\Bbb P^1 \leftarrow \bar X_n(\tau) \to E_\tau$ induced from the hyperelliptic projection and the addition map. As an application of the explicit form of the weight 10 pre-modular form $Z_4(\sigma; \tau)$, a counting formula for Lam\'e equations of degree $n = 4$ with finite monodromy is given in the appendix (by Y.-C. Chou).Comment: 32 pages. Part of content in previous versions is removed and published separately. One author is remove

Mean-Field Equations for Spin Models with Orthogonal Interaction Matrices

We study the metastable states in Ising spin models with orthogonal interaction matrices. We focus on three realizations of this model, the random case and two non-random cases, i.e.\ the fully-frustrated model on an infinite dimensional hypercube and the so-called sine-model. We use the mean-field (or {\sc tap}) equations which we derive by resuming the high-temperature expansion of the Gibbs free energy. In some special non-random cases, we can find the absolute minimum of the free energy. For the random case we compute the average number of solutions to the {\sc tap} equations. We find that the configurational entropy (or complexity) is extensive in the range T_{\mbox{\tiny RSB}}. Finally we present an apparently unrelated replica calculation which reproduces the analytical expression for the total number of {\sc tap} solutions.Comment: 22+3 pages, section 5 slightly modified, 1 Ref added, LaTeX and uuencoded figures now independent of each other (easier to print). Postscript available http://chimera.roma1.infn.it/index_papers_complex.htm

Structure preserving schemes for mean-field equations of collective behavior

In this paper we consider the development of numerical schemes for mean-field equations describing the collective behavior of a large group of interacting agents. The schemes are based on a generalization of the classical Chang-Cooper approach and are capable to preserve the main structural properties of the systems, namely nonnegativity of the solution, physical conservation laws, entropy dissipation and stationary solutions. In particular, the methods here derived are second order accurate in transient regimes whereas they can reach arbitrary accuracy asymptotically for large times. Several examples are reported to show the generality of the approach.Comment: Proceedings of the XVI International Conference on Hyperbolic Problem