Thermal noise effect

The Cahn-Hilliard theory is deterministic, ignoring the random generation of concentration fluctuations by thermal agitation. This thermal noise effect, first pointed out by Cook [14], adds a new term to the right-hand side of eq 2.4. Since the timescale of thermal agitation is much shorter than that of the slow uphill diffusive process concerned here, the added term may play a role... 

A general equation for 5(fc, t) was derived by Langer et al. [15] in 1975 (see also Langer [16]) from statistical considerations of molecular flow processes. Their formulation invokes eq 2.1 for AG and incorporates the thermal noise effect. Omitting its details, we can write the basic equation of Langer et al. as... 

Improved Analysis The above data analysis is based on two assumptions. One is that the thermal noise effect may be neglected, and the other is that the measured scattering intensity I k,t) is directly proportional to S k,t). The first assumption is considered valid for polymer mixtures, but the second one is not obtained in usual experimental setups, in which some stray light unavoidably enters into the detector for the measurement of scattering intensity. Thus, it is... 

The parameter loo in eq 2.48 is often associated with the thermal noise effect. In this consideration, Zb in loo — KSt + h is tacitly ignored. Thus, when, on the basis of a recent study of Okada and Han [28] on a d- PS VME blend, Hashimoto [19] anticipated that the thermal noise effect should be insignificant except for shallow quench depths and for k near k at which R k) vanishes, he would not have minded the term R- In actuality, however, its contribution to Zoo in the early stage of phase separation is by no means negligible, and a due analysis of spinodal decomposition data has to take the effect of Zb into account. 

The function / incorporates the screening effect of the surfactant, and is the surfactant density. The exponent x can be derived from the observation that the total interface area at late times should be proportional to p. In two dimensions, this implies R t) oc 1/ps and hence x = /n. The scaling form (20) was found to describe consistently data from Langevin simulations of systems with conserved order parameter (with n = 1/3) [217], systems which evolve according to hydrodynamic equations (with n = 1/2) [218], and also data from molecular dynamics of a microscopic off-lattice model (with n= 1/2) [155]. The data collapse has not been quite as good in Langevin simulations which include thermal noise [218]. 

Wang, G. H., Barlow, R. S., and Clemens, N. T, Quantification of resolution and noise effects on thermal dissipation measurements in turbulent non-premixed jet flames, Proc. Combust. Inst., 31, 1525, 2007. 

The viscoelastic effects on the morphology and dynamics of microphase separation of diblock copolymers was simulated by Huo et al. [ 126] based on Tanaka s viscoelastic model [127] in the presence and absence of additional thermal noise. Their results indicate that for

bulk modulus of both blocks, the area fraction of the A-rich phase remains constant during the microphase separation process. For each block randomly oriented lamellae are preferred. 

The energies of the j8 particles from most /3 emitters are very low. This, of course, leads to low-energy photons emitted from the fluors and relatively low-energy electrical pulses in the PMT. In addition, photomultiplier tubes produce thermal background noise with 25 to 30°/o of the energy associated with the fluorescence-emitted photons. This difficulty cannot be completely eliminated, but its effects can be lessened by placing the samples and the PMT in a freezer at -5 to — 8°C in order to decrease thermal noise. 

We show that application of a constant force (bias held) results in shifting the position of the ordinary SR peak together with the anticipated reduction of its height and sharpness. For the quadratic SR the situation is more complicated. There, the joint action of the thermal noise and constant bias leads to formation of a mountain-like surface over the plane of those parameters. In other words, for each given value of the bias held there exists a unique value of the noise strength that maximizes SNR and vice versa. The discovered effect can be useful, for example, for evaluation of the parameters of bistable systems through susceptibility measurements. In addition, it has to be taken into account when designing any devices where the nonlinear SR is employed. 

The studied quadratic SR of a superparamagnetic grain is caused by interaction of the periodic excitation with the thermal noise. However, there exists an analogy T —> yHHa, where Ha = 2K/Is is the particle anisotropy held, which outlines a passage from thermal to quantum fluctuations. Thus, with certain caution, one may apply the results obtained to the nonlinear SR caused by the tunnel effect at T = 0. 

The effects of fields of low strength were discussed in Section 14.6. There are still controversial aspects of this work, in particular the existence of effects when the strength of the field is less than that of thermal noise.22... 

Thermal fluctuations can contribute dominantly to the scattering intensity right after the isothermal phase separation starts [70,76], Therefore, conditions 1) and 3) must be fulfilled to ensure that the effect of thermal noise is negligible. The dynamics of phase separation can be adequately described by the mean-field model if condition 2) is satisfied. Condition 2) is a direct consequence of the Landau Ginzburg criterion [75]. Thus, one may establish prerequisites for Eqs. (27) and (33) are the conditions 1) and 3), while Eq. (34) requires conditions 2) and 3). For example, Eq. (27) and as a consequence Eq. (33) cannot be confirmed experimentally not even for small values of q if the quench depth e is too small [70]. Moreover, owing to the effect of thermal fluctuations, Eq. (33) fails at q as qc even if the Landau Ginzburg criterion is fulfilled [70,77]. Thus, in the former case condition 2) is violated whereas in the latter example conditions 1) and 3) are not satisfied. 

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