Finite-barrier corrections for multidimensional barriers in colored noise.

The usual identification of reactive trajectories for the calculation of reaction rates requires very time-consuming simulations, particularly if the environment presents memory effects. In this paper, we develop a method that permits the identification of reactive trajectories in a system under the action of a stochastic colored driving. This method is based on the perturbative computation of the invariant structures that act as separatrices for reactivity. Furthermore, using this perturbative scheme, we have obtained a formally exact expression for the reaction rate in multidimensional systems coupled to colored noisy environments.

Invariant Manifolds and Rate Constants in Driven Chemical Reactions.

Reaction rates of chemical reactions under nonequilibrium conditions can be determined through the construction of the normally hyperbolic invariant manifold (NHIM) [and moving dividing surface (DS)] associated with the transition state trajectory. Here, we extend our recent methods by constructing points on the NHIM accurately even for multidimensional cases. We also advance the implementation of machine learning approaches to construct smooth versions of the NHIM from a known high-accuracy set of its points. That is, we expand on our earlier use of neural nets and introduce the use of Gaussian process regression for the determination of the NHIM. Finally, we compare and contrast all of these methods for a challenging two-dimensional model barrier case so as to illustrate their accuracy and general applicability.

Transition state theory for activated systems with driven anharmonic barriers.

Classical transition state theory has been extended to address chemical reactions across barriers that are driven and anharmonic. This resolves a challenge to the naive theory that necessarily leads to recrossings and approximate rates because it relies on a fixed dividing surface. We develop both perturbative and numerical methods for the computation of a time-dependent recrossing-free dividing surface for a model anharmonic system in a solvated environment that interacts strongly with an oscillatory external field. We extend our previous work, which relied either on a harmonic approximation or on periodic force driving. We demonstrate that the reaction rate, expressed as the long-time flux of reactive trajectories, can be extracted directly from the stability exponents, namely, Lyapunov exponents, of the moving dividing surface. Comparison to numerical results demonstrates the accuracy and robustness of this approach for the computation of optimal (recrossing-free) dividing surfaces and reaction rates in systems with Markovian solvation forces. The resulting reaction rates are in strong agreement with those determined from the long-time flux of reactive trajectories.

Transition state theory for solvated reactions beyond recrossing-free dividing surfaces.

The accuracy of rate constants calculated using transition state theory depends crucially on the correct identification of a recrossing-free dividing surface. We show here that it is possible to define such optimal dividing surface in systems with non-Markovian friction. However, a more direct approach to rate calculation is based on invariant manifolds and avoids the use of a dividing surface altogether, Using that method we obtain an explicit expression for the rate of crossing an anharmonic potential barrier. The excellent performance of our method is illustrated with an application to a realistic model for LiNC⇌LiCN isomerization.

Transition state geometry of driven chemical reactions on time-dependent double-well potentials.

Reaction rates across time-dependent barriers are difficult to define and difficult to obtain using standard transition state theory approaches because of the complexity of the geometry of the dividing surface separating reactants and products. Using perturbation theory (PT) or Lagrangian descriptors (LDs), we can obtain the transition state trajectory and the associated recrossing-free dividing surface. With the latter, we are able to determine the exact reactant population decay and the corresponding rates to benchmark the PT and LD approaches. Specifically, accurate rates are obtained from a local description regarding only direct barrier crossings and to those obtained from a stability analysis of the transition state trajectory. We find that these benchmarks agree with the PT and LD approaches for obtaining recrossing-free dividing surfaces. This result holds not only for the local dynamics in the vicinity of the barrier top, but also for the global dynamics of particles that are quenched at the reactant or product wells after their sojourn over the barrier region. The double-well structure of the potential allows for long-time dynamics related to collisions with the outside walls that lead to long-time returns in the low-friction regime. This additional global dynamics introduces slow-decay pathways that do not result from the local transition across the recrossing-free dividing surface associated with the transition state trajectory, but can be addressed if that structure is augmented by the population transfer of the long-time returns.

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.

Communication: 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, "Persistence of transition state structure in chemical reactions driven by fields oscillating in time," 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.

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.

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.

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.

Communication: 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 Communication, 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.

Phase-space geometry of the generalized Langevin equation.

The generalized Langevin equation is widely used to model the influence of a heat bath upon a reactive system. This equation will here be studied from a geometric point of view. A dynamical phase space that represents all possible states of the system will be constructed, the generalized Langevin equation will be formally rewritten as a pair of coupled ordinary differential equations, and the fundamental geometric structures in phase space will be described. It will be shown that the phase space itself and its geometric structure depend critically on the preparation of the system: A system that is assumed to have been in existence forever has a larger phase space with a simpler structure than a system that is prepared at a finite time. These differences persist even in the long-time limit, where one might expect the details of preparation to become irrelevant.

Transition-state theory rate calculations with a recrossing-free moving dividing surface.

Two different methods for transition-state theory (TST) rate calculations are presented that use the recently developed notions of the moving dividing surface and the associated moving separatrices: one is based on the flux-over-population approach and the other on the calculation of the reactive flux. The flux-over-population rate can be calculated in two ways by averaging the flux first over the noise and then over the initial conditions or vice versa. The former entails the calculation of reaction probabilities and is closely related to previous TST rate derivations. The latter results in an expression for the transmission factor as the noise average of a stochastic variable that is given explicitly as a function of the moving separatrices. Both the reactive-flux and flux-over-population methods suggest possible new ways of calculating approximate rates in anharmonic systems. In particular, numerical simulations of harmonic and anharmonic systems have been used to calculate reaction rates based on the reactive flux calculation using the fixed and moving dividing surfaces so as to illustrate the computational advantages of the latter.

Transition state theory for laser-driven reactions.

Recent developments in transition state theory brought about by dynamical systems theory are extended to time-dependent systems such as laser-driven reactions. Using time-dependent normal form theory, the authors construct a reaction coordinate with regular dynamics inside the transition region. The conservation of the associated action enables one to extract time-dependent invariant manifolds that act as separatrices between reactive and nonreactive trajectories and thus make it possible to predict the ultimate fate of a trajectory. They illustrate the power of our approach on a driven Henon-Heiles system, which serves as a simple example of a reactive system with several open channels. The present generalization of transition state theory to driven systems will allow one to study processes such as the control of chemical reactions through laser pulses.

Extracting multidimensional phase space topology from periodic orbits.

We establish a hierarchical ordering of periodic orbits in a strongly coupled multidimensional Hamiltonian system. Phase space structures can be reconstructed quantitatively from the knowledge of periodic orbits alone. We illustrate our findings for the hydrogen atom in crossed electric and magnetic fields.

Identifying reactive trajectories using a moving transition state.

A time-dependent no-recrossing dividing surface is shown to lead to a new criterion for identifying reactive trajectories well before they are evolved to infinite time. Numerical dynamics simulations of a dissipative anharmonic two-dimensional system confirm the efficiency of this approach. The results are compared to the standard fixed transition state dividing surface that is well-known to suffer from recrossings and therefore requires trajectories to be evolved over a long time interval before they can reliably be classified as reactive or nonreactive. The moving dividing surface can be used to identify reactive trajectories in harmonic or moderately anharmonic systems with considerably lower numerical effort or even without any simulation at all.

[Transplantation of autologous adult bone marrow stem cells in patients with severe peripheral arterial occlusion disease]. / Transplantation von autologen adulten Knochenmarkstammzellen bei Patienten mit schwerer peripherer arterieller Verschlusskrankheit.

BACKGROUND: For many patients with severe peripheral arterial occlusion disease (PAOD) an interventional or surgical treatment is not feasible. The regenerative potential of adult autologous mononuclear stem cells could contribute to neoangiogenesis. PATIENTS AND METHODS: Ten patients with severe PAOD were included. The walking distance was < 200 m and no interventional or surgical treatment was possible. After harvesting of 80 ml of bone marrow the mononuclear cell fraction was separated. Thereafter, intraarterial (10 ml into the common femoral artery) and intramuscular (5 ml into the muscles of the thigh and the lower leg) transplantation of the cell suspension was performed. RESULTS: After 2 months the walking distance was enhanced significantly in all patients. Furthermore, a significant improvement of ankle-brachial index at rest, capillary-venous oxygen saturation and parameters of venous occlusion plethysmography was seen. No complications or side effects could be monitored. CONCLUSION: These results demonstrate, that the combined intraarterial and intramuscular transplantation of autologous adult bone marrow stem cells leads to a significant improvement of perfusion indices in patients with severe PAOD.

[Effects of exercise training on mobilization of BM-CPCs and migratory capacity as well as LVEF after AMI]. / Die Effekte von körperlichem Training auf die Mobilisation und funktionelle Aktivität zirkulierender Progenitorzellen aus dem Knochenmark sowie die LVEF nach akutem Myokardinfarkt.

BACKGROUND AND PURPOSE: Bone marrow-derived circulating progenitor cells (BM-CPCs) are mobilized in adult peripheral blood (PB) during the acute myocardial infarction (AMI) period and contribute to the regeneration of infarcted myocardium. In this study, the influence of physical training on the mobilization and the migratory activity of the BM-CPCs as well as on the left ventricular function (LVEF) after AMI was examined. PATIENTS AND METHODS: 26 patients with AMI were analyzed in two groups. The first group comprised 17 patients with standardized exercise training for 3 weeks 14 +/- 4 days after AMI, the second group nine control subjects without exercise training. PB concentrations of CD34/45+ and CD133/45+ were measured by FACS. The migratory activity of BM-CPCs was analyzed by migration assay. B-type natriuretic peptide (BNP) in PB and the functional investigations spiroergometry (VO2 and PaO2) and stress echocardiography (LVEF) were determined in both groups. RESULTS: A significant increase in both concentrations, CD34/45+ and CD133/45+, as well as in migratory capacity of BM-CPCs was found after 3 weeks of exercise training, which was significantly decreased 3 months after completion of exercise training. No significant difference was observed in the control group without exercise training. In the functional investigations a significant increase in VO2 as well as PaO2 was shown spiroergometrically after exercise training. There was no difference in stress echocardiographic LVEF at rest in both groups. On the other hand, interestingly, the findings showed that the increase of LVEF at peak stress was significantly higher after exercise training as compared to the control group. Moreover, a significant decrease in BNP values was found after exercise training as well as 3 months after AMI. No difference was found in the control group. CONCLUSION: This study demonstrates that exercise training for 3 weeks after AMI leads to a significant mobilization as well as increase of functional activation of BM-CPCs in humans. Moreover, regular exercise training might contribute to the positive effects on the regenerative potency after AMI.

Autologous mononuclear stem cell transplantation in patients with peripheral occlusive arterial disease.

UNLABELLED: Patients with chronic peripheral occlusive arterial disease often are not candidates for conventional revascularization procedures. Preclinical trials have shown that the transplantation of autologous bone marrow cells induces and increases the collateral vessel formation. We analyzed the clinical benefit of combined intraarterial and intramuscular transplantation of adult autologous mononuclear bone marrow stem cells in patients with lower-limb peripheral occlusive arterial disease. METHODS: Patients with severe peripheral occlusive arterial disease and a reduced walking distance (Fontaine stage II or III) were included. Bone marrow was harvested from the hip under local anesthesia and mononuclear cells were transplanted intramuscularly and intraarterially into the ischemic limb after isolation under good manufacturing practice conditions. RESULTS: After 2 months, pain-free walking distance increased 3.7-fold. Furthermore, the ankle-brachial index was significantly improved at rest and after exercise. Similar improvements could be documented by capillary-venous oxygen saturation and venous occlusion plethysmography. No side effects or complications were detected during transplantation and during time of follow-up. CONCLUSIONS: Combined intraarterial and intramuscular transplantation of autologous mononuclear bone marrow stem cells is a clinically feasible and minimally invasive therapeutic option for patients with severe, chronic peripheral occlusive arterial disease.

Stochastic transition states: reaction geometry amidst noise.

Classical transition state theory (TST) is the cornerstone of reaction-rate theory. It postulates a partition of phase space into reactant and product regions, which are separated by a dividing surface that reactive trajectories must cross. In order not to overestimate the reaction rate, the dynamics must be free of recrossings of the dividing surface. This no-recrossing rule is difficult (and sometimes impossible) to enforce, however, when a chemical reaction takes place in a fluctuating environment such as a liquid. High-accuracy approximations to the rate are well known when the solvent forces are treated using stochastic representations, though again, exact no-recrossing surfaces have not been available. To generalize the exact limit of TST to reactive systems driven by noise, we introduce a time-dependent dividing surface that is stochastically moving in phase space, such that it is crossed once and only once by each transition path.