Formulation Of Quantum Statistical Mechanics Research Articles

Quantum statistical mechanics from a Bohmian perspective

We develop a general formulation of quantum statistical mechanics in terms of probability currents that satisfy continuity equations in the multi-particle position space, for closed and open systems with a fixed number of particles. The continuity equation for any closed or open system suggests a natural Bohmian interpretation in terms of microscopic particle trajectories, that make the same measurable predictions as standard quantum theory. The microscopic trajectories are not directly observable, but provide a general, simple and intuitive microscopic interpretation of macroscopic phenomena in quantum statistical mechanics. In particular, we discuss how various notions of entropy, proper and improper mixtures, and thermodynamics are understood from the Bohmian perspective.

Quantum dynamics using path integral coarse-graining.

The vibrational spectra of condensed and gas-phase systems are influenced by thequantum-mechanical behavior of light nuclei. Full-dimensional simulations of approximate quantum dynamics are possible thanks to the imaginary time path-integral (PI) formulation of quantum statistical mechanics, albeit at a high computational cost which increases sharply with decreasing temperature. By leveraging advances in machine-learned coarse-graining, we develop a PI method with the reduced computational cost of a classical simulation. We also propose a simple temperature elevation scheme to significantly attenuate the artifacts of standard PI approaches as well as eliminate the unfavorable temperature scaling of the computational cost. We illustrate the approach, by calculating vibrational spectra using standard models of water molecules and bulk water, demonstrating significant computational savings and dramatically improved accuracy compared to more expensive reference approaches. Our simple, efficient, and accurate method has prospects for routine calculations of vibrational spectra for a wide range of molecular systems - with an explicit treatment of the quantum nature of nuclei.

Path-Integral Calculation of the Second Dielectric and Refractivity Virial Coefficients of Helium, Neon, and Argon

We present a method to calculate dielectric and refractivity virial coefficients using the path-integral Monte Carlo formulation of quantum statistical mechanics and validate it by comparing our results with equivalent calculations in the literature and with more traditional quantum calculations based on wavefunctions. We use state-of-the-art pair potentials and polarizabilities to calculate the second dielectric and refractivity virial coefficients of helium (both 3He and 4He), neon (both 20Ne and 22Ne), and argon. Our calculations extend to temperatures as low as 1 K for helium, 4 K for neon, and 50 K for argon. We estimate the contributions to the uncertainty of the calculated dielectric virial coefficients for helium and argon, finding that the uncertainty of the pair polarizability is by far the greatest contribution. Agreement with the limited experimental data available is generally good, but our results have smaller uncertainties, especially for helium. Our approach can be generalized in a straightforward manner to higher-order coefficients.

Importance of a Fully Anharmonic Treatment of Equilibrium Isotope Fractionation Properties of Dissolved Ionic Species As Evidenced by Li+(aq)

Equilibrium fractionation of stable isotopes is critically important in fields ranging from chemistry, including medicinal chemistry, electrochemistry, geochemistry, and nuclear chemistry, to environmental science. The dearth of reliable estimates of equilibrium fractionation factors, from experiment or from natural observations, has created a need for accurate computational approaches. Because isotope fractionation is a purely quantum mechanical phenomenon, exact calculation of fractionation factors is nontrivial. Consequently, a severe approximation is often made, in which it is assumed that the system can be decomposed into a set of independent harmonic oscillators. Reliance on this often crude approximation is one of the primary reasons that theoretical prediction of isotope fractionation has lagged behind experiment. A class of problems for which one might expect the harmonic approximation to perform most poorly is the isotopic fractionation between solid and solution phases. In order to illustrate the errors associated with the harmonic approximation, we have considered the fractionation of Li isotopes between aqueous solution and phyllosilicate minerals, where we find that the harmonic approximation overestimates isotope fractionation factors by as much as 30% at 25 °C. Lithium is a particularly interesting species to examine, as natural lithium isotope signatures provide information about hydrothermal processes, carbon cycle, and regulation of the Earth's climate by continental alteration. Further, separation of lithium isotopes is of growing interest in the nuclear industry due to a need for pure 6Li and 7Li isotopes. Moving beyond the harmonic approximation entails performing exact quantum calculations, which can be achieved using the Feynman path integral formulation of quantum statistical mechanics. In the path integral approach, a system of quantum particles is represented as a set of classical-like ring-polymer chains, whose interparticle interactions are determined by the rules of quantum mechanics. Because a classical isomorphism exists between the true quantum system and the system of ring-polymers, classical-like methods can be applied. Recent developments of efficient path integral approaches for the exact calculation of isotope fractionation now allow the case of the aforementioned dissolved Li fractionation properties to be studied in detail. Applying this technique, we find that the calculations yield results that are in good agreement with both experimental data and natural observations. Importantly, path integral methods, being fully atomistic, allow us to identify the origins of anharmonic effects and to make reliable predictions at temperatures that are experimentally inaccessible yet are, nevertheless, relevant for natural phenomena.

Quantum mean-field approximation for lattice quantum models: Truncating quantum correlations and retaining classical ones

In a recent work [D. Malpetti and T. Roscilde, arXiv:1605.04223] we have shown that in quantum many-body systems at finite temperature, two-point correlations can be formally separated into a thermal part, and a quantum part -- and that generically quantum correlations decay exponentially at finite temperature, with a characteristic, temperature-dependent quantum coherence length. The existence of these two different forms of correlation in quantum many-body systems suggests the possibility of formulating an approximation which affects quantum correlations only, without preventing the correct description of classical fluctuations at all length scales. Focusing on lattice boson and quantum Ising models, we make use of the path-integral formulation of quantum statistical mechanics to introduce such an approximation -- that we dub \emph{quantum mean-field} (QMF) approach, and which can be readily generalized to a cluster form (cluster QMF or cQMF). The cQMF approximation reduces to cluster mean-field theory at $T=0$, while at any finite temperature it produces a family of systematically improved, semi-classical approximations to the quantum statistical mechanics of the lattice theory at hand. Contrary to standard MF approximations, the correct nature of thermal critical phenomena is captured by any cluster size. In the two exemplary cases of the two-dimensional quantum Ising model and of two-dimensional quantum rotors, we study systematically the convergence of the cQMF approximation towards the exact result, and show that the convergence is typically linear or sub-linear in the { boundary-to-bulk ratio} of the clusters as $T\to 0$, while it becomes faster than linear as $T$ grows. These results pave the way towards the development of semi-classical numerical approaches based on an approximate, { yet} systematically improved account of quantum correlations.

From Classical to Quantum and Back: A Hamiltonian Scheme for Adaptive Multiresolution Classical/Path-Integral Simulations.

Quantum delocalization of atomic nuclei affects the physical properties of many hydrogen-rich liquids and biological systems even at room temperature. In computer simulations, quantum nuclei can be modeled via the path-integral formulation of quantum statistical mechanics, which implies a substantial increase in computational overhead. By restricting the quantum description to a small spatial region, this cost can be significantly reduced. Herein, we derive a bottom-up, rigorous, Hamiltonian-based scheme that allows molecules to change from quantum to classical and vice versa on the fly as they diffuse through the system, both reducing overhead and making quantum grand-canonical simulations possible. The method is validated via simulations of low-temperature parahydrogen. Our adaptive resolution approach paves the way to efficient quantum simulations of biomolecules, membranes, and interfaces.

Ab initio statistical mechanics of surface adsorption and desorption. II. Nuclear quantum effects

We show how the path-integral formulation of quantum statistical mechanics can be used to construct practical ab initio techniques for computing the chemical potential of molecules adsorbed on surfaces, with full inclusion of quantum nuclear effects. The techniques we describe are based on the computation of the potential of mean force on a chosen molecule and generalize the techniques developed recently for classical nuclei. We present practical calculations based on density functional theory with a generalized-gradient exchange-correlation functional for the case of H(2)O on the MgO (001) surface at low coverage. We note that the very high vibrational frequencies of the H(2)O molecule would normally require very large numbers of time slices (beads) in path-integral calculations, but we show that this requirement can be dramatically reduced by employing the idea of thermodynamic integration with respect to the number of beads. The validity and correctness of our path-integral calculations on the H(2)O/MgO(001) system are demonstrated by supporting calculations on a set of simple model systems for which quantum contributions to the free energy are known exactly from analytic arguments.

Reality of the Wigner Functions and Quantization

Here we use the extended phase space formulation of quantum statistical mechanics proposed in an earlier work to define an extended lagrangian for Wigner's functions (WFs). The extended action defined by this lagrangian is a function of ordinary phase space variables. The reality condition of WFs is employed to quantize the extended action. The energy quantization is obtained as a direct consequence of the quantized action. The technique is applied to find the energy states of harmonic oscillator, particle in the box, and hydrogen atom as the illustrative examples.

A geometric approach to the canonical reformulation of quantum mechanics

The measure of distinguishability between two neighboring preparations of a physical system by a measurement device naturally defines a line element on the preparation space of the system. We show that quantum mechanics can be derived from the invariance of this line element in the canonical formulation. We also discuss the canonical formulation of quantum statistical mechanics.

A combinatorial approach to the electron correlation problem

Starting from a path-integral formulation of quantum statistical mechanics expressed in a space of Slater determinants, we develop a method for the Monte Carlo evaluation of the energy of a correlated electronic system. The path-integral expression for the partition function is written as a contracted sum over graphs. A graph is a set of distinct connected determinants on which paths can be represented. The weight of a graph is given by the sum over exponentially large numbers of paths which visit the vertices of the graph. We show that these weights are analytically computable using combinatorial techniques, and they turn out to be sufficiently well behaved to allow stable Monte Carlo simulations in which graphs are stochastically sampled according to a Metropolis algorithm. In the present formulation, graphs of up to four vertices have been included. In a Hartree-Fock basis, this allows for paths which include up to sixfold excitations relative to the Hartree-Fock determinant. As an illustration, we have studied the dissociation curve of the N(2) molecule in a VDZ basis, which allows comparison with full configuration-interaction calculations.

Wigner approach to quantum dynamics simulations of the interacting carriers in disordered systems

AbstractThe new method for solving Wigner–Liouville's type equations and studying dynamics of quantum particles has been developed within the Wigner formulation of quantum statistical mechanics. This approach combines both molecular dynamics and Monte Carlo methods and computes traces and spectra of the relevant dynamical quantities. Considering, as an application, the quantum dynamics of an ensemble of interacting electrons in an array of random scatterers clearly demonstrates that the many‐particle interaction between the electrons can lead to an enhancement of the electrical conductivity. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Functional integral approach: a third formulation of quantum statistical mechanics.

Quantum statistical mechanics has developed primarily through two approaches, pioneered by Gibbs and Feynman, respectively. In Gibbs' method one calculates partition functions from phase-space integrations or sums over stationary states. Alternatively, in Feynman's approach, the focus is on the path-integral formulation. The Hubbard-Stratonovich transformation leads to a functional-integral formulation for calculating partition functions. We outline here the functional integral approach to quantum statistical mechanics, including generalizations and improvements to Hubbard's formulation. We show how the dimensionality of the integrals is reduced exactly, how the problem of assuming an unknown canonical transformation is avoided, how the reality of the partition function in the complex representation is guaranteed, and how the extremum conditions are simplified. This formulation can be applied to general systems, including superconductors.

Remarks on Shannon's Statistical Inference and the Second Law in Quantum Statistical Mechanics

We comment on a formulation of quantum statistical mechanics, which incorporates the statistical inference of Shannon. Our basic idea is to distinguish the dynamical entropy of von Neumann, $H = -k Tr \hat{\rho}\ln\hat{\rho}$, in terms of the density matrix $\hat{\rho}(t)$, and the statistical amount of uncertainty of Shannon, $S= -k \sum_{n}p_{n}\ln p_{n}$, with $p_{n}=< n|\hat{\rho}|n>$ in the representation where the total energy and particle numbers are diagonal. These quantities satisfy the inequality $S\geq H$. We propose to interprete Shannon's statistical inference as specifying the {\em initial conditions} of the system in terms of $p_{n}$. A definition of macroscopic observables which are characterized by intrinsic time scales is given, and a quantum mechanical condition on the system, which ensures equilibrium, is discussed on the basis of time averaging. An interesting analogy of the change of entroy with the running coupling in renormalization group is noted. A salient feature of our approach is that the distinction between statistical aspects and dynamical aspects of quantum statistical mechanics is very transparent.

Short-Range Coherence in the Energy Transfer of Photosynthetic Light-Harvesting Systems

We employ a simple model for the electronic excitations and the exciton-vibration coupling characterizing the B850 ring of the light-harvesting complex in photosynthetic bacteria to investigate the possibility of coherence in the energy transfer within the system. The structure of the equilibrium density matrix is studied using the path integral formulation of quantum statistical mechanics. The calculated mean coherence length is computed from the average root-mean-square deviation of closed imaginary time paths that are sampled via a Monte Carlo procedure. This procedure allows simultaneous examination of the effects of thermal averaging and dynamic and static disorder in a single calculation. The mean coherence length is found to be about two to three chlorophyll monomers at room temperature. The principal factor responsible for this localization is thermal averaging, although static and dynamic disorder further destabilize extended states. At low temperatures the circular arrangement of the pigments favors coherence with respect to the situation in a linear aggregate. Visual inspection of typical paths offers an intuitive picture of the extent of coherent energy delocalization in biological antenna systems.

Path integral hybrid Monte Carlo for the bosonic many-body systems

In this Letter, we propose a new simulation technique for many-body systems of bosons based on the path integral formulation of quantum statistical mechanics. The indistinguishability of the bosons was handled by importance sampling in permutation space, and the path configurations for a given permutation were sampled using a hybrid Monte Carlo method. A system of 32 bosons in a harmonic well was chosen as a model problem. Calculations were performed for the system at various temperatures. The calculated thermodynamic quantities are in excellent agreement with the exact values.

The quantum Heisenberg antiferromagnet on the square lattice

The thermodynamic behavior of the quantum Heisenberg antiferromagnet on the square lattice (QHAF) is tackled by the pure-quantum self-consistent harmonic approximation, a semiclassical method based on the path-integral formulation of quantum statistical mechanics. Results for various properties are obtained for different values of the spin and the effects of the dipolar anisotropy are considered. Furthermore, the validity of the quantum nonlinear σ model approach is discussed and shown to be inadequate to describe the behavior of the QHAF with spin value S⩾1 in the experimentally accessible temperature region.

Electron Dynamics and Anderson Localization in Wigner Formulation of Quantum Statistical Mechanics

AbstractThe new numerical approach for consideration of quantum dynamics and calculations of the average values of quantum operators and time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. The time correlation functions have been presented in the form of the integral of the Weyl's symbol of considered operators and the Fourier transform of the product of matrix elements of the dynamic propagators. For electrons in disordered systems of scatterers the numerical results have been obtained for series of the average values of the quantum operators including position and momentum dispersions, average energy, energy distribution function as well as for the frequency dependencies of tensor of electron conductivity and permittivity according to quantum Kubo formula. Zero or very small value of static conductivity have been considered as the manifestation of Anderson localization of electrons in 1D case.

Wigner approach to quantum statistical mechanics and quantum generalization molecular dynamics method. Part 2

A new numerical approach for the calculation of quantum time correlation functions in the canonical and micro-canonical ensembles has been developed within the Wigner formulation of quantum statistical mechanics. For electrons in disordered systems of scatterers the numerical results for the tensor of static electrical conductivity have been obtained according to the quantum Kubo formula in the one-electron approximation in the canonical ensemble. The second quantity considered is the flux position time correlation function characterizing the momentum flow for tunnelling particles at the barrier crossing from one potential well to another. The difficulties with studies of exponentially small effects in the temperature averaging for the canonical ensemble has been overcome by a transition to the micro-canonical ensemble with a fixed total energy. The related integral Wigner-Liouville type equations have been obtained and a numerical approach combining both molecular dynamics and Monte Carlo methods for solving this equation has also been developed. Numerical results have been obtained for the momentum flow of a particle tunnelling through the barrier dividing two potential wells in two- and three-dimensional cases.

Ab initio path integral molecular dynamics: Basic ideas

The essential ideas underlying ab initio path integral molecular dynamics and its efficient numerical implementation are discussed. In this approach the nuclei are treated as quantum particles within the path integral formulation of quantum statistical mechanics. The electronic degrees of freedom are treated explicitly based on state-of-the-art electronic structure theory. This renders ab initio simulations of quantum systems possible without recourse to model potentials. A combined extended Lagrangian for both quantum nuclei and electrons defines a dynamical system and yields molecular dynamics trajectories that can be analyzed to obtain quantum statistical expectation values of time-independent operators. The methodology can be applied to a wide range of fields addressing problems in molecular and condensed matter chemistry and physics.

On the Feynman path centroid density as a phase space distribution in quantum statistical mechanics

The phase space formulation of quantum statistical mechanics using the Feynman path centroid density offers an alternative perspective to the standard Wigner prescription for the classical-like evaluation of equilibrium and/or dynamical quantities of statistical systems. The use of this formulation has been implicit in recent work on quantum rate theories, for example, in which the centroid density distribution replaces the classical Boltzmann distribution. In order to further understand the approximations involved in this and similar transcriptions, the present work elaborates and clarifies the issue of operator ordering in a rigorous centroid-based formulation. In particular, through the use of the Weyl correspondence, a precise definition of the centroid symbol of operators and their products is presented. Though we fall short of finding the algebraic structure tantamount to that found in the Weyl symbols—of which the Wigner distribution is an example— the resulting expressions have internal consistency and are amenable to approximate evaluation through cumulant expansions.