Sampling of quantum dynamics at long time

The principle of energy conservation leads to a generalized choice of transition probability in a piecewise adiabatic representation of quantum(-classical) dynamics. Significant improvement (almost an order of magnitude, depending on the parameters of the calculation) over previous schemes is achieved. Novel perspectives for theoretical calculations in coherent many-body systems are opened.Comment: Revised versio

Long time dynamics for damped Klein-Gordon equations

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global solution is bounded. The result applies to standard energy subcritical focusing nonlinearities $|u|^{p-1} u$, 1\textless{}p\textless{}(d+2)/(d-2) as well as any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems)

On long time dynamics of perturbed KdV equations

Consider perturbed KdV equations: $$u_t+u_{xxx}-6uu_x=\epsilon f(u(\cdot)),\quad x\in\mathbb{T}=\mathbb{R}/\mathbb{Z},\;\int_{\mathbb{T}}u(x,t)dx=0,$$ where the nonlinearity defines analytic operators $u(\cdot)\mapsto f(u(\cdot))$ in sufficiently smooth Sobolev spaces. Assume that the equation has an $\epsilon$-quasi-invariant measure $\mu$ and satisfies some additional mild assumptions. Let $u^{\epsilon}(t)$ be a solution. Then on time intervals of order $\epsilon^{-1}$, as $\epsilon\to0$, its actions $I(u^{\epsilon}(t,\cdot))$ can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is $\mu$-typical

Long-time dynamics of de Gennes' model for reptation

Diffusion of a polymer in a gel is studied within the framework of de Gennes' model for reptation. Our results for the scaling of the diffusion coefficient D and the longest relaxation time tau are markedly different from the most recently reported results, and are in agreement with de Gennes' reptation arguments: D ~ 1/N^2 and tau ~ N^3. The leading exponent of the finite-size corrections to the diffusion coefficient is consistent with the value of -2/3 that was reported for the Rubinstein model. This agreement suggests that its origin might be physical rather than an artifact of these models.Comment: 5 pages, 5 figures, submitted to J. Chem. Phy

Long-time dynamics of resonant weakly nonlinear CGL equations

Consider a weakly nonlinear CGL equation on the torus~$\mathbb{T}^d$: u_t+i\Delta u=\epsilon [\mu(-1)^{m-1}\Delta^{m} u+b|u|^{2p}u+ ic|u|^{2q}u].\eqno{(*)} Here $u=u(t,x)$, $x\in\mathbb{T}^d$, $0<\epsilon<<1$, $\mu\geqslant0$, $b,c\in\mathbb{R}$ and $m,p,q\in\mathbb{N}$. Define \mbox{I(u)=(I_{\dk},\dk\in\mathbb{Z}^d)}, where I_{\dk}=v_{\dk}\bar{v}_{\dk}/2 and v_{\dk}, \dk\in\mathbb{Z}^d, are the Fourier coefficients of the function~$u$ we give. Assume that the equation $(*)$ is well posed on time intervals of order $\epsilon^{-1}$ and its solutions have there a-priori bounds, independent of the small parameter. Let $u(t,x)$ solve the equation $(*)$. If $\epsilon$ is small enough, then for $t\lesssim\epsilon^{-1}$, the quantity $I(u(t,x))$ can be well described by solutions of an {\it effective equation}: $$u_t=\epsilon[\mu(-1)^{m-1}\Delta^m u+ F(u)],$$ where the term $F(u)$ can be constructed through a kind of resonant averaging of the nonlinearity $b|u|^{2p}+ ic|u|^{2q}u$