A stochastic-Lagrangian particle system for the Navier-Stokes equations

This paper is based on a formulation of the Navier-Stokes equations developed by P. Constantin and the first author (\texttt{arxiv:math.PR/0511067}, to appear), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with $\frac{1}{N}$ times the sum over these $N$ copies. (We remark that our formulation requires one to keep track of $N$ stochastic flows of diffeomorphisms, and not just the motion of $N$ particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space \holderspace{1}{\alpha} which consists of differentiable functions whose first derivative is $\alpha$ H\"older continuous (see Section \ref{sGexist} for the precise definition). Further, we show that as $N \to \infty$ the system converges to the solution of Navier-Stokes equations on any finite interval $[0,T]$. However for fixed $N$, we prove that this system retains roughly $O(\frac{1}{N})$ times its original energy as $t \to \infty$. Hence the limit $N \to \infty$ and $T\to \infty$ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as $t \to \infty$ explicitly.Comment: v3: Typo fixes, and a few stylistic changes. 17 pages, 2 figure

Statistical Mechanics of Charged Black Holes in Induced Einstein-Maxwell Gravity

The statistical origin of the entropy of charged black holes in models of induced Einstein-Maxwell gravity is investigated. The constituents inducing the Einstein-Maxwell action are charged and interact with an external gauge potential. This new feature, however, does not change divergences of the statistical-mechanical entropy of the constituents near the horizon. It is demonstrated that the mechanism of generation of the Bekenstein-Hawking entropy in induced gravity is universal and it is basically the same for charged and neutral black holes. The concrete computations are carried out for induced Einstein-Maxwell gravity with a negative cosmological constant in three space-time dimensions.Comment: 16 pages, latex, no figure

Quasi-normal modes of Schwarzschild-de Sitter black holes

The low-laying frequencies of characteristic quasi-normal modes (QNM) of Schwarzschild-de Sitter (SdS) black holes have been calculated for fields of different spin using the 6th-order WKB approximation and the approximation by the P\"{o}shl-Teller potential. The well-known asymptotic formula for large $l$ is generalized here on a case of the Schwarzchild-de Sitter black hole. In the limit of the near extreme $\Lambda$ term the results given by both methods are in a very good agreement, and in this limit fields of different spin decay with the same rate.Comment: 9 pages, 1 ancillary Mathematica(R) noteboo

Decay of charged scalar field around a black hole: quasinormal modes of R-N, R-N-AdS and dilaton black holes

It is well known that the charged scalar perturbations of the Reissner-Nordstrom metric will decay slower at very late times than the neutral ones, thereby dominating in the late time signal. We show that at the stage of quasinormal ringing, on the contrary, the neutral perturbations will decay slower for RN, RNAdS and dilaton black holes. The QN frequencies of the nearly extreme RN black hole have the same imaginary parts (damping times) for charged and neutral perturbations. An explanation of this fact is not clear but, possibly, is connected with the Choptuik scaling.Comment: 10 pages, LaTeX, 4 figures, considerable changes made and wrong interpretation of computations correcte

Energy, Hamiltonian, Noether Charge, and Black Holes

It is shown that in general the energy ${\cal E}$ and the Hamiltonian ${\cal H}$ of matter fields on the black hole exterior play different roles. ${\cal H}$ is a generator of the time evolution along the Killing time while ${\cal E}$ enters the first law of black hole thermodynamics. For non-minimally coupled fields the difference ${\cal H}-{\cal E}$ is not zero and is a Noether charge $Q$ analogous to that introduced by Wald to define the black hole entropy. If fields vanish at the spatial boundary, $Q$ is reduced to an integral over the horizon. The analysis is carried out and an explicit expression for $Q$ is found for general diffeomorphism invariant theories. As an extension of the results by Wald et al, the first law of black hole thermodynamics is derived for arbitrary weak matter fields.Comment: 20 pages, latex, no figure

Stability analysis and quasinormal modes of Reissner Nordstr{\o}m Space-time via Lyapunov exponent

We explicitly derive the proper time $(\tau)$ principal Lyapunov exponent ($\lambda_{p}$) and coordinate time ($t$) principal Lyapunov exponent ($\lambda_{c}$) for Reissner Nordstr{\o}m (RN) black hole (BH) . We also compute their ratio. For RN space-time, it is shown that the ratio is $\frac{\lambda_{p}}{\lambda_{c}}=\frac{r_{0}}{\sqrt{r_{0}^2-3Mr_{0}+2Q^2}}$ for time-like circular geodesics and for Schwarzschild BH it is $\frac{\lambda_{p}}{\lambda_{c}}=\frac{\sqrt{r_{0}}}{\sqrt{r_{0}-3M}}$. We further show that their ratio $\frac{\lambda_{p}}{\lambda_{c}}$ may vary from orbit to orbit. For instance, Schwarzschild BH at innermost stable circular orbit(ISCO), the ratio is $\frac{\lambda_{p}}{\lambda_{c}}\mid_{r_{ISCO}=6M}=\sqrt{2}$ and at marginally bound circular orbit (MBCO) the ratio is calculated to be $\frac{\lambda_{p}}{\lambda_{c}}\mid_{r_{mb}=4M}=2$. Similarly, for extremal RN BH the ratio at ISCO is $\frac{\lambda_{p}}{\lambda_{c}}\mid_{r_{ISCO}=4M}=\frac{2\sqrt{2}}{\sqrt{3}}$. We also further analyse the geodesic stability via this exponent. By evaluating the Lyapunov exponent, it is shown that in the eikonal limit , the real and imaginary parts of the quasi-normal modes of RN BH is given by the frequency and instability time scale of the unstable null circular geodesics.Comment: Accepted in Pramana, 07/09/201