Abstract. This letter aims at justifying the stochastic equations in terms of the number density variable, which are still controversial, via complementing Dean's approach [Dean D S 1996 J. Phys. A 29 L613]. Our course is twofold: First, we demonstrate that standard manipulations straightforwardly transform the stochastic equation of density operator, derived by Dean, to the Fokker-Planck equation for the (c-number) density distribution functional P ({ρ}, t). Moreover, we verify the associated static solution of P ({ρ}, t) with the help of the conditional grand canonical partition function.
Bendy tubes: Lipid nanotubes can be aligned by a novel and simple method, whereby a fine line of a single nanotube is drawn freely by microinjecting the aqueous dispersion onto a glass plate. This technique is simple and highly adaptable; even words can be written (see picture)! The tubes can be bent and their Young's modulus measured by manipulation with optical tweezers.
Dielectric relaxation spectroscopy (1 Hz -20 GHz) has been performed on supercooled glassformers from the temperature of glass transition (Tg) up to that of melting. Precise measurements particularly in the frequencies of MHz-order have revealed that the temperature dependences of secondary β-relaxation times deviate from the Arrhenius relation in well above Tg. Consequently, our results indicate that the β-process merges into the primary α-mode around the melting temperature, and not at the dynamical transition point T ≈ 1.2Tg.PACS numbers: 64.70. Pf, 77.22.Gm, In recent years, much of the focus on glassy dynamics has been shifting to a considerably higher temperature than T g , the glass-transition one [1,2,3,4]. The topical temperature is located around T D ≡ 1.2T g , where the dynamics of supercooled liquids has been found to change fairly. So far, there have been observed the following phenomena:(i) Rössler scaling reveals that, when cooled, the Debye-Stokes-Einstein relation becomes invalid around T D [5,6]. This indicates a change of diffusion mechanism there.(ii) Stickel analysis [7] clarifies that temperature dependence of viscosity changes around T D . Therefore, in order to fit the primary α-relaxation time τ α using the Vogel-Fulcher-Tamman (VFT) relation,the coefficients (τ 0 , C, T 0 ) have to vary at T B ≈ T D . This suggests that the mechanism of slow structural-relaxation makes some alternation there.(iii) Johari-Goldstein type β-process (secondary process in the context of dielectric relaxation) [8,9,10,11,12,13] merges into the α-relaxation around T D , extrapolating the Arrhenius-type temperature dependence below T g of the β-relaxation times [6,14,15,16,17,18,19].Theoretically, the characteristic temperature T D is thought to be comparable to T C where the idealized Mode Coupling Theory (MCT) predicts a dynamical phase transition [20]. Indeed, the above first two phenomena (i, ii) can be regarded as indicators of the dynamical transition at T C . However, the MCT is irrelevant to the third one (iii), the bifurcation of α, β-modes; even the existence of the Johari-Goldstein type β-process cannot be derived.Furthermore, from experimental aspects, while the dynamical transition phenomena (i, ii) have been confirmed from either Rössler or Stickel plot definitely, the bifurcation of (iii) is inferred from the extrapolation. Actually, however, it remains an open problem as to whether the Arrhenius behavior of the β-process persists in higher temperatures near T D : the T D -decoupling of α, β-relaxations is not conclusive.This letter thus aims to investigate the secondary β-mode in high temperatures well above T g , by carrying out precisely the broad-band measurements of dielectric relaxation. Our main result is the following: as will be seen in Figures 2 and 5, the β-relaxation times deviate from the Arrhenius relation, indicating that the β-relaxation merges into the α-mode not at T D but around the Arrhenius-VFT crossover temperature T A (generally close to the melting one) where the temperatu...
More than half of a century has passed since the free energy of classical fluids defined by second Legendre transform was derived as a functional of density-density correlation function. It is now becoming an increasingly significant issue to develop the correlation functional theory that encompasses the liquid state theory, especially for glassy systems where out of equilibrium correlation fields are to be investigated. Here we have formulated a field theoretic perturbation theory that incorporates two-body fields (both of density-density correlation field and its dual field playing the role of two-body interaction potential) into a density functional integral representation of the Helmholtz free energy. Quadratic density fluctuations are only considered in the saddle-point approximation of two-body fields as well as density field. We have obtained a set of self-consistent field equations with respect to these fields, which simply reads a modified mean-field equation of density field where the bare interaction potential in the thermal energy unit is replaced by minus the direct correlation function given in the mean spherical approximation. Such replacement of the interaction potential in the mean-field equation belongs to the same category as the local molecular field theory proposed by Weeks and co-workers in a series of papers [e.g., J. M. Rodgers et al., Phys. Rev. Lett., 97, 097801 (2006); R. C. Remsing et al., P. Natl. Acad. Sci. USA, 113, 2819USA, 113, (2016]. Notably, it has been shown that even the mean-field part of the free energy functional given by the self-consistent field theory includes information on short-range correlations between fluid particles, similarly to the formulation of the local molecular field theory. The advantage of our field theoretic approach is not only that the modified mean-field equation can be improved systematically, but also that fluctuations of two-body fields in nonuniform fluids may be considered, which would be relevant especially for glass-forming liquids where heterogeneous out-of-equilibrium correlations are to be investigated.
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