Mean phase transitions

Chen J-FI and Lubensky T C 1976 Landau-Ginzburg mean-fieid theory for the nematio to smeotio C and nematio to smeotio A phase transitions Phys.Rev. A 14 1202-7... 

Semiconductor devices ate affected by three kinds of noise. Thermal or Johnson noise is a consequence of the equihbtium between a resistance and its surrounding radiation field. It results in a mean-square noise voltage which is proportional to resistance and temperature. Shot noise, which is the principal noise component in most semiconductor devices, is caused by the random passage of individual electrons through a semiconductor junction. Thermal and shot noise ate both called white noise since their noise power is frequency-independent at low and intermediate frequencies. This is unlike flicker or ///noise which is most troublesome at lower frequencies because its noise power is approximately proportional to /// In MOSFETs there is a strong correlation between ///noise and the charging and discharging of surface states or traps. Nevertheless, the universal nature of ///noise in various materials and at phase transitions is not well understood. 

For the analysis heat and mass transfer in concrete samples at high temperatures, the numerical model has been developed. It describes concrete, as a porous multiphase system which at local level is in thermodynamic balance with body interstice, filled by liquid water and gas phase. The model allows researching the dynamic characteristics of diffusion in view of concrete matrix phase transitions, which was usually described by means of experiments. 

Thermal properties of overlayer atoms. Measurement of the intensity of any diffracted beam with temperature and its angular profile can be interpreted in terms of a surface-atom Debye-Waller factor and phonon scattering. Mean-square vibrational amplitudes of surfece atoms can be extracted. The measurement must be made away from the parameter space at which phase transitions occur. 

In the inset of Fig. 9 we show the mean field frequency 0 = 0// as a function of density for T = 1. At this temperature the system undergoes a phase transition from a paramagnetic to a ferromagnetic fluid at a density whose mean field value is p mf = 0-4- For densities below this value we obtain 0 = cjq, which agrees with the frequency value of the low-order virial expansion (see Eq. (34)). For p > Pc,mF) increases with the density due to increase of the magnetization. 

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

In order to finally address the question whether our system has a reentrant phase transition, as predicted by the mean-field study the low temperature region was analyzed by the cumulant intersection finite-size scahng method described in Sec. IV A. For the rotational constant 0 = 0.6109 an... 

In this seetion of our work we present examples of the applieation of eomputer simulation methods to study ehemieally assoeiating fluids. In the first ease we eonsider the adsorption and surfaee phase transitions by means of a eonstant pressure Monte Carlo simulation. The seeond example is foeused on the problem of ehemieal potential evaluation. 

I. Jensen, H. C. Fogedby. Kinetic phase transitions in a surface-reaction model with diffusion Computer simulations and mean-field theory. Phys Rev A 2 1969-1975, 1990. 

A. Maltz, E. V. Albano. Kinetic phase transitions in dimer-dimer surface reaction models studied by means of mean-field and Monte Carlo methods. Surf Sci 277-A A-42S, 1992. 

Typical runs consist of 100 000 up to 300 000 MC moves per lattice site. Far from the phase transition in the lamellar phase, the typical equilibration run takes 10 000 Monte Carlo steps per site (MCS). In the vicinity of the phase transitions the equilibration takes up to 200 000 MCS. For the rough estimate of the equihbration time one can monitor internal energy as well as the Euler characteristic. The equilibration time for the energy and Euler characteristic are roughly the same. For go = /o = 0 it takes 10 000 MCS to obtain the equilibrium configuration in which one finds the lamellar phase without passages and consequently the Euler characteristic is zero. For go = —3.15 and/o = 0 (close to the phase transition) it takes more than 50 000 MCS for the equihbration and here the Euler characteristic fluctuates around its mean value of —48. 

Before trying to solve the master equation for growth processes by direct stochastic simulation it is usually advisable to first try some analytical approximation. The mean-field approximation often gives very good results for questions of first-order phase transitions, and at least it provides a qualitative understanding for the interplay of the various model parameters. 

Carriers and channels may be distinguished on the basis of their temperature dependence. Channels are comparatively insensitive to membrane phase transitions and show only a slight dependence of transport rate on temperature. Mobile carriers, on the other hand, function efficiently above a membrane phase transition, but only poorly below it. Consequently, mobile carrier systems often show dramatic increases in transport rate as the system is heated through its phase transition. Figure 10.39 displays the structures of several of these interesting molecules. As might be anticipated from the variety of structures represented here, these molecules associate with membranes and facilitate transport by different means. 

To summarize we have reproduced the intricate structural properties of the Fe-Co, Fe-Ni and the Fe-Cu alloys by means of LMTO-ASA-CPA theory. We conclude that the phase diagram of especially the Fe-Ni alloys is heavily influenced by short range order effects. The general trend of a bcc-fcc phase transition at lower Fe concentrations is in accordance with simple band Ailing effects from canonical band theory. Due to this the structural stability of the Fe-Co alloys may be understood from VGA and canonical band calculations, since the common band model is appropriate below the Fermi energy for this system. However, for the Fe-Ni and the Fe-Cu system this simple picture breaks down. 

Table 7.3 lists the four rules in this minimally-diluted rule-family, along with their corresponding iterative maps. Notice that since rules R, R2 and R3 do not have a linear term, / (p = 0) = 0 and mean-field-theory predicts a first-order phase transition. By first order we mean that the phase transition is discontinuous there is an abrupt, discontinuous change at a well defined critical probability Pc, at which the system suddenly goes from having poo = 0 as the only stable fixed point to having an asymptotic density Poo 7 0 as the only stable fixed point (see below). 

Rule i 4, on the other hand, has both a linear and quadratic term, so that / (p = 0) > 0 in general, and is therefore predicted to have a second order (or continuous) phase transition. Although the mean-field predictions are, of course, dimension-independent, they are expected to become exact as the dimension d —7 oo. In practice, it is often found that there exists a critical dimension dc above which the mean-field critical exponents are recovered exactly. 

Table 7.4 compares the mean-field-theory prediction for the order of the phase transition and critical probability pc to numerical results obtained by Biduax, et.al. ([bidaux89a], [bidaux89b]) by simulating dynamics on regular lattices of dimension d = I, d = 2 and d = 4. 

Ri does not show any phase transition. This should not be terribly surprising, since, in the deterministic limit, Ri exhibits either class 2 behavior (i.e. periodicity) or class 4 behavior (spatially separated propagating structures with an ill-defined statistical limit). The density p therefore has no well-defined statistical mean for p = 1 and the periodicity and/or propagating structures are rapidly destroyed (and thus p — 0) whenever p < 1. Moreover, from the above mean-field... 

Table 7.4 Oi dor of phase transition and threshold probabilities versus space dimension for rules R, ...Rn, as determined by mean-field theory and numerical calculation ([bidaux89a], [bidaux89b]).

The rapid rise in computer speed over recent years has led to atom-based simulations of liquid crystals becoming an important new area of research. Molecular mechanics and Monte Carlo studies of isolated liquid crystal molecules are now routine. However, care must be taken to model properly the influence of a nematic mean field if information about molecular structure in a mesophase is required. The current state-of-the-art consists of studies of (in the order of) 100 molecules in the bulk, in contact with a surface, or in a bilayer in contact with a solvent. Current simulation times can extend to around 10 ns and are sufficient to observe the growth of mesophases from an isotropic liquid. The results from a number of studies look very promising, and a wealth of structural and dynamic data now exists for bulk phases, monolayers and bilayers. Continued development of force fields for liquid crystals will be particularly important in the next few years, and particular emphasis must be placed on the development of all-atom force fields that are able to reproduce liquid phase densities for small molecules. Without these it will be difficult to obtain accurate phase transition temperatures. It will also be necessary to extend atomistic models to several thousand molecules to remove major system size effects which are present in all current work. This will be greatly facilitated by modern parallel simulation methods that allow molecular dynamics simulations to be carried out in parallel on multi-processor systems [115]. 

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