Finite temperature dynamical fluctuations

At finite temperature, stochastic fluctuations of the membrane due to thermal motion affect the dynamics of vesicles. Since the calculation of thermal fluctuations under flow conditions requires long times and large membrane sizes (in order to have a sufficient range of undulation wave vectors), simulations have been performed for a two-dimensional system in the stationary tank-treading state [213]. For comparison, in the limit of small deviations from a circle, Langevin-type equations of motion have been derived, which are highly nonlinear due to the constraint of constant perimeter length [213]. 

The impact of this is tremendous. No long-range order (LRO) can exist at finite temperature in one dimension no crystals, no magnets, no superconductors. Only special transitions are possible in two dimensions. The Ising model (n = 1 component) is an example [7]. The Kosterlitz-Thouless transition [8], without LRO, is another case for d = 2 and n = 2, discussed in Section V.C. The thermal fluctuations are very destructive in lower dimensions. Quantum fluctuations (i.e., those associated with the dynamics of a system) also tend to suppress LRO and can sometimes destroy it even at 0 K when the Mermin-Wagner theorem does not apply. Such is the case of the quantum spin- antiferromagnetic models [9] in one dimension. 

In the second part, we have presented a unified, comprehensive anlysis of dynamically suppressed decay and decoherence in driven TLS coupled to finite-temperature baths and undergoing random frequency fluctuations. This treatment has resulted in both principal and practical general conclusions ... 

Future applications by the VPIMD method include vibrational fluctuations of molecular clusters such as hydrogen bonded clusters. Molecular clusters characterized by weak intermolecular interactions are expected to have large anharmonicity of the potential energy surfaces. As demonstrated in the present study, the VPIMD method properly handles the anharmonicity including the case of multiple minima. Another important point is on the description of the adiabatic potential energy surfaces of molecular clusters. An improvement can be achieved by combining the VPIMD method with electronic structure calculations as in the case of the finite temperature path integral molecular dynamics [30-32], These issues will be addressed in the near future. 

This is the conceptual framework of the present technique. It is a completely general ab initio VCS-MD which can be used to simulate solids at finite temperatures and pressures. Subjected to small modifications, and in the limit of isotropic fluctuations, it maps into Andersen s dynamics[17], whose averages over trajectories correspond to those of the isoenthalpic isobaric (N,P,H) ensemble. 

The natural way to study the time dependence of Newtonian mechanics is typically based on molecular dynamics methods which, however, suffer from severe problems to ensure the correct statistical sampling at finite temperatures by using thermostats [75,76]. From a more formal point of view, it is even questionable what dynamics shall mean in a thermal system, where even under the same thermodynamic conditions trajectories typically mn differently, due to the random thermal fluctuations caused by interactions with the huge number [0(10 ) per mole] of realistically not traceable heat-bath particles. 

The simulation is performed in a grand canonical ensemble (GCE) where all microstates have the same volume (V), temperature and chemical potential under the periodic boundary condition to minimize a finite size effect [30, 31]. For thermal equilibrium at a fixed pu, a standard Metropolis algorithm is repetitively employed with single spin-flip dynamics [30, 31]. When equilibrium has been achieved, the lithium content (1 — 5) in the Li, 3 11204 electrode at a given pu is determined from the fraction of occupied sites. The thermodynamic partial molar quantities oflithium ions are theoretically obtained by fluctuation method [32]. The partial molar internal energy Uu at constant Vand T in the GCE is readily given by [32, 33]... 

Close to the phase transitions point (such as the weak first-order isotropic to nematic transition or the second-order nematic to smectic transition), the order of the low-temperature phase occurs transiently in the finite small size of fluctuations, which is called cybotactic clusters. When the correlation length of the fluctuation increases towards the transition point, the relaxation time diverges. For example, near the nematic-smectic phase transition point, it can be seen that the correlation length becomes longer towards the transition point by X-ray diffraction experiments (Fig. 10.21). In addition, the increase of the relaxation time of the fluctuations can be measured using dynamic light scattering method, which wiU be introduced in detail in Sect. 10.4.3.2. 

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