Monte Carlo renormalization-group calculation for the d=3 Ising model using a modified transformation.

We present a simple approach to high-accuracy calculations of critical properties for the three-dimensional Ising model, without prior knowledge of the critical temperature. The iterative method uses a modified block-spin transformation with a tunable parameter to improve convergence in the Monte Carlo renormalization group trajectory. We found experimentally that the iterative method enables the calculation of the critical temperature simultaneously with a critical exponent.

Thermodynamics of finite systems: a key issues review.

A little over ten years ago, Campisi, and Dunkel and Hilbert, published papers claiming that the Gibbs (volume) entropy of a classical system was correct, and that the Boltzmann (surface) entropy was not. They claimed further that the quantum version of the Gibbs entropy was also correct, and that the phenomenon of negative temperatures was thermodynamically inconsistent. Their work began a vigorous debate of exactly how the entropy, both classical and quantum, should be defined. The debate has called into question the basis of thermodynamics, along with fundamental ideas such as whether heat always flows from hot to cold. The purpose of this paper is to sum up the present status-admittedly from my point of view. I will show that standard thermodynamics, with some minor generalizations, is correct, and the alternative thermodynamics suggested by Hilbert, Hänggi, and Dunkel is not. Heat does not flow from cold to hot. Negative temperatures are thermodynamically consistent. The small 'errors' in the Boltzmann entropy that started the whole debate are shown to be a consequence of the micro-canonical assumption of an energy distribution of zero width. Improved expressions for the entropy are found when this assumption is abandoned.

Probability, Entropy, and Gibbs' Paradox(es).

Two distinct puzzles, which are both known as Gibbs' paradox, have interested physicists since they were first identified in the 1870s. They each have significance for the foundations of statistical mechanics and have led to lively discussions with a wide variety of suggested resolutions. Most proposed resolutions had involved quantum mechanics, although the original puzzles were entirely classical and were posed before quantum mechanics was invented. In this paper, I show that contrary to what has often been suggested, quantum mechanics is not essential for resolving the paradoxes. I present a resolution of the paradoxes that does not depend on quantum mechanics and includes the case of colloidal solutions, for which quantum mechanics is not relevant.

Surprising convergence of the Monte Carlo renormalization group for the three-dimensional Ising model.

We present a surprisingly simple approach to high-accuracy calculations of the critical properties of the three-dimensional Ising model. The method uses a modified block-spin transformation with a tunable parameter to improve convergence in the Monte Carlo renormalization group. The block-spin parameter must be tuned differently for different exponents to produce optimal convergence.

Continuity of the entropy of macroscopic quantum systems.

The proper definition of entropy is fundamental to the relationship between statistical mechanics and thermodynamics. It also plays a major role in the recent debate about the validity of the concept of negative temperature. In this paper, I analyze and calculate the thermodynamic entropy for large but finite quantum mechanical systems. A special feature of this analysis is that the thermodynamic energy of a quantum system is shown to be a continuous variable, rather than being associated with discrete energy eigenvalues. Calculations of the entropy as a function of energy can be carried out with a Legendre transform of thermodynamic potentials obtained from a canonical ensemble. The resultant expressions for the entropy are able to describe equilibrium between quantum systems having incommensurate energy-level spacings. This definition of entropy preserves all required thermodynamic properties, including satisfaction of all postulates and laws of thermodynamics. It demonstrates the consistency of the concept of negative temperature with the principles of thermodynamics.

Gibbs volume entropy is incorrect.

We show that the expression for the equilibrium thermodynamic entropy contains an integral over a surface in phase space, and in so doing, we confirm that negative temperature is a valid thermodynamic concept. This demonstration disproves the claims of several recent papers that the Gibbs entropy, which contains an integral over a volume in phase space, is the correct definition and that thermodynamics cannot be extended to include negative temperatures. We further show that the Gibbs entropy fails to satisfy the postulates of thermodynamics and that its predictions for systems with nonmonotonic energy densities of states are incorrect.

Efficiency and time-dependent cross correlations in multivariable Monte Carlo updating.

We show that any Monte Carlo (MC) algorithm using joint updates of more than a single variable at each step produces time-shifted correlations between variables-even if the equilibrium probabilities of the variables are independent. These spurious time-shifted correlations will affect both the magnitudes of correlation times and the values of optimal acceptance ratios. In particular, correlation times computed with local variables will not generally give the same predictions of efficiency or optimal acceptance ratios as those computed with global variables. Gelman, Roberts, and Gilks [Bayesian Statistics, edited by J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. R. M. Smith, Vol. 5 (Oxford University Press, 1996), pp. 599-607] have used a local measure of efficiency to prove a theorem for global MC updating that predicts an optimal acceptance ratio of 0.234 as the number of variables goes to infinity. We show that global measures of efficiency can produce different, and arguably more appropriate, optimal acceptance ratios. More importantly, global updating is inherently far less efficient than updating variables separately or in small groups. We suggest that previously determined optimal acceptance ratios and their implications for practical applications should be reconsidered in the context of these findings.

Monte Carlo renormalization-group analysis of percolation.

We describe a Monte Carlo renormalization group approach to the calculation of critical behavior for percolation models. This approach can be utilized to determine the renormalized bond probabilities and the values of the critical exponents. We illustrate the method for two-dimensional bond percolation, but the method is also applicable to other percolation models and other dimensions.

A model of motor performance during surface penetration: from physics to voluntary control.

The act of puncturing a surface with a hand-held tool is a ubiquitous but complex motor behavior that requires precise force control to avoid potentially severe consequences. We present a detailed model of puncture over a time course of approximately 1,000 ms, which is fit to kinematic data from individual punctures, obtained via a simulation with high-fidelity force feedback. The model describes puncture as proceeding from purely physically determined interactions between the surface and tool, through decline of force due to biomechanical viscosity, to cortically mediated voluntary control. When fit to the data, it yields parameters for the inertial mass of the tool/person coupling, time characteristic of force decline, onset of active braking, stopping time and distance, and late oscillatory behavior, all of which the analysis relates to physical variables manipulated in the simulation. While the present data characterize distinct phases of motor performance in a group of healthy young adults, the approach could potentially be extended to quantify the performance of individuals from other populations, e.g., with sensory-motor impairments. Applications to surgical force control devices are also considered.

Comparison of free energy methods for molecular systems.

We present a detailed comparison of computational efficiency and precision for several free energy difference (DeltaF) methods. The analysis includes both equilibrium and nonequilibrium approaches, and distinguishes between unidirectional and bidirectional methodologies. We are primarily interested in comparing two recently proposed approaches, adaptive integration, and single-ensemble path sampling to more established methodologies. As test cases, we study relative solvation free energies of large changes to the size or charge of a Lennard-Jones particle in explicit water. The results show that, for the systems used in this study, both adaptive integration and path sampling offer unique advantages over the more traditional approaches. Specifically, adaptive integration is found to provide very precise long-simulation DeltaF estimates as compared to other methods used in this report, while also offering rapid estimation of DeltaF. The results demonstrate that the adaptive integration approach is the best overall method for the systems studied here. The single-ensemble path sampling approach is found to be superior to ordinary Jarzynski averaging for the unidirectional, "fast-growth" nonequilibrium case. Closer examination of the path sampling approach on a two-dimensional system suggests it may be the overall method of choice when conformational sampling barriers are high. However, it appears that the free energy landscapes for the systems used in this study have rather modest configurational sampling barriers.

Adaptive integration method for Monte Carlo simulations.

We present an adaptive sampling method for computing free energies, radial distribution functions, and potentials of mean force. The method is characterized by simplicity and accuracy, with the added advantage that the data are obtained in terms of quasicontinuous functions. The method is illustrated and tested with simulations on a high density fluid, including a stringent consistency test involving an unusual thermodynamic cycle that highlights its advantages.

Feeling textures through a probe: effects of probe and surface geometry and exploratory factors.

Vibratory roughness perception occurs when people feel a surface with a rigid probe. Accordingly, perceived roughness should reflect probe and surface geometry, exploratory speed, and force. Experiments 1 and 2 compared magnitude estimation of roughness with the bare finger and two types of probes, one designed to eliminate force moments, under the subject's active control. Experiments 3 and 4 varied speed under passive control. Log magnitude was consistently a quadratic function of log spacing between elements in the surface. The location of the function's peak was related to the drop point--that is, the spacing at which the probe can just drop between elements--which is affected by probe tip diameter, element height, and speed. Other parameters of the quadratic were affected by probe type and speed.

Inverse Monte Carlo renormalization group transformations for critical phenomena.

We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two- and three-dimensional Ising models for a variety of renormalization group transformations.

Importance of multispin couplings in renormalized Hamiltonians.

We introduce a Monte Carlo approach to the calculation of more distant renormalized interactions with higher accuracy than is possible with previous methods. We have applied our method to study the effects of multispin interactions, which turn out to be far more important than commonly assumed. Even though the individual multispin interactions usually have smaller coupling constants than two-spin interactions, they can dominate the effects of two-spin interactions because they are so numerous.

Sweeny and Gliozzi dynamics for simulations of Potts models in the Fortuin-Kasteleyn representation.

We compare the correlation times of the Sweeny and Gliozzi dynamics for two-dimensional Ising and three-state Potts models, and the three-dimensional Ising model for the simulations in the percolation representation. The results are also compared with Swendsen-Wang and Wolff cluster dynamics. It is found that Sweeny and Gliozzi dynamics have essentially the same dynamical critical behavior. Contrary to Gliozzi's claim [Phys. Rev. E 66, 016115 (2002)], the Gliozzi dynamics has critical slowing down comparable to that of other cluster methods. For the two-dimensional Ising model, both Sweeny and Gliozzi dynamics give good fits to logarithmic size dependences for the correlation times; for two-dimensional three-state Potts model, their dynamical critical exponent z is 0.49+/-0.01; the three-dimensional Ising model has z=0.37+/-0.02.