arXiv subjects

Thomas Bartsch

Publications and source records attributed to Thomas Bartsch.

At least 55 records · Page 3Linked to original sources

arXiv2015

Transition State Theory for dissipative systems without a dividing surface

Transition State Theory is a central cornerstone in reaction dynamics. Its key step is the identification of a dividing surface that is crossed only once by all reactive trajectories. This assumption is often badly violated, especially when the reactive system is coupled to an environment. The calculations made in this way then overestimate the reaction rate and the results depend critically on the choice of the dividing surface. In this Letter, we study the phase space of a stochastically driven system close to an energetic barrier in order to identify the geometric structure unambiguously determining the reactive trajectories, which is then incorporated in a simple rate formula for reactions in condensed phase that is both independent of the dividing surface and exact.

physics.chem-ph↗

arXiv2015

Transition state trajectory stability determines barrier crossing rates in chemical reactions induced by time-dependent oscillating fields

When a chemical reaction is driven by an external field, the transition state that the system must pass through as it changes from reactant to product -for example, an energy barrier- becomes time-dependent. We show that for periodic forcing the rate of barrier crossing can be determined through stability analysis of the non-autonomous transition state. Specifically, strong agreement is observed between the difference in the Floquet exponents describing stability of the transition state trajectory, which defines a recrossing-free dividing surface [G. T. Craven, T. Bartsch, and R. Hernandez, Phys. Rev. E 89, 040801(R) (2014)], and the rates calculated by simulation of ensembles of trajectories. This result opens the possibility to extract rates directly from the intrinsic stability of the transition state, even when it is time-dependent, without requiring a numerically-expensive simulation of the long-time dynamics of a large ensemble of trajectories.

physics.chem-ph↗

arXiv2015

Chemical reactions induced by oscillating external fields in weak thermal environments

Chemical reaction rates must increasingly be determined in systems that evolve under the control of external stimuli. In these systems, when a reactant population is induced to cross an energy barrier through forcing from a temporally varying external field, the transition state that the reaction must pass through during the transformation from reactant to product is no longer a fixed geometric structure, but is instead time-dependent. For a periodically forced model reaction, we develop a recrossing-free dividing surface that is attached to a transition state trajectory [T. Bartsch, R. Hernandez, and T. Uzer, Phys. Rev. Lett. 95, 058301 (2005)]. We have previously shown that for single-mode sinusoidal driving, the stability of the time-varying transition state directly determines the reaction rate [G. T. Craven, T. Bartsch, and R. Hernandez, J. Chem. Phys. 141, 041106 (2014)]. Here, we extend our previous work to the case of multi-mode driving waveforms. Excellent agreement is observed between the rates predicted by stability analysis and rates obtained through numerical calculation of the reactive flux. We also show that the optimal dividing surface and the resulting reaction rate for a reactive system driven by weak thermal noise can be approximated well using the transition state geometry of the underlying deterministic system. This agreement persists as long as the thermal driving strength is less than the order of that of the periodic driving. The power of this result is its simplicity. The surprising accuracy of the time-dependent noise-free geometry for obtaining transition state theory rates in chemical reactions driven by periodic fields reveals the dynamics without requiring the cost of brute-force calculations.

physics.chem-ph↗

arXiv2015

Infinitely many global continua bifurcating from a single solution of an elliptic problem with concave-convex nonlinearity

We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form \begin{align*} \begin{aligned} -Δu &= f_λ(|x|,u,|\nabla u|) &&\text{in }Ω, u &= 0 &&\text{on }\partialΩ, \end{aligned} \end{align*} on an annulus $Ω\subset\mathbb{R}^N$, with a concave-convex nonlinearity, a special case being the nonlinearity first considered by Ambrosetti, Brezis and Cerami: $f_λ(|x|,u,|\nabla u|)=λ|u|^{q-2}u + |u|^{p-2}u$ with $1<q<2<p$. Although the trivial solution $u_0\equiv0$ is nondegenerate if $λ=0$ we prove that $(λ_0,u_0)=(0,0)$ is a bifurcation point. In fact, the bifurcation scenario is very singular: We show that there are infinitely many global continua of radial solutions $\mathcal{C}_j^\pm\subset\mathbb{R}\times\mathcal{C}^1(Ω)$, $j\in\mathbb{N}_0$ which bifurcate from the trivial branch $\mathbb{R}\times\{0\}$ at $(λ_0,u_0)=(0,0)$ and consist of solutions having precisely $j$ nodal annuli. A detailed study of these continua shows that they accumulate at $\mathbb{R}_{\ge0}\times\{0\}$ so that every $(λ,0)$ with $λ\ge0$ is a bifurcation point. Moreover, adding a point at infinity to $\mathcal{C}^1(Ω)$ they also accumulate at $\mathbb{R}\times\{\infty\}$, so there is bifurcation from infinity at every $λ\in\mathbb{R}$.

math.AP↗

arXiv2014

Ground States of a Nonlinear Curl-Curl Problem in Cylindrically Symmetric Media

We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U= Γ(x)|U|^{p-1}U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations for monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of $H(\mathrm{curl};\mathbb{R}^3)$. Under a cylindrical symmetry assumption on the functions $V$ and $Γ$ the variational problem can be posed in a symmetric subspace of $H(\mathrm{curl};\mathbb{R}^3)$. For a strongly defocusing case $\mathrm{esssup}\, Γ<0$ with large negative values of $Γ$ at infinity we obtain ground states by the direct minimization method. For the focusing case $\mathrm{essinf}\, Γ>0$ the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator $\nabla \times \nabla \times +V(x)$. Examples of cylindrically symmetric functions $V$ are provided for which this holds.

math.AP↗

arXiv2014

Bifurcations for a Coupled Schrödinger System with Multiple Components

In this paper, we study local bifurcations of an indefinite elliptic system with multiple components: \begin{equation*} \left\{\begin{array}{ll} -Δu_j + au_j = μ_ju_j^3+β\sum_{k\ne j}u_k^2u_j, u_j>0\ \ \hbox{in}\ Ω, u_j=0 \ \ \hbox{on}\ \partialΩ,\ j=1,\dots,n. \end{array} \right. \end{equation*} Here $Ω\subset{\mathbb{R}}^N$ is a smooth and bounded domain, $n\ge3$, $a<-Λ_1$ where $Λ_1$ is the principal eigenvalue of $(-Δ, H_0^1(Ω))$; $μ_j$ and $β$ are real constants. Using the positive and non-degenerate solution of the scalar equation $-Δω-ω=-ω^3$, $ω\in H_0^1(Ω)$, we construct a synchronized solution branch $\mathcal{T}_ω$. Then we find a sequence of local bifurcations with respect to $\mathcal{T}_ω$, and we find global bifurcation branches of partially synchronized solutions.

math.AP↗

arXiv2014

Critical points of the N-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations

We prove the existence of critical points of the $N$-vortex Hamiltonian $H_Ω(x_1,\ldots, x_N) =\sum\limits^N_{i=1}Γ^2_i h(x_i) + \sum\limits_{i,j=1\atop j\not= k}^N Γ_iΓ_jG(x_i,x_j)+2\sum\limits_{i=1}^NΓ_iψ_0(x_i)$ in a bounded domain $Ω\subset{\mathbb R}^2$ which may be simply or multiply connected. Here $G$ denotes the Green function for the Dirichlet Laplace operator in $Ω$, more generally a hydrodynamic Green function, and $h$ the Robin function. Moreover $ψ_0\in C^1(\overlineΩ)$ is a harmonic function on $Ω$. We obtain new critical points $x=(x_1,\dots,x_N)$ for $N=3$ or $N=4$ under conditions on the vorticities $Γ_i\in{\mathbb R}\setminus\{0\}$. These critical points correspond to point vortex equilibria of the Euler equation in vorticity form. The case $Γ_i=(-1)^i$ of counter-rotating vortices with identical vortex strength is included. The point vortex equilibria can be desingularized to obtain smooth steady state solutions of the Euler equations for an ideal fluid. The velocity of these steady states will be irrotational except for $N$ vorticFity blobs near $x_1,\dots,x_N$.

math.AP↗

arXiv2014

Persistence of transition state structure in chemical reactions driven by fields oscillating in time

Chemical reactions subjected to time-varying external forces cannot generally be described through a fixed bottleneck near the transition state barrier or dividing surface. A naive dividing surface attached to the instantaneous, but moving, barrier top also fails to be recrossing-free. We construct a moving dividing surface in phase space over a transition state trajectory. This surface is recrossing-free for both Hamiltonian and dissipative dynamics. This is confirmed even for strongly anharmonic barriers using simulation. The power of transition state theory is thereby applicable to chemical reactions and other activated processes even when the bottlenecks are time-dependent and move across space.

nlin.CD↗

arXiv2014

Periodic solutions of the N-vortex Hamiltonian system in planar domains

We investigate the existence of collision-free nonconstant periodic solutions of the $N$-vortex problem in domains $Ω\subset\mathbb{C}$. These are solutions $z(t)=(z_1(t),\dots,z_N(t))$ of the first order Hamiltonian system \[ \dot{z}_k(t)=-i\nabla_{z_k} H_Ω\big(z(t)\big),\quad k=1,\dots,N, \] where the Hamiltonian $H_Ω$ has the form \[ H_Ω(z_1,\dots,z_N) = \frac1{2π}\sum_{{j,k=1}\atop{j\ne k}}^N \log\frac1{|z_j-z_k|} - F(z). \] The function $F:Ω^N\to\mathbb{R}$ depends on the regular part of the hydrodynamic Green's function and is unbounded from above. The Hamiltonian is unbounded from above and below, it is singular, not integrable, energy surfaces are not compact and not known to be of contact type. We prove the existence of a family of periodic solutions $z^r(t)$, $0<r<r_0$, with arbitrarily small minimal period $T_r\to0$ as $r\to0$. The solutions are close to the singular set of $H_Ω$. Our result applies in particular to generic bounded domains, which may be simply or multiply connected. It also applies to certain unbounded domains. Depending on the domain there are multiple such families.

math.DS↗

arXiv2013

Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain

We find solutions $E:Ω\to\mathbb{R}^3$ of the problem \[ \left\{\begin{aligned} &\nabla\times(\nabla\times E) + λE = \partial_E F(x,E) &&\quad \text{in}Ω\\ &ν\times E = 0 &&\quad \text{on}\partialΩ\end{aligned} \right. \] on a simply connected, smooth, bounded domain $Ω\subset\mathbb{R}^3$ with connected boundary and exterior normal $ν:\partialΩ\to\mathbb{R}^3$. Here $\nabla\times$ denotes the curl operator in $\mathbb{R}^3$, the nonlinearity $F:Ω\times\mathbb{R}^3\to\mathbb{R}$ is superquadratic and subcritical in $E$. The model nonlinearity is of the form $F(x,E)=Γ(x)|E|^p$ for $Γ\in L^\infty(Ω)$ positive, some $2<p<6$. It need not be radial nor even in the $E$-variable. The problem comes from the time-harmonic Maxwell equations, the boundary conditions are those for $Ω$ surrounded by a perfect conductor.

math.AP↗

arXiv2013

Bifurcation in a multi-component system of nonlinear Schrödinger equations

We consider the system -Δu_j + a(x)u_j = μ_j u_j^3 + \be\sum_{k\ne j}u_k^2u_j, u_j>0, \qquad j=1,...,n, on a possibly unbounded domain $\Om\subset\R^N$, $N\le3$, with Dirichlet boundary conditions. The system appears in nonlinear optics and in the analysis of mixtures of Bose-Einstein condensates. We consider the self-focussing (attractive self-interaction) case $μ_1,...,μ_n > 0$ and take $\be\in\R$ as bifurcation parameter. There exists a branch of positive solutions with $u_j/u_k$ being constant for all $j,k\in{1,...,n}$. The main results are concerned with the bifurcation of solutions from this branch. Using a hidden symmetry we are able to prove global bifurcation even when the linearization has even-dimensional kernel (which is always the case when $n>1$ is odd).

math.AP↗

arXiv2012

Chaotic Dynamics in Multidimensional Transition States

The crossing of a transition state in a multidimensional reactive system is mediated by invariant geometric objects in phase space: An invariant hyper-sphere that represents the transition state itself and invariant hyper-cylinders that channel the system towards and away from the transition state. The existence of these structures can only be guaranteed if the invariant hyper-sphere is normally hyperbolic, i.e., the dynamics within the transition state is not too strongly chaotic. We study the dynamics within the transition state for the hydrogen exchange reaction in three degrees of freedom. As the energy increases, the dynamics within the transition state becomes increasingly chaotic. We find that the transition state first looses and then, surprisingly, regains its normal hyperbolicity. The important phase space structures of transition state theory will therefore exist at most energies above the threshold.

nlin.CD↗

arXiv2012

Normalized solutions of nonlinear Schrödinger equations

We consider the problem -Δu - g(u) = λu, u \in H^1(\R^N), \int_{\R^N} u^2 = 1, λ\in\R, in dimension $N\ge2$. Here $g$ is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where the associated functional is not bounded below on the $L^2$-unit sphere, and we show the existence of infinitely many solutions.

math.AP↗

arXiv2012

Multi-bubble nodal solutions for slightly subcritical elliptic problems in domains with symmetries

We study the existence of sign-changing solutions with multiple bubbles to the slightly subcritical problem $$-Δu=|u|^{2^*-2-\e}u \hbox{in}Ω, \quad u=0 \hbox{on}\partial Ω,$$ where $Ω$ is a smooth bounded domain in $\R^N$, $N\geq 3$, $2^*=\frac{2N}{N-2}$ and $\e>0$ is a small parameter. In particular we prove that if $Ω$ is convex and satisfies a certain symmetry, then a nodal four-bubble solution exists with two positive and two negative bubbles.

math.AP↗

arXiv2012

On the profile of sign changing solutions of an almost critical problem in the ball

We study the existence and the profile of sign-changing solutions to the slightly subcritical problem $$ -\De u=|u|^{2^*-2-\eps}u \hbox{in} \cB, \quad u=0 \hbox{on}\partial \cB, $$ where $\cB$ is the unit ball in $\rr^N$, $N\geq 3$, $2^*=\frac{2N}{N-2}$ and $\eps>0$ is a small parameter. Using a Lyapunov-Schmidt reduction we discover two new non-radial solutions having 3 bubbles with different nodal structures. An interesting feature is that the solutions are obtained as a local minimum and a local saddle point of a reduced function, hence they do not have a global min-max description.

math.AP↗

arXiv2012

Symmetry-breaking thermally induced collapse of dipolar Bose-Einstein condensates

We investigate a Bose-Einstein condensate with additional long-range dipolar interaction in a cylindrically symmetric trap within a variational framework. Compared to the ground state of this system, little attention has as yet been payed to its unstable excited states. For thermal excitations, however, the latter is of great interest, because it forms the "activated complex" that mediates the collapse of the condensate. For a certain value of the s-wave scatting length our investigations reveal a bifurcation in the transition state, leading to the emergence of two additional and symmetry-breaking excited states. Because these are of lower energy than their symmetric counterpart, we predict the occurrence of a symmetry-breaking thermally induced collapse of dipolar condensates. We show that its occurrence crucially depends on the trap geometry and calculate the thermal decay rates of the system within leading order transition state theory with the help of a uniform rate formula near the rank-2 saddle which allows to smoothly pass the bifurcation.

cond-mat.quant-gas↗

arXiv2012

Reaction rate calculation with time-dependent invariant manifolds

The identification of trajectories that contribute to the reaction rate is the crucial dynamical ingredient in any classical chemical reactivity calculation. This problem often requires a full scale numerical simulation of the dynamics, in particular if the reactive system is exposed to the influence of a heat bath. As an efficient alternative, we propose here to compute invariant surfaces in the phase space of the reactive system that separate reactive from nonreactive trajectories. The location of these invariant manifolds depends both on time and on the realization of the driving force exerted by the bath. These manifolds allow the identification of reactive trajectories simply from their initial conditions, without the need of any further simulation. In this paper, we show how these invariant manifolds can be calculated, and used in a formally exact reaction rate calculation based on perturbation theory for any multidimensional potential coupled to a noisy environment.

cond-mat.stat-mech↗

arXiv2012

Nonlinear Schrödinger equations near an infinite well potential

The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation \label{eq:abs1} iΦ_t(x,t) = -Δ_xΦ+V_\la(x)Φ+ f(x,Φ), \quad x\in\R^N,\ t\in\R,\tag{$NLS_\la$} where the potential $V_\la:\R^N\to\R$ is close to an infinite well potential $V_\infty:\R^N\to\R$, i. e. $V_\infty=\infty$ on an exterior domain $\R^N\setminus\Om$, $V_\infty|_\Om\in L^\infty(\Om)$, and $V_\la\to V_\infty$ as $\la\to\infty$ in a sense to be made precise. The nonlinearity may be of Gross-Pitaevskii type. A solution of \eqref{eq:abs1} with $\la=\infty$ vanishes on $\R^N\setminus\Om$ and satisfies Dirichlet boundary conditions, hence it solves \label{eq:abs2} iΦ_t(x,t) &= -Δ_xΦ+V_\la(x)Φ+ f(x,Φ), &&\quad x\in\Om,\ t\in\R Φ(x,t) &= 0 &&\quad x\in\pa\Om,\ t\in\R. \tag{$NLS_\infty$}. We investigate when a solution $Φ_\infty$ of the infinite well potential \eqref{eq:abs2} gives rise to nearby solutions $Φ_\la$ of the finite well potential \eqref{eq:abs1} with $\la\gg1$ large. Considering \eqref{eq:abs2} as a singular limit of \eqref{eq:abs1} we prove a kind of singular continuation type results.

math.AP↗