Initial growth of Boltzmann entropy and chaos in a large assembly of weakly interacting systems

We introduce a high-dimensional symplectic map, modeling a large system, to analyze the interplay between single-particle chaotic dynamics and particles interactions in thermodynamic systems. We study the initial growth of the Boltzmann entropy, S_B, as a function of the coarse-graining resolution (the late stage of the evolution is trivial, as the system is subjected to no external drivings). We show that a characteristic scale emerges, and that the behavior of S_B vs t, at variance with the Gibbs entropy, does not depend on the resolution, as far as it is finer than this scale. The interaction among particles is crucial to achieve this result, while the rate of entropy growth, in its early stage, depends essentially on the single-particle chaotic dynamics. It is possible to interpret the basic features of the dynamics in terms of a suitable Markov approximation.     

Depletion effects and gelation in a binary hard-sphere fluid

A study of the binary hard-sphere fluid with size ratio [sgrave]_B/[sgrave]_A = 0.1 is reported. Molecular dynamics and Monte Carlo simulations have been carried out over the mole fraction (x _A) range 0.002-0.1 and over the high density range where several recent authors have predicted a thermodynamic demising transition on the basis of integral equations. In this region, there is no evidence of such first-order thermodynamic phase separation, or two fluid phases. The effect of the depletion force, arising from the entropic exclusion of B spheres from between two A spheres, as x _B is increased at constant packing fraction y _A, is to cause a large increase in the partial pressure of A and the radial distribution function of A at contact, a reduction on the mobility of A, and eventually, at a sufficient x _B, the gelation of component A to an open, low coordination, amorphous structure. This gelation transition of A shows discontinuities similar to a glass transition; it can be traced back to the hard sphere glass formation as x _B approaches zero. Thermodynamic properties are reported over the range studied, and used to evaluate the predictions of current theories and the accuracy of equations of state. The Boublik—Mansoori—Carnahan—Starling—Leland equation is found to be remarkably accurate in this region, over the whole fluid range, but shows systematic deviations at high packing densities.     

Transport coefficients of hard sphere fluids

New calculations have been made of the self-diffusion coefficient D, the shear viscosity η_s, the bulk viscosity η_b and thermal conductivity λ of the hard sphere fluid, using molecular dynamics (MD) computer simulation. A newly developed hard sphere MD scheme was used to model the hard sphere fluid over a wide range up to the glass transition (～0.57 packing fraction). System sizes of up to 32 000 hard spheres were considered. This set of transport coefficient data was combined with others taken from the literature to test a number of previously proposed analytical formulae for these quantities together with some new ones given here. Only the self-diffusion coefficient showed any substantial N dependence for N < 500 at equilibrium fluid densities (ε 0.494). D increased with N, especially at intermediate densities in the range ε ～ 0.3–0.35. The expression for the packing fraction dependence of D proposed by Speedy, R. J., 1987, Molec. Phys., 62, 509 was shown to fit these data well for N ～ 500 particle systems. We found that the packing fraction ε dependence of the two viscosities and thermal conductivity, generically denoted by X, were represented well by the simple formula X/X ₀ = 1/[1 ? (ε/ε₁)]^m within the equilibrium fluid range 0 > ε > 0.493. This formula has two disposable parameters, ε and m, and X ₀ is the value of the property X in the limit of zero density. This expression has the same form as the Krieger-Dougherty formula (Kreiger, I. M., 1972, Adv. Colloid. Interface Sci., 3, 111) which is used widely in the colloid literature to represent the packing fraction dependence of the Newtonian shear viscosity of monodisperse colloidal near-hard spheres. Of course, in the present case, X _o was the dilute gas transport coefficient of the pure liquid rather than the solvent viscosity. It was not possible to fit the transport coefficient normalized by their Enskog values with such a simple expression because these ratios are typically of order unity until quite high packing fractions and then diverge rapidly at higher values over a relatively narrow density range. At the maximum equilibrium fluid packing fraction ε = 0.494 for both the hard sphere fluid and the corresponding colloidal case a very similar value was found for η_s/η_o ?30–40, suggesting that the ‘crowding’ effects and their consequences for the dynamics in this region of the phase diagram in the two types of liquid have much in common. For the hard sphere by MD, D_o/D ～ 11 at the same packing fraction, possibly indicating the contribution from ‘hydrodynamic enhancement’ of this transport coefficient, which is largely absent for the shear viscosity. Interestingly the comparable ratio for hard sphere colloids is the same.     

Multi-particle collision dynamics simulation on capturing a target sphere fuelled by chemical reaction

The directed self-propulsion of a spherical target molecule fuelled by a chemical reaction is investigated. This source–target system consists of a spherical position-fixed source molecule and a freely moving target, separated at an initial distance R₀ in solution. The irreversible chemical reaction which converts A solvent particle to B species occurs catalytically on the surface of the source molecule. This generates a non-equilibrium B solvent density gradient from the source to the target molecule. Therefore, a directed movement of the target can be achieved due to the difference of force between the target sphere and B solvent particles in all directions, and the target will finally be captured by the source. We have demonstrated the diffusiophoretic particle capture dynamics theoretically and described how the capture time and velocity can be affected by some key factors, such as chemical reaction rate, source and target molecule size, initial sphere separation, etc. The capture dynamics has also been studied by the coarse-grained mesoscopic hybrid molecular dynamics – multi-particle collision dynamics simulation. Great agreement has been found when we compare the simulated capture dynamics to the theoretical predictions. GRAPHICAL ABSTRACT     

Autocatalytic reaction dynamics in systems crowded by catalytic obstacles

Reaction and diffusion dynamics in systems crowded by catalytic obstacles are investigated using a particle-based mesoscopic simulation method. The focus of the work is on effects of correlations induced by the presence of the catalytic obstacles and solvent collective modes. As an example, a system is considered where the reaction A+C→B+C takes place on the surfaces of the C catalytic obstacles, while the autocatalytic reaction A+B→2A occurs in the bulk of the solution. It is shown that mean-field, mass-action rate laws break down and fail to describe the reaction dynamics for large volume fractions of obstacles. The influence of hydrodynamics on the reaction and diffusion dynamics is also studied.     

Exact solution of two-species ballistic annihilation with general pair-reaction probability

The reaction processA + B → ∅ is modeled for ballistic reactants on an infinite line with particle velocitiesυ _A =c andυ _B = -c and initially segregated conditions, i.e., allA particles to the left and allB particles to the right of the origin. Previous models of ballistic annihilation have particles that always react on contact, i.e., pair-reaction probabilityp = 1. The evolutions of such systems are wholly determined by the initial distributions of particles and therefore do not have a stochastic dynamics. However, in this paper the generalization is made to p< 1, allowing particles to pass through each other without necessarily reacting. In this way, theA andB particle domains overlap to form a fluctuating, finitesized reaction zone where the product ∅ is created. Fluctuations are also included in the currents ofA andB particles entering the overlap region, thereby inducing a stochastic motion of the reaction zone as a whole. These two types of fluctuations, in the reactions and particle currents, are characterised by theintrinsic reaction rate, seen in a single system, and theextrinsic reaction rate, seen in an average over many systems. The intrinsic and extrinsic behaviors are examined and compared to the case of isotropically diffusing reactants     

非均质基底表面上团簇生长的Monte Carlo模拟

Monte Carlo方法模拟了在非均质基底表面上团簇的生长过程.非均质基底由规则分布的A，B两类不同性质的物质(不同材料、不同温度等)组成.沉积粒子的初动能为E₀，在不同相基底上单位长度扩散的能耗为E_A,E_B，粒子从A经过相界扩散到B的能耗为E_AB.模拟中粒子的初动能E₀.远大于E_A,E_B，粒子从B过相界到A能耗E_BA取为0.结果表明，沉积在此类基底表面的粒子凝聚成具有分形结构的团簇.当E_AB远大于E_A,E_B时，团簇的数目、团簇的回旋半径、团簇的分形维数均随E₀/E_AB呈周期性变化. 关键词： 分形团簇 非均质基底 Monte Carlo模拟 周期性     

Influence of H-bond chains in solvents on the solubility of inert substances. A new quantitative approach.

The classic thermodynamic treatments considering self-associated liquids A as multicomponent systems are unable to explain the solubilities of substances B in these liquids by effects directly related to the presence of H-bond chains. This is not the case when the hypothesis of successive equilibria between the i-mers is abandoned and replaced by a single equilibrium between bonded and free hydroxylic protons. In the new treatment presented here two combinatorial entropies are considered : that resulting from the exchange possibilities between bonded and non-bonded protons and that originating from the exchange between A and B molecules. For large values of φ_A, the volume fraction of the associated liquid, it is shown that the latter combinatorial entropy is practically equal to that calculated in absence of self-association. The fraction γ of the free hydroxylic protons depends on the molar enthalpy of the hydrogen bond ΔH_h and on the entropy of chain-reintegration ΔS_h. The latter depends on the formal concentration F_A of the associated substance and a quantitative treatment shows that dΔS_h/dlnF_A = R (gas constant). This conclusion is experimentally confirmed by the variation of the vapor pressure with F_A in the series of the primary alcohols. The introduction of an inert substance B causes F_A to decrease and enhances the lowering of the entropy of the bonded molecules with respect to the non-bonded molecules. This creates a hydrophobic effect that is at the origin of the lowering of the solubility φ_B of B in A. As a consequence, lnφ_B at the equilibrium is diminished by an amount equal to b_Aφ_A$\text{V}$_B/$\text{V}$_A, where $\text{V}$_B and $\text{V}$_A are the formal molar volumes of both substances. b_A is a so called “structuration factor” of the solvent A that nearly equals ?1 for strongly associated solvents with single H-bond chains, like alcohols, and ?2 for solvents with double chains like water. Taking for the combinatorial entropy between A and B an expression intermediate between the classic one in mole fractions and that of Flory-Huggins in volume fractions, one derives equations that predict the solubility of alkanes in water and in alcohols. Although these equations do not contain any adjustable parameter and that the experimental solubilities φ_B cover several orders or magnitude, the agreement between prediction and experiment is remarkable.     

Griffiths inequalities for noninteractingN-vector (classical heisenberg) models and applications to interacting systems

An order relation for tensors is defined. With this ordering it is shown that in noninteractingN-vector models 〈σ_Aσ_B〉?〈σ_A〉〈σ_B〉 is positive. Applications to interacting models include a proof for the alignment of spins and the subadditivity of the free energy.     

Greens functions,Hamiltonians and modular automorphisms

We demonstrate, under circumstances that allow the construction of a G(A, B; t) = w(As_t (B))G(A, B; t) = \omega (A\sigma _t (B))     

Molecular dynamics studies of slow dynamics in random media: Type A-B and reentrant transitions

Molecular dynamics simulations of mobile particles confined in disordered immobile particles are carried out. Slow dynamics in random media are characterized by two types of dynamics: Type B dynamics for large mobile particle density and Type A dynamics for small mobile particle density. The crossover from Type A to B dynamics is studied by the mean square displacement and the density correlation function. Our results are qualitatively consistent with the results of recent numerical and theoretical studies on relevant spatially heterogeneous systems. We also investigate the effect of random matrix generation on the dynamics of mobile particles in order to examine the reentrant transition predicted by the recent mode-coupling theory. Our simulations demonstrate that the diffusion of the mobile particles largely depends on the protocol of the random matrix generation and that the reentrant transition is observed for a particular protocol.     

Correlation between the refractive index and average energy gap for simple and complex binary compounds

An empirical correlation between refractive index and corresponding average energy gap, proposed by Reddy et al, has been examined for different solids and it has been shown that the validity of such a relation is highly questionable. A new theoretical correlation is established instead of empirical correlation which is found applicable to a large number of binary A^NB^8-N type solids as well as for complex binary A₂B, AB₂, A₃B and A₃B₂ semiconductors.     

Three-dimensional nuclear dynamics in the quantized ATDHF approach

The quantized adiabatic time-dependent Hartree-Fock theory is numerically applied to the low energy large amplitude collective dynamics of heavy ion systems ranging from α + α to ¹⁶O + ¹⁶O. The problem is reduced to three successive steps. First, for the lowest mode the optimal, i.e., maximally decoupled, collective path {φ_q} is evaluated by solving a coupled set of nonlinear differential equations for the single-particle wave functions _q^(a)(r) of φ_q, depending on the collective coordinate q and three spatial coordinates. A density-dependent interaction with a direct finite range Yukawa-term is employed and three-dimensional coordinate- and momentum-grid techniques are used, including fast Fourier methods. In a second step the quantized collective Hamiltonian H_c(q, d/dq) is extracted from {φ_q} by means of generator coordinate techniques involving, besides q, a conjugate variable p. Starting from {φ} this procedure includes the numerical evaluation of the classical potential, (q), of the intertia parameter, (q), of the quantum corrections with regard to rotation, translation and collective q-motion, (q), and of the centrifugal potential. The third step consists of actually calculating the subbarrier fusion cross section by means of WKB methods applied to the collective Hamiltonian H_c(q, d/dq). The theoretical numbers are compared with results from Hartree-Fock calculations with quadrupole constraint, and with experimental data. The microscopic aspects of the dynamics, the relation to other theories, and the practical and conceptual problems arising from the quantized ATDHF theory are discussed in detail.     

Viscoelastic behaviour of fluids with steeply repulsive potentials

We analyse the shear stress, C _s(t) and pressure or ‘bulk’, C _b(t) time-correlation functions for steeply repulsive inverse power fluids (SRP) in which the particles interact via a pair potential with the analytic form, φ(r) = ε(σ/r) ⁿ , in a new approach to the understanding of their viscoelastic properties. We show analytically, and confirm by molecular dynamics simulations, that close to the hard-sphere limit both these time-correlation functions have the analytic form, C _s(t)/C _s(0) and C _b(t)/C _b(0) = 1 – T*(nt*)²+ O((nt*)⁴), where T* = k _B T/ε, is the reduced temperature, k _B is Boltzmann's constant and t* = (ε/mσ²)^½ t is the reduced time. This leads to an alternative and much simpler derivation of formulae for the shear and bulk viscosities which for the limiting case of hard spheres are numerically very close to the traditional Enskog relations. These simple relations for the finite and continuous SRP interaction are in satisfactory agreement with the essentially exact molecular dynamics simulation results for ca. n ≥ 18.     

Mean energies of hybridized valence states

A method of determining the relative signs of J _AB and J _BX in nuclear spin systems of the type AB₂X _q is described and applied to the AB₂X₃ proton spin system of 2-butyn-1-ol. Variation of the relative chemical shift v _A—v _B (through changes in temperature or in the composition of the solvent) causes certain pairs of ‘labile’ transitions in the X spectrum to coalesce on one side or other of the X spectrum. The information about the relative signs of the coupling constants is obtained by noting the two different critical values of v _A-v _B at which these coalescences occur. Since the method remains applicable even if the third coupling constant J _AX is vanishingly small, it is particularly useful in the determination of the signs of long-range spin coupling constants where the usual double resonance methods often break down. It is concluded that the long-range coupling J(H-C-C≡C-C-H) of 2-butyn-1-ol has a positive sign.     

A new expression for the combinatorial entropy of mixing in liquid mixtures

In the classic expression for the combinatorial entropy in terms of the mole fractions x_A and x_B′ the space unoccupied by the B molecules appears for them as divided in numerable parts, each belonging to an A molecule. In the Flory-Huggins expression based on the volume fractions φ_A and φ_B the space unoccupied by the B molecules is for them a continuum, not parcelled out in portions based on A standards. However in a solution containing n_B moles of B for a total number of moles n, the border of a B molecule is separated from that of the nearest B on the average by a distance equal to ((n/n_B)^1/2?1) times the mean diameter of the A molecules. Thus, there exists still in the liquid, directions for the B molecules which can be divided according to A standards. Based on this consideration, a new expression is derived for the combinatorial entropy of liquid mixtures which contains a “numerable” part besides a “continuous” part. on applying this equation for the prediction of the solubility of solid n-alkanes in liquid alkanes and neglecting non-ideality effects, the predicted values agree much better with the experimental ones than those deduced from the classic or from the Flory-Huggins expression.     

[2.2]Paracyclophane: theoretical study of its lower excited states and of the zero-field splitting parameters D

The lower excited singlet and triplet states and the zero-field splitting parameters D of [2.2]paracyclophane are studied within a semiempirical π theory which takes into account overlap effects between the two benzene rings, transanular and through-bond interaction via the methylene bridges. Whereas the singlet energies depend strongly on the through-bond interaction and the mutual polarization of σ core and π system this is not the case for the energies and zero-field splitting parameters D of the two lowest triplet states. The deformations of the benzene rings in [2.2]paracyclophane lead only to a small decrease of the excitation energies of about 0·2 eV. The D parameter can be written as a sum D = D_A + D_B + D_AB with the intrasubunit contributions D_A and D_B of the conjugated subunits A and B of the phane and an intersubunit term D_AB . We demonstrate that the deformations reduce the intrasubunit terms D_A and D_B and that they are crucial for the decrease of the D values of [2.2]paracyclophane with respect to p-xylene. The difference between the D values of the first and second triplet states is governed by the intersubunit term D_AB which has a different sign in the two states. However, this difference does not depend markedly on the transanular interaction. A further reduction of D_A and D_B in the first triplet state only is caused by transanular interaction by means of symmetrical charge-transfer terms in the wavefunction.     

Single-particle potential in a chiral approach to nuclear matter including short-range NN-terms

We extend a recent chiral approach to nuclear matter of Lutz et al.Phys. Lett. B 474, 7 (2000)) by calculating the underlying (complex-valued) single-particle potential U(p, k _f) + iW(p, k _f). The potential for a nucleon at the bottom of the Fermi sea, U(0, k _f0) = - 20.0 MeV, comes out as much too weakly attractive in this approach. Even more seriously, the total single-particle energy does not rise monotonically with the nucleon momentum p, implying a negative effective nucleon mass at the Fermi surface. Also, the imaginary single-particle potential, W(0, k _f0) = 51.1 MeV, is too large. More realistic single-particle properties together with a good nuclear-matter equation of state can be obtained if the short-range contributions of non-pionic origin are treated in mean-field approximation (i.e. if they are not further iterated with 1π-exchange). We also consider the equation of state of pure neutron matter ˉE_n(k _n) and the asymmetry energy A(k _f) in that approach. The downward bending of these quantities above nuclear-matter saturation density seems to be a generic feature of perturbative chiral pion-nucleon dynamics. Received: 19 December 2002 / Accepted: 11 February 2003 / Published online: 15 April 2003     

Solvent solute interactions probed by picosecond transient Raman spectroscopy:S 1 1,4-diphenyl-1,3-butadiene in the linear alkanes

We present theS ₁ Raman spectra of 1,4-DiPhenyl-1,3-Butadiene (DPB) in a series of linear alkanes (pentane, hexane, heptane, octane, decane, and dodecane). Bands assignable to both the 1¹ B _u and 2¹ A _g states are present, suggesting that the state we are observing in solution is a mixed state with both 1¹ B _u and 2¹ A _g character. The relative intensities of several bands associated with C_oC_o stretching motions in the 2¹ A _g and 1¹ B _u states change systematically through the solvent series. The relative intensity changes reflect a changing distribution ofs-trans conformers inS ₁ DPB as the solvent is varied. We suggest that the distribution ofs-trans conformers inS ₁ DPB controls the nature of the mixing between the 2¹ A _g and 1¹ B _u states and that the distribution of conformers is controlled by the solvent viscosity. Changes in the peak position and bandwidth of the phenyl C=C stretch with delay reflect vibrational relaxation processes inS ¹ DPB. We observe anomolous behavior in pentane that we attribute to the effect of the solvent structure on the ability of DPB to exchange energy with pentane.     

Die teleskopischen Prinzipien der Gravitationstheorie. (Machsches Prinzip,Mach-Einsteinsche Relativität der Trägheit und die Hertzsche Mechanik)

The Telescopical Principles in the Theory of Gravitation. (Machs Principle, Relativity of Inertia According to Mach and Einstein and Hertz' Mechanics) We give an explication and analytical formulation of Mach's principle of the “relativity of inertia” and of the Mach-Einstein doctrine on the determination of inertia by gravitation. These principles are whether “philosophical” nor “epistemological” postulates but well defined physical axioms with exactly analytical expressions. - The fundamental principle is the Galileian “reciprocity of motions”. According to this “generalized Galilei invariance” the principal functions of analytical dynamics (Lagrangian L and Hamiltonian H) are depending upon the differences ??_AB of the coordinate vectors ??_A and ??_B of the velocity differences ??_AB = ??_A-??_B, only. The Galileian reciprocity of motions means that whether the vectors ??_A and ??_A nor the accelerations ??_A of one particle have a physical significance. A mechanics obtaining this generalized Gailei-invariance cannot depend upon a kinematical Term \documentclass{article}\pagestyle{empty}\begin{document}$ T = \frac{1}{2}\mathop {\Sigma m_A \mathop r\nolimits_A^2}\limits_A $\end{document} in the Lagrangian. Therefore, the inertial masses of the particles must be homogeneous function of interaction potentials Φ_A,B. According to the Einsteinian equivalence of inertia and gravity these interactions have to be the Newtonian gravitation. In a universe with N mass points the Mach-Einsteinian Lagrangian for our “gravodynamics without inertia” is In such a Mach-Einstein universe the celestical dynamics becomes in the first approximation the Newtonian dynamics, in the second (the “post-Newtonian”) approximation the general relativistic Einstein effects are resulting.-However, our gravodynamics gives new effects for large masses (no gravitational collapses) and in cosmology (secular accelerations a.o.). Generally, the space of our gravodynamics is whether the Newtonian “absolute space” V₃ nor the relativistic Einstein-Minkowski world V₄ but the Hertzian configuration space V_3N of the N particles. According to the relativity of inertia the Hertzian metrics become Riemannian metrics which are homogenous functions of the Newtonian gravitational potentials. .