Fluctuations, critical

In statistical mechanics, the Helmholtz energy, is calculated through the canonical partition function Z 

Experimentally, it is well established that asymptotically close to the critical point all physical properties obey simple power laws. The universal powers in these laws are called critical exponents, the values of which can be calculated from RG recursion relations. The phenomenological approach that interrelates the critical power laws is called scaling theory. In particular, the isochoric heat capacity diverges at the vapour-liquid critical point of one-component fluids along the critical isochore as 

Accurate light-scattering experiments have shown that the correlation length of fluctuations diverges at the critical point of a fluid along the critical isochore as 

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Easier V, Schweiss P, Meingast C, Obst B, Wuhl H, Rykov A I and Tajima S 1998 3D-XY critical fluctuations of the thermal expansivity in detwinned YBa2Cu30y g single crystals near optimal doping Phys. Rev. Lett. 81 1094-7... 

In general, with the normalization of the critical fluctuation, the initial spatial spectrum of the concentration fluctuation corresponding to the wave number components, A and Ay, is given as... 

Figure 45. Semilog plot of the pit-growth current, J vs. V = 50 mV, [NiCU = 5 mol in 3, (NaCIJ = 1000 mol m 3. T=300 K, J0 is the current component shown in Eq. (Ill), which becomes unstable at the minimum state and Iq is the growth factor of the pits expressed by Eq. (112). (Reprinted from M. Asanuma and R. Aogaki, Nonequilibrium fluctuation theory on pitting dissolution. III. Experimental examinations on critical fluctuation and its growth process in nickel dissolution, J. Chem. Phys. 106,9944, 1997, Fig. 14. Copyright 1997, American Institute of Physics.)...

I2H2O as a function of the reciprocal temperature. The points are data obtained from fits of the Mdssbauer spectra (Fig. 6.6). The broken curve is a fit to the Einstein model for a Raman process. The dotted curve corresponds to a contribution from a direct process due to interactions between the electronic spins and low-energy phonons associated with critical fluctuations near the phase transition temperature. (Reprinted with permission from [32] copyright 1979 by the Institute of Physics)... 

Veatch, S.L., Cicuta, P., Sengupta, P., Honerkamp-Smith, A., Holowka, D., Baird, B. Critical fluctuations in plasma membrane vesicles. ACS Chem. Biol. 2008, 3, 287-93. 

Veatch, S.L., Soubias, O., Keller, S.L., Gawrisch, K. Critical fluctuations in domain-forming lipid mixtures. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 17650-5. 

Close to the transition temperature Tg, the dynamics of a spin glass system will be governed by critical fluctuations, but critical fluctuations are also of importance on experimental timescales quite far from Tg. At temperatures both below and above Tg, length scales shorter than the coherence length of the... 

At the time of writing, the only evidence for critical fluctuations near the consolute point known to us comes from the work of Damay (1973). The thermopower of Na-NH3 plotted against T at the critical concentration is shown in Fig. 10.21. We conjecture that this behaviour is due to long-range fluctuations between two metallic concentrations, and that near the critical point, where the fluctuations are wide enough to allow the use of classical percolation theory, the... 

Figure 27 presents values of W(T) calculated by Dietl et al. (2001a) in comparison to the experimental data of Shono et al. (2000). Furthermore, in order to establish the sensitivity of the theoretical results to the parameter values, the results calculated for a value of Xc that is 1.8 times larger are included as well. The computed value for low temperatures, W = 1.1 /xm, compares favorably with the experimental finding, W = 1.5 fim. However, the model predicts a much weaker temperature dependence of IV than observed experimentally, which Dietl et al. (2001a) link to critical fluctuations, disregarded in the mean-field approach. 

Remark. A great deal of attention has been paid in recent years to non-equilibrium stationary processes that are unstable and also extended in space. They can give rise to different phases that exist side by side, so that translation symmetry is broken. The name dissipative structures has been coined for them, and the prime examples are the Benard cells and the Zhabotinski reactions, but they also occur in biology and meteorology. However, these are features of the macroscopic equations. They are only relevant for fluctuation theory inasmuch as the fluctuation becomes very large at the point where the instability sets in. The critical fluctuations in XIII.5 are an example. There are many more varieties, in particular in the case of more variables. 

According to RG theory [11, 19, 20], universality rests on the spatial dimensionality D of the systems, the dimensionality n of the order parameter (here n = 1), and the short-range nature of the interaction potential 0(r). In D = 3, short-range means that 0(r) decays as r p with p>D + 2 — tj = 4.97 [21], where rj = 0.033 is the exponent of the correlation function g(r) of the critical fluctuations [22] (cf. Table I). Then, the critical exponents map onto those of the Ising spin-1/2 model, which are known from RG calculations [23], series expansions [11, 12, 24] and simulations [25, 26]. For insulating fluids with a leading term of liquid metals [27-29] the experimental verification of Ising-like criticality is unquestionable. 

There are other scenarios for an apparent mean-field criticality [15, 17]. The most likely one is crossover from asymptotic Ising behavior to mean-field behavior far from the critical point, where the critical fluctuations must vanish. For the vicinity of the critical point, Wegner [43] worked out an expansion for nonasymptotic corrections to scaling of the general form... 

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

It should be emphasized that the comparatively large change obtained in more recent work is mainly caused by the application of finite-size scaling. Under these circumstances, one certainly needs to reconsider how far the results of analytical theories, which are basically mean-field theories, should be compared with data that encompass long-range fluctuations. For the van der Waals fluid the mean-field and Ising critical temperatures differ markedly [249]. In fact, an overestimate of Tc is expected for theories that neglect nonclassical critical fluctuations. Because of the asymmetry of the coexistence curve this overestimate may be correlated with a substantial underestimate of the critical density. 

MC simulations can reflect the nonclassical critical fluctuations if the simulation box is sufficiently large or if special techniques are applied to analyze the fluctuations. Simulations for simple nonionic models such as the square-well fluid (SCF) [52] show that there is indeed a good chance to study details of criticality. As noted, MC simulations have also been profitably exploited... 

Extension of the classical Landau-Ginzburg expansion to incorporate nonclassical critical fluctuations and to yield detailed crossover functions were first presented by Nicoll and coworkers [313, 314] and later extended by Chen et al. [315, 316]. These extensions match Ginzburg theory to RG theory, and thus interpolate between the lower-order terms of the Wegner expansion at T -C Afa and mean-field behavior at f Nci-... 

At first, one would tend to reconsider conventional crossover due to mean-field criticality associated with long-range interactions in terms of the refined theories. Conventional crossover conforms to the first case mentioned—that is, small u with the correlation length of the critical fluctuations to be larger than 0. However, in the latter case one expects smooth crossover with slowly and monotonously varying critical exponents, as observed in nonionic fluids. Thus, the sharp and nonmonotonous behavior cannot be reconciled with one length scale only. 

Actually, MC simulations should reflect the nonclassical critical fluctuations as well, thus allowing us to identify the critical exponents and the... 

M. Corti and V. Degiorgio. Micellar properties and critical fluctuations in aqueous-solutions of non-ionic amphiphiles. J. Phys. Chem., 85(10) 1442-1445, 1981. 

A rep < 1, Des < 1, the nucleation dynamics is stochastic in nature as a critical fluctuation in one, or more, order parameters is required for the development of a nucleus. For DeYep > 1, Des < 1 the chains become more uniformly oriented in the flow direction but the conformation remains unaffected. Hence a thermally activated fluctuation in the conformation can be sufficient for the development of a nucleus. For a number of polymers, for example PET and PEEK, the Kuhn length is larger than the distance between two entanglements. For this class of polymers, the nucleation dynamics is very similar to the phase transition observed in liquid crystalline polymers under quiescent [8], and flow conditions [21]. In fast flows, Derep > 1, Des > 1, A > A (T), one reaches the condition where the chains are fully oriented and the chain conformation becomes similar to that of the crystalline state. Critical fluctuations in the orientation and conformation of the chain are therefore no longer needed, as these requirements are fulfilled, in a more deterministic manner, by the applied flow field. Hence, an increase of the parameters Deiep, Des and A results into a shift of the nucleation dynamics from a stochastic to a more deterministic process, resulting into an increase of the nucleation rate. 

The transport properties of a near-critical system contain an enhancement or a reduction due to critical fluctuations in addition to the contributions of molecular transport processes which are strictly a function of the thermodynamic state. Therefore, the transport coefficients in the critical region are usually... 

Table IV Values of Parameters Obtained from Theory of Critical Fluctuations...

For magnetic systems, NMR is a useful technique in the investigation of the behavior of the magnetic moments and the critical phenomena (order-disorder, critical fluctuations)34,35. ... 

In any case the spin-Peierls transition is driven by one-dimensional pre-transitional structural fluctuations [46]. Such fluctuations start to develop at some temperature TF above TsP. The effect of these critical fluctuations is to induce a local pairing of the spins which leads to an observable deviation of the magnetic susceptibility x below TF from the general Bon-... 

However, in the case of MEM(TCNQ)2, which is considered as one of the most representative spin-Peierls materials, with TsP = 19 K, the results are quite at variance with the normal behavior described above [46]. For this material, critical fluctuations are also observed correctly, by x-rays, below Tp = 40 K, but they are only of a three-dimensional nature. Moreover, these fluctuations do not produce any detectable effect below TF on the Bonner-Fisher dependence of the magnetic susceptibility. Consequently, this law is perfectly followed, with J = 53 K, down to 19 K [17,18,46]. Some earlier comments on this point have also been given by Schulz [50]. 

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