Finite temperature random phase

In the next paper [160], Villain discussed the model in which the local impurities are to some extent treated in the same fashion as in the random field Ising model, and concluded, in agreement with earlier predictions for RFIM [165], that the commensurate, ordered phase is always unstable, so that the C-IC transition is destroyed by impurities as well. The argument of Villain, though presented only for the special case of 7 = 0, suggests that at finite temperatures the effects of impurities should be even stronger, due to the presence of strong statistical fluctuations in two-dimensional systems which further destabilize the commensurate phase. 

Another development in the quantum chaos where finite-temperature effects are important is the Quantum field theory. As it is shown by recent studies on the Quantum Chromodynamics (QCD) Dirac operator level statistics (Bittner et.al., 1999), nearest level spacing distribution of this operator is governed by random matrix theory both in confinement and deconfinement phases. In the presence of in-medium effects... 

If the impurity potential is smooth, the process of scattering on them proceeds quasi-classically. In this case no real scattering takes place and the impurity effect may be reduced to the appearance of a random phase of the electron wave function. As has been shown by Zawadowski (1), such impurities do not affect the thermodynamics of the one-dimensional system, in which, however, no phase transitions exist. The finite temperature of the transition arises due to three-dimensional effects which establish the coherent state in the whole volume. The impurities cause the phase shift on each thread, and, as a result, the coherence drops and the transition temperature diminishes. 

An anlysis of this question meets complications since in a purely one-dimensional substance the fluctuations exclude the existence of an order parameter (i.e., a phase transition) at finite temperatures. The fluctuations can be suppressed by means of the intercahin hopping but the impurities may prevent this process. The study of the influence of a smooth random potential (the potential (l) in our notations) in the work ] shows that what concerns fluctuations both types of transitions are suppressed by impurities, i.e., there is no gain on this way. One must have in mind however, that the fluctuation are imposed on the purely one-dimensional pairing and the absence of the latter means definitely no pairing even with hopping. 

It is presumed that both the a and (3 phases should form a single phase at perfect equilibrium. The obscure a—p transition at 24K should stem from residual crystallites of the p phase due to a finite cooling rate from liquid O2. Marked time dependence is one of the characteristics of random magnetism, and therefore the higher temperature solid phase (> 24 K) should show random magnetism. As the X value of the lower temperature phase (<24K) decreases with the decrease of temperature, it should be antiferromagnetic. 

It is not clear, whether the experimentally observed random local freezing of the deuterons in the O-D—O bonds in deuteron glasses corresponds to a true thermodynamic phase transition or whether one deals with a dynamic phenomenon which only seems static because of the finite observation time of the experimental techniques. The recently observed42 splitting between the field-cooled and zero-field dielectric susceptibilities below an instability temperature Tf seems to speak for the occurrence of an Almeida-Thouless-like thermodynamic phase transition in deuteron glasses. It is well known that ID NMR and EPR allow a direct measurement of the Edwards-Anderson order parameter qEA only on time scales of 10 3-10 8 s and 2D exchange NMR possibly seems to be a better technique for such slow motions. 

The temperature at which the phase transition occurs is called the critical temperature or Tg. Most, but not all, magnetic phase transitions are continuous , sometimes called second order . From a microscopic point of view, such phase transitions follow a scenario in which, upon cooling from high temperature, finite size, spin-correlated, fractal like, clusters develop from the random, paramagnetic state at temperatures above Tg, the so-called critical regime . As T Tg from above, the clusters grow in size until at least one cluster becomes infinite (i.e. it extends, uninterrapted, throughout the sample) in size at Tg. As the temperature decreases more clusters become associated with the infinite cluster until at T = 0 K all spins are completely correlated. 

An outstanding problem concerns itself with the structure of a hard sphere phase. This is a special instance of the more. general difficulty of the specification of the structure of infinitely extended random media. These questions will perhaps be the subject of a future mathematical discipline-stochastic geometry. The pair correlation function g(r), even if it is known, hardly suffices to specify uniqudy the stochastic metric properties of a random structure. For a finite N and V finite) system in equilibrium in thermal contact with a heat reservoir at temperature T, the density in the configuration space of the N particles [Eq. (2)]... 

C, within the one-phase channel. The temperature is chosen at the hydrophile-lipophile balance (HLB) temperature for a salinity of 0.49%. The SANS data taken with an oil-water pontrast are analyzed by using a random-wave model with an appropriate spectral function. The spectral function is an inverse eighth-order polynomial in wave number k, containing three length scales 1/a, 1/b, and 1/c, and has finite second and fourth moments. This three-... 

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