There is an intense debate in the recent literature about the correct generalization of Maxwell's velocity distribution in special relativity. The most frequently discussed candidate distributions include the Jüttner function as well as modifications thereof. Here we report results from fully relativistic one-dimensional molecular dynamics simulations that resolve the ambiguity. The numerical evidence unequivocally favors the Jüttner distribution. Moreover, our simulations illustrate that the concept of ''thermal equilibrium'' extends naturally to special relativity only if a many-particle system is spatially confined. They make evident that ''temperature'' can be statistically defined and measured in an observer frame independent way. DOI: 10.1103/PhysRevLett.99.170601 PACS numbers: 05.70.ÿa, 02.70.Ns, 03.30.+p At the beginning of the last century, it was commonly accepted that the one-particle velocity distribution of a dilute gas in equilibrium is described by the Maxwellian probability density function (PDF)[m: rest mass of a gas particle; v: velocity; T k B ÿ1 : temperature; k B : Boltzmann constant; d: space dimension; throughout, we adopt natural units such that the speed of light c 1]. When Einstein [1,2] had formulated the theory of special relativity (SR) in 1905, Planck and others noted immediately that f M is in conflict with the fundamental relativistic postulate that velocities cannot exceed the light speed c. A first solution to this problem was put forward by Jüttner [3]. Starting from a maximum entropy principle, he proposed the following relativistic generalization of Maxwell's PDF:[Z J Z J m; J ; d : normalization constant; E m v m 2 p 2 1=2 : relativistic particle energy; p mv v : momentum with Lorentz factor v 1 ÿ v 2 ÿ1=2 , jvj < 1]. Jüttner's distribution (2) became widely accepted among theorists during the first threequarters of the 20th century [4 -8]-although a rigorous microscopic derivation is lacking due to the difficulty of formulating a relativistically consistent Hamilton mechanics of interacting particles [9][10][11][12][13]. Doubts about the Jüttner function f J began to arise in the 1980s, when Horwitz et al. [14,15] proposed a ''manifestly covariant'' relativistic Boltzmann equation, whose stationary solution differs from Eq. (2) and, in particular, predicts a different mean energy-temperature relation in the ultrarelativistic limit T ! 1 [16]. Since then, partially conflicting results and proposals from other authors [17][18][19][20][21] have led to an increasing confusion as to which distribution actually represents the correct generalization of the Maxwellian (1). For example, a recently discussed alternative to Eq. (2) is the ''modified'' Jüttner function [18,19] The distribution (3) can be obtained, e.g., by combining a maximum relative entropy principle and Lorentz symmetry [20]. Compared with f J at the same parameter values J MJ & 1=m, the modified PDF f MJ exhibits a significantly lower particle population in the high energy tail because of the additional 1=E pre...

We study, within the spin-boson dynamics, the synchronization of a quantum tunneling system with an external, time-periodic driving signal. As a main result, we find that at a sufficiently large system-bath coupling strength (i.e., for a friction strength > 1) the thermal noise plays a constructive role in yielding forced synchronization. This noise-induced synchronization can occur when the driving frequency is larger than the zero-temperature tunneling rate. As an application evidencing the effect, we consider the charge transfer dynamics in molecular complexes. DOI: 10.1103/PhysRevLett.97.210601 PACS numbers: 05.60.Gg, 05.40.ÿa, 05.45.Xt, 82.20.Gk The study of the different versions of synchronization appearing in nonlinear classical systems has gained importance over the past decade [1][2][3][4]. A special class of problems is provided by noise-induced forced synchronization in driven bistable nonlinear systems [2,5,6]. Here a stochastic phase process can be associated with the jumping events between two domains of attraction. The locking of the average frequency of the phase process to that of the external driving and the smallness of the phase-diffusion coefficient in a corresponding interval of noise strengths are the fingerprints of such noise-induced forced synchronization [7][8][9]. Another manifestation of the rich dynamics of such systems is stochastic resonance (SR) [10], which recently has been generalized to the quantum regime [11][12][13]. Its experimental realization on the level of a nanomechanical quantum memory element is now feasible [14]. Although synchronization and SR are related, the existence of SR does not necessarily imply a (phase) synchronization, as emphasized in Ref. [5]. The extension of noiseinduced synchronization into the realm of quantum physics has not been considered thus far. This latter task presents a challenge which, apart from prominent academic interest, also comprises a great potential for nanoscience with beneficial applications ranging from quantum control to quantum information processing. With this work, we undertake a first step in this direction.Dissipative quantum tunneling changes radically the physics of classical synchronization. At zero temperature, the system can only tunnel towards its lowest energy state when a biasing dc signal is applied. As the bias periodically changes its sign due to the action of a driving field, tunneling causes the particle to move periodically towards its corresponding lowest energy state, as long as the driving period is much longer than the typical time scale for tunneling. Consequently, one expects that the system may synchronize when driven by a periodic, e.g., rectangular-shaped, signal. By contrast, in the absence of thermal noise, synchronization in overdamped classical bistable systems driven by subthreshold signals fails as no overbarrier transitions occur.Two interesting questions now emerge: What is the effect of the generally deteriorating thermal quantum noise at finite temperatures on synchronization? How does...

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 analogue simulation results by Gingl et al. in Refs. [18,19].

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 part), 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 non-monotonic behavior of the SNR vs. the noise strength. The analytical results agree well with those obtained from numerical solutions of the Langevin equation.The phenomenon of Stochastic Resonance (SR) attracts ever growing interest due to its multi-facetted relevance for a variety of noise-induced features in physics, chemistry, and the life sciences [1,2,3,4,5]. Several SRquantifiers have been used to characterize the response of a noisy system to the action of time-periodic external forces. In particular, the non-monotonic behavior of the output signal-to-noise ratio (SNR) with the strength of the noise has been used widely. A dimensionless quantity that measures the "quality" of the response with respect to the input signal is the SR-gain defined as the ratio of the output SNR over the input SNR. Ideally, one would wish to obtain the characteristic amplification of the SR phenomenon [2,6] and, simultaneously, SR-gains larger than unity. For superthreshold sinusoidal input signals SR-gains larger than unity have been reported before [7]. In recent analog [8,9,10] and numerical [11,12] simulations of noisy bistable systems driven by subthreshold multifrequency input forces, surprisingly large SR-gains larger than unity have been established.In order to clarify the conditions under which these anomalous large SR-gains occur, it would be interesting to propose simplified models, amenable to analytical treatment, which describe this rich behavior of the response. A detailed proof that SR-gains larger than unity are incompatible with Linear Response Theory (LRT) has been presented in [11]. Thus, any theoretical explanation of the simultaneous existence of SR and anomalous large gains is rooted in the response beyond LRT.The main focus of this work is to present such a simplified two-state description of the nonlinear dynamics of a noisy, symmetric bistable system driven by a rectangular subthreshold signal. A two-state description of SR has been considered previously in the pioneering work by McNamara and Wiesenfield for sinusoidal input signals [13]. In clear contrast to their work, however, we will here not linearize the transition probabilities in the strength of the applied force. In doing so, we put forward explicit analytical expressions for the driven population probabilities, the nonlinear correlation function (its coherent and incoherent part), the SNR and the SR-gain. These novel nonlinear response results come forth solely because the rectangular signal -in contrast to sinusoidal d...

The effect of a high-frequency signal on the FitzHugh-Nagumo excitable model is analyzed. We show that the firing rate is diminished as the ratio of the high-frequency amplitude to its frequency is increased. Moreover, it is demonstrated that the excitable character of the system, and consequently the firing activity, is suppressed for ratios above a given threshold value. In addition, we show that the vibrational resonance phenomenon turns up for sufficiently large noise strength values. DOI: 10.1103/PhysRevE.73.061102 PACS number͑s͒: 05.40.Ϫa, 02.50.Ϫr, 87.19.La, 05.90.ϩm Nonlinear noisy systems have been studied with ever growing interest due to the applicability to the modeling of a great variety of phenomena of relevance in physics, chemistry, and the life sciences ͓1,2͔. Perhaps, the simplest nonlinear noisy system studied in the literature is the bistable system, which has been used successfully to illustrate the phenomenon of stochastic resonance. In addition, the spiking activity of a neuron has been theoretically studied using nonlinear excitable noisy models ͓3͔, among them, the FitzHugh-Nagumo ͑FHN͒ system ͓4͔ being one of the most utilized due to its simplicity. Besides, networks of FHN units have been considered as simplified models useful in the description of both the cardiac tissue ͓5-8͔ and reactiondiffusion chemical systems ͓9,8͔.Recently, it has been shown, both theoretically and experimentally, that the addition of a high-frequency ͑HF͒ signal results in an improvement of the stochastic resonance in a bistable optical system ͓10͔. Nevertheless, many experimental studies have suggested that HF ͑nonionizing͒ fields may damage several structural and functional properties of neuronal membranes in single cells, as well as cause a number of negative physiological effects on typically excitable media such as cardiac tissue ͓11͔. It has also been found that certain HF signals are able to suppress the steady directed motion in a ratchet model ͓12͔. Thus, HF signals seem to play a twofold role, positive or negative, depending on the nonlinear system under consideration.In this paper we study the influence of a HF field on a rather simple excitable model, such as the archetypal FHN system. This system is governed by the following equations ͓13͔:In the context of neurophysiology, v͑t͒ is called the voltage variable, and w͑t͒ the recovery variable. Here S͑t͒ is an external forcing of period T, ͑t͒ a noisy term, and ⌫͑t͒ is an added HF signal which will be specified later on. The "neuronal" noise ͑t͒ is assumed to be an unbiased Gaussian white noise with the autocorrelation function ͗͑t͒͑s͒͘ =2D␦͑t − s͒. The values of the model parameters define the dynamical regimes of the system, and are discussed below. Let us focus our attention on the deterministic FHN model ͑D =0͒ in the absence of a HF signal ͓⌫͑t͒ =0͔. In the case of a time-independent external signal S͑t͒ = S 0 , the excitable character of the FHN model relies on the existence of a threshold value of this signal, S H , at which a Hopf bifurca...

Dynamical systems often contain oscillatory forces or depend on periodic potentials. Time or space periodicity is reflected in the properties of these systems through a dependence on the parameters of their periodic terms. In this paper we provide a general theoretical framework for dealing with these kinds of systems, regardless of whether they are classical or quantum, stochastic or deterministic, dissipative or nondissipative, linear or nonlinear, etc. In particular, we are able to show that simple symmetry considerations determine, to a large extent, how their properties depend functionally on some of the parameters of the periodic terms. For the sake of illustration, we apply this formalism to find the functional dependence of the expectation value of the momentum of a Bose-Einstein condensate, described by the Gross-Pitaewskii equation, when it is exposed to a sawtooth potential whose amplitude is periodically modulated in time. We show that, by using this formalism, a small set of measurements is enough to obtain the functional form for a wide range of parameters. This can be very helpful when characterizing experimentally the response of systems for which performing measurements is costly or difficult. It often happens that a system-physical or otherwise-can be described using a model that includes one or several periodic functions, with the same or different periodicities. Whether these functions represent external oscillatory forces, modulating amplitudes, or space-periodic potentials is immaterial for our forthcoming discussion, as are the specific details of the underlying dynamics, which can be either deterministic or stochastic, classical or quantum, dissipative or nondissipative, linear or nonlinear, etc. For our purposes, the only relevant feature that all these systems have in common is that their properties depend in a certain way on the periodicities, amplitudes, and relative phases of these functions.Examples of these types of systems abound in the literature. Because of their ability to describe a wide variety of phenomena, the most significant ones are probably the periodically driven systems [1]. To name but a few instances, these driven systems have proven to be useful in the study of stochastic resonance [2], vibrational resonance [3], classical and quantum stochastic synchronization [4], opinion formation processes [5], coherent destruction of tunneling [6], dynamical localization and delocalization [7], ratchet effect [8], and atomic quantum motors [9]. It is worth mentioning that some of the above examples (e.g., the ratchet effect) include both time-and space-periodic functions.

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.

We investigate the role of noise in the phenomenon of stochastic synchronization of switching events in a rocked, overdamped bistable potential driven by white Gaussian noise, the archetype description of Stochastic Resonance. We present a new approach to the stochastic counting process of noise-induced switching events: starting from the Markovian dynamics of the nonstationary, continuous particle dynamics one finds upon contraction onto two states a non-Markovian renewal dynamics. The output frequency is determined as the velocity of the underlying discrete phase dynamics. The phenomenon of noise-assisted phase synchronization is investigated in terms of an effective, instantaneous phase diffusion. The theory is applied to rectangular-shaped rocking signals versus increasing input-noise strengths. Precise numerical simulations corroborate very favorably our analytical results. The novel theoretical findings are also compared with prior findings.