Classical phase transitions

Osterloh and Jante [53] reported that classical phase transitions from II( —) III II( + ) were observed using PO-EO sulfates in combination with a blend of stock tank oil and a light fraction from the crude oil, probably in a salinity scan. 

The purpose of this chapter is twofold In the following two sections. Phase Transitions and Critical Behavior and Quantum vs. Classical Phase Transitions, we give a concise introduction into the theory of quantum phase transitions, emphasizing similarities with and differences from classical thermal transitions. After that, we point out the computational challenges posed by quantum phase transitions, and we discuss a number of successful computational approaches together with prototypical examples. However, this chapter is not meant to be comprehensive in scope. We rather want to help scientists who are taking their first steps in this field to get off on the right foot. Moreover, we want to provide experimentalists and traditional theorists with an idea of what simulations can achieve in this area (and what they cannot do,. .. yet). Those readers who want to learn more details about quantum phase transitions and their applications should consult one of the recent review articles or the excellent textbook on quantum phase transitions by Sachdev. ... 

A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

Helium Purification and Liquefaction. HeHum, which is the lowest-boiling gas, has only 1 degree K difference between its normal boiling point (4.2 K) and its critical temperature (5.2 K), and has no classical triple point (26,27). It exhibits a phase transition at its lambda line (miming from 2.18 K at 5.03 kPa (0.73 psia) to 1.76 K at 3.01 MPa (437 psia)) below which it exhibits superfluid properties (27). 

The transition from a ferromagnetic to a paramagnetic state is normally considered to be a classic second-order phase transition that is, there are no discontinuous changes in volume V or entropy S, but there are discontinuous changes in the volumetric thermal expansion compressibility k, and specific heat Cp. The relation among the variables changing at the transition is given by the Ehrenfest relations. 

The computation of quantum many-body effects requires additional effort compared to classical cases. This holds in particular if strong collective phenomena such as phase transitions are considered. The path integral approach to critical phenomena allows the computation of collective phenomena at constant temperature — a condition which is preferred experimentally. Due to the link of path integrals to the partition function in statistical physics, methods from the latter — such as Monte Carlo simulation techniques — can be used for efficient computation of quantum effects. 

Of the variety of quantum effects which are present at low temperatures we focus here mainly on delocalization effects due to the position-momentum uncertainty principle. Compared to purely classical systems, the quantum delocalization introduces fluctuations in addition to the thermal fluctuations. This may result in a decrease of phase transition temperatures as compared to a purely classical system under otherwise unchanged conditions. The ground state order may decrease as well. From the experimental point of view it is rather difficult to extract the amount of quantumness of the system. The delocahzation can become so pronounced that certain phases are stable in contrast to the case in classical systems. We analyze these effects in Sec. V, in particular the phase transitions in adsorbed N2, H2 and D2 layers. 

With increasing values of P the molar volume is in progressively better agreement with the experimental values. Upon heating a phase transition takes place from the a phase to an orientationally disordered fee phase at the transition temperature where we find a jump in the molar volume (Fig. 6), the molecular energy, and in the order parameter. The transition temperature of our previous classical Monte Carlo study [290,291] is T = 42.5( 0.3) K, with increasing P, T is shifted to smaller values, and in the quantum limit we obtain = 38( 0.5) K, which represents a reduction of about 11% with respect to the classical value. 

The central quantity is the order parameter as a function of temperature (see Fig. 13). The phase transition temperature Tq of the classical system can be located around 38 K. At high temperatures, the quantum curve of the order parameter merges with the classical curve, whereas it starts to deviate below Tq. Qualitatively, quantum fluctuations lower the ordering and thus the quantum order parameter is always smaller than its classical counterpart. The inclusion of quantum effects results in a nearly 10% lowering of Tq (see Fig. 13). 

Furthermore, one can infer quantitatively from the data in Fig. 13 that the quantum system cannot reach the maximum herringbone ordering even at extremely low temperatures the quantum hbrations depress the saturation value by 10%. In Fig. 13, the order parameter and total energy as obtained from the full quantum simulation are compared with standard approximate theories valid for low and high temperatures. One can clearly see how the quasi classical Feynman-Hibbs curve matches the exact quantum data above 30 K. However, just below the phase transition, this second-order approximation in the quantum fluctuations fails and yields uncontrolled estimates just below the point of failure it gives classical values for the order parameter and the herringbone ordering even vanishes below... 

One prominent example of rods with a soft interaction is Gay-Berne particles. Recently, elastic properties were calculated [89,90]. Using the classical Car-Parrinello scheme, the interactions between charged rods have been considered [91]. Concerning phase transitions, the sohd-fluid equihbria for hard dumbbells that interact additionally with a quadrupolar force was considered [92], as was the nematic-isotropic transition in a fluid of dipolar hard spherocylinders [93]. The influence of an additional attraction on the phase behavior of hard spherocylinders was considered by Bolhuis et al. [94]. The gelation transition typical for clays was found in a system of infinitely thin disks carrying point quadrupoles [95,96]. In confined hquid-crystalline films tilted molecular layers form near each wall [97]. Chakrabarti has found simulation evidence of critical behavior of the isotropic-nematic phase transition in a porous medium [98]. 

In 1978, Bryan [11] reported on crystal structure precursors of liquid crystalline phases and their implications for the molecular arrangement in the mesophase. In this work he presented classical nematogenic precursors, where the molecules in the crystalline state form imbricated packing, and non-classical ones with cross-sheet structures. The crystalline-nematic phase transition was called displacive. The displacive type of transition involves comparatively limited displacements of the molecules from the positions which they occupy with respect to their nearest neighbours in the crystal. In most cases, smectic precursors form layered structures. The crystalline-smectic phase transition was called reconstitutive because the molecular arrangement in the crystalline state must alter in a more pronounced fashion in order to achieve the mesophase arrangement [12]. 

Integration of the phase density over classical phase space corresponds to finding the trace of the density matrix in quantum mechanics. Transition to a new basis is achieved by unitary transformation... 

The infinite potential barrier, shown schematically in figure 10 corresponds to a superselection rule that operates below the critical temperature [133]. Above the critical temperature the quantum-mechanical superposition principle applies, but below that temperature the system behaves classically. The system bifurcates spontaneously at the critical point. The bifurcation, like second-order phase transformation is caused by some interaction that becomes dominant at that point. In the case of chemical reactions the interaction leads to the rearrangement of chemical bonds. The essential difference between chemical reaction and second-order phase transition is therefore epitomized by the formation of chemically different species rather than different states of aggregation, when the symmetry is spontaneously broken at a critical point. 

The finite-temperature field theory has been the most popular approach to equilibrium phase transitions (L. Dolan et.al., 1974). The effective potential of quantum fluctuations around a classical background provides a convenient tool to describe phase transitions. The symmetry breaking or restoration mechanism can be illustrated by a scalar field model with broken symmetry... 

Before the phase transition (t < 0), the measures for decoherence and classical correlation are exactly found... 

Therefore, we conclude that the long wavelength mode neither decoheres nor is classically correlated before the phase transition. However, after the phase transitions, the unstable long wavelength mode becomes classical, gaining both quantum decoherence and classical correlation. Thus an order parameter appears from long wavelength modes (S.P. Kim et.al., 2000 2002 2001). 

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