There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The Weighted Histogram Analysis Method (WHAM), an extension of Ferrenberg and Swendsen's Multiple Histogram Technique, has been applied for the first time on complex biomolecular Hamiltonians. The method is presented here as an extension of the Umbrella Sampling method for free-energy and Potential of Mean Force calculations. This algorithm possesses the following advantages over methods that are currently employed: (1) It provides a built-in estimate of sampling errors thereby yielding objective estimates of the optimal location and length of additional simulations needed to achieve a desired level of precision; (2) it yields the “best” value of free energies by taking into account all the simulations so as to minimize the statistical errors; (3) in addition to optimizing the links between simulations, it also allows multiple overlaps of probability distributions for obtaining better estimates of the free-energy differences. By recasting the Ferrenberg–Swendsen Multiple Histogram equations in a form suitable for molecular mechanics type Hamiltonians, we have demonstrated the feasibility and robustness of this method by applying it to a test problem of the generation of the Potential of Mean Force profile of the pseudorotation phase angle of the sugar ring in deoxyadenosine. © 1992 by John Wiley & Sons, Inc.

The weighted histogram analysis method for free-energy calculations on biomolecules. I: The method

Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling

http://experimentationlab.berkeley.edu/sites/default/files/AFMImages/butt_AFM.pdf

Force measurements with the atomic force microscope: Technique, interpretation and applications

Implicit Solvation Models: Equilibria, Structure, Spectra, and Dynamics.

A Lennard‐Jones fluid confined between two planar hard walls is simulated using grand canonical Monte Carlo, and capillary evaporation is found for liquid subcritical bulk states. General methods are given for simulating a metastable fluid beyond coexistence. For the systems studied, the liquid and the gas phases coexist in equilibrium at a separation of ∼5 diam, the spinodal cavitation separation is at ∼4 diam, and the spinodal condensation separation is at ≳15 diam. The interaction pressure between the walls is found to be attractive and increases rapidly as the spinodal separation is approached. On the equilibrium liquid branch, the net pressure still appears significantly larger than the van der Waals attraction at separations of ∼10 diam. A simple analytic theory is given, which relates the force to the approach of the separation‐induced phase transition. It is suggested that this is the microscopic origin of the measured attractions between hydrophobic surfaces in water.

Cavitation of a Lennard‐Jones fluid between hard walls, and the possible relevance to the attraction measured between hydrophobic surfaces

The structure of a model colloidal ferrofluid, the dipolar hard-sphere fluid, at low temperature has been investigated using Monte Carlo simulations. Extensive particle association into chainlike and ringlike clusters is observed at low density. The structure factors have been calculated, and are analyzed with the aid of simple scaling arguments. We describe the progression of fluid structures from the low-density associated phase, to the high-density liquid phase. This paper may be of help in obtaining an experimental observation of a fluid-fluid transition in colloidal ferrofluids.

Structure and scattering in colloidal ferrofluids

In this paper we examine the dielectric and structural properties of hard polarizable multipolar models for liquid water. The theoretical results were obtained by solving the self-consistent mean field (SCMF) approximation together with the reference hypernetted-chain (RHNC) theory. The dielectric constants are in good agreement with experiment over a large range of temperatures and pressures. The rather poor agreement between the radial distribution functions determined for our water-like fluids at 25°C and those measured for liquid water is discussed.

The solution of the reference hypernetted-chain approximation for water-like models

Molecular-dynamics simulation results are reported for systems of strongly interacting dipolar soft spheres. Calculations have been carried out along two isotherms and the structure of the liquid-crystal and solid phases obtained is described in detail. It is found that in addition to the ferroelectric nematic phase we previously reported [Phys. Rev. Lett. 68, 2043 (1992)], liquid crystals with columnar order can also be obtained. The model freezes to form a ferroelectric solid which is shown to have a tetragonal I crystal structure. The influence of different boundary conditions upon the simulation results is also discussed.

Ferroelectric liquid-crystal and solid phases formed by strongly interacting dipolar soft spheres

Methods of calculating the first two terms in the density expansion of the bridge function are given. The closure to the Ornstein–Zernike equation is now exact to two orders in density beyond the hypernetted‐chain or Percus–Yevick approximations. The bridge function is resummed as a Pade approximant, and the results for hard spheres are relatively accurate over the whole density regime. The closure is shown to yield physically reasonable results for highly asymmetric mixtures. Infinitely dilute hard sphere solutes with diameters up to 30 times that of the hard‐sphere solvent are also considered. The Derjaguin approximation for rescaling the force between spheres to the interaction free energy between planes is examined and found to give the dominant curvature correction.

G. N. Patey

Papers

Cavitation of a Lennard‐Jones fluid between hard walls, and the possible relevance to the attraction measured between hydrophobic surfaces

Structure and scattering in colloidal ferrofluids

The solution of the reference hypernetted-chain approximation for water-like models

Ferroelectric liquid-crystal and solid phases formed by strongly interacting dipolar soft spheres

Hypernetted‐chain closure with bridge diagrams. Asymmetric hard sphere mixtures