A study of entangled systems in the many‐body signed particle formulation of quantum mechanics

Recently a new formulation of quantum mechanics has been introduced, based on signed classical field‐less particles interacting with an external field by means of only creation and annihilation events. In this article, we extend this novel theory to the case of many‐body problems. We show that, when restricted to spatial finite domains and discrete momentum space, the proposed extended theory is equivalent to the time‐dependent many‐body Wigner Monte Carlo method. In this new picture, the treatment of entangled systems comes naturally and, therefore, we apply it to the study of quantum entangled systems. The latter is represented in terms of two Gaussian wave packets moving in opposite directions. We introduce the presence of a dissipative background and show how the entanglement is affected by different (controlled) configurations.     

Theory for non-equilibrium statistical mechanics

This paper reviews a new theory for non-equilibrium statistical mechanics. This gives the non-equilibrium analogue of the Boltzmann probability distribution, and the generalization of entropy to dynamic states. It is shown that this so-called second entropy is maximized in the steady state, in contrast to the rate of production of the conventional entropy, which is not an extremum. The relationships of the new theory to Onsager's regression hypothesis, Prigogine's minimal entropy production theorem, the Langevin equation, the formula of Green and Kubo, the Kawasaki distribution, and the non-equilibrium fluctuation and work theorems, are discussed. The theory is worked through in full detail for the case of steady heat flow down an imposed temperature gradient. A Monte Carlo algorithm based upon the steady state probability density is summarized, and results for the thermal conductivity of a Lennard-Jones fluid are shown to be in agreement with known values. Also discussed is the generalization to non-equilibrium mechanical work, and to non-equilibrium quantum statistical mechanics. As examples of the new theory two general applications are briefly explored: a non-equilibrium version of the second law of thermodynamics, and the origin and evolution of life.     

A statistical mechanics description of environmental variability in metabolic networks

Many of the chemical reactions that take place within a living cell are irreversible. Due to evolutionary pressures, the number of allowable reactions within these systems are highly constrained and thus the resulting metabolic networks display considerable asymmetry. In this paper, we explore possible evolutionary factors pertaining to the reduced symmetry observed in these networks, and demonstrate the important role environmental variability plays in shaping their structural organization. Interpreting the returnability index as an equilibrium constant for a reaction network in equilibrium with a hypothetical reference system, enables us to quantify the extent to which a metabolic network is in disequilibrium. Further, by introducing a new directed centrality measure via an extension of the subgraph centrality metric to directed networks, we are able to characterise individual metabolites by their participation within metabolic pathways. To demonstrate these ideas, we study 116 metabolic networks of bacteria. In particular, we find that the equilibrium constant for the metabolic networks decreases significantly in-line with variability in bacterial habitats, supporting the view that environmental variability promotes disequilibrium within these biochemical reaction systems.     

The statistical mechanics of two-dimensional vesicles

Recent work on the statistical mechanics of model membranes and vesicles in two dimensions by Leibler, Singh and the author [l,2] is briefly summarized.     

Eigenvalues of a statistical mechanics matrix

We consider the problem of finding the spectrum of an n × n matrix which arises in the study of a certain model of long-range interactions in a one-dimensional statistical mechanics system. Our analysis exhibits a curious resemblance of the suitably normalized distribution of eigenvalues to the Marčenko–Pastur law in the limit n → ∞.     

Modeling mercury porosimetry using statistical mechanics

We consider mercury porosimetry from the perspective of the statistical thermodynamics of penetration of a nonwetting liquid into a porous material under an external pressure. We apply density functional theory to a lattice gas model of the system and use this to compute intrusion/extrusion curves. We focus on the specific example of a Vycor glass and show that essential features of mercury porosimetry experiments can be modeled in this way. The lattice model exhibits a symmetry that provides a direct relationship between intrusion/extrusion curves for a nonwetting fluid and adsorption/desorption isotherms for a wetting fluid. This relationship clarifies the status of methods that are used for transforming mercury intrusion/extrusion curves into gas adsorption/desorption isotherms. We also use Monte Carlo simulations to investigate the nature of the intrusion and extrusion processes.     

Semiclassical statistical mechanics of hard-body fluid mixtures

The thermodynamic properties of semiclassical hard-body fluid mixtures are studied. Explicit expressions are given for the free-energy, equation of state and virial coefficients of the classical hard convex-body fluid mixtures. The numerical results are discussed under different conditions. The agreement with the exact data is good in all cases. The first-order quantum corrections are also studied. The quantum effects depend on the condition, shape parameters L11* and L22*, and concentrations x1 and x2 in general and increase with an increase of packing fraction eta, in particular.     

Thermodynamics and statistical mechanics of multilayer adsorption

The multilayer adsorption models of Brunauer-Emmett-Teller and Guggenheim-Anderson-de Boer are reconsidered. The relationship between the fitting parameters and the physical parameters of the equation is discussed. The preexponential factors of the parameters are shown to be in general far different from unity, contrary to a widespread use. A thermodynamical derivation illuminates the hypothesis on which the multilayer sorption equation is dependent and frees it from too restrictive hypothesis usually taken as necessary for its validity. Equations are derived for the number fraction of sorption sites occupied by different numbers of molecules. The Guggenheim-Anderson-de Boer equation is shown to imply incomplete occupation (jamming) of the first sorption layer at saturation.     

Semiclassical statistical mechanics of two-dimensional hard-body fluids

The problem of calculating the thermodynamic properties of two-dimensional semiclassical hard-body fluids is studied. Explicit expressions are given for the first-order quantum corrections to the free energy, equation of state, and virial coefficients. The numerical results are calculated for the planar hard dumbbell fluid. Significant features are the increase in quantum corrections with increasing eta and increasing L*=L/sigma(0).     

A variational principle in Wigner phase-space with applications to statistical mechanics

We consider the Dirac-Frenkel variational principle in Wigner phase-space and apply it to the Wigner-Liouville equation for both imaginary and real time dynamical problems. The variational principle allows us to deduce the optimal time-evolution of the parameter-dependent Wigner distribution. It is shown that the variational principle can be formulated alternatively as a "principle of least action." Several low-dimensional problems are considered. In imaginary time, high-temperature classical distributions are "cooled" to arrive at low-temperature quantum Wigner distributions whereas in real time, the coherent dynamics of a particle in a double well is considered. Especially appealing is the relative ease at which Feynman's path integral centroid variable can be incorporated as a variational parameter. This is done by splitting the high-temperature Boltzmann distribution into exact local centroid constrained distributions, which are thereafter cooled using the variational principle. The local distributions are sampled by Metropolis Monte Carlo by performing a random walk in the centroid variable. The combination of a Monte Carlo and a variational procedure enables the study of quantum effects in low-temperature many-body systems, via a method that can be systematically improved.     

Bohmian mechanics with complex action: a new trajectory-based formulation of quantum mechanics

In recent years there has been a resurgence of interest in Bohmian mechanics as a numerical tool because of its local dynamics, which suggest the possibility of significant computational advantages for the simulation of large quantum systems. However, closer inspection of the Bohmian formulation reveals that the nonlocality of quantum mechanics has not disappeared-it has simply been swept under the rug into the quantum force. In this paper we present a new formulation of Bohmian mechanics in which the quantum action, S, is taken to be complex. This leads to a single equation for complex S, and ultimately complex x and p but there is a reward for this complexification-a significantly higher degree of localization. The quantum force in the new approach vanishes for Gaussian wave packet dynamics, and its effect on barrier tunneling processes is orders of magnitude lower than that of the classical force. In fact, the current method is shown to be a rigorous extension of generalized Gaussian wave packet dynamics to give exact quantum mechanics. We demonstrate tunneling probabilities that are in virtually perfect agreement with the exact quantum mechanics down to 10(-7) calculated from strictly localized quantum trajectories that do not communicate with their neighbors. The new formulation may have significant implications for fundamental quantum mechanics, ranging from the interpretation of non-locality to measures of quantum complexity.     

A statistical model for instable thermodynamical systems

A generic model is presented for statistical systems which display thermodynamic features in contrast to our everyday experience, such as infinite and negative heat capacities. Such system are instable in terms of classical equilibrium thermodynamics. Using our statistical model, we are able to investigate states of instable systems which are undefined in the framework of equilibrium thermodynamics. We show that a region of negative heat capacity in the adiabatic environment, leads to a first order like phase transition when the system is coupled to a heat reservoir. This phase transition takes place without a phase coexistence. Nevertheless, all intermediate states are stable due to fluctuations. When two instable system are brought in thermal contact, the temperature of the composed system is lower than the minimum temperature of the individual systems. Generally, the equilibrium states of instable system cannot be simply decomposed into equilibrium states of the individual systems. The properties of instable system depend on the environment, ensemble equivalence is broken.     

Semigroups and the density matrix formulation of quantum mechanics

A formulation of quantum mechanics is presented based on the theory of semigroups and the associated enveloping algebras of functions defined on countable subsemigroups. The existence of a unique *-involution is not assumed. The fundamental elements of a semigroup are identified with experimental precedures for the separation of subensembles from a given ensemble of experimental systems. Observables are represented as elements of enveloping algebras, and ensembles as density matrices within an enveloping algebra. The statistical properties of ensembles are expressed in terms of traces defined on the semigroup and its enveloping algebras. The elements and generators of the Poincaré group can be defined and interpreted in the usual way. A variety of applications is described, in which the theory of the density matrix plays an essential or effective role. Advantages associated with the resulting freedom from the limitations of Hilbert space are illustrated.     

Size-distribution of droplets in emulsions by statistical mechanics calculation

This report is concerned with theoretical demonstration of the spontaneous emulsification which has been observed in a soft contact of nitrobenzene with water without surfactant [K. Aoki, M. Li, J. Chen, T. Nishiumi, Electrochem. Commun. 11 (2009) 239]. The demonstration is based on the model of spherical oil droplets with any size in equilibrium. The droplets are composed of the smallest droplets, the total number of which is given. An assembly of small droplets has larger surface energy than that of large ones because the surface energy is proportional to the surface area. The former has larger configurational entropy than the latter because the number of small droplets is bigger than that of the large ones. Since the free energy is determined by the competition between the surface energy and the entropy, it is not clear which assembly has lower free energy. This question was solved numerically here by statistical mechanics calculation of the size distributions, which contained only a parameter of the surface energy. The results of the computation at small number of droplets were used for deriving approximate equations for extremely large number of droplets. The size distribution was localized both to the smallest and the largest droplets. The diameter of the largest droplet was estimated from the dynamics in which coalescence by diffusion of droplets is disturbed by gravitational convection. The size then predicted was of the order of micrometer, being close to experimental values.     

A molecular-orbital-based molecular mechanics approach to study conjugated systems

The Allinger's MM2 program has been extended to calculate conjugated systems including N, O, and S hetero atoms.     

Quantum statistical mechanics of closed‐ring molecular chains

A model for a closed‐ring unhindered three‐dimensional macromolecular chain, based on Quantum Mechanics, is presented. Upon starting from an exact non‐relativistic Hamiltonian operator, we integrate out all electronic degrees of freedom, in the Born‐Oppenheimer framework, giving rise to an effective vibro‐rotational Hamiltonian for the chain. Then, assuming a harmonic oscillator‐like vibrational potential between nearest‐neighbour atoms, the integration of the atomic radial degrees of freedom is carried in the limit of high frequencies. Thus, all bond lengths become fixed, including the one which makes the chain to become a closed ring. This formulation leads to a specific Hamiltonian for the remaining angular variables of the closed‐ring chain, and constitutes an alternative in comparison with standard Gaussian models, which do not. Use is made of a variational inequality by Peierls to find an approximate quantum partition function for the angular variables of the system. We then proceed to obtain approximately another representation for the angular partition function in the classical limit. Several features of the classical partition function are discussed.     

Explicit factorization of external coordinates in constrained statistical mechanics models

If a macromolecule is described by curvilinear coordinates or rigid constraints are imposed, the equilibrium probability density that must be sampled in Monte Carlo simulations includes the determinants of different mass-metric tensors. In this work, the authors explicitly write the determinant of the mass-metric tensor G and of the reduced mass-metric tensor g, for any molecule, general internal coordinates and arbitrary constraints, as a product of two functions; one depending only on the external coordinates that describe the overall translation and rotation of the system, and the other only on the internal coordinates. This work extends previous results in the literature, proving with full generality that one may integrate out the external coordinates and perform Monte Carlo simulations in the internal conformational space of macromolecules.     

A variational formulation of the rate equations for closed chemical systems

This paper presents a variational formulation of the kinetic equations corresponding to a closed chemical system.

---

, .

     

Theory of cluster self-organization for crystal-forming systems: A model of transition from disodered to ordered systems

The current state of ideas concerning the self-organization of crystal-forming systems where long-range order spontaneously appears in the arrangement of nanolevel structural units of any nature (micro- and macromolecules or atomic clusters) that initially existed in a dynamic state as a chaotic mixture is considered. Three partially overlapping stages of self-organization of a system accepted in physical models of “order-disorder” kinetic transitions are matched to those used in supramolecular chemistry. An algorithmically constructed model of transition from disordered to hierarchically ordered systems is considered. The geometrical and topological modeling of density fluctuations of n-atomic species (clusters) An in a crystal-forming medium is carried out. A specific set of An clusters with block-diagonal connectivity matrices is recognized. These types of clusters (S ₃⁰), having “sectional” or “hierarchic” partition, are defined as precursors of crystal structures that are capable of evolving most rapidly to give rise to a long-range order in structures. For an S ₃⁰ ring cluster shaped as a triangle, geometrical and topological modeling is carried out for all of the eight topologically and symmetrically possible types of S ₃¹ primary chains built of S ₃⁰ using theory of one-dimensional symmetry groups. Thirty three structural variants of morphologically and topologically different types of S ₃² micronets described by two-dimensional groups of symmetry are considered. Algorithms are presented for combinatorial and topological analysis to search for precursor clusters and restore a three-dimensional net of covalent and noncovalent bonds in a crystal structure by the matrix (cluster) self-assembly mechanism. The model advanced is universal. Examples of self-assembly of a series of cluster-assembled structures of AB₂ alloys of the unique Friauf-Laves family (which counts in 1400 of binary and ternary compounds) are given: for MgCu₂ (cF24) (with its superstructures of ZrCu ₅ and MgSnCu₄ types), MgZn₂ (hP12), and MgNi₂ (hP24) (from AB₂ or A₂B + B₃ three-atom clusters); for ZrZn₂₂ icosahedral structures (from a suprapolyhedral cluster built of a ZrZn₁₆ Friauf polyhedron and two ZnZn₁₂ icosahedra); NaCd₂ (from one A cluster with 61 atoms and two B clusters with 63 atoms); and for a bimolecular compound C₇₈H₃₀ (which is formed of fullerene C₆₀ and a C₂₈H₃₀ molecule). The scenario of formation of self-curving nets with icosahedral symmetry is considered: to form a B₁₂ icosahedron from two isomers with n = 3, a C₂₀ dodecahedron from two isomers with n = 5, and C₆₀ fullerene from pentagonal clusters with n = 5.