Fluctuations of the Order Parameter

The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

In the previous section we assumed that disorder results in random fluctuations of the order parameter around some average value A<). Such an approach is, essentially, a mean field treatment of the lattice. It requires sufficiently strong interchain interactions, whose role is to establish a coherence between the phases of the order parameter in different chains. 

The mean-field SCFT neglects the fluctuation effects [131], which are considerably strong in the block copolymer melt near the order-disorder transition [132] (ODT). The fluctuation of the order parameter field can be included in the phase-diagram calculation as the one-loop corrections to the free-energy [37,128,133], or studied within the SCFT by analyzing stability of the ordered phases to anisotropic fluctuations [129]. The real space SCFT can also applied for a confined geometry systems [134], their dynamic development allows to study the phase-ordering kinetics [135]. 

In order to demonstrate how fluctuations of the order parameter may affect thermodynamic quantities let us calculate the contribution of fluctuations... 

Comparing the mean field (12) and the fluctuation (15) contributions to the specific heat (in the low and high temperature limiting cases one may use Eqs. (22), (24)) we may estimate the fluctuation temperature < Tc, at which the contribution of fluctuations of the order parameter becomes to be as important as the mean field one (so called Ginzburg - Levanyuk criterion),... 

In the condensed matter physics one usually performs calculations in the high temperature limit. In this limit one neglects the time (frequency) dependent terms considering quasi-static thermal fluctuations of the order parameter. Then the fluctuation contribution is determined with the help of the functional... 

So far we have discussed the specific behavior of fluctuations of the order parameter at fixed temperature and density. There are also fluctuations of the temperature and the local quark density. They are statistically independent quantities [19], < 5T5pq >= 0, and their mean squares are... 

Anomalous behavior of fluctuations might manifest itself in the event-by-event analysis of the heavy ion collision data. In small (L) size systems, L < , (zero dimension case would be L order parameter to the specific heat is still increased, as we have mentioned, see [15]. The anomalous behavior of the specific heat may affect the heat transport. Also kinetic coefficients are substantially affected by fluctuations due to the shortening of the particle mean free paths, as the consequence of... 

As it is known [5], the intensity of the scattered light gives us an information about the system s disorder, e.g., presence therein of pores, impurities etc. Since macroscopically liquid is homogeneous, critical opalescence arises due to local microscopic inhomogeneities - an appearance of small domains with different local densities. In other words, liquid is ordered inside these domains but still disorded on the whole since domains are randomly distributed in size and space, they appear and disappear by chance. Fluctuations of the order parameter have large amplitude and involve a wide spectrum of the wavelengths (which results in the milk colour of the scattered light). 

Consider now the fluctuations of the order parameter in the system possessing the chemical reaction this problem could be perfectly illustrated by computer simulations on lattices. We start with the bimolecular A + B -y 0 reaction discussed above, and first of all froze particle diffusion. Let the recombination event happen instantly when a pair AB of dissimilar particles occupies the nearest lattice sites (assume lattice to be squared). Immobile particles enter into reaction as a result of their creation with the equal probabilities in empty lattice sites from time to time a newly created particle A(B) finds itself nearby pre-created B(A) and they recombine. (Since this recombination event is instant, the creation rate is of no importance.) This model describes, in particular, Frenkel defect accumulation in solids under... 

How could we take into account the fluctuations of the order parameter Let us return to the well-studied example of the gas-liquid system. A general equation of the state of gases and liquids proved in statistical physics [9] has a form p = nk T - n2G(x) where G(x) is some integral containing the interaction potential of particles and the joint correlation function x(r). Therefore, the equation for the long-range order parameter n contains in itself the functional of the intermediate-order parameter x r)-... 

The transition from a stable steady-state solution observed at large p to the oscillatory regime assumes the existence of the critical value of the parameter pc, which defines the point of the kinetic phase transition as p > pc, the fluctuations of the order parameter are suppressed and the standard chemical kinetics (the mean-field theory) could be safely used. However, if p < pc, these fluctuations are very large and begin to dominate the process. Strictly speaking, the region p pc at p > pc is also fluctuation-controlled one since here the fluctuations of the order parameter are abnormally high. 

In the mean-field approximation, the possibility for fluctuations of the order parameter is forbidden. Therefore, the last term in Eq. (6) is zero and Hinx reduces to the mean-field Hamiltonian. 

Scattering and turbidity. The non-analytical divergences at critical points result from fluctuations of the order parameter, which can be observed by scattering experiments. The intensity I of single scattering in binary systems is determined by the concentration fluctuations, which in a rather good approximation are described by the Ornstein-Zernike equation,... 

The grand-canonical ensemble is particularly well suited for studies of liquid-vapor phase coexistence (i) Fluctuations of the order parameter, i.e., the density, are efficiently relaxed. Since the density is not conserved, spatial fluctuations do not decay via slow diffusion of polymers but relax much faster through insertion/deletion moves. In the grand-canonical ensemble one controls the temperature, T, the volume, V, and the chemical potential, p,... 

A peculiar feature of Bose condensates and superfluids is that their low-energy excitations correspond to collective modes, which can be described as fluctuations of the order parameter [106]. For uniform dilute boson gases, with an effective interparticle interaction potential F(r — r ) = g8(r — r ), where g = Anfisa/m is the coupling constant, the excitations (characterized by energies e (fe)) are given by the Bogoliobov spectrum [136]... 

This neglect of fluctuations in general is not warranted. One can recognize this problem in the framework of Landau s theory itself. This criterion named after Ginzburg (1960) considers the mean square fluctuation of the order parameter in a coarse graining volume Ul and states that Landau s theory is selfconsistent if this fluctuation is much smaller than the square of the order parameter itself,... 

In a purely one-dimensional case an infinitesimal impurity can cause as first stated by Mott and Twose, a localization of electrons and an infinite resistance at T=0. The electronic phase Transitions cannot happen in a purely one-dimensional system because of the fluctuations of the order parameter. 

Peierls transitions and this produces a principle way to change the properties of materials The influence of impurities on the interchain hopping probability is also essential since the hopping is necessary to suppress the fluctuations of the order parameter [6],... 

Near the critical point, the relevant scale for the concentration variation is set by the order parameter mcocx, Eqs. (14), (15). The condition for the validity of mean-field theory (for T < Tc) now is that the mean square fluctuation of the order parameter in a volume region of linear dimension L over which we average, Eq. (92), must be small in comparison with the order parameter square itself,... 

However, for all second order phase transitions it is well-known that one must pay attention to fluctuations of the order parameter - taking them into account often invalidates Landau-like theories [74]. This also happens here (Fig. 42, right part) it turns out that no longer any second order transition occurs at all, rather we encounter a fluctuation-induced first order transition [58, 327]. 

At any finite temperature, the oil/water interface is not flat but is roughened due to the presence of thermally excited capillary waves. The spectrum of capillary waves can be calculated by considering small fluctuations of the order parameter field, (I>(r) = d>(r) -f f/(r), around the planar interface. By an expansion of the free energy functional to second order in r, one finds the energy [42,96]... 

The equilibrium order determined within the mean-field theory is perturbed due to thermal fluctuations which give rise to collective excitations. Except in the close vicinity of the phase/structural transitions, the thermal fluctuations of the order parameter can be assumed small, and the free energy of the fluctuations can be considered a correction to the mean-field free energy. In such a case, the fluctuations of the liquid-crystaUine order are described consistently by a harmonic Hamiltonian of the form... 

The name pseudo-Casimir force for the fluctuation-induced interaction is due to the analogy with the Casimir effect [63] at T = 0 quetntum fluctuations of the electromagnetic field in a cavity yield a weak yet measurable attraction between the walls of the cavity. Because the force between the walls is determined by a derivative of the free energy of a system rather than by a derivative of its energy, a similar effect is expected above absolute zero where the interaction is not just due to quantum but also due to thermal fluctuations. In liquid crystals, the fluctuation-induced interaction is due to thermal fluctuations of the order parameter field instead of the electromagnetic field. 

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