Nonequilibrium noise

The mean-square amplitude of nonequilibrium noise, quite in contrast to that of equilibrium thermal noise, may reach rather high values for instance, hundreds of millivolts during anodic polarization of semiconductors (Parkhutik and Timashev, 2000). 

One of the major reasons for the development of nonequilibrium noise in electrochemical systems is the inhomogeneity (micro- or macro-heterogeneity) of electrode surfaces. For this reason, the analysis of electrochemical noise proved... 

The first source of nonequilibrium noise, described as early as 1918 (23) (in fact 10 years earlier than Johnson noise), was shot noise that stems from the discrete nature of charge transfer. The current spectral density, Sj(/), of this noise is white (independent of frequency /) up to frequencies of the order of the inverse time of elementary charge transfer and is given by... 

Nonequilibrium noise generated by carrier-mediated ion transport was studied in lipid bilayers modified by tetranactin (41). As expected, deviations of measured spectral density from the values calculated from the Nyquist formula 1 were found. The instantaneous membrane current was described as the superposition of a steady-state current and a fluctuating current, and for the complex admittance in the Nyquist formula only a small-signal part of the total admittance was taken. The justification of this procedure is occasionally discussed in the literature (see, for example, Tyagai (42) and references cited therein), but is unclear. 

Fluctuation-dissipation theorem, transition state trajectory, white noise, 203—207 Fluctuation theorem, nonequilibrium thermodynamics, 6—7... 

Indirect evidence of nonequilibrium flucmations due to CRRs in structural glasses has been obtained in Nyquist noise experiments by Ciliberto and co-workers. In these experiments a polycarbonate glass is placed inside the plates of a condenser and quenched at temperatures below the glass transition temperature. Voltage fluctuations are then recorded as a function of time during the relaxation process and the effective temperature is measured ... 

Averaging over sin e excitations is required to clear the output signal of intrinsic thermal fluctuations. These can be described by the thermal bath mean kinetic energy kgT, and if we call N the total number of single excitations performed, we expect for the angular velocity < > a noise of the order of [kgT/ N — as compared to the nonequilibrium excited average... 

Another very important feature of the stochastic equations considered here, when they are subjected to RMT analysis, is their resemblance to the general formalism arrived at in the thermodynamics of nonequilibrium processes this suggests an analogy between the effects of multiplicative noise and the continuous flux of energy which maintains the systems far from equilibrium. This is considered the main characteristic of self-organizing living systems and means that multiplicative stochastic models could take on a new and fundamentally important role. 

Alternatively, one could use SLLOD equations to do direct simulations, such as shear a system under planar Couette flow and measure the shear stress. As we have already discussed, this approach has been used successfully to calculate a host of transport properties. It is important to remember, however, that direct simulation is often unable to simulate realistic materials at experimentally accessible shear rates. At low shear rates, the nonequilibrium response becomes small compared to the magnitude of the equilibrium fluctuations that naturally arise. The extremely small signal-to-noise ratio would demand prohibitively long simulations before any meaningful answers could be obtained. 

Excitable systems as considered here are many particle systems far from eqnilibrium. Hence variables as voltage drop (neurons), light intensity (lasers) or densities (chemical reactions) are always subject to noise and fluctations. Their sources might be of quite different origin, first the thermal motion of the molecules, the discreteness of chemical events and the quantum uncertainness create some unavoidable internal fluctuations. Bnt in excitable systems, more importantly, the crucial role is played by external sources of fluctuations which act always in nonequilibrium and are not counterbalanced by dissipative forces. Hence their intensity and correlation times and lengths can be considered as independent variables and, subsequently, as new control parameters of the nonlinear dynamics. 

Indeed, even nonequilibrium systems do not necessarily show measurable excess noise and, thus, deviate from relation 1. An appropriate example that is relevant to the subject is a capillary channel that contains a stream of electrolyte maintained by an external pressure difference. Measurements on several aqueous polymer solutions with added electrolytes performed at up to 5000 dyn/cm2 shear stresses and zero external voltage showed that measurable excess noise can be observed only for non-Newtonian solutions exhibiting elasticity (19, 20). Similar results were obtained for colloid suspensions... 

Finally, neither the effect of external noise, which affects nonequilibrium transitions in chemical and biological systems (Lefever, 1981 Horsthemke Lefever, 1984 Lefever Turner, 1986), nor the stochastic aspects of these transitions (Nicolis, Baras Malek-Mansour, 1984) are considered - with the exception of the glycolytic system (chapter 2). Such a simplification, justified in the first approximation by the absence of systematic noise in the biological systems considered, permits us to avoid complicating from the outset the analysis of systems whose kinetics is already complex. 

This observation is fundamental, reveling the fact that the characteristic energies of the Fokker-Planck equation are reactive energies, and in the final, nonequilibrium energies. This aspect is directly correlated with the nonequilibrium character specific for the Fokker-Planck equation while modeling open systems (driven by drift diffusion and factors, stochastic noise, etc.). Moreover, if the analytical solution of the eigen-values for the Schrodinger equation with the potential ) is considered, the consecrated expression is obtained ... 

Other phenomena are interesting from the noise point of view. They related to ion transport across membranes/ " equilibrium and nonequilibrium kinetic systems/ nerve membrane noise, " and membrane current fluctuations from ionic channels (Na channels and K channels in axons) in stationary or nonstationary states.Some of these studies have been described in extended reviews. 

Y. Chen, Differentiation Between Equilibrium and Nonequilibrium Kinetic Systems by Noise Analysis, Biophys. J. 21, 279-285 (1978). 

Horsthemke, W. Lefever, R. (1984a). Finite size effects and external noise in nonequilibrium systems, Phys. Letters, 106A, 10-12. 

However, at that time there still existed considerable healthy skepticism regarding the existence of nonperiodic behavior in well-controlled nonequilibrium chemical reactions. After all, nonperiodic behavior can arise from fluctuations in stirring rate or flow rate, evolution of gas bubbles from the reaction, spatial inhomogeneities due to incomplete mixing, vibrations in the stirring motor, fluctuations in the amount of bromide and dissolved oxygen in the feed, and so on. Any experimental data, no matter how well a system is controlled, will contain some noise arising from fluctuations in the control parameters therefore, it is reasonable to ask "Will noise, always present in experiments, inevitably mask deterministic nonperiodic behavior (chaos) "... 

Stochastically driven systems exhibit a variety of interesting nonequilibrium effects. These have been recently reviewed by HORSTHEMKE and LEFEVER [1] and also addressed by other authors in this workshop. In this contribution we focus our attention on the role played by the internal fluctuations of a system driven by an external noise [2,3,4]. External noise effects are usually studied in the thermodynamic limit in which internal fluctuations become negligible. This procedure assumes that the external driving noise completely dominates the fluctuations in the system. Nevertheless, a framework in which internal and external fluctuations are simultaneously considered is necessary to calculate finite size effects. Within such a framework a better understanding of the physical contents of "noise induced transition" phenomena [1] is obtained by investigating how changes in a stationary distribution induced by external noise are smoothed out by internal fluctuations. A major novel outcome of the unified theory of internal and external fluctuations presented here is the existence of "crossed-fluctuation" contributions which couple the two independent sources of randomness in the system. 

The influence of external noise on nonequilibrium systems can advance or delay the onset of oscillatory behaviour. Without drawing much attention, this phenomenon was apparently first described in the field of radio engineering [1-3]. KUTZNETSOV et al. [ 1] in a paper on the valve oscillator remark that the amplitude of the oscillations tends to zero if the intensity of the noise exceeds a certain threshold. Conversely, studying numerically the effect of substrate input noise on an oscillatory enzymatic reaction, HAHN et al. [4] found that it may induce quasi-periodic behaviour under conditions where oscillations do not occur according to the deterministic equations. 

Xg is the homogeneous macroscopic steady state of the system, F and Q being the macroscopic chemical rate and chemical noise strength respectively, and the sum over 1 runs over all nearest neighbors. The last term in the l.h.s. of (8) is a source term for spatial correlations. Explicit calculations [211 show that it is proportional to the net material flux, Jg, through the system at the steady state that is, ultimately, to the nonequilibrium constraint acting on the system. 

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