M. Morillo

On the effects of solvent and intermolecular fluctuations in proton transfer reactions

We present a theory of proton transfer reactions which incorporate the modulation of the proton’s potential surface by intermolecular vibrations and the effect of coupling to solvent degrees of freedom. The proton tunnels between states corresponding to it being localized in the wells of a double minimum potential. The resulting tunnel splitting depends on the intermolecular separation. The solvent response to the proton’s charge is modeled as that of a damped oscillator, allowing the introduction of friction effects in the solvent dynamics. The rate of transfer is evaluated by perturbation theory in the level splitting. We find that typically the intermolecular and solvent contributions enhance the rate relative to the values obtained in their absence. This effect is evident at low temperature where friction can enhance the rate by increasing opportunity for solvent tunneling. At high temperature the intermolecular motion enhances the rate by sampling over a distribution of tunnel splittings.

Solvent effects on proton‐transfer reactions

We present a theory of solvent effects on the rate of intramolecular proton‐transfer (IPT) reactions. The proton tunnels between two vibrational levels of a double minimum potential. The proton’s coupling to the solvent is modeled with an oscillator bath, appropriate to reactions where a charge interacts with many solvent molecules. The rate is evaluated by use of the Golden Rule; the perturbation is the level splitting. The IPT rate constant has several limiting expressions, one of which has an activated form. The activation energy is related to the medium reorganization energy, and provides a mechanism to slow the IPT reaction. Since reorganization energies are small in nonpolar and large in polar solvents, the rate is expected to be smaller in the latter class of solvents. Isotopic substitution is predicted to only affect the prefactor of the rate expression. Another regime is obtained for smaller reorganization energies where the solvent dynamics, as described by a dielectric relaxation time, are important. Comparison is made with recent experimental studies of IPT in solution.

The transition from nonadiabatic to solvent controlled adiabatic electron transfer kinetics: The role of quantum and solvent dynamics

We analyze the transition from nonadiabatic to solvent controlled adiabatic electron transfer kinetic behavior with emphasis on the inverted regime. By viewing the electron transfer process as dynamical motion on a donor surface followed by possible crossing to the acceptor surface, we are able to obtain a simple consecutive reaction expression for the overall rate in terms of rate constants for motion on the surfaces and crossing motion. When the crossing between surfaces is not localized to a point, we find that there are frictional effects on this crossing rate constant, in contrast to the localized case. Use of typical electron transfer parameters shows that these friction effects will only be in evidence for the inverted regime of electron transfer kinetics.

Control of proton‐transfer reactions with external fields

The possibility of controlling the tunneling of a proton in a condensed phase with the use of static or time varying external fields, which couple to the transition dipole moment of the tunneling proton, is investigated. Starting from a Hamiltonian, an equation of motion describing the tunnel dynamics of the proton as a stochastically modulated, externally driven, two‐level system is derived under suitable restrictions. For external fields that satisfy a precise connection between frequency and amplitude, whereby the resulting Floquet eigenvalues (quasienergies) are degenerate, tunneling can be suppressed in the absence of the medium. With the medium present, we examine the consequences to this tunnel suppression. Static fields, if sufficiently strong, can also suppress tunneling. Expressions are derived for the effect of a static external field on the medium‐influenced, tunnel‐rate constant. The rate constant can be enhanced or decreased, depending on the sizes of the medium‐reorganization energy and ext...

Two-State Theory of Nonlinear Stochastic Resonance

An amenable, analytical two-state description of the nonlinear population dynamics of a noisy bistable system driven by a rectangular subthreshold signal is put forward. Explicit expressions for the driven population dynamics, the correlation function (its coherent and incoherent parts), the signal-to-noise ratio (SNR), and the stochastic resonance (SR) gain are obtained. Within a suitably chosen range of parameter values this reduced description yields anomalous SR gains exceeding unity and, simultaneously, gives rise to a nonmonotonic behavior of the SNR vs the noise strength. The analytical results agree well with those obtained from numerical solutions of the Langevin equation.

Gain in stochastic resonance: precise numerics versus linear response theory beyond the two-mode approximation.

In the context of the phenomenon of stochastic resonance (SR), we study the correlation function, the signal-to-noise ratio (SNR), and the ratio of output over input SNR, i.e., the gain, which is associated to the nonlinear response of a bistable system driven by time-periodic forces and white Gaussian noise. These quantifiers for SR are evaluated using the techniques of linear response theory (LRT) beyond the usually employed two-mode approximation scheme. We analytically demonstrate within such an extended LRT description that the gain can indeed not exceed unity. We implement an efficient algorithm, based on work by Greenside and Helfand (detailed in the Appendix), to integrate the driven Langevin equation over a wide range of parameter values. The predictions of LRT are carefully tested against the results obtained from numerical solutions of the corresponding Langevin equation over a wide range of parameter values. We further present an accurate procedure to evaluate the distinct contributions of the coherent and incoherent parts of the correlation function to the SNR and the gain. As a main result we show for subthreshold driving that both the correlation function and the SNR can deviate substantially from the predictions of LRT and yet the gain can be either larger or smaller than unity. In particular, we find that the gain can exceed unity in the strongly nonlinear regime which is characterized by weak noise and very slow multifrequency subthreshold input signals with a small duty cycle. This latter result is in agreement with recent analog simulation results by Gingl et al. [ICNF 2001, edited by G. Bosman (World Scientific, Singapore, 2002), pp. 545-548; Fluct. Noise Lett. 1, L181 (2001)].

Microcanonical quantum fluctuation theorems

Previously derived expressions for the characteristic function of work performed on a quantum system by a classical external force are generalized to arbitrary initial states of the considered system and to Hamiltonians with degenerate spectra. In the particular case of microcanonical initial states, explicit expressions for the characteristic function and the corresponding probability density of work are formulated. Their classical limit as well as their relations to the corresponding canonical expressions are discussed. A fluctuation theorem is derived that expresses the ratio of probabilities of work for a process and its time reversal to the ratio of densities of states of the microcanonical equilibrium systems with corresponding initial and final Hamiltonians. From this Crooks-type fluctuation theorem a relation between entropies of different systems can be derived which does not involve the time-reversed process. This entropy-from-work theorem provides an experimentally accessible way to measure entropies.

Subthreshold stochastic resonance: rectangular signals can cause anomalous large gains.

The main objective of this work is to explore aspects of stochastic resonance (SR) in noisy bistable, symmetric systems driven by subthreshold periodic rectangular external signals possessing a large duty cycle of unity. Using a precise numerical solution of the Langevin equation, we carry out a detailed analysis of the behavior of the first two cumulant averages, the correlation function, and its coherent and incoherent parts. We also depict the nonmonotonic behavior versus the noise strength of several SR quantifiers such as the average output amplitude, i.e., the spectral amplification, the signal-to-noise ratio, and the SR gain. In particular, we find that with subthreshold amplitudes and for an appropriate duration of the pulses of the driving force, the phenomenon of stochastic resonance is accompanied by SR gains exceeding unity. This analysis thus sheds light on the interplay between nonlinearity and the nonlinear response, which in turn yields nontrivial unexpected SR gains above unity.

Stochastic resonance: Theory and numerics

We address the phenomenon of stochastic resonance in a noisy bistable system driven by a time-dependent periodic force (not necessarily sinusoidal) and in its two-state approximation. Even for driving forces with subthreshold amplitudes, the behavior of the system response might require a nonlinear description. We introduce analytical and numerical tools to analyze the power spectral amplification and the signal-to-noise ratio in a nonlinear regime. Our analysis shows the importance of the effects of the driving force on the system fluctuations in a nonlinear regime. These effects can be usefully exploited to achieve high quality output signals with gains larger than unity, which is impossible within a linear regime.

Self-synchronization of populations of nonlinear oscillators in the thermodynamic limit

A population of identical nonlinear oscillators, subject to random forces and coupled via a mean-field interaction, is studied in the thermodynamic limit. The model presents a nonequilibrium phase transition from a stationary to a time-periodic probability density. Below the transition line, the population of oscillators is in a quiescent state with order parameter equal to zero. Above the transition line, there is a state of collective rhythmicity characterized by a time-periodic behavior of the order parameter and all moments of the probability distribution. The information entropy of the ensemble is a constant both below and above the critical line. Analytical and numerical analyses of the model are provided.