Formulation Of Statistical Mechanics Research Articles

Canonical Thermal Pure Quantum State

A thermal equilibrium state of a quantum many-body system can be represented by a typical pure state, which we call a thermal pure quantum (TPQ) state. We construct the canonical TPQ state, which corresponds to the canonical ensemble of the conventional statistical mechanics. It is related to the microcanonical TPQ state, which corresponds to the microcanonical ensemble, by simple analytic transformations. Both TPQ states give identical thermodynamic results, if both ensembles do, in the thermodynamic limit. The TPQ states corresponding to other ensembles can also be constructed. We have thus established the TPQ formulation of statistical mechanics, according to which all quantities of statistical-mechanical interest are obtained from a single realization of any TPQ state. We also show that it has great advantages in practical applications. As an illustration, we study the spin-1/2 kagome Heisenberg antiferromagnet.

Electron Microscopy: A Versatile Tool in Nanoworld

Microscopy is the characterization of objects smaller than what can be seen with the naked human eye, and from its inception, optical microscopy has played a seminal role in the development of science. In the 1660s, Robert Hooke first resolved cork cells and thereby discovered the cellular nature of life. Robert Browns 1827 observation of the seemingly random movement of pollen grains [ led to the understanding of the motion that still bears his name, and ultimately to the formulation of statistical mechanics. The contributions of optical microscopy continue into the present, even as the systems of interest approach nanometer size. What makes optical microscopy so useful is the relatively low energy of visible light: in general, it does not irreversibly alter the electronic or atomic structure of the matter with which it interacts, allowing observation of natural processes in situ. Moreover, light is cheap, abundant, and can be manipulated with common and relatively inexpensive laboratory hardware.Science and technology ever seek to build structures of progressively smaller size. This effort at miniaturization has finally reached the point where structures and materials can be built through atom-by-atom engineering. Typical chemical bonds separate atoms by a fraction of a nanometer (109 m), and the term nanotechnology has been coined for this emerging area of development. By manipulating the arrangements and bonding of atoms, materials can be designed with a far vaster range of physical, chemical and biological properties than has been previously conceived. But how to characterize the relationship between starting composition, which can be controlled, with the resulting structure and properties of a nanoscale-designed material that has superior and unique performance? Microscopy is essential to the development of nanotechnology, serving as its eyes and hands.

Models for Type VI Adsorption Isotherms from a Statistical Mechanical Formulation

In this study, two theoretical expressions for the Type VI isotherm in the IUPAC classification are presented. The formulation of these new expressions is based on a rigorous statistical mechanical description. The expressions allow for the estimation of physicochemical parameters within the theoretical model. The proposed models allow a good correlation to Type VI experimental isotherms taken from the literature such as adsorption isotherms for adsorption of methane onto MgO (100) and onto an exfoliated graphite surface. The parameters intervening in the adsorption process have been deduced directly from experimental adsorption isotherms by numerical simulation. The theoretical expressions provide an understanding and interpretation of the adsorption isotherms at the microscopic level.

Glueballs and the Yang-Mills plasma in aT-matrix approach

The strongly coupled phase of Yang-Mills plasma with arbitrary gauge group is studied in a $T$-matrix approach. The existence of lowest-lying glueballs, interpreted as bound states of two transverse gluons (quasi-particles in a many-body set up), is analyzed in a non-perturbative scattering formalism with the input of lattice-QCD static potentials. Glueballs are actually found to be bound up to 1.3 $T_c$. Starting from the $T$-matrix, the plasma equation of state is computed by resorting to Dashen, Ma and Bernstein's formulation of statistical mechanics and favorably compared to quenched lattice data. Special emphasis is put on SU($N$) gauge groups, for which analytical results can be obtained in the large-$N$ limit, and predictions for a $G_2$ gauge group are also given within this work.

Schrödinger equation revisited

The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. We obtain the Schrödinger equation from a mathematical identity by a slight generalization of the formulation of classical statistical mechanics based on the Hamilton-Jacobi equation. This approach brings out most clearly the fact that the linearity of quantum mechanics is intimately connected to the strong coupling between the amplitude and phase of a quantum wave.

Structural fluctuation of protein in water around its native state: A new statistical mechanics formulation

A new statistical mechanics formulation of characterizing the structural fluctuation of protein correlated with that of water is presented based on the generalized Langevin equation and the 3D-reference interaction site model (RISM)/RISM theory of molecular liquids. The displacement vector of atom positions, and their conjugated momentum, are chosen for the dynamic variables for protein, while the density fields of atoms and their momentum fields are chosen for water. Projection of other degrees of freedom onto those dynamic variables using the standard projection operator method produces essentially two equations, which describe the time evolution of fluctuation concerning the density field of solvent and the conformation of protein around an equilibrium state, which are coupled with each other. The equation concerning the protein dynamics is formally akin to that of the coupled Langevin oscillators, and is a generalization of the latter, to atomic level. The most intriguing feature of the new equation is that it contains the variance-covariance matrix as the "Hessian" term describing the "force" restoring an equilibrium conformation, which is the second moment of the fluctuation of atom positions. The "Hessian" matrix is naturally identified as the second derivative of the free energy surface around the equilibrium. A method to evaluate the Hessian matrix based on the 3D-RISM/RISM theory is proposed. Proposed also is an application of the present formulation to the molecular recognition, in which the conformational fluctuation of protein around its native state becomes an important factor as exemplified by so called "induced fitting."

Statistical Mechanical Formulation and Simulation of Prime Factorization of Integers

We propose a new formulation of the problem of prime factorization of integers. With replica exchange Monte Carlo simulation, the behavior which is seemed to indicate exponential computational hardness is observed. But this formulation is expected to give a new insight into the computational complexity of this problem from a statistical mechanical point of view.

Kinetic derivation of a quasi-hydrodynamic system of equations on the base of nonextensive statistics

- Applying the formalism of the Tsallis nonadditive statistical mechanics, a derivation of hydrodynamic and quasi-hydrodynamic equations is considered based on the generalized BGK kinetic equation. In particular, those equations are intended for construction of more appropriate macroscopic and mesoscopic models of transport systems related to the so-called anomalous systems where the corresponding phase space possesses a complicated (fractal) structure. The classic Boltzmann-Gibbs statistics (or Maxwell’s distribution of velocities in the case of kinetic theory) is violated in such systems and hence the additivity of extensive variables related to it does not hold either. The application of the approach developed in this paper does not change the structure of hydrodynamic and quasihydrodynamic equations, but the modified thermal and calorific state equations and also the transfer coefficients contain two additional free parameters, which are the parameter q of the non-additivity of the system and the fractional dimension D of the phase space. Along with the smoothing parameter t, these parameters can be determined empirically in each particular case from statistical or experimental data, which allows us to simulate the actual variable traffic situation within a continual approach both in regular and crisis cases.

Conceptual difficulties with the q-averages in non-extensive statistical mechanics

The q-average formalism of nonextensive statistical mechanics proposed in the literature is critically examined by considerations of several pedagogical examples. It is shown that there exist a number of difficulties with the concept of q-averages.

Formulation of the Hellmann–Feynman theorem for the “second choice” version of Tsallis’ thermostatistics

An approach to formulating the Hellmann–Feynman theorem within the “second choice” formalism of non-extensive statistical mechanics is considered. For the state of thermal equilibrium, we derive a relation of Hellmann–Feynman type between the derivative of the non-extensive free energy with respect to the external parameter and the quantum statistical q-average of the derivative of the Hamilton operator. We also give a proper extension for an arbitrary observable commuting with the Hamiltonian. Some reasons for the usefulness of new formulas are discussed.

Five popular misconceptions about osmosis

Osmosis is the flow of solvent across a semipermeable membrane from a region of lower to higher solute concentration. It is of central importance in plant and animal physiology and finds many uses in industry. A survey of published papers, web resources, and current textbooks reveals that numerous misconceptions about osmosis continue to be cited and taught. To clarify these issues, we re-derive the thermodynamics of osmosis using the canonical formalism of statistical mechanics and go on to discuss the main points that continue to lead to misunderstandings.

Covariant Statistical Mechanics and the Stress-Energy Tensor

After recapitulating the covariant formalism of equilibrium statistical mechanics in special relativity and extending it to the case of a nonvanishing spin tensor, we show that the relativistic stress-energy tensor at thermodynamical equilibrium can be obtained from a functional derivative of the partition function with respect to the inverse temperature four-vector β. For usual thermodynamical equilibrium, the stress-energy tensor turns out to be the derivative of the relativistic thermodynamic potential current with respect to the four-vector β, i.e., T(μν)=-∂Φ(μ)/∂β(ν). This formula establishes a relation between the stress-energy tensor and the entropy current at equilibrium, possibly extendable to nonequilibrium hydrodynamics.

Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. II. Adsorption of atomic and molecular fluids in a porous material

We propose to apply expanded Wang-Landau simulations to study the adsorption of atomic and molecular fluids in porous materials. This approach relies on a uniform sampling of the number of atoms and molecules adsorbed. The method consists in determining a high-accuracy estimate of the grand-canonical partition function for the adsorbed fluids. Then, using the formalism of statistical mechanics, we calculate absolute and excess thermodynamic properties relevant to adsorption processes. In this paper, we examine the adsorption of argon and carbon dioxide in the isoreticular metal-organic framework (IRMOF-1). We assess the reliability of the method by showing that the predicted adsorption isotherms and isosteric heats are in excellent agreement with simulation results obtained from grand-canonical Monte Carlo simulations. We also show that the proposed method is very efficient since a single expanded Wang-Landau simulation run at a given temperature provides the whole adsorption isotherm. Moreover, this approach provides a direct access to a wide range of thermodynamic properties, such as, e.g., the excess Gibbs free energy and the excess entropy of adsorption.

Quantum particle statistics on the holographic screen leads to modified Newtonian dynamics

Employing a thermodynamic interpretation of gravity based on the holographic principle and assuming underlying particle statistics, fermionic or bosonic, for the excitations of the holographic screen leads to modified Newtonian dynamics (MOND). A connection between the acceleration scale ${a}_{0}$ appearing in MOND and the Fermi energy of the holographic fermionic degrees of freedom is obtained. In this formulation the physics of MOND results from the quantum-classical crossover in the fermionic specific heat. However, due to the dimensionality of the screen, the formalism is general and applies to two-dimensional bosonic excitations as well. It is shown that replacing the assumption of the equipartition of energy on the holographic screen by a standard quantum-statistical-mechanics description wherein some of the degrees of freedom are frozen out at low temperatures is the physical basis for the MOND interpolating function $\stackrel{\texttildelow{}}{\ensuremath{\mu}}$. The interpolating function $\stackrel{\texttildelow{}}{\ensuremath{\mu}}$ is calculated within the statistical mechanical formalism and compared to the leading phenomenological interpolating functions, most commonly used. Based on the statistical mechanical view of MOND, its cosmological implications are reinterpreted: the connection between ${a}_{0}$ and the Hubble constant is described as a quantum uncertainty relation; and the relationship between ${a}_{0}$ and the cosmological constant is better understood physically.

Ion Binding Sites and Their Representations by Reduced Models

The binding of small metal ions to complex macromolecular structures is typically dominated by strong local interactions of the ion with its nearest ligands. Progress in understanding the molecular determinants of ion selectivity can often be achieved by considering simplified reduced models comprised of only the most important ion-coordinating ligands. Although the main ingredients underlying simplified reduced models are intuitively clear, a formal statistical mechanical treatment is nonetheless necessary in order to draw meaningful conclusions about complex macromolecular systems. By construction, reduced models only treat the ion and the nearest coordinating ligands explicitly. The influence of the missing atoms from the protein or the solvent is incorporated indirectly. Quasi-chemical theory offers one example of how to carry out such a separation in the case of ion solvation in bulk liquids, and in several ways, a statistical mechanical formulation of reduced binding site models for macromolecules is expected to follow a similar route. However, there are also important differences when the ion-coordinating moieties are not solvent molecules from a bulk phase but are molecular ligands covalently bonded to a macromolecular structure. Here, a statistical mechanical formulation of reduced binding site models is elaborated to address these issues. The formulation provides a useful framework to construct reduced binding site models, and define the average effect from the surroundings on the ion and the nearest coordinating ligands.

Computing a Knot Invariant as a Constraint Satisfaction Problem

We point out the connection between mathematical knot theory and spin glass/search problem. In particular, we present a statistical mechanical formulation of the problem of computing a knot invariant; p-colorability problem, which provides an algorithm to find the solution. The method also allows one to get some deeper insight into the structural complexity of knots, which is expected to be related with the landscape structure of constraint satisfaction problem.

Equilibrium-reduced density matrix formulation: Influence of noise, disorder, and temperature on localization in excitonic systems

An exact method to compute the entire equilibrium reduced density matrix for systems characterized by a system-bath Hamiltonian is presented. The approach is based upon a stochastic unraveling of the influence functional that appears in the imaginary time path integral formalism of quantum statistical mechanics. This method is then applied to study the effects of thermal noise, static disorder, and temperature on the coherence length in excitonic systems. As representative examples of biased and unbiased systems, attention is focused on the well-characterized light harvesting complexes of FMO and LH2, respectively. Due to the bias, FMO is completely localized in the site basis at low temperatures, whereas LH2 is completely delocalized. In the latter, the presence of static disorder leads to a plateau in the coherence length at low temperature that becomes increasingly pronounced with increasing strength of the disorder. The introduction of noise, however, precludes this effect. In biased systems, it is shown that the environment may increase the coherence length, but only decrease that of unbiased systems. Finally it is emphasized that for typical values of the environmental parameters in light harvesting systems, the system and bath are entangled at equilibrium in the single excitation manifold. That is, the density matrix cannot be described as a product state as is often assumed, even at room temperature. The reduced density matrix of LH2 is shown to be in precise agreement with the steady state limit of previous exact quantum dynamics calculations.

Generalizing Planck’s distribution by using the Carati–Galgani model of molecular collisions

Classical systems of coupled harmonic oscillators are studied using the Carati–Galgani model. We investigate the consequences for Einstein’s conjecture by considering that the exchange of energy in molecular collisions follows the Lévy type statistics. We develop a generalization of Planck’s distribution admitting that there are analogous relations in the equilibrium quantum statistical mechanics of the relations found using the nonequilibrium classical statistical mechanics approach. The generalization of Planck’s law based on the nonextensive statistical mechanics formalism is compatible with our analysis.

Fisher Information and Quantum Mechanics

A suggestive relation links Fisher' information measure (FIM)Iand Schrodinger equation (SE). The connection is based upon the fact that the constrained minimization of I leads to a SE. This, in turn, is the origin of intriguing relationships between various aspects of SE, on the one hand, and the formalism of statistical mechanics derived from Jaynes's maximum entropy principle (MaxEnt), on the other one. The link entails the existence of a Legendre transform structure underlying the SE, which allows for the emergence of two first-order differential equations that must, respectively, be satisfied by i) the Fisher measure and ii) the SE energy eigenvalues. The complete A) I-solution and B) energysolution are both obtained bypassing the SE and, furthermore, linked by the Legendre structure.

Ergodic Hypothesis and Equilibrium Statistical Mechanics in the Quantum Mechanical World View

In this paper, we study and answer the following fundamental problems concerning classical equilibrium statistical mechanics: 1): Is the principle of equal a priori probabilities indispensable for equilibrium statistical mechanics? 2): Is the ergodic hypothesis related to equilibrium statistical mechanics? Note that these problems are not yet answered, since there are several opinions for the formulation of equilibrium statistical mechanics. In order to answer the above questions, we first introduce measurement theory (i.e., the theory of quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics. And we propose the measurement theoretical foundation of equili-brium statistical mechanics, and further, answer the above 1) and 2), that is, 1) is “No”, but, 2) is “Yes”.