Mark Dykman

Large fluctuations and optimal paths in chemical kinetics

The eikonal approximation (instanton technique) is applied to the problem of large fluctuations of the number of species in spatially homogeneous chemical reactions with the probability density distribution described by a master equation. For both autocatalytic and nonautocatalytic reactions, the analysis of the distribution about a stable stationary state and of the transitions between coexisting stable states comes, to logarithmic accuracy, to the analysis of Hamiltonian dynamics of an auxiliary dynamical system. The latter can be done explicitly in a few cases, including one‐species systems, systems with detailed balance, and systems close to the bifurcation points where the number of the stable states changes. In the last case, the fluctuations display universal features, and, for saddle‐node bifurcation points, the logarithm of the probability of escape from the metastable state (per unit time) is proportional to the distance to the bifurcation point (in the parameter space) raised to the power 3/2. We compare the eikonal approximation for the stationary distribution of a master equation to Monte Carlo numerical solutions for two chemical two‐variable systems with multiple stationary states, where none of the cited restrictions exists. For one of the systems in the pattern of optimal paths we observe caustics emanating from the saddle point.

Transport and current reversal in stochastically driven ratchets

Abstract We present analytic results for the current in a system moving in an arbitrary periodic potential and driven by weak Gaussian noise with an arbitrary power spectrum which are valid to order ( t c t r ) 2 , where t c is the largest characteristic time of the noise, and t r is the characteristic intrawell relaxation time. The dependence of the current on the shape of the potential, and on the shape of the power spectrum of the noise is illustrated. It is demonstrated that the direction of the current is opposite when the power spectrum of the noise has a minimum or maximum at zero frequency. A simple physical mechanism for this behavior is suggested. The behavior of the system in the limit of slow noise ( t c ⪢ t r ) is also discussed.

Nonconventional Stochastic Resonance

It is argued, on the basis of linear response theory (LRT), that new types of stochastic resonance (SR) are to be anticipated in diverse systems, quite different from the one most commonly studied to date, which has a static double-well potential and is driven by a net force equal to the sum of periodic and stochastic terms. On this basis, three new nonconventional forms of SR are predicted, sought, found, and investigated both theoretically and by analogue electronic experiment: (a) in monostable systems; (b) in bistable systems with periodically modulated noise; and (c) in a system with coexisting periodic attractors. In each case, it is shown that LRT can provide a good quantitative description of the experimental results for sufficiently weak driving fields. It is concluded that SR is a much more general phenomenon than has hitherto been appreciated.

Linear Response Theory in Stochastic Resonance

The susceptibility of an overdamped Markov system fluctuating in a bistable potential of general form is obtained by analytic solution of the Fokker-Planck equation (FPE) for low noise intensities. The results are discussed in the context of the LRT theory of stochastic resonance. They go over into recent results of Hu et al. [Phys. Lett. A 172 (1992) 21] obtained from the FPE for the case of a symmetrical potential, and they coincide with the LRT results of Dykman et al. [Phys. Rev. Lett. 65 (1990) 2606; JETP Lett. 52 (1990) 144; Phys. Rev. Lett. 68 (1992) 2985] obtained for the general case of bistable systems.

Stochastic resonance: Linear response and giant nonlinearity

The response of a bistable noise-driven system to a weak periodic force is investigated using linear response theory (LRT) and by analogue electronic experiment. For quasithermal systems the response, and in particular its increase with increasing noise intensityD, are described by the fluctuationdissipation relations. For smallD the low-frequency susceptibility of the systemχ(ω) has been found in explicit form allowing for both forced oscillations about the states and periodic modulation of the probabilities of fluctuational transitions between the states. It is shown, both theoretically and experimentally, that a phase lagφ between the force and the response passes through a maximum whenD is tuned through the range where stochastic resonance (SR) occurs. A giant nonlinearity of the response is shown to arise for smallD and small frequencies of the driving force. It results in the signal induced by a sinusoidal force being nearly rectangular. The range of applicability of LRT is established.

Noise-induced linearisation

It is found that the response of a nonlinear dynamical system can be linearised, and its frequency dispersion diminished, by the addition of external noise of sufficient intensity. Taking as an example an overdamped bistable system driven by a low-frequency periodic field, this noise-induced linearisation is investigated through analogue electronic experiments. The wider implications are considered.

Power spectra of noise-driven nonlinear systems and stochastic resonance

The results of recent experimental and theoretical investigations of the spectral densities of fluctuations (SDFs) of noise-driven nonlinear dynamical systems are reviewed. Emphasis is placed on the analysis of the shapes and intensities of peaks in the SDFs. Three different types of phenomena are considered. First, the SDFs of a class of monostable underdamped nonlinear systems, in which the variation of eigenfrequency with energy is nonmonotonic, are investigated. It is shown that they exhibit zero-dispersion peaks and noise-induced spectral narrowing, as well as zero-frequency peaks. Secondly, it is demonstrated that systems bistable in an external periodic field can exhibit supernarrow spectral peaks within the range of a kinetic phase transition. Finally, recent results in stochastic resonance (SR) are reviewed, including phase shifts, giant nonlinearities for weak noise, SR for periodically modulated noise intensity, and high-frequency SR for periodic attractors.

Noise-Induced Spectral Narrowing in Nonlinear Oscillators

The spectral densities of the fluctuations of noise-driven underdamped nonlinear oscillators are discussed with particular reference to the large class of systems whose eigenfrequencies vary nonmonotonically with energy. It is shown by analogue electronic experiments and theoretically that, astonishingly, the widths of their spectral peaks can sometimes decrease with increasing noise intensity.

Stochastic resonance in a passive bistable system

The responses of an all-optical bistable system and an analog model of the Brownian motion in the symmetric Duffing potential to a weak periodic force in the presence of noise are investigated. The appearance of a stochastic resonance in both cases is explained in the theory of a linear response.

Fluctuational transitions and critical phenomena in a periodically driven nonlinear oscillator subject to weak noise

Fluctuation‐induced transitions between coexisting periodic attractors in a periodically driven nonlinear oscillator have been investigated theoretically and by analogue electronic experiment. Calculations and measurements of the corresponding activation energies are in good agreement, and have enabled the position of the kinetic phase transition (KPT) line to be established over its full range.