Canonical Distribution Function Research Articles

Disclosure of the True Essence of Quantum Theory on the Basis of Scientific Philosophy

It is well known that in 1900, after Planck obtained the basic equation of quantum theory, there came a moment when it became necessary to obtain a theoretical justification of this equation. In 1911, based on the possibility of Gibbs' canonical distribution function (CDF), he solved this problem partially: he obtained a justification for the second multiplier of his equation. Then on the basis of the analysis of these results he stated that henceforth in the role of the basic concept of quantum theory there is a necessity to accept the possibility of the concept of the quantum of action. However, Poincaré at the same congress criticized these new ideas. For he recognized the value of the idea of the quantum of action in solving only frequent problems. Therefore, he further, in his article entitled "On the Theory of Quanta", made the following conclusion: in his opinion, the main essence of quantum theory is connected with the concept of the quantum of energy [1]. Thus, he wanted to say that in the future the main results of Planck's quantum theory have the possibility to substantiate only if the possibility of the concept of energy quantum is taken as more general. In this paper an attempt is made to prove that it is really so.

Development of a gyrokinetic hyperbolic solver based on discontinuous Galerkin method in tokamak geometry

A hyperbolic solver is developed for the gyrokinetic equation in tokamak geometry. Aiming a whole device modeling of fusion plasma surrounded by first walls of tokamak device, an unstructured spatial mesh is introduced. The discontinuous Galerkin (DG) method is used to discretize the gyrokinetic equation on the mesh and test various numerical elements for the discretization. Based on the conservations of physical quantities such as mass, kinetic energy, and toroidal canonical angular momentum in an axisymmetric configuration of toroidal plasma, we investigate the effects of basis functions for the DG method on the numerical solutions. With proper choices of the basis functions and spatial grid resolutions, the conservations of the key physical quantities are shown to be satisfied nearly to machine accuracies in the simplified circular magnetic geometry. Even in realistic tokamak geometry with machine wall boundaries, it is shown that a good conservation property can be demonstrated if the flux across boundaries of a test domain is carefully accounted for. Also, the invariance of the canonical Maxwellian distribution function in time is well satisfied with the developed solver. The effect of weighting functions for the basis is investigated too. Overall, the Maxwellian weighted basis shows a similar conservation property with the polynomial basis. On the other hand, the Maxwellian weighted basis shows better performance in resolving small scale structures in velocity space, which can be utilized to set up an efficient basis set to simulate fine structures with less computational costs. The parallelization of the newly developed solver is also reported. Employing MPI for the parallelization, the solver shows good performances up to a few thousand CPU cores.

Sampling, rates, and reaction currents through reverse stochastic quantization on quantum computers

The quest for improved sampling methods to solve statistical mechanics problems of physical and chemical interest proceeds with renewed efforts since the invention of the Metropolis algorithm, in 1953. In particular, the understanding of thermally activated rare-event processes between long-lived metastable states, such as protein folding, is still elusive. In this case, one needs both the finite-temperature canonical distribution function and the reaction current between the reactant and product states, to completely characterize the dynamic. Here we show how to tackle this problem using a quantum computer. We use the connection between a classical stochastic dynamics and the Schroedinger equation, also known as stochastic quantization, to variationally prepare quantum states allowing us to unbiasedly sample from a Boltzmann distribution. Similarly, reaction rate constants can be computed as ground state energies of suitably transformed operators, following the supersymmetric extension of the formalism. Finally, we propose a hybrid quantum-classical sampling scheme to escape local minima, and explore the configuration space in both real-space and spin hamiltonians. We indicate how to realize the quantum algorithms constructively, without assuming the existence of oracles, or quantum walk operators. The quantum advantage concerning the above applications is discussed.

Thermal imaging and multivariate techniques for characterizing and screening wheat genotypes under water stress condition

The present study was undertaken to understand the combined impact of thermal imaging and multivariate techniques to characterize and screen different wheat genotypes under water stress condition. Ten wheat genotypes namely HD-2987, HDR-77, PBW-343, PBW-175, HD-2967, HD-2781, HD-2985, HD-3043, C-306 and PBW-502 were grown under two irrigation conditions (I and Io corresponding to IW/CPE ratio of 1.0 and 0.4, respectively). The water stress conditions in wheat genotypes were characterized thermo-graphically using canopy temperature-based stress indices, namely crop water stress index (CWSI), stomatal conductance index (IG) and stomatal resistance index (CSI3). Simultaneously, plant biophysical parameters like normalised difference vegetation index (NDVI), leaf area index (LAI), transpiration, stomatal conductance, photosynthesis, were measured at peak vegetative stage along with yield and its attributes (panicle number, tiller number, 1000 seed weight, crop yield, total dry matter, harvest index) were observed. These parameters were analyzed using multivariate techniques namely principal component analysis (PCA), hierarchical cluster analysis (HCA) and discriminant analysis (DA) to develop a new index called normalized water stress tolerance index (NWSTI), to group wheat genotypes depending on their ability to tolerate water stress. Based on these multivariate analysis, wheat genotypes were classified as tolerant (C-306, HD-3043, HD-3987 HD-3985 and HD-2781), moderately tolerant (HDR-77, PBW-175 and PBW-502) and sensitive (HD-2967, PBW-343) to water stress with NWSTI score of >0.66, 0.33–0.66 and <0.33, respectively. Structural matrix from DA analysis showed that biophysical parameters like stomatal conductance, harvest index, photosynthesis, transpiration, NDVI and panicle number have significant (p < 0.001) role in discriminating wheat genotypes into three different groups of water stress tolerance. Distinct location of group centroid in the canonical distribution function further signifies (p < 0.001) the truthfulness of the HCA based dendrogram prepared using the normalized values of crop biophysical parameters. A significantly higher correlation (R2 , p < 0.001) between NWSTI and the thermal image-based stress indices (that characterize the instantaneous crop condition for the ten wheat genotypes), prove the usefulness of NWSTI as a stress index. In future, the newly developed water stress index NWSTI, may play a potential role in the screening and selection of drought tolerant wheat genotypes.

Thermodynamic properties of the solid and metal electrons in the nonextensive quantum statistics with a nonextensive parameter transformation

We study the thermodynamic properties of solid and metal electrons in the nonextensive quantum statistics with a nonextensive parameter transformation. First we study the nonextensive grand canonical distribution function and the nonextensive quantum statistics with a parameter transformation. Then we derive the generalized Boson distribution and Fermi distribution in the nonextensive quantum statistics. Further we study the thermodynamic properties of solid and metal electrons in the nonextensive quantum system, including the generalized Debye models, the generalized internal energies, the generalized capacities and chemical potential. We derive new expressions of these thermodynamic quantities, and we show that they all depend significantly on the nonextensive parameter and in the limit they recover to the forms in the classical quantum statistics. These new expressions may be applied to study the new characteristics in some nonextensive quantum systems where the long-range interactions and/or long-range correlations play a role.

Computation of shock wave structure using a simpler set of generalized hydrodynamic equations based on nonlinear coupled constitutive relations

Generalized hydrodynamic equations were originally proposed to describe the rarefied non-equilibrium flows beyond the Navier–Stokes–Fourier (NSF) equations by constructing a non-equilibrium canonical distribution function for the mesoscopic Boltzmann equation. Subsequently, nonlinear coupled constitutive relations (NCCR) were developed under the adiabatic assumption, the Eu’s closure, and Myong’s simplification. NCCR+ was also proposed to include the omitted terms in Myong’s simplification. The goal of this paper is to assess the improvements in the accuracy due to NCCR+ and the influence of the bulk viscosity in one-dimensional steady shock wave structure for monatomic and diatomic gases. In order to solve NCCR+ equations, a coupled solution process based on the time-independent method for non-conserved variables is employed, which is different from the previous uncoupled solution process used for the NCCR equations. Shock structures in argon and nitrogen are calculated up to Mach 50, where the shock profile, inverse shock thickness, asymmetry parameter, and temperature–density separation distance are validated by DSMC and available experimental measurements. The results show that NCCR+ could not provide much improvement in accuracy compared to NCCR but adds to the computational cost, suggesting that Myong’s simplification used in NCCR is satisfactory. Both NCCR and NCCR+ perform better than NSF in computing the one-dimensional shock wave structure at high Mach numbers. It is also shown that the bulk viscosity has significant influence on the accuracy of prediction of the shock wave structure in a diatomic gas using both NSF and NCCR. Stokes’ hypothesis in conventional NSF is valid only for flows in a monatomic gas and for very low Mach number flows in a diatomic gas. Additionally, it is also found that the viscosity exponent s = 0.81 in variable-hard-sphere model provides a good fit with the experimental data for shock wave in argon using NCCR.

Molecular dynamics coupled with a virtual system for effective conformational sampling.

An enhanced conformational sampling method is proposed: virtual-system coupled canonical molecular dynamics (VcMD). Although VcMD enhances sampling along a reaction coordinate, this method is free from estimation of a canonical distribution function along the reaction coordinate. This method introduces a virtual system that does not necessarily obey a physical law. To enhance sampling the virtual system couples with a molecular system to be studied. Resultant snapshots produce a canonical ensemble. This method was applied to a system consisting of two short peptides in an explicit solvent. Conventional molecular dynamics simulation, which is ten times longer than VcMD, was performed along with adaptive umbrella sampling. Free-energy landscapes computed from the three simulations mutually converged well. The VcMD provided quicker association/dissociation motions of peptides than the conventional molecular dynamics did. The VcMD method is applicable to various complicated systems because of its methodological simplicity. © 2018 Wiley Periodicals, Inc.

Multi-dimensional virtual system introduced to enhance canonical sampling.

When an important process of a molecular system occurs via a combination of two or more rare events, which occur almost independently to one another, computational sampling for the important process is difficult. Here, to sample such a process effectively, we developed a new method, named the "multi-dimensional Virtual-system coupled Monte Carlo (multi-dimensional-VcMC)" method, where the system interacts with a virtual system expressed by two or more virtual coordinates. Each virtual coordinate controls sampling along a reaction coordinate. By setting multiple reaction coordinates to be related to the corresponding rare events, sampling of the important process can be enhanced. An advantage of multi-dimensional-VcMC is its simplicity: Namely, the conformation moves widely in the multi-dimensional reaction coordinate space without knowledge of canonical distribution functions of the system. To examine the effectiveness of the algorithm, we introduced a toy model where two molecules (receptor and its ligand) bind and unbind to each other. The receptor has a deep binding pocket, to which the ligand enters for binding. Furthermore, a gate is set at the entrance of the pocket, and the gate is usually closed. Thus, the molecular binding takes place via the two events: ligand approach to the pocket and gate opening. In two-dimensional (2D)-VcMC, the two molecules exhibited repeated binding and unbinding, and an equilibrated distribution was obtained as expected. A conventional canonical simulation, which was 200 times longer than 2D-VcMC, failed in sampling the binding/unbinding effectively. The current method is applicable to various biological systems.

Thermodynamics and galactic clustering with a modified gravitational potential

Based on thermodynamics, we study the galactic clustering of an expanding Universe by considering the logarithmic and volume (quantum) corrections to Newton's law along with the repulsive effect of a harmonic force induced by the cosmological constant ($\mathrm{\ensuremath{\Lambda}}$) in the formation of the large scale structure of the Universe. We derive the $N$-body partition function for extended-mass galaxies (galaxies with halos) analytically. For this partition function, we compute the exact equations of states, which exhibit the logarithmic, volume, and cosmological constant corrections. In this setting, a modified correlation (clustering) parameter (due to these corrections) emerges naturally from the exact equations of state. We compute a corrected grand canonical distribution function for this system. Furthermore, we obtain a deviation in differential forms of the two-point correlation functions for both the point-mass and extended-mass cases. The consequences of these deviations on the correlation function's power law are also discussed.

Enhancement of canonical sampling by virtual-state transitions

A novel method was developed to enhance canonical sampling. A system is divided into virtually introduced sub-states, called "virtual states," which does not exist in reality. The configuration sampling is achieved by a standard canonical sampling method, the Metropolis Monte Carlo method, and confined in a virtual state for a while. In contrast, inter-virtual state motions are controlled by transition probabilities, which can be set arbitrarily. A simple recursive equation was introduced to determine the inter-virtual state transition probabilities, by which the sampling is enhanced considerably. We named this method "virtual-system coupled canonical Monte Carlo (VcMC) sampling." A simple method was proposed to reconstruct a canonical distribution function at a certain temperature from the resultant VcMC sampling data. Two systems, a one-dimensional double-well potential and a three-dimensional ligand-receptor binding/unbinding model, were examined. VcMC produced an accurate canonical distribution much more quickly than a conventional canonical Monte Carlo simulation does.

Phase space view of quantum mechanical systems and Fisher information

Pennini and Plastino showed that the form of the Fisher information generated by the canonical distribution function reflects the intrinsic structure of classical mechanics. Now, a quantum mechanical generalization of the Pennini–Plastino theory is presented based on the thermodynamical transcription of the density functional theory. Comparing to the classical case, the phase-space Fisher information contains an extra term due to the position dependence of the temperature. However, for the special case of constant temperature, the expression derived bears resemblance to the classical one. A complete analogy to the classical case is demonstrated for the linear harmonic oscillator.

A new interpretation of X-ray reflectivity in real space for low contrast multilayer systems I. Mathematical algorithm and numerical simulations

It is shown that X-ray specular reflectivity may be described in terms of canonical distribution functions (CDFs) pj(z) which is a probability to find an element of sort j at a depth z from the sample surface. The problem reduces to determine K CDFs, where K is the number of elements in the multilayer sample. Using the properties of the canonical functions, we have introduced the interface function pint(z) and use it as one unknown function at solving the integral equation. The integral Fredholm equation of the first kind belongs to the class of ill-posed problems and for solving it needs special methods. We use the Tikhonov regularization method. This method is applied to study the low contrast multilayer systems.

On the high Mach number shock structure singularity caused by overreach of Maxwellian molecules

The high Mach number shock structure singularity arising in moment equations of the Boltzmann equation was investigated. The source of the singularity is shown to be the unbalanced treatment between two high order kinematic and dissipation terms caused by the overreach of Maxwellian molecule assumption. In compressive gaseous flow, the high order stress-strain coupling term of quadratic nature will grow far faster than the strain term, resulting in an imbalance with the linear dissipation term and eventually a blow-up singularity in high thermal nonequilibrium. On the other hand, the singularity arising from unbalanced treatment does not occur in the case of velocity shear and expansion flows, since the high order effects are cancelled under the constraint of the free-molecular asymptotic behavior. As an alternative method to achieve the balanced treatment, Eu's generalized hydrodynamics, consistent with the second law of thermodynamics, was revisited. After introducing the canonical distribution function in exponential form and applying the cumulant expansion to the explicit calculation of the dissipation term, a natural platform suitable for the balanced treatment was derived. The resulting constitutive equation with the nonlinear factor was then shown to be well-posed for all regimes, effectively removing the high Mach number shock structure singularity.

Grassmann phase space theory and the Jaynes–Cummings model

The Jaynes–Cummings model of a two-level atom in a single mode cavity is of fundamental importance both in quantum optics and in quantum physics generally, involving the interaction of two simple quantum systems—one fermionic system (the TLA), the other bosonic (the cavity mode). Depending on the initial conditions a variety of interesting effects occur, ranging from ongoing oscillations of the atomic population difference at the Rabi frequency when the atom is excited and the cavity is in an n-photon Fock state, to collapses and revivals of these oscillations starting with the atom unexcited and the cavity mode in a coherent state. The observation of revivals for Rydberg atoms in a high-Q microwave cavity is key experimental evidence for quantisation of the EM field. Theoretical treatments of the Jaynes–Cummings model based on expanding the state vector in terms of products of atomic and n-photon states and deriving coupled equations for the amplitudes are a well-known and simple method for determining the effects.In quantum optics however, the behaviour of the bosonic quantum EM field is often treated using phase space methods, where the bosonic mode annihilation and creation operators are represented by c-number phase space variables, with the density operator represented by a distribution function of these variables. Fokker–Planck equations for the distribution function are obtained, and either used directly to determine quantities of experimental interest or used to develop c-number Langevin equations for stochastic versions of the phase space variables from which experimental quantities are obtained as stochastic averages. Phase space methods have also been developed to include atomic systems, with the atomic spin operators being represented by c-number phase space variables, and distribution functions involving these variables and those for any bosonic modes being shown to satisfy Fokker–Planck equations from which c-number Langevin equations are often developed. However, atomic spin operators satisfy the standard angular momentum commutation rules rather than the commutation rules for bosonic annihilation and creation operators, and are in fact second order combinations of fermionic annihilation and creation operators. Though phase space methods in which the fermionic operators are represented directly by c-number phase space variables have not been successful, the anti-commutation rules for these operators suggest the possibility of using Grassmann variables—which have similar anti-commutation properties. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of phase space methods in quantum optics to treat fermionic systems by representing fermionic annihilation and creation operators directly by Grassmann phase space variables is rather rare. This paper shows that phase space methods using a positive P type distribution function involving both c-number variables (for the cavity mode) and Grassmann variables (for the TLA) can be used to treat the Jaynes–Cummings model. Although it is a Grassmann function, the distribution function is equivalent to six c-number functions of the two bosonic variables. Experimental quantities are given as bosonic phase space integrals involving the six functions. A Fokker–Planck equation involving both left and right Grassmann differentiations can be obtained for the distribution function, and is equivalent to six coupled equations for the six c-number functions.The approach used involves choosing the canonical form of the (non-unique) positive P distribution function, in which the correspondence rules for the bosonic operators are non-standard and hence the Fokker–Planck equation is also unusual. Initial conditions, such as those above for initially uncorrelated states, are discussed and used to determine the initial distribution function. Transformations to new bosonic variables rotating at the cavity frequency enable the six coupled equations for the new c-number functions–that are also equivalent to the canonical Grassmann distribution function–to be solved analytically, based on an ansatz from an earlier paper by Stenholm. It is then shown that the distribution function is exactly the same as that determined from the well-known solution based on coupled amplitude equations.In quantum–atom optics theories for many atom bosonic and fermionic systems are needed. With large atom numbers, treatments must often take into account many quantum modes—especially for fermions. Generalisations of phase space distribution functions of phase space variables for a few modes to phase space distribution functionals of field functions (which represent the field operators, c-number fields for bosons, Grassmann fields for fermions) are now being developed for large systems. For the fermionic case, the treatment of the simple two mode problem represented by the Jaynes–Cummings model is a useful test case for the future development of phase space Grassmann distribution functional methods for fermionic applications in quantum–atom optics.

An optimized replica exchange molecular dynamics method

We introduce a new way to perform swaps between replicas in replica exchange molecular dynamics simulations. The method is based on a generalized canonical probability distribution function and flattens the potential of mean force along the temperature coordinate, such that a random walk in temperature space is achieved. Application to a Go model of protein A showed that the method is more efficient than conventional replica exchange. The method results in a constant probability distribution of the replicas over the thermostats, yields a minimum round-trip time between extremum temperatures, and leads to faster ergodic convergence.

On the completeness of description of an equilibrium canonical ensemble using a reduceds-particle distribution function

In this paper it is shown that for a classical equilibrium canonical ensemble of molecules with sufficientlysmall s-body interaction, the full Gibbs distribution can be uniquely expressed in terms of a reduceds-particle distribution function. This means that whenever the number of particlesN and the volumeV of such a system arefixed, the reduced s-particle distribution function contains as much information about theequilibrium system as the canonical Gibbs distribution function. The latter isrepresented as an absolutely convergent power series relative to the reduceds-particle distribution function. As an example, a linear term of this expansion iscalculated. It is also shown that reduced distribution functions of order less thans do not possess such a property and, to all appearances, do not contain all of theinformation about the system under consideration.

Relativistic transformation of the canonical distribution function in relativistic statistical mechanics

We deduce a relativistic transformation of the canonical distribution function from basic principles; that is, from relativistic canonical transformations of dynamical variables. The thermodynamics which is obtained from such a distribution coincides with the recent proposal put forward by Ares de Parga et al. [G. Ares de Parga, B. López-Carrera, F. Angulo-Brown, J. Phys. A 38 (2005) 2821] of a relativistic renormalized thermodynamics. The invariances of the canonical distribution and the partition function are obtained. By noticing a mistake committed by Balescu while dealing with the partition function of an ideal gas, the consistency of the theory is showed. The Maxwell distribution function of an ideal gas moving with a constant velocity with respect to a reference frame is calculated. A natural asymmetry appears as a difference with the distribution function at the same temperature but at rest with respect to the same reference frame. The quantum case and the covariance of the theory are analyzed.

Equilibrium theory for a particle pulled by a moving optical trap

The viscous drag on a colloidal particle pulled through solution by an optical trap is large enough that on experimentally relevant time scales the mechanical force exerted by the trap is equal and opposite the viscous drag force. The rapid mechanical equilibration allows the system to be modeled using equilibrium theory where the effects of the energy dissipation (thermodynamic disequilibrium) show up only in the coordinate transformations that map the system from the laboratory frame of reference, relative to which the particle is moving, to a frame of reference in which the particle is, on average, stationary and on which the stochastic dynamics is governed by a canonical equilibrium distribution function. The simple equations in the stationary frame can be analyzed using the Onsager-Machlup theory for stochastic systems and provide generalizations of equilibrium and near equilibrium concepts such as detailed balance and fluctuation-dissipation relations applicable to a wide range of systems including molecular motors, pumps, and other nanoscale machines.

The nature of A1–L1 0 ordering transitions in alloy nanoparticles: A Monte Carlo study

Equilibrium A1–L1 0 ordering transitions in alloy nanoparticles are investigated from Monte Carlo simulations based on a lattice-model framework with parameters chosen to be representative of the FePt system. The present simulations yield results for the effects of size and surface segregation upon equilibrium ordering temperatures in good agreement with previous Monte Carlo studies and the most recent experimental measurements for FePt nanoparticles. Specifically, for nanoparticle diameters in the range of ∼4 nm the calculated equilibrium ordering transition is estimated to lie well above the temperatures commonly employed in the annealing of FePt nanoparticles, and at such annealing temperatures the magnitude of the size-induced reduction in the volume-averaged long-range order parameter is estimated to be of the order of 20%. In this work a detailed analysis of the nature of the L1 0 ordering transition in confined nanoparticle geometries is presented based upon results obtained for surface concentrations, order parameter profiles and calculated canonical distribution functions. It is shown that the main results obtained from this and previous Monte Carlo simulation studies, namely the size dependence of the volume-averaged long-range order parameters and ordering temperatures, and the continuous nature of the ordering transition, can be understood as being natural manifestations of a disordering mechanism that is surface induced.

Nonequilibrium effective temperature of superfluid vortex tangle

An effective nonequilibrium temperature in counterflow superfluid turbulence is proposed, as a parameter characterizing a canonical probability distribution function of vortex orientation, and relating the diffusion coefficient of vortex lines to the vortex friction through an Einstein relation.