arXiv subjects

Robert H. Swendsen

Publications and source records attributed to Robert H. Swendsen.

18 recordsLinked to original sources

arXiv2017

Resolving the debate about proposed expressions for the classical entropy

Despite well over a century of effort, the proper expression for the classical entropy in statistical mechanics remains a subject of debate. The Boltzmann entropy (calculated from a surface in phase space) has been criticized as not being an adiabatic invariant. It has been suggested that the Gibbs entropy (volume in phase space) is correct, which would forbid the concept of negative temperatures. An apparently innocuous assumption turns out to be responsible for much of the controversy, namely, that the energy $E$ and the number of particles $N$ are given exactly. The true distributions are known to be extremely narrow (of order $1/\sqrt{N}$), so that it is surprising that this is a problem. The canonical and grand canonical ensembles provide alternative expressions for the entropy that satisfy all requirements. The consequences are that negative temperatures are thermodynamically valid, the validity of the Gibbs entropy is limited to increasing densities of states, and the completely correct expression for the entropy is given by the grand canonical formulation. The Boltzmann entropy is shown to provide an excellent approximation in almost all cases.

cond-mat.stat-mech↗

arXiv2017

The surprising convergence of the Monte Carlo renormalization group for the d=3 Ising Model

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

cond-mat.stat-mech↗

arXiv2017

Detecting Multi-Spin Interactions in the Inverse Ising Problem

While the usual goal in Monte Carlo (MC) simulations of Ising models is the efficient generation of spin configurations with Boltzmann probabilities, the inverse problem is to determine the coupling constants from a given set of spin configurations. Most recent work has been limited to local magnetic fields and pair-wise interactions. We have extended solutions to multi-spin interactions, using correlation function matching (CFM). A more serious limitation of previous work has been the uncertainty of whether a chosen set of interactions is capable of faithfully representing real data. We show how our confirmation testing method uses an additional MC simulation to detect significant interactions that might be missing in the assumed representation of the data.

cond-mat.dis-nn↗

arXiv2016

Finite thermal reservoirs and the canonical distribution

The microcanonical ensemble has long been a starting point for the development of thermodynamics from statistical mechanics. However, this approach presents two problems. First, it predicts that the entropy is only defined on a discrete set of energies for finite, quantum systems, while thermodynamics requires the entropy to be a continuous function of the energy. Second, it fails to satisfy the stability condition ($ΔS / ΔU < 0$) for first-order transitions with both classical and quantum systems. Swendsen has recently shown that the source of these problems lies in the microcanonical ensemble itself, which contains only energy eigenstates and excludes their linear combinations. To the contrary, if the system of interest has ever been in thermal contact with another system, it will be described by a probability distribution over many eigenstates that is equivalent to the canonical ensemble for sufficiently large systems. Novotny et al. have recently supported this picture by dynamical numerical calculations for a quantum mechanical model, in which they showed the approach to a canonical distribution for up to 40 quantum spins. By simplifying the problem to calculate only the equilibrium properties, we are able to extend the demonstration to more than a million particles.

cond-mat.stat-mech↗

arXiv2016

The definition of the thermodynamic entropy in statistical mechanics

A definition of the thermodynamic entropy based on the time-dependent probability distribution of the macroscopic variables is developed. When a constraint in a composite system is released, the probability distribution for the new equilibrium values goes to a narrow peak. Defining the entropy by the logarithm of the probability distribution automatically makes it a maximum at the equilibrium values, so it satisfies the Second Law. It is also satisfies the postulates of thermodynamics. Objections to this definition by Dieks and Peters are discussed and resolved.

cond-mat.stat-mech↗

arXiv2016

Comparison of canonical and microcanonical definitions of entropy

For more than 100 years, one of the central concepts in statistical mechanics has been the microcanonical ensemble, which provides a way of calculating the thermodynamic entropy for a specified energy. A controversy has recently emerged between two distinct definitions of the entropy based on the microcanonical ensemble: (1) The Boltzmann entropy, defined by the density of states at a specified energy, and (2) The Gibbs entropy, defined by the sum or integral of the density of states below a specified energy. A critical difference between the consequences of these definitions pertains to the concept of negative temperatures, which by the Gibbs definition, cannot exist. In this paper, we call into question the fundamental assumption that the microcanonical ensemble should be used to define the entropy. Our argument is based on a recently proposed canonical definition of the entropy as a function of energy. We investigate the predictions of the Boltzmann, Gibbs, and canonical definitions for a variety of classical and quantum models, including models which exhibit a first-order phase transition. Our results support the validity of the concept of negative temperature, but not for all models with a decreasing density of states. We find that only the canonical entropy consistently predicts the correct thermodynamic properties, while microcanonical definitions of entropy, including those of Boltzmann and Gibbs, are correct only for a limited set of simple models.

cond-mat.stat-mech↗

arXiv2015

Negative temperatures and the definition of entropy

The concept of negative temperature has recently received renewed interest in the context of debates about the correct definition of the thermodynamic entropy in statistical mechanics. Several researchers have identified the thermodynamic entropy exclusively with the "volume entropy'' suggested by Gibbs, and have further concluded that by this definition, negative temperatures violate the principles of thermodynamics. We disagree with these conclusions. We demonstrate that volume entropy is inconsistent with the postulates of thermodynamics for systems with non-monotonic energy densities, while a definition of entropy based on the probability distributions of macroscopic variables does satisfy the postulates of thermodynamics. Our results confirm that negative temperature is a valid extension of thermodynamics.

cond-mat.stat-mech↗

arXiv2015

Magnetic ground state of semiconducting transition metal trichalcogenide monolayers

Layered transition metal trichalcogenides with the chemical formula $ABX_3$ have attracted recent interest as potential candidates for two-dimensional magnets. Using first-principles calculations within density functional theory, we investigate the magnetic ground states of monolayers of Mn- and Cr-based semiconducting trichalcogenides. We show that the second and third nearest-neighbor exchange interactions ($J_2$ and $J_3$) between magnetic ions, which have been largely overlooked in previous theoretical studies, are crucial in determining the magnetic ground state. Specifically, we find that monolayer $\text{CrSiTe}_3$ is an antiferromagnet with a zigzag spin texture due to significant contribution from $J_3$, whereas $\text{CrGeTe}_3$ is a ferromagnet with a Curie temperature of 106 K. Monolayers of Mn-compounds ($\text{MnPS}_3$ and $\text{MnPSe}_3$) always show antiferromagnetic Neel order. We identify the physical origin of various exchange interactions, and demonstrate that strain can be an effective knob for tuning the magnetic properties. Possible magnetic ordering in the bulk is also discussed. Our study suggests that $\text{ABX}_3$ can be a promising platform to explore 2D magnetic phenomena.

cond-mat.mtrl-sci↗

arXiv2015

The thermodynamic entropy of a macroscopic quantum system is a continuous function of energy

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 also 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 also demonstrates the consistency of the concept of negative temperature with the principles of thermodynamics.

cond-mat.stat-mech↗

arXiv2015

The Gibbs "volume" entropy is incorrect

In recent papers, several authors have claimed that a definition of the thermodynamic entropy in terms of the logarithm of a volume in phase space, originally suggested by Gibbs, is the only valid definition. Arguing from the Gibbs entropy, these authors claim that thermodynamics cannot be extended to include negative temperatures. However, the Gibbs entropy fails to satisfy the postulates of thermodynamics, leading to serious errors. In particular, predictions of the Gibbs entropy for systems with non-monotonic energy densities are incorrect. We show that the correct expression for the equilibrium entropy contains an integral over a surface in phase space, and negative temperature is a valid thermodynamic concept.

cond-mat.stat-mech↗

arXiv2006

Comparison of free energy methods for molecular systems

We present a detailed comparison of computational efficiency and precision for several free energy difference ($ΔF$) methods. The analysis includes both equilibrium and non-equilibrium approaches, and distinguishes between uni-directional and bi-directional 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 $ΔF$ estimates as compared to other methods used in this report, while also offering rapid estimation of $ΔF$. 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 uni-directional, ``fast-growth'' non-equilibrium 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.

physics.bio-ph↗

arXiv2004

Replica Monte Carlo Simulation (Revisited)

In 1986, Swendsen and Wang proposed a replica Monte Carlo algorithm for spin glasses [Phys. Rev. Lett. 57 (1986) 2607]. Two important ingredients are present, (1) the use of a collection of systems (replicas) at different of temperatures, but with the same random couplings, (2) defining and flipping clusters. Exchange of information between the systems is facilitated by fixing the tau spin (tau=sigma^1σ^2) and flipping the two neighboring systems simultaneously. In this talk, we discuss this algorithm and its relationship to replica exchange (also known as parallel tempering) and Houdayer's cluster algorithm for spin glasses. We review some of the early results obtained using this algorithm. We also present new results for the correlation times of replica Monte Carlo dynamics in two and three dimensions and compare them with replica exchange.

cond-mat.stat-mech↗

arXiv2002

Comments on 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 prepresentation. 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 (cond-mat/0201285), 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 two-dimensional three-state Potts model, their dynamical critical exponent z is 0.49(1); the three-dimensional Ising model has z = 0.37(2).

cond-mat.stat-mech↗

arXiv2002

Binary tree summation Monte Carlo method for Potts models

We give a new sampling algorithm for the Potts model based on the Fortuin-Kasteleyn transformation. The method produces independent samples and sums up a large number of configurations for each sweep. The partition function and thermodynamic averages for all values of the temperature can be computed from a single run. We compare the results with exact 2D Ising model.

cond-mat.stat-mech↗

arXiv2001

Transition Matrix Monte Carlo Method

We present a formalism of the transition matrix Monte Carlo method. A stochastic matrix in the space of energy can be estimated from Monte Carlo simulation. This matrix is used to compute the density of states, as well as to construct multi-canonical and equal-hit algorithms. We discuss the performance of the methods. The results are compared with single histogram method, multi-canonical method, and other methods. In many aspects, the present method is an improvement over the previous methods. PACS numbers: 02.70.Tt, 05.10.Ln, 05.50.+q. Keywords: Monte Carlo method, flat histogram, multi-canonical ensemble.

cond-mat.stat-mech↗

arXiv1999

Transition Matrix Monte Carlo

Although histogram methods have been extremely effective for analyzing data from Monte Carlo simulations, they do have certain limitations, including the range over which they are valid and the difficulties of combining data from independent simulations. In this paper, we describe an complementary approach to extracting information from Monte Carlo simulations that uses the matrix of transition probabilities. Combining the Transition Matrix with an N-fold way simulation technique produces an extremely flexible and efficient approach to rather general Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv1998

Transition Matrix Monte Carlo Reweighting and Dynamics

We study an induced dynamics in the space of energy of single-spin-flip Monte Carlo algorithm. The method gives an efficient reweighting technique. This dynamics is shown to have relaxation times proportional to the specific heat. Thus, it is plausible for a logarithmic factor in the correlation time of the standard 2D Ising local dynamics.

cond-mat.stat-mech↗

arXiv1995

Rotationally symmetric ordered phase in the three-state antiferromagnetic Potts model

We investigate the three-state antiferromagnetic Potts model on a simple cubic lattice with a cluster flipping Monte Carlo simulation algorithm in the temperature region below the transition into disorder at T_{c1}. We find both the well established broken-sublattice-symmetry (BSS) phase at low temperature and a new, rotationally symmetric phase at higher temperature, but below T_{c1}. The properties of the second phase and the transition temperature to the BSS phase are in disagreement with recent cluster-variation and Monte Carlo simulation results, but in agreement with simulations by Kolesik and Suzuki.

cond-mat↗