Quantum Statistical Mechanics in Classical Phase Space. III. Mean Field Approximation Benchmarked for Interacting Lennard-Jones Particles

A Monte Carlo computer simulation algorithm in classical phase space is given for the treatment of quantum systems. The non-commutativity of position and momentum is accounted for by a mean field approach and instantaneous effective harmonic oscillators. Wave function symmetrization is included at the dimer and double dimer level. Quantitative tests are performed against benchmarks given by Hernando and Van\'i\v{c}ek (2013) for spinless neon--parahydrogen, modeled as interacting Lennard-Jones particles in a one dimensional harmonic trap. The mean field approach is shown to be quantitatively accurate for high to moderate temperatures $\beta \hbar \omega_\mathrm{LJ} < 7$, and moderate densities, $\rho \sigma \approx 1$. Results for helium show that at the lowest temperature studied, the average energy is about 4\% lower for bosons than for fermions. It is argued that the mean field algorithm will perform better in three dimensions than in one, and that it will scale sub-linearly with system size.Comment: 9 pages, 7 figures, 23 equations, 18 reference

Expansion for Quantum Statistical Mechanics Based on Wave Function Symmetrization

An expansion for quantum statistical mechanics is derived that gives classical statistical mechanics as the leading term. Each quantum correction comes from successively larger permutation loops, which arise from the factorization of the symmetrization of the wave function with respect to localized particle interchange. Explicit application of the theory yields the full fugacity expansion for the quantum ideal gas, and the second fugacity coefficient for interacting quantum particles, which agree with known results. Compared to the Lee-Yang virial cluster expansion, the present expansion is expected to be more rapidly converging and the individual terms appear to be simpler to evaluate. The results obtained in this paper are intended for practical computer simulation algorithms for terrestrial condensed matter quantum systems.Comment: Quantum Physics. 40 pages. Version 2 clarifies Sec. IIB and App.

Quantum Monte Carlo in Classical Phase Space. Mean Field and Exact Results for a One Dimensional Harmonic Crystal

Monte Carlo simulations are performed in classical phase space for a one-dimensional quantum harmonic crystal. Symmetrization effects for spinless bosons and fermions are quantified. The algorithm is tested for a range of parameters against exact results that use 20,000 energy levels. It is shown that the singlet mean field approximation is very accurate at high temperatures, and that the pair mean field approximation gives a systematic improvement in the intermediate and low temperature regime. The latter is derived from a cluster mean field approximation that accounts for the non-commutativity of position and momentum, and that can be applied in three dimensions.Comment: 10 pages, 5 figs, 4 sections. Total 1

Quantum Statistical Mechanics. I. Decoherence, Wave Function Collapse, and the von Neumann Density Matrix

The probability operator is derived from first principles for an equilibrium quantum system. It is also shown that the superposition states collapse into a mixture of states giving the conventional von Neumann trace form for the quantum average. The mechanism for the collapse is found to be quite general: it results from the conservation law for a conserved, exchangeable variable (such as energy) and the entanglement of the total system wave function that necessarily follows. The relevance of the present results to the einselection mechanism for decoherence, to the quantum measurement problem, and to the classical nature of the macroscopic world are discussed.Comment: 12 page

Quantum Statistical Mechanics. III. Equilibrium Probability

Given are a first principles derivation and formulation of the probabilistic concepts that underly equilibrium quantum statistical mechanics. The transition to non-equilibrium probability is traversed briefly.Comment: 22 page

Quantum Statistical Mechanics in Classical Phase Space. Expressions for the Multi-Particle Density, the Average Energy, and the Virial Pressure

Quantum statistical mechanics is formulated as an integral over classical phase space. Some details of the commutation function for averages are discussed, as is the factorization of the symmetrization function used for the grand potential and for the multi-particle density. Three binary choices (eight routes) for the average energy are shown to be mutually consistent. An expression for the phase space function that gives the average virial pressure is derived.Comment: 10 page

How to Measure Forces when the Atomic Force Microscope shows Non-Linear Compliance

A spreadsheet algorithm is given for the atomic force microscope that accounts for non-linear behavior in the deflection of the cantilever and in the photo-diode response. In addition, the data analysis algorithm takes into account cantilever tilt, friction in contact, and base-line artifacts such as drift, virtual deflection, and non-zero force. These are important for accurate force measurement and also for calibration of the cantilever spring constant. The zero of separation is determined automatically, avoiding human intervention or bias. The method is illustrated by analyzing measured data for the silica-silica drainage force and slip length.Comment: 20 pages, 11 figure

Quantum Statistical Mechanics in Classical Phase Space. V. Quantum Local, Average Global

One-particle energy eigenfunctions are used to obtain quantum averages in many particle systems. These are based on the effective local field due to fixed neighbors in classical phase space, while the averages account for the non-commutativity of the position and momentum operators. Used in Monte Carlo simulations for a one-dimensional Lennard-Jones fluid, the results prove more reliable than a high temperature expansion and a harmonic local field approach, and at intermediate temperatures agree with benchmark numerical results. Results are presented for distinguishable particles, fermions, and bosons.Comment: 12 pages, 6 figures, 3 appendece

Design of Chemotaxis Devices Using Nano-Motors

Several designs for micro-devices for chemotaxis based on nano-motors are proposed. The nano- or micro-motors are the conventional Janus rods or spheres that are powered by the catalytic reaction of fuels such as hydrogen peroxide. It is shown how these can be linked to make a device that can follow a concentration gradient of the fuel. The feasibility of assembling the devices using micromanipulation or metallic deposition is discussed. A possible design principle is suggested for a device that follows the concentration gradient of an analyte other than the fuel.Comment: 4 pages, 8 figure

More Reliable Measurements of the Slip Length with the Atomic Force Microscope

Further improvements are made to the non-linear data analysis algorithm for the atomic force microscope [P. Attard, arXiv:1212.3019v2 (2012)]. The algorithm is required when there is curvature in the compliance region due to photo-diode non-linearity. Results are obtained for the hydrodynamic drainage force, for three surfaces: hydrophilic silica (symmetric, Si-Si), hydrophobic dichlorodimethylsilane (symmetric, DCDMS-DCDMS), and hydrophobic octadecyltrichlorosilane (asymmetric, Si-OTS). The drainage force was measured in the viscous liquid di-n-octylphthalate. The slip-lengths are found to be 3nm for Si, 2nm for DCDMS, and 2nm for OTS, with an uncertainty on the order of a nanometer. These slip lengths are a factor of 4--15 times smaller than those obtained from previous analysis of the same raw data [L. Zhu et al., Langmuir, 27, 6712 (2011). Ibid, 28, 7768 (2012)].Comment: 18 pages, 11 figures (improved discussion of cantilever drag